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

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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 of Basel who introduced me to the group. Jens Feder, Torstein Jossang and Paul Meakin were my scientific advisors. Without their help, support, and encouragement, I could not have written this thesis. All three of them never stopped to impress me by their deep insight into the mysteries of physics and their unconven ­ tional ways of thinking. Perhaps the most important aspect of their teaching is the open mind, the enthusiasm and the positive attitude they display towards the new and the exciting. At times, they also did not fail to let me struggle on my own; certainly a useful experience. Many other people have contributed to the writing of this thesis. I thank Aleksander Birovl- jev with whom I shared an office and had the pleasure to collaborate. His experimental work forms part of the basis of this thesis. Vidar Frette also played an important role during my first years in the group. Whenboth of them had graduated, H&kon Amundsen took over and started a new series of experiments that inspired me. I am grateful for the chance to work with all of them! Other people I had or have collaboration with include Ragrthild Halvorsrud, Anne Vedvik, Jonn-Erik Farmen, Thomas Rage, Thomas Walmann, and Anders-Malthe Sorenssen. I enjoyed working with all of them very much, and I am impressed by their skills and insight. Ragnhild introduced me to the world of slime molds, and we enjoyed ourselves watching them grow. Anne has become a very close friend of mine. Thomas W kept up our growing computer net­ work and I thank him for his efforts and for knowing so much about computers. Thor Engoy, Kim Christensen, Terje Johnsen, Finn Boger and Liv Furuberg are all former graduate students that were always willing to share their knowledge and to whom I could look up. H&kon, Anne, Ragnhild and Paul also read an early version of this thesis and helped me to improve the text. Among the lectures I heard, I remember very well those of Yuri Galperin and those of Am- non Aharony. Both are excellent teachers, and I thank them for their great efforts. Knut-Jorgen Mctloy was working on the same floor, and the work by him and by his students had a lot of influence on my thinking. Financial help was always most welcome. I am greatly indebted to my home Kanton Basel-Landschaft for repeatedly offering generous scholarships. Additional support came from the Jubilaumsstiftung der Basellandschaftlichen Kantonalbank. From my guest country Nor ­ way I was supported generously by VISTA, a research cooperation between The Norwegian Academy of Science and Letters and Den Norske Stats Oljeselskap a.s. (STATOIL). Quite un­ expectedly, I was fortunate to receive a scholarship from The Norwegian State Department of Foreign Affairs through The Research Council of Norway (NFR), as a result of an agreement between Norway and Switzerland on cultural exchange. I greatly appreciate the work of our office manager Inger Lauvstad who keeps the overview even in stressful moments. I am also grateful for the work of the cleaning staff at the Depart­ ment. Doing research would be impossible without their efforts. During the thesis work I shared flats with the following persons: Marianne, Margrete, Stein,

IX Terje, Gunnar, Magne, Jorunn, Thomas, Astrid, Johannes, Christine, Ragnhild and Jonn-Arild. They put up with my language problems and we became friends. I received a lot of moral sup­ port and became introduced to Norwegian lifestyle. With great interest I followed the hot de­ bate about the proposed Nowegian EU-membership that started long before I came and is still going on - about half of my flat-mates were pro, and the other half were contra. Thank you all for a great time! Most of the thesis work was done at a time in which people went to war in Europe, only a few hours by plane away. I am permanently aware of the privilege to be allowed to live and work in the peaceful town of Oslo.

Oslo, January 10,1997 Geri Wagner

x List of Symbols

a Levy index K Permeability r Saturation k Relative permeability Aij Difference of the (/-coor ­ k Wave vector in Fourier dinates of the ith and the space jth site L Lattice size V Viscosity l Length 9 Contact angle m Mass density Critical exponents Af Distribution of sizes e Correlation length P Pressure n Migration balance Pc Capillary pressure dif­ 7T Number density ference P Density V Migration probability <7 Interfacial tension in simulations of fluid A, C, fAN Prefactor or Constant migration in one dimen­ Bo Bond number sion b Aperture P Invasion threshold C(r) Two-point density corre ­ P' Withdrawl threshold in lation function simulations of fluid mi­ Ca Capillary number gration D, Dh Fractal dimensionality Q, q, q' Random number d Euclidean dimension R, r Radius /(%) Scaling function r Space vector 9 Gradient representing the rd Displacement vector accelaration of gravity S(|k|) Power spectral density g Acceleration of gravity- s Size (number of occupied vector sites) of a cluster or a frag ­ H Hurst exponent or rough ­ ment ness exponent U Flow velocity h Height V Volume lx Pore-scale invasion mech­ w Relative width anism in imbibition a, y, z Space coordinates

XI

CONTENTS

Contents

Summary v

Deutsche Zusammenfassung vii

Preface ix

List of Symbols xi

1 Overview 1 1.1 Content of the Thesis...... 1 1.2 Matters of Interest...... 2 1.3 Slow Two Phase Flow and Percolation Theory ...... 2 1.4 Slow Two Phase Flow in Industry and Technology ...... 3

2 Slow Two-Phase Flow in Porous Media and in Fractures 5 2.1 Physics of Slow Displacement ...... 5 2.1.1 Capillary Pressure...... 5 2.1.2 Capillary Pressure Hysteresis ...... 5 2.1.3 Displacement Steps...... 6 2.1.4 Presence of Buoyancy ...... 6 2.1.5 Macroscopic Description ...... 7 2.2 Displacement Mechanisms I...... 8 2.2.1 Water Flood Experiments ...... 8 2.2.2 Pore-level Mechanisms...... 9 2.2.3 Hierarchy of Displacement Mechanisms...... 10 2.2.4 Fluid Topology ...... 11 2.3 Displacement Mechanisms II ...... 11 2.3.1 Drainage Experiments ...... 12 2.3.2 Experiments on the Secondary Migration of Oil...... 12 2.3.3 Pore-level Mechanisms...... 13

3 Invasion Percolation 15 3.1 The Invasion Percolation Model ...... 15 3.1.1 Description of the Model ...... 15 3.1.2 Fractal Dimensionality and Trapping ...... 16 3.1.3 Including Gravity and Heterogenities ...... 17 3.2 Strengths and Weaknesses of the Invasion Percolation Model ...... 18 3.2.1 Invasion Percolation Compared to Experiments ...... 18 3.2.2 Limits of Invasion Percolation ...... 19 3.3 Earlier Modified IP Models and Models with IP as a Limiting Case...... 20 3.4 Other Percolation Models ...... 21 3.4.1 Percolation Models for Imbibition ...... 22 3.4.2 Percolation Models for Drainage ...... 23 3.4.3 A Percolation Model for Drainage and Imbibition ...... 23

4 Invasion Percolation and Beyond 25 4.1 Extending Invasion Percolation to Simulate Imbibition ...... 25 4.1.1 A Site-Bond IP Model for Imbibition ...... 26 4.1.2 Discussion of the Site-Bond IP Model for Imbibition ...... 28 4.1.3 Applying the Site-Bond IP Model to Simulate Secondary Imbibition . . 31 4.1.4 Related IP-like Models for Imbibition ...... 33 4.2 Extended Invasion Percolation Models for Fluid Migration ...... 34 4.2.1 A Site-Bond IP Model for Fluid Migration ...... 35

xiii CONTENTS

4.2.2 Discussion of the Site-Bond IP Model for Fluid Migration ...... 36 4.2.3 Applying the Site-Bond IP Model to Simulate Migration and Fragmenta ­ tion of IP Clusters...... 37 4.2.4 A Site IP Model for Fluid Migration ...... 39 4.2.5 Discussion of the Site IP Model for Fluid Migration ...... 40 4.2.6 Applying the Site IP Model to Simulate Migration and Fragmentation of IP clusters...... 40 4.2.7 Applying the Site IP Model to Simulate Secondary Migration Through Porous Rock ...... 42

5 Invasion Percolation in Correlated Disordered Media 45 5.1 Invasion Percolation in Correlated Porous Media...... 45 5.1.1 Observations of Heterogeneous Permeability Distributions ...... 45 5.1.2 Earlier Simulations of One- and Two-Phase Flow through Heterogeneous Porous Media...... 46 5.1.3 Invasion Percolation on Lattices With Multifractal Threshold Distributions 47 5.2 Invasion Percolation in Single Fractures...... 49 5.2.1 Observations of Fracture Aperture Fields...... 49 5.2.2 Observations of Two-Phase Flow in Fractures ...... 50 5.2.3 Earlier Percolation Models of Two-Phase Fluid Flow in Single Fractures 50 5.2.4 Invasion Percolation on Self-Affine Topographies ...... 51 5.2.5 Invasion Percolation in a Fractal Fracture...... 53 5.2.6 Applying the Site IP Model to Simulate Secondary Migration of Oil Through Single Fractures...... 55 5.2.7 Applying the Site IP Model to Simulate Drainage in a Single Fracture With Fluid Re-Distribution ...... 56

6 Scaling Issues 59 6.1 Fragmentation of Invasion Percolation Clusters...... 59 6.2 Migration and Fragmentation of Invasion Percolation Clusters...... 60 6.3 Migration and Fragmentation of Clusters in One Dimension ...... 61 6.4 Migration of Invasion Percolation Clusters at Constant Gradient...... 61 6.5 Scaling of Invasion Percolation Clusters on Correlated Threshold Maps...... 62 6.6 Scaling Laws in Practice...... 64

References 65

List of Papers 75

Papers 77

xiv 1.1 Con ten t of the Thesis

1 Overview This thesis describes simulations of the slow ments is not yet well understood. However, displacement of one fluid by another fluid in the microscopic mechanisms that control the a complex, random porous medium and in displacement processes have been studied ex­ a single fracture. Multi-phase displacement tensively. Section 2 presents an overview over processes are of relevance in many branches of the mechanisms of slow displacement in nar­ technology, including oil reservoir engineer ­ row pores. ing and waste management. When the phases One of the simplest and yet most accurate are characterized by different physical prop ­ models used to describe the slow displace­ erties, such as viscosity, density, or interfacial ment of a wetting fluid by a non-wetting fluid tension, the displacement process cannot be in a random porous medium is the invasion described easily; and when the displacement percolation model. The model is conceptu ­ takes place in a porous medium with non- ally attractive and relatively well understood. uniform properties the process becomes very In section 3 the invasion percolation model is complex. introduced and various modifications of the The study is restricted to two-phase flow model are discussed. Section 2 and section 3 in the quasi-static limit in which viscous forces are introductory and intended to provide a can be neglected. Many underground flows background for the reader. take place in this regime, most notably the sec­ Despite the merits of the invasion perco ­ ondary migration of oil. In this process, oil is lation model to account for the displacement transported from a source rock through very of wetting fluid by non-wetting fluid, the re­ large volumes of water-saturated carrier rock verse process cannot be described in a simi­ until it becomes trapped in a reservoir. Typical lar satisfactory manner. The simulation of this migration rates in the secondary migration of process requires the development of exten­ oil are less than 1 meter per year. 1 sions to the invasion percolation model. The Some of the simulations described in this extensions may include mechanisms to sim­ thesis were accompanied by experimental ulate fluid migration, fragmentation and co ­ studies carried out by Aleksander Birovl- alescence. An important example of immis­ jev, Vidar Frette, and H&kon Amundsen at cible fluid migration is the secondary migra ­ the University of Oslo. In these cases, the tion of oil in which oil is migrating through comparison of experiment and simulation porous rock that is saturated with water. Fig­ provided an assessment of the quality of the ure 1 shows a scanning electron photomicro ­ simulation. Various references to relevant graph of a rock sample to illustrate the com ­ experimental work are given in the text. plexity of the boundary conditions of the mi­ The core of the thesis is formed by the pa­ gration process. The secondary migration of pers PI - P8, reproduced in the last section. In oil through porous rock was studied using the preceding sections an introduction to the simplified variants of the simulation models. issues treated in the papers is presented. Com ­ The assumptions and considerations behind pared to the papers, the notation was slightly the models used in the thesis work are the sub­ changed when appropriate. ject of section 4. The application of the models and results from the simulations are described 1.1 Content of the Thesis in papers PI - P6. Natural sedimentary rocks are heteroge ­ When one fluid displaces another in a dis­ neous structures. The heterogeneities exist ordered, complex medium such as a porous over a wide range of spatial scales and af­ rock, a complicated pattern begins to form - fect the underground flow. Some rocks are here the regions that are occupied by the one fractured and the fluid flow is concentrated fluid, there the regions that are occupied by onto a network of fractures. The hetero ­ the other fluid. The displacement process oc ­ geneities introduce additional complexity into curs far from thermodynamic equilibrium and the problem of slow two-phase flow. Studies is irreversible. In general, the overall structure of flow through heterogeneous porous media of the patterns resulting from such displace­ and through a single fracture based on the in-

1 1 OVERVIEW

of the process. A computer simulation may serve as a prototypical conceptual model of thought, even if all features of the experimen ­ tal patterns cannot be reproduced. With all of the computer models presented here, the growth and development of one or more fluid clusters that were surrounded by an immiscible second fluid was studied. The geometry of the clusters and the dynamics of the growth in a random medium was in the fo ­ cus of interest. Typical questions to ask were: How big are the clusters? Do they have a typ ­ ical size? Do they have fractal properties? Do FIGURE 1: Scanning electron micrographs of a the properties change in the course of the dis­ Berea sandstone rock sample with magnifications placement? Is there a simple relationship be­ given by the bar scales in microns. From Wardlaw tween the properties of the simulated medium and Taylor. 2 and the cluster geometry? vasion percolation model are described in sec­ tion 5. The studies are further discussed in pa­ 1.3 Slow Two Phase Flow and Per­ pers P6, P7, and PS. Section 5 also contains colation Theory brief discussions of experimental and compu ­ tational studies related to the thesis work. Transport processes in random media are re­ Most of the thesis work dealt with the lated to percolation theory. 3'4 The general for ­ study of the scaling properties of the struc­ mulation of the percolation problem is con ­ tures observed in simulations of two-phase cerned with elementary geometrical objects flow. A brief overview over the results ob ­ (spheres, sticks, sites, bonds, etc.) placed at tained is presented in section 6. This section random in a (/-dimensional lattice or contin ­ is intended to put the various issues treated uum of infinite size. Objects are connected if in the previous sections in a common context the distance between points belonging to dif­ based on the principles of percolation theory. ferent objects is less than a connectivity radius R. Percolation theory deals with the size and 1.2 Matters of Interest structure of the clusters formed by objects that are connected to each other. If the density tt The understanding of the immiscible displace­ of connecting objects exceeds a threshold 7rc, ment of two fluids in a porous or fractured an infinite cluster of connecting objects spans medium and of numerous other natural pat­ the space. - The infinite space may represent a tern formation processes is still at an early porous of fractured medium that is saturated stage. Important questions that have not been with one phase, and the objects may repre­ completely answered include: What is the sent pores or fractures in the medium that are right language to describe the shapes that ap­ filled with a second phase. Flow of the sec­ pear? Which features are common to different ond phase through the medium is possible if patterns observed in similar processes? What the concentration of pores or fractures filled role is played by heterogeneities in time and in with the phase exceeds the threshold concen ­ space? How "random" are the patterns, and tration. to what extent are the shapes pre-determined The percolation problem describes the by the medium and by the history of the pro ­ simplest possible phase transition with non ­ cess? trivial critical behavior. The transition is In the present work various computer purely geometrical in nature and can be models are discussed that were developed described in terms of fractal geometry. Al­ with the aim of simulating pattern growth though the rules that govern the connectivity processes arising in slow displacements. The of elementary objects are well defined, the construction of an algorithmic model can be structure of the percolation cluster at the a helpful step to arrive at an understanding threshold concentration is far from being

2 1.4 Slow Two Phase Flow in Industry and Technology

groundwater hydrology, soil science, and nu­ clear waste management. The understanding of such transport processes addresses many of today's energy and environmental concerns. Understanding is required on length scales ranging from 1 -100 pm for flow on the pore level, through 1 -100 cm for flow through core samples, to 1 - 100 km for flow through geo ­ logic reservoir bodies. 7 The first investigations of slow two-phase flow in porous media8,9 were motivated by agronomical needs to study the properties of soil. Soil is partially saturated with water, and Figure 2: Illustration of the basic percolation the distribution of water changes constantly problem. Given a connectivity rule and a ran ­ due to precipitation transport and evapora ­ dom pattern of occupied sites (black) on a lattice of tion processes. sites, what are the properties of the clusters of oc ­ Oil and gas, which amount for 39 % and cupied sites? Is there a spanning cluster such that 22 %, respectively, of current (1993) energy a walker can cross the lattice, if the walker is re­ consumption, are stored in porous or frac­ stricted to walking on occupied sites? In the figure, tured reservoir rocks. Significant quantities of a fraction r = 0.5 of the sites was occupied. oil are produced from fields widely separated from known oil sources. The mechanics of oil migration and accumulation in sedimentary understood. Correspondingly, the current de­ basins remain controversial. 10 - In primary oil scriptions of two-phase flow in random media recovery a reservoir produces oil under high focus on phenomenological approaches, in internal pressure. In secondary oil recovery, a the absence of a fundamental theory. reservoir is flooded with water to displace and Percolation models are qualitative models drive the oil in the reservoir to a producing of two-phase flow in which a random medium well. In enhanced secondary oil recovery, the is represented by a two- or three-dimensional reservoir is flooded with polymers. In all these lattice, see Fig. 2. The sites or bonds of the lat­ processes slow multiphase flow plays an im­ tice represent regions in the medium that are portant role. Better insight into underground either completely filled with one fluid, or com ­ flow would be of value in oil exploration, in pletely filled with the other fluid. the assessment of reservoirs, and in the opti ­ The displacement process is modeled mization of economic oil recovery. based on a Monte Carlo procedure such that Deep caverns and mines may be used the simulation is of stochastic nature. Percola ­ as future storing sites for high-level nuclear tion models cannot account for viscous effects, waste. Groundwater entering the repository and there is no parameter that corresponds and carrying radioactive waste and gases to to the viscosity ratio of displacing fluids. The the surface is a major potential hazard. The use of percolation models is thus restricted safety of underground disposal depends on to studies of displacement processes at low the groundwater flow patterns and the trans­ capillary numbers, similar to the use of IP port properties of the rock surrounding the models. Percolation-based models can be ex­ repository. tended to simulate three-phase displacement processes in porous media.5,6 Chemical spills from industrial sites cause a concentrated input of chemicals into the ground, often occurring in a very short pe­ 1.4 Slow Two Phase Flow in Indus­ riod of time. The degree of environmental try and Technology contamination of soil and water resources de­ pends on the underground transport. Pre­ The slow simultaneous flow of two immis­ dictions of contaminant migration are often cible fluids in porous and/or fractured me­ based on the results of field-scale tracer exper ­ dia is of relevance in many branches of tech­ iments and are dependent on an understand­ nology, including oil reservoir engeneering, ing of the flow and displacement processes in

3 1 OVERVIEW the contaminated soil. by air in narrow pores. Controlling the drying Dry wood is mechanically stronger than process is necessary to obtain large quantities untreated wood, it is lighter, and it is a bet­ of dry wood of good quality. Other materials ter thermal isolator. The drying process may that require drying include agricultural prod ­ be regarded as the slow displacement of water ucts, ceramics, and paper.

4 2.1 Physics of Slow Displacement

2 Slow Two-Phase Flow in Porous Media and in Fractures In very slow displacement processes, vis­ the wetting fluid by an amount Pc to sustain cous forces can be neglected, and the forces the curvature of the interface. The pressure governing the displacement are the capillary difference (called capillary pressure for short) forces and the buoyancy forces. Capillary is given by the Young-Laplace equation forces arise if the fluids that displace each other have a finite interfacial tension. Buoy ­ = + (2.1) ancy forces arise if the two fluids have dif­ ferent densities and the displacement is not where a is the surface tension of the two fluids confined to the horizontal direction. Sec­ with respect to each other, and R\ and % are tion 2.1 presents the basic physics governing the principal radii of curvature at any point slow fluid-fluid displacement processes in the on the interface. In a pore of circular cross- presence of buoyancy. section with radius r, the radius of curvature The mechanisms of displacement of a non ­ of the interface between two immiscible fluids wetting fluid by a wetting fluid are different is Ri = R2 =| rj cos 9 |. From Eq. (2.1), a from the mechanisms of the displacement of non-wetting fluid displaces a wetting fluid if a wetting fluid by a non-wetting fluid. Sec­ its pressure Pnw exceeds the pressure Pw of the tions 2.2 and 2.3 summarize the dispacement wetting fluid by the threshold capillary pres­ mechanisms for the two cases. sure 2(7 cos 9 (2.2) 2.1 Physics of Slow Displacement Pc = r in Pores The displacement is reversed if the pres­ The interface between two immiscible fluids sure of the wetting fluid is increased such that in a porous medium or a fracture meets the Pw - Pc > Pnw ■ The contact angle 6 is usu­ solid surface of the matrix at a contact angle 9, ally not constant but depends upon whether see Fig. 3. The wetting phase is the phase in the three-phase line of contact is advancing or which 6 < 90°. receding over the solid surface (contact angle hysteresis), 11,12 due to adsorption processes involved in the displacement process.

2.1.2 Capillary Pressure Hysteresis A porous or fractured medium is character­ ized by its pressure-saturation curve relat­ Figure 3: Illustration of the wetting phe­ ing the saturation of the medium with one nomenon. Phase A is wetting with respect to phase to the capillary pressure. The pressure- Phase B such that the contact angle 9Ab is greater saturation curve reflects a second hysteresis than 90°. Phase A is non-wetting with respect to effect, as shown in Fig. 4 a. At a given relative Phase C and 6Ac < 90°. saturation Fnro (0 < Fnro < 1) of the medium with the non-wetting fluid, the equilibrium The phenomenon of capillary pressure pressure difference Pc = Pnw — Pw between across the fluid-fluid interfaced is briefly the two fluids depends on the initial condi ­ described in sections 2.1.1 and 2.1.2. General tions and the displacement process. Capillary descriptions of displacement processes in pressure hysteresis is due to the irregular pore porous media are given in sections 2.1.3,2.1.4, morphologies that characterize random me­ and 2.1.5. dia, and breaks the thermodynamic reversibil­ ity of the displacement processes. 2.1.1 Capillary Pressure Capillary pressure hysteresis may be par­ tially understood by considering the bistabil­ In equilibrium, the pressure of the non ­ ity of an air-water interface in a non-uniform wetting fluid must exceed the pressure of capillary of the type shown in Figs. 4 b and

5 2 SLOW TWO-PHASE FLOW IN POROUSMEDIA AND IN FRACTURES

possible values of the mean interface curva­ ture correspond to the maximum and mini­ mum value of the capillary pressure sustained by the interface.11

2.1.3 Displacement Steps If the pressure of the displacing fluid is var­ Figure 4: Illustration of capillary pressure ied, a stable interface configuration can be­ hysteresis. Part (a) shows a schematic pressure- come unstable. The displacement occurs in saturation curve for a porous medium after steps (called Haines jumps after Haines8 ). Haines ,9 relating the saturation Fto of the medium Figure 5 illustrates the motion of the inter­ with wetting fluid (at constant pressure Pw of face in a model capillary system consisting of the wetting fluid) to the pressure Pnw of the two tubes that are initially filled with water. non-wetting fluid. The upper branch (A) shows When the pressure of the air at the inlets of the relationship for the displacement of a wetting the tubes is increased, water is displaced re­ fluid by an invading non-wetting fluid. The lower versibly and the interfaces in both tubes move branch (B) shows the relationship for the opposite smoothly (Fig. 5 a). The boundary condition process. Parts (b) and (c) illustrate the bistability at the enlargement A of the tube cross section of a fluid-fluid interface in a non-uniform tube cause the interface to become unstable. The after Morrow. 12 If the tube is initially filled with interface passes the bulbous region instanta­ water (shaded) and slowly raised, the air (white) neously and stabilizes at B (Fig. 5 b). The dis­ displaces the water from above and the equilibrium placed water is redistributed such that the wa­ interface may be located in the narrow section ter volume is conserved during the interface above the bulbous region (b). If the tube is initially jump. The pressure of the air decreases dur­ filled with air and slowly lowered, water imbibes ing the jump and must increase again to con ­ the tube from below and the interface may be tinue the displacement process (Fig. 5 c). The located at the lower end of the bulbous region. remaining water can be displaced under re­ versible conditions until the system again be­ comes unstable as the lower bulb drains spon ­ c.12 In equilibrium, the wetting fluid rises to taneously. a height In the pore space of a porous medium, in­ 2

6 2.1 Physics of Slow Displacement

(a) (b) (c) (d)

of invading air displacement

FIGURE 5: Infinitesimally slow displacement of water (shaded) by air (white) in a model capillary system after Morrow. 12 (a) Water is removed reversibly from both tubes until the interface meets the enlarge ­ ment in tube cross section at A. (b) The interface becomes unstable and moves to a stable position in the vicinity of B. (c) Water is removed reversibly. Part (d) shows a schematic plot of the air pressure against the volume of invaded air. eter 2r is given by a pressure gradient VP is expressed through Pb = 2Ayo|g|r , (2.4) Darcy's law as15 where Ap is the density difference, and g the (U) = -&VP-p|g|z), (2.6) acceleration due to gravity. The capillary pres­ sure across a typical pore is given by Eq. (2.2). where g, rj, p and z are, respectively, the accel­ The interplay between the capillary forces and eration due to gravity, the viscosity, the den­ the buoyancy forces is expressed by the di­ sity of the fluid, and the unit vector in the ver­ mensionless Bond number tical direction. The entity K is called the per­ _ Pb _ A/>|g|r 2 (25) meability of the medium. Sahimi16 gives an Pc cr overview over the history of Eq. (2.6) and its the ratio between the buoyancy pressure and derivation. the capillary pressure across a typical pore. Equation (2.6) may be generalized to two- If one fluid is injected into a porous or frac­ phase flow by introducing relative permeabil ­ tured medium that is saturated with a sec­ ities kw and knw for the wetting and the non ­ ond fluid, buoyancy either acts against (stabi­ wetting phase, respectively: lizes) or acts with (destabilizes) the displace­ ment process. - Many displacement proces ­ (U*) = _ p„|g|z) , (2.7) ses in nature are driven by buoyancy. One *hu example is the secondary migration of oil in which oil is transported to a reservoir. In the (U„*) = - Kh ""I""F (VP - p»*|g|z) . (2.8) process of secondary migration, hydrocarbon 7]nw fluids (oil and gas) are transported from the source rocks in which they were generated, to The relative permeabilities kw and knw vary a trap or reservoir in which they are found, from 0 to 1 and depend on the relative satu­ many millions of years later (see Fig. 6). The rations 0 < F < 1 of the medium with the main driving force for the secondary migra ­ two phases, as well as on the saturation his­ tion of hydrocarbons is buoyancy. 13 tory. The concept of relative permeabilities is based on a number of assumptions that are 2.1.5 Macroscopic Description of Two- discussed by Blunt et al 17 The practical ap­ Phase Flow plication of the concept is discussed by Heav­ iside.18 Relative permeabilities are an engi ­ The average velocity vector (U) of a fluid that neering tool whose usefulness remains con ­ is flowing through a porous medium due to troversial. 19 Relative permeabilities are used

7 2 SLOW TWO-PHASE FLOW IN POROUS MEDIA AND IN FRACTURES

Structure in detail using micromodels. 20-"36 Micromod ­ els are idealized horizontal two-dimensional fccumuiated porous media in which buoyancy effects are suppressed. The models have a network of flow channels and can be used to observe, on the pore level, displacement processes. Migration Most of the micromodels used today con ­ sist of a cover plate sealed on top of a base plate in which a network of pores and pore necks has been etched.7,37 The base plate is constructed by etching a silica glass using hydrofluoric acid, or by photoetching nylon Accumulated 011 from which replicas in epoxy resin are cast.

Area of Reservoir Section 2.2.1 presents experiments on the Mature slow displacement of a non-wetting fluid by Generation 'Source Source Rock Rock a wetting fluid in porous models. Detailed observations of displacement mechanisms are figratibn reported in section 2.2.2. The "hierarchy" of the various displacement mechanisms and the role of fluid topology in the imbibition process are described in sections 2.2.3 and 2.2.4.

FIGURE 6: Map view (a) and cross section AA' 2.2.1 Water Flood Experiments (b) of oil migration in a geological formation af ­ Mattax and Kyte 20 constructed a micromodel ter Dembicki and Anderson. 14 The reservoir rock consisting of glass with 350 x 350 intercon ­ is overlain by sealing sediments and underlain by nected capillary tubes. The micromodel was source rock. Oil migrates from the area of mature, filled with oil and water was injected to study generating source rock along narrow, restricted the mechanisms of water flooding. In a water pathways to accumulation. The vertical scale is ex­ flood, water is slowly injected into a oil reser­ aggerated in part (b). voir, and the displaced oil is recovered. The quantity of interest is the amount of residual in reservoir simulators to describe the macro ­ oil that is surrounded by water (trapped) and scopic flow of two phases through inhomo ­ cannot be mobilized at low water flow rates. geneous rock. For this reason, a considerable Wardlaw and McKellar21 built various fraction of experimental and numerical stud­ glass models with ducts and pores laid out on ies of slow two-phase flow focuses mainly on a square grid. The number of ducts protrud ­ the determination of relative permeabilities of ing from a pore (the coordination number) selected fluids in various conditions. was either 4 or 8. The models were saturated At a given saturation, the relative perme­ with (non-wetting) mercury from one end ability of a fluid depends on the fluid configu ­ with a vacuum applied at the other. The ration in the porous medium. Since the fluid pressure of the mercury was slowly reduced. configuration is characterized by hysteresis The mercury was found to withdraw first loops, the relative permeability depends on from some of the smallest ducts and pores. the history of the fluid. Thereafter the mercury withdrew completely from entire domains consisting of both large and small ducts and pores. The amount 2.2 Mechanisms in Slow Displace ­ of residual mercury did not depend on the ment of Non-Wetting Fluids by coordination number but increased strongly Wetting Fluids as the ratio of pore sizes to duct sizes (the pore size aspect ratio) was increased. Local The slow immiscible displacement of a non ­ non-random heterogeneities in the models wetting fluid by a wetting fluid (called imbibi­ were found to cause trapping of mercury. tion in most of the literature) has been studied Chatzis et al. 22 constructed glass micro-

8 Figure 4

(d)

Volume of invading air

Figure 5

Figure 7

Figure 8 Figure 9 Figure 10

0.1 0.4 0.8 0.8 0.9 0.1 0.4 0.8 0.8 0.9 0.1 0.4 0.8 0.8 0.9

0.8 0.7 0.4 0.7 0.5 0.8 0.7 0.4 0.7 0.5 0.8 0.5 •j Inj. 0.9 0.8 0.9 | Site 0.9 0.8 0.9 0.9 0.8 0.4 0.5 0.7 0.2 0.1 0.4 0.5 0.7 0.2 0.1 0.4 0.5 0.7 0.2 0.1

0.5 0.2 0.8 0.4 0.7 0.5 0.2 0.8 0.4 0.7 0.5 0.2 0.8 0.4 0.7

(a) (b) (c)

Figure 13

Figure 15 2.2 Displacement Mechanisms I

(a) (b) (C) J I____I I___ I J L J L hji ir n!r n„r

Figure 7: The four dominant mechanisms in the displacement of a non-wetting fluid (white) by a wet­ ting fluid (shaded) after Lenormand et al.25 (a) Piston-like flow in a pore neck, (b) Snap-off in a pore neck, (c) II displacement in a pore, (d) 12 displacement in a pore. During snap-off, the meniscus in a duct is saddle-shaped (the two radii of curvature have opposite signs). models and varied the coordination num­ fluid, the displacement becomes either stabi­ ber, the absolute sizes of the pores, and the lized (leading to a flat fluid-fluid interface) or pore size aspect ratio. Some of the mod ­ destabilized at high capillary numbers. els were prepared with macroscopic hetero ­ geneities while others were prepared with all 2.2.2 Pore-level Mechanisms pores and all ducts of equal size. In water flood experiments, the models were saturated Lenormand et al. 25 used transparent polyester with oil and the amount of residual oil in resin to prepare small networks of rectangu ­ the models was studied. For a given model, lar ducts with 135 intersections representing repetitions of the experiments generally did pores. Different displacement mechanisms not precisely reproduce the displacement pat­ could be identified, depending on the fluid terns, and the sequences of pore-fillings were pressures, the pore geometry, and the topol ­ different. ogy of the fluid-fluid interface. In equilib ­ Vizika et al. 33,35 reported on the mecha­ rium, the pressure difference Pnw — Pw be­ nisms of disconnection and entrapment of the tween the pressure Pnw of the non-wetting non-wetting fluid using glass micromodels fluid (air) and the pressure Pw of the wetting with several thousand pores that were con ­ fluid (oil) was equal to the threshold capil­ nected by a grid of rectangular ducts. In one lary pressure Pc given by the Young-Laplace of the models studied, the ducts formed an equation (2.1). If Pw was increased, the inter­ angle of 45° with the flow direction whereas face moved either "piston-like" in a reversible the ducts were parallel or perpendicular to the manner (Fig. 7 a), or an irreversible "snap-off" flow direction in the other models. A non ­ occurred (Fig. 7 b). wetting fluid saturating the model was slowly displaced by a wetting fluid that entered the Piston-like motion and snap-offs Snap-off model with a constant flow rate. The capillary refers to the motion of the interface in a duct number such that non-wetting fluid in the duct is sepa­ rated into two disconnected regions. The wet­ ting fluid covered the surface of the model with a film. If Pw was increased, the film expressing the ratio between viscous forces swelled and the non-wetting fluid in a duct and capillary forces across a pore of typical could become displaced out of the duct. When size was in the range of 10~7 (low flow rate) Pnw — Pw became less than a snap-off thresh­ to 10-4 (high flow rate). In Eq. (2.9), rj de­ old pressure Ps, the film swelled and the en­ notes the viscosity of the invading (here wet­ tire cross-section of the duct was filled with ting) fluid, and |U| denotes the flow velocity the wetting fluid. The interface snapped and of the invading fluid. the duct was filled completely with the wet­ If the flow rate of the displacing wet­ ting fluid. - The snap-off threshold pressure ting fluid is increased, viscous forces con ­ was always less than the threshold capillary trol the displacement above the capillary pressure Pc. Snap-off displacement occurred forces. 30,31,34,38 Depending on the ratio of the only if both intersections at the ends of the viscosities of the wetting and the non-wetting duct were filled with the non-wetting fluid

9 2 SLOW TWO-PHASE FLOW IN POROUS MEDIA AND IN FRACTURES

filled with the non-wetting fluid could be­ come imbibed in a type-"I2" displacement step (Figs. 7 d and 9). The non-wetting fluid in the intersection was displaced as Pw in­ -'4\ creased. When the fluid-fluid interface moved through the intersection (position 3 in Fig. 9), the curvature of the interface decreased. An instability occurred as Pw was further in­ FIGURE 8: Behavior of the fluid-fluid interface creased and the interface touched one of the during type-11 displacement of a non-wetting fluid edges forming the pore (position 4). The non ­ (white) by a wetting fluid (shaded) in an intersec­ wetting fluid in the two remaining pores could tion of ducts after Lenormand et al.25 be displaced in a piston-like process if Pw was not reduced or Pnw was not increased. - Other configurations of the fluid-fluid interface at in­ and piston-type displacement was not possi ­ tersections were found to be very stable. ble for topological reasons.

II and 12 displacement If three of the four 2.2.3 Hierarchy of Displacement Mecha­ ducts forming an intersection were filled with nisms the wetting fluid, the non-wetting fluid in the Lenormand and Zarcone 26 found that in resin intersection could be displaced in a type-"Il" micromodel networks consisting of pores that displacement step ((Figs. 7 c and 8). The digit were connected by rectangular ducts, the four n in the terminology "In" refers to the number dominant displacement mechanisms piston, of ducts that are occupied by the non-wetting snap-off, II and 12 occurred at different dis­ fluid prior to the step. The pressure Pw of placement pressures. Air was used as a non ­ the wetting fluid was increased such that the wetting fluid and oil was used as a wetting capillary pressure Pc = Pnw — Pw was re­ fluid. The threshold displacement pressures duced and the curvature of the interface be­ depended on the contact angle 9 and on the tween the two fluids decreased (position 2 in pore geometry. For constant displacing pres­ Fig. 8). An instability occurred when the inter­ sure Pw, piston-like displacement was possi ­ face no longer touched the walls of the ducts ble at a higher pressure Pnw of the withdraw­ and moved rapidly into the pore (position 3). ing non-wetting fluid than the other mecha­ As the curvature of the interface increased, nisms. In networks with small pores (low pore Pw exceeded the equilibrium pressure Pnw - size aspect ratio) the snap-off threshold pres­ Pcr and a piston-like displacement of the non ­ sure of the non-wetting fluid was less than wetting fluid in the fourth duct (position 4) the threshold pressures for II and 12. Conse ­ followed. quently, II and 12 displacement occurred be­ An intersection with two adjacent ducts fore that snap-off occurred. In networks with large pores (high pore size aspect ratio), snap- off in pore necks occurred before that II and 12 steps occurred. Chen and Koplik 28 used micromodels constructed from molded epoxy resin with I 4/ 16 pores of random diameter laid out on a square grid and connected by random-sized U-shaped ducts. The models were filled with •• ’ 5 air, and (wetting) oil was injected through one duct. The capillary number was in the range of 10-6 « Ca « 10-3. The entire displacement process was documented in a FlGURE 9: Behavior of the fluid-fluid interface series of photographs. At high flow rates of during type-12 displacement of a non-wetting fluid the displacing oil, piston-like filling of ducts (white) by a wetting fluid (shaded) in an intersec­ was favored in models with low pore size tion of ducts after Lenormand et al.25 aspect ratio. Little or no trapping of air was

10 2.3 Displacement Mechanisms II observed. At lower flow rates snap-off events pect ratio. The model consisted of 80 bulbous were common, and trapping of air bubbles pores of different sizes laid out on a square was observed. Using models with high pore grid and connected by tubes with different size aspect ratio, snap-off occurred even at diameters. The model was filled with (non ­ high flow rates. wetting) mercury that was slowly withdrawn Li and Wardlaw29 studied the capillary and displaced by (wetting) air. The frequency pressures controlling piston-like and snap-off of displacement steps of different types and displacement in very simple glass micromod ­ the sizes of pores occupied by mercury at the els consisting of a number of pores that were beginning and end of mercury withdrawal connected to an inlet and an outlet in parallel was obtained from photographs. through capillary tubes. Displacement exper ­ A decreasing frequency of pore-emptying iments were conducted using various pairs of steps that could be characterized by their immiscible fluids showing different amounts topologic types in the order II > 12 > I2A > of contact angle hysteresis. Piston-like, type- 13 > 14 was found. Here, I2A refers to pores 12, and snap-off displacement steps were ob ­ with two collinear adjacent tubes filled with served. Several types of snap-off could be dis­ mercury, 13 to pores with three adjacent tubes tinguished, see Fig. 10. filled with mercury, and 14 to pores with all The occurrence of snap-off depended four adjacent tubes filled with mercury, re­ strongly on the pore size aspect ratio, and spectively. The majority of the pores in which to a much lesser extent on the contact angle. the mercury was displaced had small diame­ In models with high pore size aspect ratio, ters, compared to the ones containing residual snap-off occurred in some of the tubes before mercury. piston-like displacement in adjacent pores From the experiments it was concluded could occur such that the displacement pro ­ that the fluid topology at each stage had cess was influenced by the size and shape of more influence on the movement of the fluid- both the pores and the tubes, and the arrange ­ fluid interface than did the sizes of individual ment of both pores and tubes with respect pores. For constant pressure Pnw of the non ­ to the direction of the advancing displac­ wetting fluid, type-11 displacement steps oc­ ing wetting fluid. Earlier interface motions curred at a much lower wetting fluid pressure determined which displacement steps were Pw than type-12 steps. The displacement effi­ possible at later stages, and the sequence and ciency was expected to be high in comparable position of snap-off and 12 displacement steps porous media since the non-wetting fluid was determined the amount and the location of forced to withdraw first from "dead end" (or trapped non-wetting fluid. "dangling end") branches in the continuous cluster of pores filled with the non-wetting 2.2.4 Fluid Topology fluid, through type-11 steps. The non-wetting fluid could be displaced since the continuous Wardlaw and Li32 analyzed the role of fluid cluster provided a transport path to an out ­ topology further in imbibition experiments let. Only at later stages, when the pressure of using a micromodel with a low pore size as- the non-wetting fluid was further reduced to empty the larger pores, type-12 displacement steps occurred that could fragment the contin ­ uous cluster, and a fraction of the non-wetting fluid could become trapped. The local con ­ figuration of the interface or fluid topology of a given pore could change during the imbibi­ tion of adjacent pores, see Fig. 11.

2.3 Mechanisms in Displacement of FlGURE 10: Different types of snap-off steps in Wetting Fluids by Non-Wetting the displacement of a non-wetting fluid (white) by a wetting fluid (shaded) in a concave pore after Li Fluids and Wardlaw. 29 (a) Snap-off in a tube, (b) Snap- The slow displacement of a wetting fluid by a off in a pore, (c) Snap-off at a pore-tube junction. non-wetting fluid in a porous medium (called

11 2 SLOW TWO-PHASE FLOW IN POROUS MEDIA AND IN FRACTURES

Figure 11: Displacement of non-wetting fluid (white) by wettingfluid (shaded) from pores with type-11 topology causes non-wetting fluid to withdraw by type-11 steps from pores which previously had type- 12, type-13, and type-14 topology (after Wardlaw and Li32). The labels indicate the topology of the local interface configuration, (a) Configuration at the beginning of the displacement, (b) Configuration after the withdrawal of non-wetting fluid from three pores, (c) Configuration after the withdrawal from six pores. drainage in most of the literature) has been iment41 was studied and found to be consis ­ studied in similar detail as the opposite imbi­ tent with the invasion percolation model with bition process. 23,25,27'28 '30'31,34,39 '41 Compared trapping (see section 3). to the imbibition process, the fluid-fluid in­ As in imbibition, viscous forces come into terfaces observed in displacements of a wet­ play and the regime of capillary-dominated ting fluid by a non-wetting fluid tend to be flow is left if the flow rate of the displacing irregular and exhibit structures at all scales non-wetting fluid is increased.31,34,38 The dis­ down to the pore scale. Sections 2.3.1 and 2.3.2 placement process also depends strongly on present experiments on the slow displacement the viscosity ratio of the two fluids. If the vis­ of a wetting fluid by a non-wetting fluid in po ­ cosity yw of the wetting fluid is much higher rous models. Section 2.3.3 presents detailed than the viscosity rjnw of the non-wetting fluid, observations of displacement mechanisms in the pressure drop in the injected non-wetting drainage processes. fluid is negligible, and the displacement be­ comes destabilized. If yw is much lower than 2.3.1 Drainage Experiments Pnwt the pressure drop in the displaced wet­ ting fluid is negligible, and the displacement Chatzis and Dullien23 built capillary micro ­ becomes stabilized. models of pore doublets and of pore multi- plets (pore doublets connected in series). The 2.3.2 Experiments on the Secondary Migra ­ models were made of glass and filled with a tion of Oil wetting fluid (water), and a non-wetting fluid (n-decane) was injected at one end. The dis­ Of particular interest in industry-related ap­ placement and trapping of the wetting fluid plications is the study of the migration of a was studied using doublets of various diam­ non-wetting fluid through a porous medium eters. that is saturated with a wetting fluid. Pioneer ­ Lenormand and Zarcone 40 filled polyester ing migration experiments were carried out by resin micromodels containing 42,000 capillary Hubbard. 42 Catalan et al ,43 reported labora ­ ducts of random size with a wetting fluid (oil tory experiments on the secondary migration or a water-sucrose solution) and displaced the of oil using long glass columns packed with wetting fluid by a non-wetting fluid (water- glass beads or sand. The migration was found sucrose solution or air), at constant low flow to take place along restricted pathways while rate (Ca < 11 x 10-9 ). The non-wetting fluid an imbibition front formed at the bottom of the was found to form very thin fingers ("cap­ zone representing the source rock. illary fingering"). Similar fingering was ob ­ A similar experimental setup was used served at Ca = 3 x 10“6 if the displacing non ­ earlier by Dembicki and Anderson 14 who wetting fluid was more viscous than the wet­ used columns that were filled with water-wet ting fluid. The geometry of the cluster of quartz sand or water-wet sand-sized dolomite non-wetting fluid obtained in a similar exper­ grains. The migration of crude oil through

12 2.3 Displacement Mechanisms II

a distance at least seven times larger than the radius of the pore at the narrowest part. This snap-off mechanism can therefore only occur in pores with high contrast between pore size and pore neck size (high pore size aspect ra­ tio). Lenormand et al. 25 used polyester resin mi­ cromodels with small networks of intercon ­ necting rectangular ducts that were filled with a wetting oil. A non-wetting fluid (air) was in­ jected slowly into the model and displaced the oil. The non-wetting fluid was stopped by the Figure 12: Displacement pattern observed by throat at the entrance of a duct or a pore neck Thomas and Clouse 44 in a study of the secondary until its pressure Pnw exceeded the pressure of migration of oil, using a porous model of 100 x the wetting fluid Pw by a value equal to the 52 x 2.5 cm inclined at an angle of 5°. Oil (dark capillary pressure Pc (Eq. (2.1)). structure at the left-hand boundary of the model) is UPnw-Pc > Pw, the displacement process slowly rising from the lower left comer to the up­ in a duct (a pore neck) occurred piston-like. per right comer and migrates through the water- When the fluid-fluid interface reached the ad­ saturated sand (white) in the model. jacent pore or intersection of ducts, the curva­ ture of the fluid-fluid interface decreased such that the capillary pressure Pc across the inter­ the water-saturated columns suggested that face was reduced. The interface moved into the secondary migration process was very effi­ and passed the pore rapidly without collaps ­ cient, in the sense that a very small amount of ing. oil was lost as residual saturation in the pore space. In two-dimensional networks, some ducts Hirsch and Thompson 19 studied exper ­ that were filled with the wetting fluid became imentally the invasion of oil into water- surrounded by the non-wetting fluid. The saturated rocks in the range of very small wetting fluid in these pores and pore necks Bond numbers (Eq. 2.5)) between 10“5 and was trapped and could not be displaced. In 5 x 10“2. Addressing specifically the problem some cases, the wetting fluid flowed slowly of secondary migration, the importance of through the film of wetting fluid along the pore size distribution and buoyancy was edges of the ducts, so that wetting fluid from emphasized. trapped regions was transported to regions Thomas and Clouse 44 attempted to rep­ that were connected to the outlet by a path of resent a time span of 100,000 years by ob ­ ducts filled with the wetting fluid. The non ­ serving the migration of oil through a sand- wetting fluid filled the pores and pore necks filled water-saturated model for 231 days (see that were emptied in this manner. The flow Fig. 12). The migration of oil appeared to be a rate of the leak mechanism was very low and very rapid transport process on geologic time depended on the viscosity ratio of the two dis­ scales, and the transport rate may be limited placing fluids. by the oil release from the source rock rather Chen and Koplik 28 filled epoxy resin mi­ than by the capacity of the carrier bed. cromodels with 16 pores of random diameter connected by ducts of random diameter with (wetting) oil. The oil withdrew as the oil pres­ 2.3.3 Pore-level Mechanisms sure Pw was reduced. (Non-wetting) Air at at­ Roof 39 studied the drainage of a glass constric ­ mospheric pressure entered the models at the tion of approximately toric shape representing opposite entrance. The capillary number was a single pore. The pore was filled with (wet­ in the range of 10~6 to 10-3. ting) water diluted with alcohol, and a (non ­ Slow motion of the fluid-fluid interface in wetting) mineral oil of the same density as the the ducts and more rapid motion in the pores diluted water was slowly injected. The non ­ was observed. When the air had reached the wetting fluid was observed to snap off when oil outlet, no more oil could be displaced and the interface had protruded into the pore by the experiment was stopped. The process

13 2 SLOW TWO-PHASE FLOW IN POROUSMEDIA AND IN FRACTURES was reproducible at low flow rates, with the rate was increased, the reproducibility was motion at any time restricted to the meniscus lost and viscous forces affected the sequence in the largest available duct or pore. The dis­ of pore fillings. An significant fraction of the placing air always choose the widest available oil remained trapped in the network in all channel during advancement. As the flow cases.

14 3.1 The Invasion Percolation Model

3 Invasion Percolation In this thesis various variants of the the ulated displacement patterns and of trapping invasion percolation model were used to rules are briefly discussed in section 3.1.2. The simulate the slow displacement of one fluid simulation of slow displacement processes in by another in a random medium. Section 3.1 the presence of gravity is outlined in sec­ presents the invasion percolation model, and tion 3.1.3. section 3.2 discusses strengths and weak­ nesses of the model. Section 3.3 presents a variety of models that are closely related to 3.1.1 Description of the Model invasion percolation. The last section brings a Wardlaw and Taylor 2 studied capillary pres­ brief overview over other types of percolation sure curves of rock samples by mercury injec­ models that were used in simulations of tion. As a theoretical model, a hexagonal net­ two-phase flow in disordered media. work of idealized ducts and pores was con ­ The invasion percolation model maybe re­ structed. The pores were of equal size whereas garded as the archetype of network simula­ the ducts were assigned random sizes, ex­ tors. 45 Network simulators are tools to study pressed by a random number ranging from 1 two-phase flow in random media.31,46""50 In (smallest) to 6 (largest). The network was sup­ network models, the pore space of a porous posed to be filled with wetting fluid (air). The medium is subdivided into two components: "pressure" required for a non-wetting fluid pores and throats. The pores contain most of (mercury) to invade a duct was given by the the open volume available to the fluids, and random number assigned to the duct. Pores the throats serve as constrictions that sepa­ at the end of a duct that became filled by the rate the pores. Initially, the network is satu­ non-wetting fluid were filled instantaneously. rated with one fluid, and a few pores are filled The non-wetting fluid was allowed to invade with the second (displacing) fluid. The dis­ the network through the left edge only. As placement process is simulated by computing the pressure of the non-wetting fluid was var­ pressure values at the pores and by adjust­ ied from 6 (lowest pressure) to 1 (highest pres­ ing the fluid-fluid interface in small steps ac­ sure), the progressive invasion of pores and cording to the pressure field. Network sim­ ducts was studied. ulators allow the inclusion of viscosity and The model provides an accurate represen­ density effects into the simulation. The inva­ tation of the displacement of a wetting fluid sion percolation model is essentially a simpli­ by a non-wetting fluid in a porous medium in fied network simulator in which the pores and the limit of vanishing viscous forces (see sec­ throats combine to sites and the calculation of tion 2.3.3). In this case capillary forces prevent the pressure field is dropped since flow in the the non-wetting fluid from invading a narrow quasi-static limit is considered in which vis­ duct spontaneously. The capillary forces in­ cous forces vanish. crease as a duct becomes more narrow. Con ­ 3.1 The Invasion Percolation Model sequently, the displacing non-wetting fluid al­ ways chooses the widest available channel The invasion percolation model provides a during advancement. simple algorithm for the simulation of the Lenormand, 52 Chandler et al., 53 and slow displacement of a wetting fluid by a non ­ Wilkinson and Willemsen51 generalized the wetting fluid in a porous medium. The term model and pointed out its close relation to the "invasion percolation" was coined by Wilkin­ percolation problem. In the simplest form of son and Willemsen51 and refers to the connec ­ "invasion percolation" (IP), a lattice of sites is tion of the model to the general percolation considered. Each site i is assigned a random problem. The term is now also used to refer to number pi called the "invasion threshold". the physical drainage process in which a wet­ Initially, all sites are occupied with "de­ ting fluid is slowly displaced by a non-wetting fender" fluid. An injection site is chosen and fluid. filled with "invader" fluid. The algorithm The model is introduced in section 3.1.1. then consists in repeating the following three The issues of fractal dimensionality of the sim­ steps:

15 3 INVASION PERCOLATION

0.1 0.4 0.8 0.8 0.9

0.8 0.7 0.7 0.5 Inj. 0.9 0.9 0.8 Site 0.4 0.5 0.7 0.2 0.1

0.5 0.2 0.8 0.4 0.7

(a) (b) (c)

Figure 13: Illustration of the first steps in the growth of an IP cluster (shaded sites). The invasion thresholds of the empty sites are indicated. The site that will be invaded next is framed, (a) IP cluster after the first invasion step, (b) IP cluster after the second invasion step, (c) IP cluster after the third invasion step.

1. Identify all defender sites that are adja­ eolation threshold. 54-56 Figure 14 shows an IP cent to invader sites. cluster generated in this manner. - The algo ­ rithm is well suited to simulate the displace­ 2. Among these perimeter sites, find the ment of a wetting fluid by a non-wetting fluid one with the lowest threshold p,. in a porous medium, at constant (infinitesi­ 3. Fill this site with invader fluid. mal) flow rate. Each site of the lattice rep­ resents a pore of the medium. The random Figure 13 illustrates the procedure. The al­ invasion thresholds p; represent the capillary gorithm ends when the lattice boundary is threshold pressure Pc required for an invad­ reached by the invader fluid. The sites that ing non-wetting fluid to displace a wetting become filled by the invader fluid during the fluid from the pores. In each invasion step, a simulation form a connected cluster with nu­ new pore is filled with the invading fluid. merous holes and loops. Invasion percolation can also be studied The cluster of invader fluid generated in on a lattice of bonds, each representing a nar­ this fashion appears to be equivalent to an or ­ row pore neck in a porous medium. Simi­ dinary infinite percolation cluster at the per- larly, the porous medium can be represented by a site-bond lattice in which sites represent­ ing pores are connected by bonds representing ducts or pore necks (see section 4), in analogy to the pores and throats of conventional net­ work simulators.

3.1.2 Fractal Dimensionality and Trapping IP clusters have fractal properties. An IP cluster growing in a three-dimensional lat­ tice of sites has the fractal dimensionality D to 2.5.51 In clever displacement experi ­ ments using three-dimensional random po ­ 256 lattice units rous media, Clement et al. 57 and Frette et al. 5& measured the fractal dimensionality of struc­ tures of non-wetting fluid that were formed FIGURE 14: An invasion percolation cluster during the slow displacement of a wetting grown on a lattice of 256 x 256 sites. The cluster fluid. The analysis yielded fractal dimension ­ started to grow in the center of the lattice (arrow). alities consistent with the value obtained from Growth was stopped when the lattice boundary IP model simulations. was reached. No trapping rule (see section 3.1.2) In a two-dimensional scenario, the very was applied. low compressibility of the fluids may be taken

16 3.1 The Invasion Percolation Model

If g is positive, the invasion thresholds Pi tend to decrease as the structure reaches sites with greater heights representing more jj elevated parts of the medium. The IP clus­ % % ter formed during the simulation is oriented I along the direction of the gradient (destabi­ lized IP). If g is negative, the IP cluster tends to % fill out the lower parts of the lattice since the % invasion thresholds increase with increasing height. Compared to standard invasion perco ­ lation (g = 0), the front width of the IP cluster is reduced (stabilized IP). Figure 15: Illustration of the effect of the trap ­ Figure 17 illustrates these cases. The IP ping rule. The IP cluster (shaded sites) can invade cluster shown in Fig. 17 (a) was grown in the the site A since it is connected to the boundary of absence of a gradient (g = 0). The cluster is the lattice by a path consisting of nearest-neighbor a self-similar fractal on all length scales up to defender sites (dotted line). If any of the hatched the size of the system, containing holes of all sites becomes invaded, A is trapped. sizes. The IP cluster shown in Fig. 17 (b) was grown in the presence of a stabilizing gradi ­ ent (g < 0). The cluster does not contain large into account by including a "trapping rale", holes and its front is irregular only on short see Fig.15. 51,53 After each step, the unoccu ­ length scales; on length scales, comparable to pied perimeter sites that are not connected the size of the lattice, the front is stable. In con ­ to the surrounding "infinite" reservoir of de­ trast, the IP cluster shown in Fig. 17 (c) was fender fluid by a path consisting of steps be­ tween nearest-neighbor defender sites are re­ grown in the presence of a destabilizing gradi ­ moved from the list of invadable sites. Such ent (g > 0). The IP cluster formed a chain con ­ sisting of irregular, fractal "blobs". The chain a rale changes the global growth pattern of the IP cluster and reduces the fractal dimen­ is oriented in the direction of the gradient vec­ sionality from 91/48 to D ~ 1.82 in two di­ tor. mensions .51,59 This value is in agreement with In the presence of constant acceleration measurements by Lenormand and Zarcone 41 due to gravity, the Bond number characteriz­ and Frette et al. 5S of the fractal dimensional ­ ing a displacement in a porous medium can ity of structures of non-wetting fluid obtained vary spatially due to heterogeneities in the in displacement experiments in two-dimen ­ medium. Heterogeneities in the porous me­ sional random porous media. dia can be modeled by varying the mean value (Q) of the random variable in Eq. (3.1) in dif­ ferent regions of the lattice (see section 5.1). 3.1.3 Including Gravity and Heterogenities The effects of gravity can be included by im­ posing a gradient on the invasion thresholds assigned to the sites or bonds of the lattice.60-62 The threshold p,- of the ith site is given by

Pi — Qi 9Vi ; (T.l) where Qi is a random number uniformly dis­ tributed over the range 0 <£?,-< 1, % is the height of the site above a fixed level, FIGURE 16: Illustration of the assignment of inva ­ and y is the "gradient" representing the effects sion thresholds in simulations of gradient IP. The of the acceleration due to gravity acting on invasion threshold p, of the ith site is the sum the density difference between the two fluids. of a random number and a gradient contribution Figure 16 illustrates the concept. For conve ­ (Eq. (3.1)). The grey scale represents the invasion nience, the magnitude |p| of the gradient can thresholds of the sites, with the darkest shades rep­ be interpreted as the "Bond number" Bo char­ resenting the highest thresholds. The direction of acterizing the displacement (see section 2.1.4). the gradient is indicated (arrow).

17 3 INVASION PERCOLATION

Section 3.2.1 presents comparisons of ex­ perimental displacement processes and sim­ ulated displacement processes using the IP model. Issues that limit the use of the IP model are briefly discussed in section 3.2.2.

3.2.1 Invasion Percolation Compared to Ex­ periments Since its introduction, the invasion percola ­ tion model has been submitted to a number of experimental tests. As mentioned in sec­ tion 2.3.1 and in section 3.1.2, Lenormand and Zarcone 41 compared the structure of clusters of non-wetting fluids observed in displace­ ment experiments to the structure of IP clus­ ters. The patterns were found to agree in terms of the fractal dimensionality. In subse­ quent experiments, Lenormand et al. 30'31 stud­ ied slow two-phase displacements using large micromodels and a range of capillary num­ bers and viscosity contrasts characterizing the displacements. In the limit of very low capil­ lary numbers, the displacement patterns were found, from visual inspection, to be very sim­ ilar to IP clusters. A measurement of the fractal dimensional ­ ity of clusters of non-wetting fluid in a three- dimensional porous medium was reported by Clement et al., 57,63 as mentioned in sec­ Figure 17: Displacement patterns generated by tion 3.1.2. In these experiments, (non-wetting) two-dimensional invasion percolation models with molten Woods metal was injected slowly into trapping. The non-wetting fluid (black) was in­ a porous medium consisting of crushed glass. jected at the bottom faces of the lattices. Part a The geometry of the cluster of Woods metal shows a pattern generated by a site invasion per­ was studied by analyzing cuts through the colation model with zero gradient. Part b shows a cluster after the metal had hardened. The frac­ pattern generated with a (stabilizing) gradient of tal dimensionality of the cluster was found to 10~2 and part c shows a pattern generated with a be in qualitative agreement with the one of IP (destabilizing) gradient of -10~2. The direction of clusters in three dimensions. the gradient is indicated (arrow). Displacement experiments in three dimen­ sions in the absence of gravity effects were conducted by Frette et al. 58 Buoyancy forces 3.2 Strengths and Weaknesses of were excluded by using immiscible fluids the Invasion Percolation Model with the same density. The experimental cell consisted of a random packing of glass beads Any simulation of physical reality is dwarfed with the same index of refraction as the sat­ by the overwhelming richness of nature. Sim­ urating wetting fluid. The non-wetting fluid ulations are limited by constraints regarding was injected through a tube into the center computational power and algorithmic com ­ of the cell. For very slow displacement rates, plexity. It is thus surprising and fortuitous projections of the cluster of non-wetting fluid that the simple invasion percolation algorithm were found to be similar to projections of IP captures the essential characteristics of a pro ­ clusters in three dimensions. cess of such diversity as the displacement of Shaw65 carried out drying experiments us­ one fluid by another in a random medium. ing refraction index-matched materials. The

18 3.2 Strengths and Weaknesses of the Invasion Percolation Model

Figure 18: Comparison of patterns observed in slow displacement experiments and in gradient IP sim­ ulations. (a) and (b) Projections of structures of non-wetting fluid that was migrating slowly through a three-dimensional porous medium at Bo = 0.4 and -0.046, respectively, (c) and (d) Projections of simu­ lated IP clusters obtained with gradients g = 0.027 and 0.003, respectively. From Meakin et al. 64 process of evaporation was regarded as a form profiles measured in invasion percolation of immiscible displacement, characterized by simulations. counterflow of the displaced fluid (water) in the same pore as the displacing fluid (air). The 3.2.2 Limits of Invasion Percolation drying process was compared to invasion per­ colation without trapping, and good agree ­ Relative Permeabilities A major interest in ment was found. simulations of slow two-phase flow is the Birovljev et al. 66 studied the influence of determination of relative permeabilities that buoyancy on slow two-phase flow, using im­ may serve as input parameters for reservoir miscible fluids of different density and two- simulators (see section 2.1.5). However, it is dimensional porous cells that consisted of a not obvious how to extract relative perme­ monolayer of glass beads. The cells were abilities from IP simulations. The general ap­ filled with a wetting fluid and tilted by an an­ proach is the following: gle such that the magnitude of the buoyancy forces could be varied. The experiments were 1. For each phase (invader fluid or de­ simulated using a gravity invasion percola ­ fender fluid), assign a conductance to tion model, and good agreement was found. each occupied site (bond). Regions that Hulin et al .61 and Meakin et al. 62'64'67 studied are not occupied by the phase have zero displacement experiments in the presence of conductance. buoyancy forces using three-dimensional po ­ 2. Impose a unit pressure drop AP across rous cells. The displacement patterns were the lattice. compared quantitatively to gravity invasion percolation models and found to be similar 3. Find the overall hydraulic conductance (see Figure 18). of the occupied sites (bonds) by solving Good quantitative agreement between the Laplace equation V2P = 0 with the the gravity invasion percolation model in appropriate boundary conditions. three dimensions and displacement experi ­ ments using bead packs was also reported On a two-dimensional lattice, this procedure by Chaouche et al. 6& In subsequent stud­ may lead to difficulties since only one phase ies,69 the saturation profiles measured in can span the lattice at each stage, The relative displacement experiments in two and three permeability of a phase is then either 0 or 1. In dimensions were compared to corresponding the presence of a de-stabilizing gradient, both

19 3 INVASION PERCOLATION the invader fluid and the defender fluid may The geometrical complexity of natural ran­ span the lattice in the direction of the gradi ­ dom media is thus not captured by the inva­ ent (see section 3.1.3), and more realistic val­ sion percolation model. While it is possible to ues for the relative permeabilities may be ob ­ carry out simulations using more complicated tained. On a three-dimensional lattice, both lattices such as Voronoi polyhedra, the sim­ phases may span the lattice under all circum­ plicity of regular lattices is convenient for the stances. A discussion of the calculation of rela­ programmer. Using regular lattices does not tive permeabilities from percolation models is jeopardize the quality of the simulation due given by Blunt ef al. 17 to the close relationship of IP to the percola ­ tion problem. Percolation theory is concerned with universal properties of the geometrical Failure to Model Imbibition and Viscous objects and patterns considered. The overall Flow The IP model in its simplest form properties of a percolation pattern are not af­ cannot account for the various displacement fected by the microscopic structure of the un­ mechanisms that may occur during the dis­ derlying lattice but depend only on the dimen­ placement of a non-wetting fluid by a wetting sionality of the lattice.73 fluid (see section 2.2.2). Extensions of the IP Not only the arrangement of pores in nat­ model with the aim to capture the physics of ural porous media is complicated, but also imbibition processes are the subject of section the shapes and sizes. The IP model is based 4. However, even drainage processes are not on the assumption that the invasion of the possible without flow of both fluids. Fluid non-wetting fluid through a pore throat into a flow usually leads to viscous forces, as ideal given pore is determined by a single constric ­ fluids with zero viscosity do exist only close tion at the narrowest part of the pore throat so to 0 K (superfluid He). In the IP model, vis­ that the geometry of the remaining part of the cous forces are disregarded by definition. This throat is irrelevant. The sizes of the constric ­ circumstance limits the application of the IP tions may be distributed according to some model to scenarios in which the displacement distribution that is unknown in many cases. process occurs very slowly and the viscous In the IP model, the distribution of invasion forces become vanishingly small, as pointed thresholds may approximate the size distribu­ out above. tion of natural constrictions. However, only the order (rank) of thresholds is important; the Film Flow In the slow displacement of a shape of the distribution does not influence wetting fluid by a non-wetting fluid, flow the invasion process. At each stage, the site of the wetting fluid through films along the with the lowest threshold is invaded, indepen­ edges of the porous matrix was observed (see dent of the magnitude of the threshold. section 2.3.3). This type of flow is not captured For the same reason, the IP model does not by the IP model since pores are assumed to be account for the effects of variations of the con ­ occupied completely by either the invading or tact angle 6. The invasion threshold assigned the defending fluid. Attempts to model film- to a given site represents the capillary pres­ flow are discussed in section 4. sure Pc required for the invasion of a circular pore. The capillary pressure Pc depends on 6 through a constant factor (see Eq. (2.2)). The Irregularity of Natural Porous Media Nat­ inclusion of a contact angle parameter would ural porous media are, of course, not charac­ thus not affect the simulated pattern since the terized by regular arrangements of pores, as magnitude of the invasion thresholds would assumed in the invasion percolation model. be changed but not the global order of thresh­ The pore geometry of geomaterials (in which olds. most of the natural displacement processes addressed by the model take place) is irreg ­ ular and complex .2,70-72 The pore coordina ­ 3.3 Earlier Modified IP Models and tion number indicating the number of near­ Models with IP as a Limiting est neighbor pores with respect to a given Case pore varies from pore to pore. In contrast, the pore coordination number is constant for Time Scale Invasion percolation models all "pores" (sites) on a simple regular lattice. have no intrinsic time scale. In most experi-

20 3.4 Other Percolation Models merits the injection rate is constant such that model produced irregular patterns similar the in equal periods of time an equal amount IP clusters known from invasion percolation of pore space is filled. The pore volume models. In the wetting limit (9 — 0°), patterns may vary greatly, and the number of pores with - on a large scale - flat interfaces were that are invaded in equal periods of time is obtained. The algorithm differs from IP since then not constant. M&loy et al. 74 introduced it requires the calculation of the microscopic a modified site-bond IP model in which a configuration of the interface, and since it at­ capacitive volume was associated with each tempts to represent displacement at constant invaded site. The sites were connected to pressure, rather than at constant (infinitesi­ each other by bonds. The "pressure" of the mal) flow rate. invader fluid was increased at a constant rate. After each invasion process, the pressure dropped as invader fluid was redistributed Cross-over to DLA Fernandez et al. 80 in the invaded bonds and sites. Bonds and modeled fluid-fluid displacement in two sites that were adjacent to a newly invaded dimensions by a modified diffusion-limited- region could be invaded instantaneously if aggregation (DLA) algorithm in which the invader pressure was sufficiently high. random "capillary forces" were taken into If no further invasion step was possible, the account. Each site on a square lattice repre­ pressure increased again as time elapsed. sented a region of space and was assigned a random number representing the capil­ lary pressure characterizing the region. The Drying Prat75,76 reported on a two- Laplace equation with the boundary condi ­ dimensional model of drying under isother ­ tions given by the random capillary pressures mal conditions. Each bond on a lattice was was solved on the lattice, using a DLA al­ assigned a size and was initially filled with gorithm. The magnitude of the capillary a wetting fluid. Non-wetting fluid invaded pressures was related to the capillary number through one edge of the lattice. The remaining Ca characterizing a physical displacement three edges were impenetrable. The wetting process. In the limiting case of very low Ca, fluid "evaporated" and the evaporation flux the displacement patterns were similar to from a region of wetting fluid was assumed invasion percolation clusters. to pass entirely through the widest bond at the interface between the two fluids. In each step, the non-wetting fluid invaded the bond Directed Invasion Percolation De Arcange- among all the bonds at the interface that lis and Henman81 presented a gradient IP carried the highest evaporation flux. model in which a radial gradient was imposed on the invasion thresholds assigned to the bonds on a square lattice. "Directed IP" with­ Role of Contact Angle Cieplak et al. 77 ~79 pre­ out trapping was studied in which the bond sented a model of capillary displacement in that was invaded in an invasion step had the a two-dimensional porous medium consist ­ lowest threshold among all unoccupied bonds ing of an array of disk of random radii. The that were located to the North or East (in two medium was assumed to be saturated with dimensions) of an adjacent invaded bond. The one fluid and a second immiscible fluid was injected. The wetting angle 6 characterizing idea was extended by Herrmann et al.82 who studied a directed percolation model for the the combination of fluids was a tunable pa­ penetration of a non-wetting fluid into soil rameter of the model. At a fixed simulated that becomes damaged in the process. A sec­ capillary pressure Pc, a stable interface be­ ond gradient was introduced corresponding tween the two fluids consisted of a set of arcs to the effect of a cloud of microcracks arising between consecutive disks. Each arc attached besides the central flow path. to both disks at the proper angle 9. The radii of the arcs were all equal and inversely pro ­ portional to Pc (see Eq. (2.2)). When the pres­ 3.4 Other Percolation Models sure of the invading fluid was increased, some of the arcs became unstable, and the inter­ Apart from invasion percolation, various face advanced in a forward step to the nearest other percolation models were developed disk. In the nonwetting limit (9 = 180°), the to simulate slow two-phase flow in ran­

21 3 INVASION PERCOLATION dom media. To complete this section, some with pi < p with the invader fluid, where p percolation models are briefly discussed here. was a threshold. When the threshold was in­ Percolation models are qualitative models creased from p to p + dp, only those sites were of two-phase flow in which a random medium invaded in addition that were connected to the is represented by a two- or three-dimensional boundary of the lattice by a path consisting lattice. The sites or bonds of the lattice rep­ of steps between nearest-neighbor defender resent regions in the medium that are either fluid sites. The defender fluid in the regions completely filled with one fluid, or completely that were surrounded by the invader fluid was filled with the other fluid. trapped and could not be "displaced". The displacement process is modeled Pyrak-Nolte et al. 85 used an approach based on a Monte Carlo procedure so that the closely related to both percolation with trap­ simulation is of stochastic nature. Percolation ping and invasion percolation to simulate models cannot account for viscous effects, the slow invasion of a wetting fluid into a and there is no parameter that corresponds single fracture that was initially filled with to the viscosity ratio of displacing fluids. The a non-wetting fluid. Each site i of a two- use of percolation models is thus restricted dimensional lattice represented a region of to studies of displacement processes at low the fracture with aperture 6, . The distribution capillary numbers, similar to the use of IP of apertures was obtained from a statistical models. model. The wetting fluid invaded through Models for the simulation of imbibition one edge of the lattice and occupied all sites and of drainage are briefly reviewed in sec­ that were adjacent to the invader front and tions 3.4.1 and 3.4.2, respectively. A percola ­ that had an apertures b, < b, where b was tion model for the simultaneous invasion of a threshold given by the "pressure" of the both drainage and imbibition is reviewed in invader fluid. The non-wetting (defender) section 3.4.3. fluid withdrew through the opposite edge of the lattice. When the wetting fluid pressure (or b) increased, additional sites could be 3.4.1 Percolation Models for Imbibition invaded by the wetting fluid unless the sites were completely surrounded by the wetting Lenormand and Zarcone 83 used a two- dimensional square lattice of bonds and sites fluid such that the non-wetting fluid could to simulate the slow imbibition of a planar not withdraw. flow cell. Initially the network was empty Barabasi et al .86,87 studied rough fronts oc ­ (drained). Bonds at random positions in curring in slow imbibition of water into pa­ the lattice were filled, one after one. These per in the presence of gravity and presented bonds represented the most narrow throats in a model for the growth of the fronts. The the cell. After each bond filling step, it was sites on a lattice were either "blocked " or "un­ checked if the filled bond was adjacent to a blocked" for the imbibing invader fluid. Un­ second filled bond. In this case the site at the blocked sites could be invaded if they were ad­ intersection of the two adjacent bonds, and jacent to invaded regions. Blocked sites could the two remaining bonds connected to the be invaded if an unblocked site in the same intersection, were filled. Eventually, these column on the lattice with greater height was filling processes induced new bond and site invaded. The model could be mapped onto fillings, leading to a nucleated cluster growth. the directed percolation problem. The random filling of bonds modeled snap-off Blunt et al. 17 studied a percolation model displacement steps, and the filling of sites for imbibition using a cubic lattice of sites that modeled 12-type displacement steps (see were connected by bonds. Each site repre­ section 2.2.2). sented a spherical pore of constant size, and Dias and Wilkinson 84 studied a percola ­ each bond represented a pore throat with a ra­ tion process in which a "trapping rule" (see dius n that varied according to a distribution also section 3.1.2) was included. All sites on /(r>). The imbibition process was assumed to a lattice were initially filled with the defender be controlled by throat filling through snap- fluid, and each site was assigned a random offs, and by II pore filling steps. Initially, all number p,-. The saturation of the lattice with sites and bonds were filled with the defend­ the invader fluid was increased by filling sites ing non-wetting fluid. The tubes were filled

22 3.4 Other Percolation Models

one at a time, starting with the thinnest and and increased slowly such that the interface then the next thinnest and so on. This process moved at only one column during a step. The modeled the filling of throats through snap-off geometry of the interface was shown to de­ steps, i.e., film flow of the wetting fluid that velop a self-affine character and to have un­ coated the surface of the pores and throats. At usual scaling properties. any stage in the simulation, if an empty site had five (all but one) bonds connected to it that were filled with the invading wetting fluid, 3.4.2 Percolation Models for Drainage the site and the last bond were invaded too. A Hirsch and Thompson 19 modeled the sec­ trapping rule was included by filling of bonds ondary migration of oil in a three-dimensional and sites only if the displaced defender fluid porous medium. In the scenario considered, could escape to the outlet. a non-wetting fluid slowly entered the bottom Bragard and Lebon 88 simulated capillary face of a porous medium that was initially sat­ rise in a porous medium on the sub-pore level. urated with a wetting fluid. At each point The porous medium was modeled by a square in time, capillary equilibrium was reached. lattice in which some of the sites were filled The pressure in the non-wetting fluid was per­ randomly with "matter" to represent the solid mitted to rise to the value necessary to over ­ matrix of the medium. The remaining sites come capillary pressure. A cubic lattice of were filled with non-wetting fluid. The void bonds represented the pore necks in a porous sites at the bottom of the lattice were filled medium with cylindrical geometry. At the be­ with wetting fluid. Capillary rise of the wet­ ginning of the simulation, all bonds were filled ting fluid in the medium was simulated by fill­ with the wetting fluid, and a random number ing additional void sites with the wetting fluid representing the size of a pore neck was as­ that were below or at the same height as ad­ signed to each bond. Non-wetting fluid could jacent wetting fluid sites. Propagation of the invade through the bottom face of the lattice. wetting fluid in the vertical direction was pos ­ At a given pressure of the non-wetting fluid, sible up to a height given by an equation sim­ all bonds at the front of the invaded regions ilar to Eq. (2.3), where the width of the "cap­ with sufficiently large size were invaded. The illary tubes" in the medium was determined invasion steps represented the slow displace­ by the number of void sites in a horizontal row ment at constant pressure. The pressure of the between two matrix sites. non-wetting fluid was increased in small steps Roux and Hansen89 studied the "Robin until the propagating front had reached the Hood" model 90 for the pinning of an interface top of the lattice. Gravity was included in the by impurities of random strength and applied simulation. the model to the simulation of imbibition in a heterogeneous porous medium. The model is related to more general models of interfacial 3.4.3 A Percolation Model for Drainage and growth in a random medium.91-93 The fluid- Imbibition fluid interface was discretized and defined by a single-valued function yt — y(xi), where Vidales et al.94 used a Be the lattice of bonds Xi denoted the z'th column of a square lattice and sites to study the capillary pressure hys ­ of sites. At each point a threshold pi was as­ teresis in porous media. The Bethe lattice signed to represent a pinning force. The pin­ was chosen since it allows a comparison to be ning force was able to support the pressure made with analytical results. Invasion perco ­ P of the invading wetting fluid if p, > P + lation on such a lattice was studied previously (yi-i - 2yi + yi+1), where the expression in by Nickel and Wilkinson .54 Topologically, the brackets represented the local curvature of the Bethe lattice is of infinite dimensionality and interface. The thresholds were chosen at ran­ does not represent a physical system. dom from a uniform distribution. If the pres­ In the beginning of the simulation, sizes sure could not be supported at the z'th column, were assigned to the sites and bonds of the lat­ the interface was moved by increasing p, by an tice. The lattice represented a correlated po ­ amount chosen at random, and a new thresh­ rous medium in the sense that the size of ev­ old value pi was picked. After each step P ery site had to be greater or equal to any of its was reduced to the lowest sustainable value three adjacent bonds (see section 5.1).

23 3 INVASION PERCOLATION

On the particular geometry of the Be the percolation cluster. lattice, both the drainage process and the im­ To simulate imbibition at a pressure Pnw of bibition process may be considered to be a the (receding) non-wetting fluid, all sites with pure percolation problem. During the simula­ a radius rs less than rs{Pnw ) were assumed to tion, the pressure Pw of the wetting fluid was become filled with the wetting fluid. The crit­ assumed to be constant and was set to zero. ical site radius rs {Pnw ) was given by Eq. (2.2). The pressure of the non-wetting fluid Pnw was Complete imbibition was possible if the in­ equal to the capillary pressure Pc. vaded sites formed a percolation cluster. To simulate drainage at a pressure Pnw of By using known expressions for the bond the non-wetting fluid, all bonds with a radius percolation probability and the site percola ­ greater than rb(Pnw ) were assumed to be in­ tion probability on the Bethe lattice, pressure- vaded by the non-wetting fluid. Adjacent sites saturation relationships for drainage and for had a greater radius rs and became invaded imbibition could be computed. The model ex­ by default. The critical bond radius rb{Pnw ) hibited capillary pressure hysteresis loops that was given by Eq. (2.2). Complete drainage depended on the degree of correlations im­ was possible if the invaded bonds formed a posed on the lattice.

24 4.1 Extending Invasion Percolation to Simulate Imbibition

4 Invasion Percolation and Beyond

The invasion percolation model captures gravity, the fragments may migrate on their accurately the capillary dominated displace­ own through the pore space. The migration ment of a wetting fluid by a non-wetting fluid includes, on the pore scale, both drainage and in a porous medium. Physical examples for imbibition processes. Section 4.2 presents two this scenario may be the slow injection of air extended IP models capable of simulating mi­ into a porous rock that is saturated with wa­ gration processes. The models were used in ter, or the migration of oil through water- papers P2, P3, P4, P5 and P6. saturated layers of gravel. A number of ex­ 4.1 Extending Invasion Percolation perimental tests of the invasion percolation to Simulate Imbibition model were listed in section 3.2.1. More ex­ otic applications of the invasion percolation Imbibition in networks of intersecting ducts model include the study of the traveling sales­ is characterized by four dominant mecha­ man problem .95 nisms, as described in section 2.2.2. The non ­ However, many displacement processes of wetting fluid may withdraw from ducts ei­ practical interest involve the displacement of ther "piston-like" or as a "snap-off". Displace­ a non-wetting fluid by a wetting fluid, i.e., the ments in the pores formed by the intersec­ opposite of the process the invasion percola ­ tions can occur by "II" and "12" mechanisms. tion model was designed for. Attempts to sim­ The sequence of displacement steps that occur ulate the imbibition process by means of the during an imbibition process is characterized original IP model were not successful.96 This by a hierarchy. For constant pressure of the is not surprising since the microscopic mech­ non-wetting fluid II steps occur at lower wet­ anisms of the displacement of a non-wetting ting fluid pressure than 12 steps, and piston ­ fluid by a wetting fluid in narrow pores are like steps occur at lower wetting fluid pressure different from the mechanisms of drainage than snap-offs. (see section 2). In imbibition, a variety of dis­ In random porous media (packings of placement mechanisms may occur depending glass beads, packings of crushed glass, po ­ on the pore geometries and the local configu ­ rous rock) the distinction between pores and ration of the fluid-fluid interface. Simulation pore necks may not always be clear and the of this type of displacement must take the var­ ious pore-scale mechanisms into account. In section 4.1 an extended IP model is dis­ cussed that were designed to simulate imbibi­ tion processes. The model was tuned to repro ­ duce displacement processes in porous mod ­ els consisting of a monolayer of glass beads. An application of the model is reported in pa­ pers PI and P2. In section 4.1 are also two IP-like models by Glass and Yarrington 97 and by Blunt and Scher,98 respectively, briefly re­ viewed. These models were constructed inde­ pendently of the model used in papers PI and P2. The invasion percolation model assumes Figure 19: Illustration of the mapping of a po ­ perfect connectivity of the invading fluid, rous medium onto a model of sites and bonds. Part leading to the formation of a single cluster (a) shows a schematical drawing of a cross-section of invader fluid. Experimental studies indi­ of a rock sample with pores and pore throats. In cate that this assumption is not always jus­ site-bond IP models, the rock is represented bp a tified. During the invasion of a non-wetting regular lattice of sites and bonds (b). The map ­ fluid into a pore space that is saturated with a ping is based on the assumption that percolating wetting fluid fragmentation may occur. If the fluid structures (shaded regions) have similar prop ­ invasion is driven by an external force such as erties.

25 4 INVASION PERCOLATION AND BEYOND

distinction between the various displace­ 12, piston and snap-off. The incorporation ment mechanisms becomes arbitrary, see of II and 12 pore invasion mechanisms was Fig. 19. Nevertheless, it may be a good start­ based on site invasion thresholds that were ing point to assume that imbibition in a ran­ not invariant quantities, as in standard IP, but dom medium is controlled by the same mech­ changed dynamically in the course of the sim­ anisms as imbibition in a regular micromodel. ulation. The incorporation of piston and snap- Pores that have all but one or two of their adja­ off throat invasion mechanisms required the cent pore throats filled with the wetting fluid use of a site-bond IP model in which the pores may then become invaded by the wetting fluid in the porous medium were represented as through II steps or 12 steps, respectively. Pore sites on a two-dimensional square lattice and throats that are filled with the non-wetting the throats were represented as bonds con ­ fluid and that connect a pore filled with the in­ necting the sites. vading wetting fluid to a pore filled with the The displacement of non-wetting fluid receding non-wetting fluid may become in­ from a large pore is a major barrier for the vaded in a piston-like displacement. Finally, invading wetting fluid due to the small capil­ pore throats in a region that is filled with the lary pressure difference required for the non ­ non-wetting fluid may become invaded by the wetting fluid to sustain the fluid-fluid inter­ wetting fluid in a snap-off process, due to film face. The displacement process was thus as­ flow along the surface of the pore space. sumed to be governed by the site invasion The correctness of these assumptions can steps. In the beginning of a simulation, a ran­ be tested by comparing experimental dis­ dom number q was assigned to each site, serv­ placement patterns to simulated displace­ ing as a measure for the size of the pore repre­ ment patterns. The adoption of an order of sented by the site. The random numbers were well-defined displacement mechanisms in the distributed uniformly on the unit interval. modeling is related to the fact that, for conve ­ Both the wetting (invader) and the non ­ nience, the simulations are carried out using wetting (defender) fluid were assumed to be regular lattices representing regular arrange ­ incompressible. Only defender fluid at sites ments of pores. Hence the models simulate that were connected to the outlet by a path of displacements in porous media with a con ­ nearest-neighbor defender fluid sites, linked stant pore coordination number, correspond ­ by defender fluid bonds, could be displaced. ing to the micromodels giving rise to the defi­ Similarly, only defender fluid at bonds that nition of the various mechanisms. The gener ­ were connected to the outlet by a path of de­ alization of the simulation results to random fender fluid sites and bonds could be dis­ porous media is based on the universality of placed. percolation patterns (see also section 3.2.2). Initially, all sites and all bonds were filled Section 4.1.1 presents a site-bond IP model with the defender fluid. Some sites and pores designed to simulate slow imbibition pro ­ at the inlet were filled with the invader fluid. cesses. The assumptions on which the model In each simulation step, all invadable sites is based are discussed in section 4.1.2. Sec­ tion 4.1.3 presents an application of the model and bonds were identified. Invadable sites in which an imbibition process occurring sub­ were sites that were filled with the defender fluid, that were adjacent to invaded bonds, sequent to a drainage process is studied. The and that could be filled without violating the last section presents brief reviews of two sim­ trapping rule. Invadable bonds were bonds ulation models that share common features that were filled with the defender fluid and with the models used in the thesis work. The that could be invaded without violating the models discussed in section 4.1.4 were devel­ trapping rule. Among the invadable sites and oped independently of the thesis work and are bonds, the site or the bond with the lowest in­ presented to put the work in perspective. vasion threshold was chosen and invaded. If several sites or bonds had the same minimal 4.1.1 A Site-Bond IP Model for Imbibition threshold, one site or bond was chosen at ran­ dom among them. In each invasion step, the In papers PI and P2, a site-bond IP model invader fluid displaced the defender fluid ei­ for imbibition was used that incorporated ther from a bond by piston or snap-off pro ­ the dominant displacement mechanisms II, cesses, or from a site by II or 12 processes.

26 4.1 Extending Invasion Percolation to Simulate Imbibition

Figure 20 illustrates the invasion threshold vasion threshold p,- of the ith bond was set ac- assignments. In the absence of gravity, the in- cording to the following rule:

{C if the z'th bond connected an invaded site with a defender site (piston-like invasion). 1 + C if the ith bond connected two defender sites to each other (snap-off).

The parameter C was a constant that was used old p, of the ith site was set according to the for notational convenience. In paper PI, C = following rule: 0. In paper P2, C = — 1. The invasion thresh-

%/2 + C if the ith site had only one adjacent defender bond (II invasion), Pi - 1/2 + <7i/2 + C if the ith site had only two adjacent (4.2) defender bonds forming a right angle (12 invasion).

Sites with other configurations of adjacent after the site invasion. bonds could not be withdrawn. The constant At each stage, the maximum invasion C in Eqs. (4.1) and (4.2) is of no relevance for threshold pm of all the bonds and sites in­ the simulated patterns in the context of simu­ vaded so far was recorded, pm represented lated imbibition. The role of C becomes appar ­ the pressure of the invader fluid and was con ­ ent in simulations of simultaneous drainage tinuously increasing from the initial value and imbibition processes (see section 4.2.1). (Pm = C), at which only piston-like bond Since the threshold for piston-like invasion invasion was possible, up to the final value of bonds was minimal, all defender bonds ad­ (pm — C +1) at which bond invasion by snap- jacent to a newly invaded site became invaded off became possible. by default in sequences of bond invasion steps The threshold assignment rules Eqs. (4.1)

piston-like snap-off bond invasion bond invasion

! ysssssss/ssssssssssJ////WS/W/SJ^ ^

C+1/2 Invasion Threshold

FIGURE 20: Schematical histogram of the distribution of thresholds pfor the four displacement mecha­ nisms piston, snap-off, 11 and 12 in the simulation of imbibition.

27 4 INVASION PERCOLATION AND BEYOND

Outlet Outlet Outlet Outlet Outlet Outlet

FIGURE 21: Illustration of a possible sequence of invasion steps in simulations of imbibition using the site-bond IP model. The displaced non-wetting fluid (white) escapes through the outlet (arrows), (a) Type-11 site invasion of wetting fluid (black) occurs with C < p < C + 1/2. (b) Piston-like bond inva ­ sion occurs at the minimum invasion threshold p — C. (c) Two type-12 site invasion steps occur with C + l/2 C). be set according to the following rule:

28 4.1 Extending Invasion Percolation to Simulate Imbibition

Outlet Outlet Outlet Outlet Outlet Outlet

-i-i-e-i «, -i-i-i-i-

Figure 21: (cont.) (d) The site invasion steps enable numerous piston-like bond invasions at the min­ imum invasion threshold p = C. (e) A type-12 site invasion occurs with C +1/2 < p < C +1. The invasion disconnects a cluster consisting of two sites and one bond, (f) A piston-like bond invasion follows. The disconnected cluster of non-wetting fluid is trapped and cannot be displaced.

{Aqi + C if the tth bond connected an invaded site with a defender site (piston-like invasion). (4.3)

1 — Aqi +C if the ith bond connected two defender sites to each other (snap-off). where 0 < A < 1 is a model parameter. Nu­ also section 4.1.4). However, in the case in merous other formulations of Eq. (4.1) are con ­ which the threshold distributions for the vari­ ceivable. However, there is no experimental ous mechanisms do not overlap (as expressed support for the validity of such a rule, and it is by Eq. (4.2)), the dependency vanishes. Thus not obvious which values should be assigned Eq. (4.2) represents the most general formula ­ to additional model parameters such as A. For tion and does not introduce a "hidden" addi­ the sake of simplicity, A = 0 was chosen in pa­ tional model parameter. A formulation of the pers PI and P2. More general formulations of model in which the thresholds for respectively the model are explored in paper P2. II and 12 displacement steps are distributed The fifth assumption is supported by ex­ with complete overlap is studied in paper P2. perimental evidence, as far as the ranking of The sixth assumption is based on both ex­ the two displacement mechanisms II and 12 is perimental observations (see section 2.2.2) and concerned (see section 2.2.4). The magnitudes by the desire to keep the model as simple as of the threshold capillary pressures depend on possible. From a topological point of view, the the size and shape of both the pores and the inclusion of higher order displacement mech­ adjacent pore throats, and on the contact an­ anisms as I2A and 13 steps is not necessary to gle. The threshold capillary pressures for ei­ simulate imbibition, provided that II and 12 ther of the two mechanisms are represented by steps are possible and that the piston-like in­ the site invasion thresholds used in the model. vasion of bonds that are adjacent to invaded In standard IP, the global order of the thresh­ sites occurs by default, as in the present case. olds determines the displacement pattern, and Figure 22 shows various stages in two sim­ the magnitude of the thresholds is irrelevant ulations of imbibition processes. The fluid- (see section 3.2.2). fluid interface at each stage is characterized In the extended form of IP used here, the by linear segments. From the initial config ­ displacement pattern is, in general, depen­ urations shown in Figs. 22 (a) and (e), II site dent on the magnitudes of the thresholds for displacement steps were topologically impos ­ both the II and 12 displacement steps (see sible in the absence of bond snap-offs. The

29 4 INVASION PERCOLATION AND BEYOND

anI::?::!::!::::

Figure 22: Stages in the simulation of imbibition using the site-bond IP model described in sec­ tion 4.1.1. The wetting fluid (black) entered through the bottom face of the lattices. Initially, the non­ wetting fluid (white) covered the entire lattice. Part (a) shows the initial stages in a simulation using a lattice that contained 20 x 40 sites, and part (e) shows the initial stages in a simulation using a lattice that contained 100 x 200 sites. Parts (b) - (d) and (f) - (h) show later stages in the simulations. The invasion of the wetting fluid occurred along the borders of the lattices since the only sites that could be displaced at each stage were the sites with at least two adjacent invaded bonds (comer sites). only sites that could be displaced at each stage be expected to be equal to the maximum in­ were the comer sites at the beginning or at vasion threshold assigned to the sites, since a the end of a linear interface segment. Since small number of sites with very high thresh­ the total number of comer sites was small (in­ old is sufficient to effectively pin the interface. dependent of the system size), the interface The maximum site invasion threshold was was "pinned" by a handful of comer sites with equal to the invasion threshold assigned to high invasion threshold. Most of the displace­ bonds that were invaded by snap-off. ment steps occurred at the latest stages of the Snap-off did not occur in the simulations simulation when sites with very high thresh­ since all sites could be invaded at pffi < C + olds were invaded. Once a pinning comer 1 and the remaining bonds could be invaded site was invaded, a sequence of subsequent 12 piston-like. In principle, this apparent weak­ site and bond piston-like displacement steps ness of the model could be remedied by us­ could occur. ing a broader distribution of bond invasion In a typical simulation of imbibition, all thresholds, as pointed out above. However, sites and bonds were initially filled with there are scenarios in which snap-off invasion the non-wetting fluid, and the wetting fluid processes do play a role even if very high invaded through the inlet. The maximum thresholds are used. The poor performance of the model shown in Fig. 22 is largely a result threshold pm among all the thresholds of the invaded sites and bonds (representing the of the initial conditions that were applied. In pressure of the invading wetting fluid) in­ the next section an application of the model is creased slowly up to the breakthrough value described in which the maximum pressure of the invading wetting fluid increased up to the pffl at which the invader fluid had reached final value pm = 1 + C so that snap-off pro ­ the outlet. For infinite system size, pffl may cesses participated in the simulation.

30 4.1 Extending Invasion Percolation to Simulate Imbibition

4.1.3 Applying the Site-Bond IP Model to bonds nor sites could be invaded. Hence, Simulate Secondary Imbibition snap-off was a displacement mechanism that participated in the simulation. The development of the model to its present As an initial step in the simulation, the in­ form was motivated by experimental stud­ vasion of air into the experimental cell was ies of imbibition occurring subsequent to a simulated using the same lattice of sites and drainage process. The description of the ex­ bonds that was used to simulate the imbibi­ periments and the corresponding simulations tion process. For the sake of completeness, are the subject of papers PI and P2. A de­ this part of the simulation is described briefly tailed description of the experimental proce ­ at this point. A detailed description is given in dure was given by Birovljev." In the ex­ paper P2. The drainage process was assumed periments, a monolayer of glass beads sand­ to be governed by the narrowest constrictions wiched between two parallel glass plates was of the cell corresponding to the bonds on the used as the quasi two-dimensional experi­ site-bond lattice. To start a simulation, a ran­ mental porous cell. The cell had two inlets, dom number Q was assigned to each bond. one at one edge of the cell and the other at the The random number Q, served as a measure opposite edge. The cell was filled with a wet­ for the size of the pore throat represented by ting glycerol/water mixture and positioned the ith bond. The numbers were distributed horizontally. One inlet was connected to a non-uniformly from 0 to 1. The shape of the reservoir filled with the wetting fluid, and the distribution was not relevant for the simula­ other inlet was connected to the atmosphere. tion pattern (see section 3.2.2). Figure 1 in paper PI shows the setup. The experiment was started by lowering the fluid In the drainage process, air is represented reservoir, leading to a reduction of the wetting by the invader fluid and the water/glycerol fluid pressure Pw. Air (non-wetting) invaded mixture is represented by the defender fluid. the cell through the inlet, and the wetting fluid All sites and bonds of the lattice were filled was displaced through the other inlet. The with the defender fluid, with the exception of invading air formed an invasion percolation ­ bonds and sites at one edge of the lattice that like pattern. The displacement was then re­ were filled with the invader fluid. The op ­ versed by lifting the reservoir to increase Pw posite edge of the lattice represented the de­ fender fluid reservoir. The invader fluid ad- and the air was forced back (secondary imbi­ bition). The experiment was terminated when no more air could be displaced. The secondary imbibition process was simulated using the site-bond IP model de­ scribed in section 4.1.1. In a typical sequence of invasion steps, bond and site invasion by piston and by II steps occurred until no more defender fluid at sites with three adjacent invader fluid bonds was present. Then a site invasion by an 12 step occurred. This always enabled a new series of II and piston invasion steps to take place. In the later stages neither sites that were invadable by II nor sites that were invadable by 12 nor bonds that were invadable piston-like existed. Then a snap-off invasion of an invadable bond occurred, FlGURE 23: Initial and final stage in the simula ­ bringing the invader fluid pressure up to the tion of secondary imbibition using the site-bond IP final value pm = 1 + C. Since the invasion model described in section 4.1.1. The non-wetting thresholds for snap-off bond invasion were fluid (white) entered at the top face of the lattices. equal, one of the invadable bonds was cho ­ Part (a) shows an IP cluster at breakthrough before sen at random. The snap-off could enable the beginning of the secondary imbibition. Part (b) further site invasions and piston-like bond show the stage at the end of the secondary imbibi­ invasions allowing the displacement process tion when no more displacement was possible. The to continue until, at the final stage, neither lattice contained 20 x 40 sites.

31 4 INVASION PERCOLATION AND BEYOND

invader fluid. Figure 23 shows two stages in the simulation of secondary imbibition, using a small lattice of size 20 x 40 sites for clarity and a lattice of size 200 x 400 sites. In the simulation of drainage, invariable sites were sites that were filled with the de­ fender fluid and were connected to the edge of the lattice representing the defender fluid reservoir by a path consisting of steps between nearest-neighbor defender sites. Invariable bonds were bonds that were filled with the de­ fender fluid and that were connected to the de­ fender fluid reservoir by a path of defender fluid sites and bonds. The remaining sites and bonds that were filled with the defender fluid FIGURE 23: (coni). Initial and final stage in the were trapped and could not be invaded. simulation of secondary imbibition similar to the The trapping rules were formulated to in­ ones shown in Fig. 23. The non-wetting fluid clude the flow of wetting defender fluid by (white) entered at the top face of the lattices. Part film flow along the surface of the porous (c) shows an IP cluster at breakthrough before the matrix.26 Each defender fluid bond had six beginning of the secondary imbibition. Part (d) nearest-neighbor bonds that could be part of a shows the stage at the end of the secondary imbibi­ path that connected the bond to the reservoir. tion when no more displacement was possible. The Collinear defender fluid bonds could form a lattice contained 200 x 400 sites. connected path even if the sites adjacent to the bonds were filled with the invader fluid. In vanced stepwise, invading either a bond or a this way, "fjords" consisting of defender fluid site in each step. In each step, among all in­ bonds could penetrate the cluster of invader variable sites and bonds the site or the bond fluid. Similar phenomena were observed in with the lowest invasion threshold p was in­ vaded. The invasion threshold pi of the zth bond was given simply by

Pi = Qi , (4.4) and the invasion threshold pi of the ith site was set to a constant value:

Pi = 0 . (4.5)

Equations (4.4) and (4.5) imply that the in­ vasion of a bond connected to a defender site led to the immediate invasion of the site. The invasion process was thus governed by bond invasions, as intended. Due to this rule the drainage model was equivalent to stan­ dard invasion percolation (with trapping) on a bond lattice. FIGURE 24: Illustration of the trapping rules used The simulation was terminated when the in the simulation of secondary imbibition. The sim­ cluster of invader fluid had reached the de­ ulated non-wetting fluid (white) entered and left fender fluid reservoir. The IP cluster served the lattice through the inlet row (arrows) and could then as a starting condition for the subsequent trap clusters of wetting fluid (black). The wetting simulation of secondary imbibition, by as­ fluid site A and the bond a are trapped. The wet­ suming that the sites and bonds of the IP clus­ ting fluid bonds b-e are not trapped since they are ter were filled with defender fluid, and that neighbors to the untrapped wetting fluid bonds f, the remaining sites and bonds were filled with g, and h, respectively.

32 4.1 Extending Invasion Percolation to Simulate Imbibition

the experiments." Figure 24 illustrates the fluid had to overcome to occupy the ith pore trapping rules. A different approach to mod ­ was then obtained using Eq. (2.2), inserting eling the film flow of wetting fluids was pre­ the effective radius r[lK The threshold was sented by Blunt et al., 17 using a site-bond IP thus not given by the random variable rp'1 as­ model. Sites could become trapped ("trap­ signed at the beginning, but a dynamic en­ ping in pores") but bonds could not become tity that was a function of the random vari­ trapped. able. As a result, the invasion of sites with sev­ eral invaded neighbor sites was preferred over 4.1.4 Related IP-like Models for Imbibition the invasion of sites with only one invaded neighbor site. The dynamic variation of the in­ In this section two simulation models are de­ vasion threshold was called "facilitation" by scribed that have features in common with the Glass and Yarrington. To include the effects of extended IP model described in section 4.1.1. gravity, a constant gradient g was imposed on Both models described in this section aim to all thresholds p,-, as described in section 3.1.3. take into account the hierarchical order of In simulations of vertical downward infil­ mechanisms that characterizes imbibition pro ­ tration of water into a column of initially dry cesses (see section 2.2.3). The model by Glass sand, Glass and Yarrington observed the for ­ and Yarrington 97 is an extended site IP model mation of a single vertically oriented macro ­ whereas the model by Blunt and Scher98 is scopic "finger" of invading water. Similar a more general network simulator that can fingers could be observed in simulations us­ reproduce invasion percolation-like displace­ ing the standard IP model. However, the ex­ ment patterns. tended IP model yielded fingers with a struc­ ture that was much different from the (frac­ Glass and Yarrington introduced a site tal) finger structure of standard IP. With facil­ invasion percolation model designed to itation, the finger structure and finger width simulate the immiscible displacement of a depended on the pore size distribution, in non-wetting fluid by a wetting fluid within pronounced contrast to standard IP (see sec­ tion 3.2.2). a porous network, under the influence of If the pore size distribution was suffi­ gravity .97 A two-dimensional network was represented by a regular array of two- ciently narrow, the effect of facilitation dom ­ inated the invasion process. At almost any dimensional pores (sites) connected to each other by throats. The throats connected to a stage, the invader fluid preferred to invaded sites adjacent to sites that had been invaded pore were always assumed to fill when the pore filled and did not affect the invasion earlier in the process. The invader fluid formed a finger with varying width and with process. Quasi-three-dimensional networks were also studied. a smooth front. If the pore size distribution To begin a simulation, a pore radius r'p1 was sufficiently broad, the variability in pore geometry dominated the invasion process and was assigned randomly to each site (i) of the facilitation was a secondary effect. The macro ­ lattice. The invasion threshold p, controlling scopic finger was meandering in the direc­ the invasion of the ith site was given by an ef­ tion of the gradient. At intermediate pore size fective radius r7 '1 — , where is distributions, the effect of gravity dominated a factor that depended on the state of the ad­ and a nearly straight finger of almost constant jacent sites (representing adjacent pores and width formed that was oriented parallel to the throats). For the two-dimensional network gradient. and a perfectly wetting invading fluid (con­ tact angle 0 = 0), if two, three, or Blunt and Scher presented a network simu­ four (in the absence of trapping) adjacent sites lator that was designed to represent the film were filled. If only one adjacent site was filled, flow of wetting fluid along the surface of r'y = 2rp\ For finite contact angle 9, the the porous matrix in immiscible displacement value r was obtained by multiplying with a processes (see sections 2.2.2 and 2.3.3). A di­ trigonometric factor in addition. mensional pore space was represented by a The invasion threshold p< representing the two- or three-dimensional array of angular capillary pressure that the invading wetting pores (sites) connected by ducts (bonds). The

33 4 INVASION PERCOLATION AND BEYOND sizes of the pores and ducts were selected at (see section 3.4.1), was observed using narrow random and distributed uniformly. size distributions of pores and ducts. The wetting fluid could invade a pore by an type-in step, with the digit n referring to the number of ducts that were occupied by 4.2 Extended Invasion Percolation the non-wetting fluid prior to the step. For Models for Fluid Migration each pore and each of the type-in mechanisms, In the experiments on secondary imbibition the mean radius of curvature Rn of the fluid- described in section 4.1.3 and papers PI and fluid interface in the pore was calculated sep­ P2, the IP-like cluster of air that was formed arately, using a set of random numbers and a after the initial drainage process disintegrated set of input parameters. The threshold pres­ into numerous fragments. Fragmentation sure that the wetting fluid had to overcome in could occur after pore invasion steps other order to fill a pore was given by pc = 2a/R n . than type -11 steps and after pore throat inva­ Ducts could be invaded either piston-like or sion steps other than piston-like throat inva­ by a snap-off mechanism. The threshold pres­ sion. The fragments were entirely surrounded sure was equal to 2a fr and a/r for piston-like and could not be displaced by the imbibing invasion and snap-off invasion, respectively. wetting fluid. If subjected to an external force The size r of ducts and pores with rectangu ­ such as gravity, the fragments may have mi­ lar cross-section was given by the inscribed ra­ grated through the cell. Gravity forces could dius of the cross-section. be imposed on the fragments by tilting the ex­ None of the pores and ducts was entirely perimental cell. filled by the non-wetting fluid; the wetting Experiments with tilted cells are the sub­ fluid was assumed to occupy the comers of ject of papers P2 and P3. In these experiments, the pore space. The pressure of the non ­ an IP-like cluster of air was formed, using an wetting fluid was zero. The pressure drop experimental cell that was positioned horizon ­ across the ducts that were partially occupied tally. The cell was then tilted slowly, leading to by the wetting fluid was calculated using a increasing buoyancy forces. When the forces generalized form of Poisseuille's law. In re­ were sufficiently high, the air cluster started gions that were occupied completely by the in­ to migrate through the cell. During the pro ­ vading wetting fluid, the fluid pressure was cess, the cluster disintegrated into IP-like frag ­ equal everywhere. ments. Initially, the pore space was saturated with If a "bubble" of non-wetting fluid is mi­ the non-wetting fluid. The wetting fluid was grating slowly through a porous medium sat­ injected from one edge or face of the pore urated with a wetting fluid, drainage and im­ space. The pressure at the inlet was increased bibition processes may occur simultaneously, in small steps during the simulation. If the on the pore scale. At the advancing end of pressure of the wetting fluid exceeded the the bubble, the wetting fluid is displaced; at threshold pressure Fluid to fill pores and ducts the receding end of the bubble, the wetting ahead of the completely invaded region was fluid displaces the non-wetting. The process supplied by flow along the comers of the may be simulated on the basis of a site-bond IP ducts and pores. In some simulations, flow model by combining the mechanisms for the along the comers was stopped. simulation of imbibition and the mechanisms Depending on the model parameters, dif­ for the simulation of drainage outlined in sec­ ferent flow regimes were observed. In the ab­ tions 4.1.1 and 4.1.3. In sections 4.2.1 - 4.2.3, sence of film flow, the wetting fluid filled the an extended site-bond IP model for the simu­ smallest pores adjacent to already filled ducts lation of the migration and fragmentation of a and an invasion percolation-like displacement non-wetting fluid in a porous medium is intro ­ pattern was observed. If the pore size distri­ duced and applied. bution was narrow, fluid topology was crucial, For general studies of migration through and flat frontal advance was observed as an 13- random media, the site-bond simulation type invasion step was followed by a cascade model may appear too complex, and sim­ of I2-tupe steps. In the presence of film flow, plifications may be attempted. A simplified nucleated faceted cluster growth, as in the per­ site IP model for the simulation of migration colation model by Lenormand and Zarcone 83 and fragmentation processes is introduced

34 4,2 Extended Invasion Percolation Models for Fluid Migration

Figure 25: Illustration of a possible sequence of migration steps in the simulation of fluid migration using the site-bond IP model. The driving buoyancy force acts in the positive vertical direction (shaded arrow). Untrapped defender fluid sites and bonds are not indicated, (a) Migration of a cluster of invader fluid (black) by simultaneous type-11 site withdrawal and piston-like bond withdrawal and invasion of a new site and a new bond, (b) Migration of a cluster by simultaneous type-12 site withdrawal and piston­ like bond withdrawal and invasion of a new site and a new bond. As a side effect, trapped defender fluid bonds (hatched bonds) become untrapped, (c) Migration of a cluster by piston-like bond withdrawal and invasion of a new bond, leading to coalescence with a second cluster. In the case of sufficiently strong buoyancy forces, snap-off withdrawal of bonds (encircled bond) may be preferred at this stage, (d) Migra­ tion of a cluster by simultaneous type-11 site withdrawal and piston-like bond withdrawal and invasion of a new site and a new bond. As a side effect, a trapped region (hatched site) becomes untrapped. and discussed in sections 4.2.4 and 4.2.5. drawal of invader fluid from a site or a bond Sections 4.2.6 and 4.2.7 present applications of and the invasion of a new site or bond. Each the model, including a study of the secondary migration step consisted either in the simul­ migration of oil in a homogeneous porous taneous migration of a site and an adjacent medium. bond, the migration of a site, or the migration of a bond. The invasion of a new site or a new bond was modeled as a drainage process, us­ 4.2.1 A Site-Bond IP Model for Fluid Mi­ ing the invasion thresholds given in Eqs. (4.4) gration and (4.5). The withdrawal from an invaded In papers P3 and P2, a site-bond IP model was site or an invaded bond was modeled as an used to simulate the migration of an IP cluster­ imbibition process, using the thresholds given like structure through a random medium. The in Eqs. (4.1) and (4.2) with C = -1. model is described briefly in paper P3 and in The migration was assumed to be opposed more detail in paper P2. In this section the by the capillary forces required to invade model is introduced with emphasis on the kin­ new regions and to withdraw from new re­ ship to the site-bond IP model for imbibition gions, represented by invasion thresholds and presented in section 4.1.1. by "withdrawal thresholds", and driven by The model simulated the migration of a buoyancy forces, represented by a gradient cluster of (non-wetting) invader fluid through parameter g. A migration step was possible if a porous medium that was saturated with the thresholds were outweighted by the buoy ­ (wetting) defender fluid. The porous medium ancy forces. The invasion of the iih site or was represented by a lattice of sites and bonds. bond and the withdrawal from the jth site or A cluster consisted of a connected structure of bond was possible if invaded sites and bonds. Ft = jOi+p'j - gAij < 0 , (4.6) The migration of a cluster was modeled as a sequence of migration steps. Figure 25 illus­ where pi, pf and A,-j denote the invasion trates a possible sequence of migration steps. threshold of the ith site or bond, the with­ Migrating a site or a bond implied the with­ drawal threshold of the jth site or bond, and

35 4 INVASION PERCOLATION AND BEYOND

Figure 25: (cont.) (e) Migration of a cluster by type-12 site withdrawal and invasion of a new site that is connected to the cluster by an invaded bond, (f) and (g) Migration of a cluster by piston-like bond withdrawal and invasion of a new bond, (h) Migration of a cluster by type-11 site withdrawal and piston-like bond withdrawal and invasion of a new site and a new bond. the separation of the ith site or bond from the fragments. The number of invaded sites and jth site or bond in the direction of the gradi ­ bonds in of a fragment remained constant dur­ ent, respectively. At each stage, the combi ­ ing the simulation if the fragment did not un­ nation of sites and bonds was identified that dergo coalescence or fragmentation. Never­ yielded the global minimum of II. If no combi ­ theless, the sizes of the pores and pore throats nation could be found that fulfilled the migra ­ represented by the sites and bonds would be tion condition Eq. (4.6), the gradient parame­ different in a real system, as the random as­ ter g was increased. signment of the numbers q and Q to the sites The withdrawal of invader fluid from sites and to the bonds implies. In the physical ex­ or from bonds could lead to fragmentation periment, all fluid-fluid interface meniscii of of the migrating cluster. The cluster frag ­ a migrating fragment were of course advanc­ ments could migrate on their own and coa ­ ing or receding during migration. The volume lesce. When searching for the global mini­ of the fragment was conserved exactly but not mum of Eq. (4.6), the destination site or des­ necessarily the number of invaded pores and tination bond i was required to be adjacent to throats. the cluster or the fragment that contained the Equation (4.6) expresses the concept of source site or source bond j. driving buoyancy forces and opposing capil­ The trapping rule described in sec­ lary forces. However, the form of the equation tion 4.1.3 was implemented and the volume is misleading. Since the withdrawal thresh­ of the trapped regions of defender fluid was olds were negative, the capillary forces re­ conserved separately. Trapping of defender quired to withdraw the (non-wetting) invader fluid could occur both within a single cluster fluid from a pore or from a site were counted fragment and in-between several fragments. negative, i.e., the capillary forces associated A detailed description of the trapping rule is with withdrawal were effectively driving the given in paper P2. migration process (along with the buoyancy forces). This issue is discussed in section 4.2.3 in the context of an application of the model. 4.2.2 Discussion of the Site-Bond IP Model for Fluid Migration When searching for the global minimum of II in Eq. (4.6), it was not necessary to com ­ The model was based on the site-bond IP pare all invariable sites and bonds i with all model for imbibition described in section 4.1.1 invaded sites and bonds j. As pointed out in and relied on the assumptions listed in sec­ the description, invader fluid from a site or a tion 4.1.2. An additional assumption was bond belonging to a given fragment could, of the implicit volume conservation of migrating course, only be transported to a site or a bond

36 4.2 Extended Invasion Percolation Models for Fluid Migration

at the perimeter of the fragment. In the simu­ fragment. Fragmentation occurred if the size lations, the minimum of II was computed of the new fragment was less than the size of with respect to all possible migration steps as­ the original fragment. In this case the new sociated to the A-th fragment. The global min­ fragment(s) were identified and added to the imum of II was found among the values list of fragments. When searching 11^1 it was not necessary to compare all invariable sites and bonds i at the perimeter of the kih fragment with all invaded 4.2.3 Applying the Site-Bond IP Model to sites and bonds of the Mh fragment. Equa ­ Simulate Migration and Fragmenta ­ tion (4.6) may be written in the form tion of IP Clusters

n =Pi+ Pj - 9Vi + 9Vj < 0 , (4.7) Papers P2 and P3 describe experiments in which the migration and fragmentation of an where ?/,- and yj denote the ^-coordinate of IP cluster-like structure was studied. The ex­ the ith site or bond on the lattice and the y- perimental setup was similar to the one used coordinate of the jth site or bond, respectively. in the studies of secondary imbibition (sec­ The y-axis was oriented along the direction of tion 4.1.3). An IP cluster-like structure was the gradient such that A,y = y, — %. Equa ­ obtained by slowly injecting air into a two- tion (4.7) consists of two terms that are associ ­ dimensional porous medium that was filled ated with the destination site or bond (the ith with a wetting water/glycerol mixture. The site or bond) and of two terms that are associ ­ experimental cell was positioned horizontally. ated with the source site or bond (the jth site When the cluster had reached a suitable size, or bond), and there is no coupling. At each the cell was sealed and slowly tilted. As stage, it was sufficient to find the ith perime­ the buoyancy forces increased, the air cluster ter site or bond that yielded the minimum of started to migrate by invading new pores and the contribution />,- — gy,, i.e., a site or a bond pore throats at its "upper" end and simulta­ that combined a low invasion threshold with neously abandoning pores and throats at its a high y-coordinate. In a second pass, the jth "lower" end. During the migration process site or bond of the fragment could be identi­ the cluster fragmented. The cluster fragments fied that yielded a minimum of the contribu ­ migrated on their own through the cell and tion pi +gyj, i.e., an invaded site or an invaded had again an IP cluster-like structure. bond that combined a low withdrawal thresh­ The experiments were simulated using the old with a low y-coordinate. site-bond IP model presented in section 4.2.1. If the Ath fragment underwent migration, In the beginning of the simulation, all sites the search for the minimal value II was re­ and bonds of the lattice were filled with the peated to identify the next migration step of (wetting) defender fluid. One site in the cen­ the fragment. If the fragment disintegrated ter of the lattice represented the injection point into smaller fragments ki and k2 during the and was filled with the (non-wetting) invader step, II^1) and were found separately. fluid. An IP cluster was formed by invad­ In each migration step, both fragmentation ing additional sites and bonds adjacent to the and coalescence with another fragment could invaded region, in the same manner as the occur. Coalescence occurred if the site or bond simulated drainage process described in sec­ that was invaded during a step was adjacent tion 4.1.3. to a bond or a site that belonged to a another When the cluster had reached a suitable fragment. In this case all sites and bonds be­ size, the invasion process was terminated. longing to the first fragment were added to the The gradient parameter g was increased in second, while the first was removed from the small steps, representing the tilting of the cell. list of fragments. After each increment, all possible migration The management of fragmentation re­ steps were carried out, beginning with the quired a considerable computational effort. one yielding the lowest value of II. Migra ­ After each migration step, a "burning" al­ tion steps could lead to fragmentation of the gorithm 73 was used to identify the invaded cluster. The fragments could migrate on their sites and bonds that were connected to the own if a valid migration step was possible. site or the bond that was invaded in the step. Fragmentation and coalescence of migrating The connected sites and bonds formed a fragments was handled as described in sec-

37 4 INVASION PERCOLATION AND BEYOND

-r

(c)

§

Figure 26: four stages in the simulation of fluid migration using the site-bond IP model described in section 4.2.1. The non-wetting fluid (white) entered through an injection site in the lower part of the lattice. The driving buoyancy forces were oriented in the positive vertical direction. Parts (a) and (e) show IP clusters formed at the beginning of the migration. The lattices contained 20 x 40 sites and 200 x 400 sites, respectively. The non-wetting fluid covered 40 sites and 4000 sites, respectively. Parts (b), (c), (d), (f), (g) and (h) show intermediate stages in the migration at g = 0.027,0.025, 0.1, 0.0018, 0.0044, and 0.0072, respectively. At the final stages, the non-wetting fluid reached the lattice boundary. tion 4.2.2. The simulation was terminated withdrawal of invader fluid from the "lower" when migrating fragments had reached the ends of migrating fragments were negative. upper end of the lattice, or when the gradient The lowest withdrawal threshold C was as­ parameter had reached an upper limit. signed to piston-like withdrawal from bonds, In each migration step, invader fluid was and the highest threshold l + C was assigned transported either from a bond to a new bond to snap-off withdrawal from bonds (Eqs. (4.1) or from a site to a new site. Favorable mi­ and (4.2)), and C = —1 was used. The gration steps involving fluid transport from a thresholds assigned to type -11 and type -12 bond to anew bond included piston-like with­ withdrawals from sites were between these drawals of invader fluid and the invasion of limits. In the migration experiments, it was a new bond with low invasion threshold. Fa­ not possible to measure the capillary pres­ vorable migration steps involving fluid trans­ sure associated to the withdrawal of the air port from a site to a new site included type -11 from a pore or a throat. In the simulations, withdrawal of invader fluid. Figure 26 shows the programmer is at liberty to change the four stages in the simulation of fluid migra ­ rules of the simulated world. When shifting tion, using a small lattice containing 20 x 40 the value of the constant C to obtain posi ­ sites for clarity, and a lattice containing 200 x tive withdrawal thresholds, the simulated pat­ 400 sites. The original IP cluster contained 40 terns did not show compelling similarities to sites and 4000 invaded sites, respectively. the experimental displacement patterns. Nat­ As pointed out in section 4.2.2, the with­ urally, migration occurred only at relatively drawal thresholds used in the simulation of high gradient values (representing strong tilt-

38 4.2 Extended Invasion Percolation Models for Fluid Migration mg of the cell) and numerous tiny cluster frag ­ ments were formed, in contrast to the experi ­ mental observation. On the other hand, pressure measure­ ments could be carried out in the experiments on secondary imbibition described in sec­ tion 4.1.3. These measurements are discussed further in paper P2. From the measurements it was concluded that most of the imbibition steps took place already at a wetting fluid pressure Pw lower than the pressure of the wetting fluid at the beginning of the exper ­ iment (when the wetting fluid occupied the FlGURE 27: Illustration of a possible sequence of entire cell and was displaced by the invading migration steps in the simulation of fluid migra ­ non-wetting fluid). When simulating imbibi­ tion using the site IP model. The driving buoy ­ tion in the type of porous cells used in these ancy force was oriented in the positive vertical di­ experiments, the use of negative withdrawal rection (shaded arrow). Un trapped defender fluid thresholds is thus justified. sites are not indicated, (a) Migration of a cluster of invader fluid (black) without fragmentation or coalescence, (b) Migration of a cluster leading to 4.2.4 A Site IP Model for Fluid Migration coalescence with a second cluster. The migrating cluster fragments. As a side effect, a trapped re­ It is possible to radically simplify the extended gion (hatched site) becomes untrapped, (c) Simul­ IP model for fluid migration presented and taneous fragmentation of a cluster and migration discussed in sections 4.2.1 and 4.2.2. The of part of the cluster. simplification reduces the model to a site IP model, at the expense of the detailed modeling of the various displacement mechanisms. The nal buoyancy forces. The capillary forces were site IP model for fluid migration presented in represented by invasion thresholds and by this section was introduced in Ref. 100 and withdrawal thresholds. The invasion thresh­ used in papers P4, P5, and P6. All of these pa­ old pi of the ith site was given simply by pers contain descriptions of the model. The notations used in those descriptions differ Pi = Qi ■ (4.8) slightly from the one used here. The withdrawal threshold p[ of the ith site The simulation model used a regular lat­ was given equally simply by tice of sites representing the pores of a po ­ rous medium. Two random numbers g, and Pi =

39 4 INVASION PERCOLATION AND BEYOND could migrate themselves and coalesce. In tion under the influence of increasing buoy ­ some simulations, a trapping rule was in­ ancy forces described in section 4.2.3. This cluded such that defender fluid sites that were study is reported in Ref. 100. Being aware of not connected to the boundary of the lattice the deficiencies of the model, an exact repre­ by a path consisting of nearest-neighbor de­ sentation of the physical reality was not ex­ fender fluid sites were trapped. Trapped sites pected. could not be invaded, to conserve the volume The simulation was carried out using a of trapped regions. Figure 27 shows a possible square lattice of sites, with two random num­ sequence of migration steps. bers q and Q assigned to each site. All sites were initially filled with the defender fluid 4.2.5 Discussion of the Site IP Model for and one site in the center of the lattice was Fluid Migration filled with the invader fluid, representing the injection point. The standard IP algorithm It is not hard to recognize the similarities be­ with trapping was applied to form an IP clus­ tween the site-bond IP model presented in sec­ ter. When the cluster had reached a suitable tion 4.2.1 and the site IP model presented in size, the buoyancy-driven migration of the section 4.2.4. In both cases the model was cluster was simulated in the same manner as based on the concept of step-wise migration described in section 4.2.3. Figure 28 shows with simultaneous drainage and imbibition snapshots from a typical simulation. displacement steps on the pore level and con ­ The same sort of simulation may be car­ servation of the invaded volume. The forces ried out using a three-dimensional cubic lat­ included in the modeling were capillary forces tice of sites.101 In three dimensions, trapping and gravity forces. of wetting fluid is very rare. Consequently, the Compared to the more sophisticated site- trapping rule was discarded. Figure 29 shows bond IP model, the site IP model modeled the snapshots from a tyical simulation. Experi­ various imbibition mechanisms in a rudimen­ mentally, a study of two-phase displacement tary manner. The withdrawals of the non ­ in a porous medium with buoyancy forces wetting fluids did not depend on the local con ­ slowly increasing from zero to a finite value figuration of the fluid-fluid interface but were could be conducted using a centrifuge. On controlled by constant threshold values rep­ the computer, the study of such a scenario resenting constant capillary pressures. The provides an even simpler example of complex withdrawal thresholds were positive. fragmentation dynamics (due to the absence The drainage steps were simulated by the of the trapping rule and its global implica­ filling of a site with invader fluid. In the tions). absence of bonds, random constant thresh­ olds were assigned to the sites to represent The studies of fluid migration in two drainage capillary pressures. In a migration and three dimensions showed that the struc­ step, the driving buoyancy forces had to over ­ tures observed in the simulations could be come both the invasion threshold of the in­ described by scaling laws similar to those ob ­ vaded site and the withdrawal thresholds of served in experimental studies of two-phase the abandoned sites. flow (see section 6). In view of the crudity of The additional simplification of a single the migration model, this fact is surprising. displacement mechanism, equal for both dis­ In paper P4, a one-dimensional version of placement of the invader fluid and displace­ the model is studied in which a "string" of ment of the defender fluid, lends attractive invader fluid moves along a line of defender symmetry to the model and reduces the com ­ fluid sites, each characterized by random putational effort. However, it also moves the invasion thresholds and withdrawal thresh­ model further away from physical reality. olds. In this scenario, the migration process is amenable to theoretical analysis. 4.2.6 Applying the Site IP Model to Simu­ In a different and more realistic scenario late Migration and Fragmentation of of fluid migration, the driving gradient force IP clusters is constant and acting throughout the entire process. Such a scenario may represent typi ­ The site IP model described in section 4.2.4 cal gravity-destabilized drainage experiments was used to study the scenario of fluid migra ­ (see section 3.2.1). Figure 30 shows a pattern

40 4.2 Extended Invasion Percolation Models for Fluid Migration

FIGURE 28: Patterns formed at six stages during a typical simulation of fluid migration in two dimen ­ sion with increasing buoyancy forces. The buoyancy forces were oriented in the positive vertical direc ­ tion. Part (a) shows the initial IP cluster with a mass of 2000. Parts (b), (c), (d), (e) and (f) show the system after all migration has ceased at gradients g of 0.01, 0.05, 0.1, 0.2, and 0.5, respectively. Black sites represent invader fluid and shaded sites represent trapped defender fluid. From Meakin et al.J0° obtained in a simulation of the migration of a the lattice. After each invasion step, all pos ­ non-wetting fluid through a porous medium sible migration steps were carried out, using that was saturated with a wetting fluid. A con ­ the site IP model for fluid migration described stant pressure gradient g was imposed to rep­ in section 4.2.4. The figure should be com ­ resent the effects of gravity, and non-wetting pared with the experimental structures shown fluid was injected at a site near the bottom of in Figs. 18 (a) and (b). Simulations of fluid mi-

41 4 INVASION PERCOLATION AND BEYOND

Figure 30: Displacement pattern obtained by simulating fluid migration at constant gravity forces. The non-wetting fluid (shaded) was injected near the bottom face of the lattice. The black struc­ ture represents the projection of the pattern, com ­ parable to the experimentally observed projections of patterns shown in Fig. 17. The gradient had a magnitude of g = 0.01. The direction of the gradi ­ ent is indicated (arrow).

driving force is usually constant. In particu­ lar, buoyancy-driven underground flow as the secondary migration of oil takes place at con ­ stant hydrostatic pressure gradients. Applica ­ tions of the site IP model for fluid migration in FIGURE 29: Patterns formed at five stages dur ­ which constant driving buoyancy forces were ing a typical simulation of fluid migration in three simulated are the subject of papers P5 and P6. dimensions with increasing buoyancy forces. The buoyancy forces were oriented in the upward di ­ Oil and gas are believed to be the result of rection. Part (a) shows the initial IP cluster with the decomposition of organic sediments - veg ­ a mass of 2000. Parts (b), (c), (d) and (e) show etable and animal organisms that lived during the system after all migration has ceased at gradi ­ previous geologic ages - under the influence of heat and pressure. This theory is supported ents g of 0.05,0.1,0.2, and 0.5, respectively. From by much geological evidence. Oil and gas are Meakin et al.202 commonly found in sedimentary basins. The reservoirs where oil is found today were filled gration at constant driving force in two di­ by hydrocarbon fluids that were generated in mensions are the subject of the next sec­ beds which contain large amounts of organic tion 4.2.7 and are discussed further in pa­ matter (generation source rocks). In some per P5. cases, the source rocks maybe located as much as 500 km away from the reservoir .102™106 4.2.7 Applying the Site IP Model to Sim­ Source rocks may overpressure as hy ­ ulate Secondary Migration Through drocarbons fill the pore space. The pressure Porous Rock buildup eventually causes microfracturing that allows transport of confined water and The computational study of scenarios with hydrocarbons out of the source rock. The changing buoyancy forces described in sec­ petroleum may migrate out of fine-grained tion 4.2.6 may give general insight into the dy ­ source beds into coarse-grained or fractured namics of transport processes in random me­ reservoir rocks (called "primary migration" dia. However, in practical applications the in some texts). Secondary migration is the

42 4.2 Extended Invasion Percolation Models for Fluid Migration

FIGURE 31: Simulation of the secondary migration of oil using a variant of the site IP model for fluid migration described in section 4.2.4. Fluid migration though a layer was simulated by increasing the thresholds in the central layer by a factor of 10. The threshold values of the sites are indicated by the grey shades at the boundaries of the lattice. Invader fluid (white) was injected through a site at the bottom of the lattice and driven in the vertical direction by a gradient of g = 0.09. The withdrawal thresholds and the invasion thresholds ranged from 0 to 1 and from 0 to 0.1, respectively. The lattice size was 128 x 128 x 128 sites. movement of petroleum, after leaving the stant value g > 0. A site in the center of the lat­ source bed, through porous, permeable tice represented the injection point. The injec­ reservoir rocks and nonsealing faults and tion point site was tilled with the invader fluid fractures. The primary forces causing migra ­ representing petroleum that escaped from the tion of petroleum are buoyancy and capillary, source bed. and hydraulic potential gradients generated The simulation began by invading a new primarily by flow of meteoric water. Even­ site adjacent to the invaded region. Among all tually, petroleum accumulates in a trap (the adjacent sites, the one with the lowest effec­ reservoir) when the upward movement is tive invasion threshold was invaded. Hie ef­ restricted by natural barriers. Reviews of the fective invasion threshold pf^1 of the ith site physics of secondary migration are given by was given by Schowalter 13 and by England et al ,104 pfff] =Pi~ 9Vi , (4.11) The process of secondary migration was simulated using the site IP model presented in where j/,- denotes the height of the ith site section 4.2.4. Figure 31 shows the final stage in above the injection point. Equation (4.11) is, of a simulation in which the migration through course, quite similar to Eq. (3.1). The concept a porous layer was studied. The simulations of effective invasion thresholds is introduced of secondary migration of oil were carried out here only for notational consistency with sec­ using a regular lattice of sites, all of them ini­ tion 4.2.4. tially filled with the defender fluid represent­ If valid migration steps were possible ac­ ing water. The sites of the lattice represented cording to the migration condition Eq. (4.10), the pores of the rock through which the oil was the invader fluid was migrating in the direc­ migrating. Two random numbers % and Qi tion of the gradient, beginning with the step were assigned to the ith site and characterized that yielded the lowest value II. When no fur­ the geometry of the pore represented by the ther migration steps were possible, an addi­ site. The gradient parameter was set to a con ­ tional site was invaded. Only defender fluid

43 4 INVASION PERCOLATION AND BEYOND sites that were connected to the injection point the paper is the finding that the trapping rule by a path consisting of nearest-neighbor in­ plays a crucial role in the formation of the vader sites could be invaded at this stage. structure. Figures 3 and 4 in paper P5 show Among all invadable sites, the one with the typical displacement patterns. The geome ­ lowest invasion threshold was invaded. try of the simulated structures could be un­ The migration steps could lead to fragmen ­ derstood in terms of standard percolation and tation of the growing IP cluster. The fragments invasion percolation theory. Paper P6 ad­ could migrate on their own in the direction dresses specifically the problem of secondary of the gradient. Migrating fragments eventu­ migration and presents qualitative computa ­ ally became stuck since each fragmentation re­ tional studies of secondary migration based duced the extension of the fragments in the di­ on the site IP model. These studies were rection of the gradient. If no more migration carried out using both two-dimensional and steps were possible, a fragment was "frozen" three-dimensional lattices using various algo ­ and could become mobilized only if undergo ­ rithms to achieve a heterogeneous distribu­ ing coalescence with another migrating frag ­ tion of thresholds (see section 5.1.3). An ap­ ment. plication of the model to study fluid migration Paper P5 describes a study of this model on through a single fracture is discussed in sec­ a two-dimensional lattice. Hie main result of tion 5.2.6.

44 5.1 Invasion Percolation in Correlated Porous Media

5 Invasion Percolation in Correlated Disordered Media The invasion percolation model described low thresholds. The various regions on the in section 3 was designed to simulate slow lattice represent regions of the medium with two-phase flow in porous media that are ho ­ large pores and regions with small pores, re­ mogeneous on all scales much larger than the spectively. pore scale. The homogeneity is reflected by However, the exact nature of the hetero ­ the threshold assignment procedure: All sites geneity distributions is not clear. In sec­ on the lattice are assigned a random num­ tion 5.1.1 geologic observations of correlated ber drawn from the same distribution. How ­ transport properties are summarized. In sec­ ever, the pore sizes in geological fields clearly tion 5.1.2 are briefly discussed various early are strongly correlated on large scales, and or recently published simulation models that there are indications that correlations also ex­ were used to study flow in correlated porous ist on scales down to the pore scale.107,108 media. A popular method to generate mul­ Flow through media with heterogeneous per­ tifractal correlations is to use iterative algo ­ meability distributions can be simulated by a rithms. Multifractal threshold distributions variety of algorithms. Section 5.1 addresses were used in paper P6 and are the subject of studies of invasion percolation using multi­ section 5.1.3. fractal threshold distributions. Fractured rocks represent extreme cases 5.1.1 Observations of Heterogeneous Per­ of inhomogeneous porous media and are a meability Distributions common feature of the upper portions of the Earth's crust. The magnitude and direction Many hydrology data sets are known to have of underground flow is affected by the pres­ a fractal character.109 The amount of annual ence of fractures. A necessary first step to flows in rivers was shown to be not a sta­ understand flow through a fracture network tistically independent random variable but to is to understand flow patterns in single frac­ depend on the sequence of previous annual tures. The apertures in a single fracture are flows .110 Geological sediments were formed strongly correlated in the sense that points by water flow and deposition of grains. It may with a given aperture tend to be surrounded be expected that the properties of sediments by other points with similar apertures. Simu­ are fractally distributed in space, since their lations based on IP models of slow two-phase statistics are determined largely by the pro ­ flow through a single fracture are discussed in cesses which formed them.111 Section 5.2. Hewett111 postulated a description of the 5.1 Invasion Percolation in Corre ­ statistics of permeability fluctuations by frac­ lated Porous Media tional Brownian motion (fBm).112 In one di­ mension, the Brownian process describes the Heterogeneities play an important role in trace B(t) of a Brownian particle (a random underground flow because they may lead walker) in the time t. The trace is character­ to channeling of the flow (concentration of ized by the scaling law the flow onto a small fraction of the porous medium). These heterogeneities range in B(bt) = 61/2B(t) . (5.1) scale from the microscopic roughness and diagenetic decorations on individual grains to In Eq. (5.1), the equality holds in a statistical length scales of the order of many kilometers sense. (the size of a sedimentary basin). Fractional Brownian motion (fBm) is a gen ­ As pointed out in section 3.1.3, the effects eralization of Brownian motion in which the of heterogeneities on slow two-phase flow motion of the particle is either persistent or may be simulated on the basis of the IP model. anti-persistent (steps in a given direction are The permeability variations can be mapped on more likely to be followed by another step in the invasion thresholds and the lattice repre­ the same direction, or more likely to be fol ­ senting the porous medium is separated into lowed by a step in the opposite direction). Fig­ regions with high thresholds and regions with ure 32 shows a set of fBm traces. A fBm trace

45 5 INVASION PERCOLATION IN CORRELATED DISORDERED MEDIA

-0.5 - H=0.9

0.0 - •0.5 ■ H=0.5

-0.5 - H=0.1

FIGURE 32: Plots offBm traces y(x) characterized by H = 0.1, H = 0.5, and H — 0.9, respectively.

Bh (t) follows the scaling law As in fBm, the Hurst exponent quantifies the degree of spatial correlation. The Levy index BH(bt) = bHB(t) . (5.2) 0 < a < 2 quantifies the degree of spatial A fBm trace is a self-affine fractal and can variability, with a = 2 recovering fBm. fLm be partially characterized by its power spec­ is based on Levy-stable distributions in the tral density S(|k|) decaying as113 sense that the difference between two values 1/1(21) and 1/2(22) on a trail y(x) representing S(|k|) ~ |k|-<1+2/f) . (5.3) fLm in one dimension is distributed accord ­ ing to a symmetric Levy distribution, with the Here, k denotes a wave vector in Fourier width parameter determined by the separa­ space, and the tilde denotes proportionality tion |2i — 22|. The Levy distribution does not in a statistical sense in the asymptotic limit have a closed analytic form except for special ("scales as"). The Hurst exponent or rough ­ cases and is characterized by power-law tails ness exponent H varies from 0 to 1, with (in contrast to the Gaussian distribution which if = 0.5 corresponding to ordinary Brownian decays exponentially). motion. fBm is based on the Gaussian distri­ McCauley et a i}08 ,iu,u 8 analyzed the per­ bution in the sense that the difference y\ — y2 meability spectra of well logs from North Sea between two values 1/1(21) and 1/2(23) on a sediments. The spectra were found to dis­ trace y(x) representing fBm in one dimension play multifractal properties. Multifractal data is Gaussian distributed in the limit of large sets require an infinite number of parame­ separations — x2\, with the variance deter­ ters to be fully specified, in contrast to sim­ mined by |$i — 22|. ple ("monofractal") fractal sets.112 Theoretical The permeability distribution of layers in and numerical studies of multifractal perme­ a well log as a function of depth was shown ability fields were presented by Saucier and to have fractal properties .111,114 fBm may be Muller.119-122 used to stochastically interpolate permeabil ­ ity fields that are partly known from borehole 5.1.2 Earlier Simulations of One- and Two- data. Phase Flow through Heterogeneous A generalization of the method was pre­ Porous Media sented by Painter et al. 115-117 who analyzed vertical and horizontal fluctuations in perme­ Giordano et al. i9 studied numerically and ex­ ability and porosity in sedimentary forma ­ perimentally the effects of permeability vari­ tions. The distribution of heterogeneities in ations on miscible displacement processes. A reservoir rocks was described by fractional network simulator was used to simulate one- Levy motion (fLm). fLm is parametrized by phase flow through heterogeneities within a the Levy index a and the Hurst exponent H. layer. Chan and Yortsos 123 reported a numer­

46 5.1 Invasion Percolation in Correlated Porous Media ical study of the saturation of heterogeneous porous media, based on continuum equations and the concept of relative permeabilities. The heterogeneities considered included fBm per­ meability variations. Chaouche et al. 6& conducted displacement experiments in three dimensions in porous cells with a permeability gradient. The dis­ placements were simulated using an IP model. The same group studied numerically correla ­ tions in the saturation profile of IP clusters growing on lattices with long-range threshold correlations. 69 Vidales et al. l2A presented simulations of ^------256 lattice units------slow two-phase flow in correlated porous me­ dia using site-bond percolation models and FIGURE 33: Simulation of the secondary migra ­ site-bond IP models. The correlations were in­ tion of oil using the site IP model for fluid migra ­ duced by the constraint that a bond size be less tion described in section 4.2.4 and a layered mul­ or equal to the size of the smaller site to which tifractal threshold map that was tilted by 20° and it was connected. Compared to uncorrelated had two impermeable vertical boundaries (black lattices, changes in the IP cluster growth pat­ lines). The invasion threshold values of the sites are tern were observed. 125 indicated by the grey scale, with the darkest shades Painter et al. 116 simulated water floods in a representing the sites with the highest thresholds. heterogeneous reservoir using a network sim­ A fault in the center region was simulated by dis ­ ulator. The reservoir was assumed to have a placing the two halfs. Invader fluid (white) was in­ permeability map characterized by fLm. Pa­ vading from the bottom and driven in the vertical terson et al.45,126 studied site IP models in two direction by a gradient g = 0.005. The lattice size and in three dimensions using a variety of cor ­ was 256 x 256 sites. related threshold maps. The maps had het­ erogeneities corresponding to fLm and to fBm (see also section 5.2.4). Multifractal maps (see of size 2x2. section 5.1.3) were also considered. The threshold pi assigned to the ith site was then given by the product of all random 5.1.3 Invasion Percolation on Lattices With numbers assigned to the site in the course of Multifractal Threshold Distributions the procedure. The resulting threshold map was spatially correlated and had multifractal Meakin127-129 studied the IP model with and properties. The map could also be regarded without trapping using two-dimensional as the projection of a hierarchical self-affine square lattices of sites and a multifractal surface in the limiting case of H —> O,130 the threshold map. The thresholds were assigned thresholds representing the surface elevations by an iterative procedure. In a first step, a (see paper P7). - In these studies the fractal di­ lattice containing 2n x 2" sites was divided mensionality D of the IP cluster was reported into four quadrants, and four different ran­ to be little or not at all affected by the spatial dom numbers q\, q2, qz and q4 were assigned correlations. The findings are consistent with to all the sites in quadrants 1, 2, 3 and 4, the measurements of D described in paper P7. respectively. In the next step, each of the Invasion percolation on lattices with mul­ quadrants of the lattice was divided into four tifractal threshold maps was also studied re­ subquadrants, and new random numbers q\, cently by Paterson et al.,45/126 using the itera­ <72 , qz and q4 were assigned to all the sites tive algorithm described in Ref. 129. In the in the subquadrants 1, 2, 3 and 4 in each case of IP with trapping, the fractal co-dimen ­ quadrant. This procedure continued down sionality 2 — D characterizing the resulting to smaller and smaller scales until at the last IP clusters was very close to the one corre ­ stages the random numbers qi, q2, qz and q4 sponding to IP with trapping on uncorrelated were assigned to the individual sites in blocks lattices. In IP without trapping, a higher co-

47 5 INVASION PERCOLATION IN CORRELATED DISORDEREDMEDIA

Figure 34: Two views from a simulation of the secondary migration of oil using the site IP model for fluid migration described in section 4.2.4 and a layered three-dimensional multi-fractal threshold map that was tilted by 55°. The invasion threshold values of the sites at the boundaries of the lattice are indi ­ cated by the grey scale, with the darkest shades representing the sites with the highest thresholds. Invader fluid (white) was injected from the bottom and driven in the upward direction by a gradient g = 0.005. The lattice size was 128 x 128 x 128 sites. dimensionality was measured than in corre ­ the lattice was divided into horizontal stripes, sponding simulations on uncorrelated lattices, each having the width of the entire lattice and implying that the clusters had a more compact a height of 1, measured in units of lattice sites. structure. At this stage, each of the strips was divided In paper P6 qualitative simulations of the into two halfs of size 2n ~1 x 1, and two random secondary migration of oil through a porous numbers qi^ and qkj 1 out of two new distri­ medium with a layered, multifractal perme­ butions were assigned to all the sites in half 1 ability distribution are presented. The site IP and 2 in the ith strip, respectively. Each half of model for fluid migration described in sec­ a strip was then further divided into sub-halfs, tions 4.2.4 and 4.2.7 was applied and lattices and new random numbers qi^ and qi^ were with multifractal threshold maps were used. assigned to all the sites in half 1 and 2 in each Figures 33 and 34 show stages in simulations half in the zth strip, respectively. This proce ­ of this type. dure continued down to smaller and smaller The threshold maps were generated using scales until each strip was divided into parts a random multiplicative algorithm similar to that included only one site each. The thresh­ that described above. However, to represent a old that was assigned to the jth site was given strongly layered reservoir, the procedure was by the product of all random numbers that carried out in a different order. First the lat­ were assigned to the site in the course of the tice containing 2" x 2" sites was divided hori ­ procedure. zontally into two halfs of size 2” x 2n_1 each, On a two-dimensional lattice, the proce ­ and random numbers q\ and qi out of two dif­ dure led to strongly layered threshold maps, ferent distributions were assigned to all sites depending on the distributions of the ran­ in half 1 and 2, respectively. The distributions dom numbers. Since cuts in different direc­ used were uniform and differed in their lower tions had different multifractal scaling prop ­ and upper bounds. In the next step each of erties, the threshold maps could be consid ­ the halfs was further divided horizontally into ered to be multi-affine. Three-dimensional sub-halfs, and new random numbers q\ and q^ layered threshold maps were obtained by a out of the two distributions were assigned to straightforward generalization of the proce ­ all sites in sub-halfs 1 and 2 in each half, re­ dure. Vertical faults in a layered porous spectively. This procedure was iterated until medium could be simulated by displacing

48 5.2 Invasion Percolation in Single Fractures

Figure 35: Schematical illustration of the fluid-fluid interface in a narrow fracture. The curvature of the interface is determined by the fracture aperture field b(x, y).

parts of the threshold map relatively to each The regions are then mapped onto a lattice of other. Inclination of the layers could be simu­ sites, assigning one region to each site. The lated by a simple linear transformation of the fracture is approximated by the array of void map. spaces given by the sites and their correspond ­ ing apertures. In effect, each site represents 5.2 Invasion Percolation in Single an idealized fracture region consisting of two parallel plates ("parallel plate model" 131). The Fractures invasion of the fracture by a non-wetting fluid The original invasion percolation model de­ is simulated by invading the sites of the lat­ scribed in section 3.1 was designed to sim­ tice and controlled by invasion thresholds as­ ulate capillary-dominated two-phase flow in signed to the sites. The invasion threshold of porous media. The physical invasion of a the ith site is given through Eq. (5.4) by the in­ given pore corresponds to the simulated in­ verse of frj. vasion of the corresponding site. During the In this approach, the aperture field b(x, y) physical invasion process the capillary pres­ determines the displacement pattern. Studies sure difference Pc = Pnw - Pw must exceed of fracture aperture fields are summarized in a threshold value that is determined by the section 5.2.1, and observations of two-phase maximum curvature of the fluid-fluid inter­ flow in a single fracture are summarized in face and represented by the invasion thresh­ section 5.2.2. Earlier and more recent simu­ old of the site. lations of two-phase flow in single fractures Capillary forces may also be expected to based on percolation models are briefly dis­ play a dominant role also in flow through nar­ cussed in section 5.2.3. row fractures. In fractures, the fluid-fluid in­ The validity of the invasion percolation terface is not broken into numerous menisdi model in the context of flow in fractures was separated by a solid matrix but is continu ­ verified experimentally by Amundsen,132 us­ ous. In a region with aperture b, the first ing artificial fracture models and comparing principal radius of curvature of the interface the displacement patterns with IP simula­ is given by the aperture and the contact an­ tions. Simulations that are related to this work gle, i?i — |6/(2 cos (9) | (see Fig. 35). The sec­ are discussed in section 5.2.4 and are reported ond radius %, oriented in parallel to the frac­ in paper P7. Invasion percolation using aper­ ture plane, is much larger than R\. Using ture fields that were obtained by displacing Eq. (2.1), the capillary pressure that the non ­ fBm surfaces relatively to each other is the wetting fluid must overcome to invade the re­ subject of section 5.2.5 and paper P8. Ex­ gion is given by tended IP models capable of simulating fluid migration through a single fracture are dis­ cussed in sections 5.2.6 and 5.2.7. for R2^> Ri. 5.2.1 Observations of Fracture Aperture An IP model of two-phase flow in a frac­ Fields ture can be constructed in a straightforward way. The fracture is discretized into regions, Bianchi and Snow 133 measured the aperture each characterized by its mean aperture of fractures in crystalline rock by covering the

49 5 INVASION PERCOLATION IN CORRELATED DISORDERED MEDIA fractures with a porous film that was satu­ 5.2.2 Observations of Two-Phase Flow in rated with a fluorescent liquid penetrant. The Fractures apertures were found to be log-normal dis­ tributed. Gentier et al ,134 studied the void Flow patterns in natural fractures are difficult space of a fractured rock by means of an epoxy to observe, because rock is not transparent. casting technique, and presented a qualitative Nicholl et al. 147 injected water into an initially study of the topology of the void space. Pig- dry fracture sample measuring approximately gott and Derek135 measured the pressure of 2200 cm2 in surface area and disassembled the a flowing fluid at nodes in a fractured rock fracture at the end of the experiment to study sample and deduced the aperture distribution the wetted regions. The flow pattern develop ­ by simulating the flow of the fluid through ing during gravity-driven injection was stud­ the fracture. The simulations were based on ied qualitatively. Instable fingering of the wet­ the "cubic law"136 relating the rate of lam­ ting fluid front in the direction of the accelera­ inar flow of a single phase through a frac­ tion due to gravity and channeling due to het­ ture region to the third power of the aper­ erogeneities in the fracture void was found. ture in the region. Johns et al. 137 reported the Pyrak-Nolte et al. 85 made a mold of the use of X ray computed tomography to mea­ two surfaces of a natural fracture in granite sure non-invasively fracture apertures in crys ­ measuring 52 mm in diameter. The mold was talline rock and found a log-normal distribu­ made using Wood's metal, a Bismuth-based tion of apertures. metal with a low melting point. The molds of each surface were filled with expoy. Af­ Brown et al. 138 used a mechanical profiler ter the epoxy solidified, the molds with epoxy scanning along lines in different directions on were placed in boiling water to remove the two rough surfaces bounding a natural frac­ Wood's metal. The two casts of the fracture ture. The power spectral density of the mea­ surface were placed together to form a trans­ sured surface traces showed power-law be­ parent fracture. The epoxy model was satu­ havior in agreement with Eq. (5.3). From these rated with a non-wetting fluid, and a wetting measurements, it was concluded that fracture fluid was injected into the fracture. Trapping topographies have fractal properties. 139 The of the non-wetting fluid was observed during topographies of the two surfaces had spatial the displacement process. correlations: points of contact or small aper­ Nicholl and Glass148,149 constructed artifi­ tures were likely to be surrounded by other cial fractures consisting of two textured glass points of contact or small apertures, while plates in close contact. Flow patterns and the points of large apertures were likely to be sur­ relative permeability of the fracture model for rounded by other points of large aperture. a wetting fluid were studied by establishing a Self-affine fractals defined by fBm were found constant flow of the fluid through the model, to be useful models for fracture surfaces.140,141 which initially was filled with air. Amund­ In a supplementary study, 142 the two surfaces sen132 used a similar system consisting of a bounding a fracture were found to be corre ­ textured glass plate and a plane glass plate lated, based on profile scans of various natural to study the slow displacement of a wetting fracture samples. fluid by an invading non-wetting fluid (see On much larger scales, Power et aZ.143,144 section 5.2.7). Even simpler models consist ­ measured the power spectral densities of nat­ ing of two parallel glass plates were reported ural fault surfaces including surfaces of the by Tors aster150 and by Fourar et a /.151,152 In a San Andreas fault. Over 11 decades of wave­ different approach, Amundsen132 used an ar­ length, power-law scaling indicating fractal tificial fracture model consisting of a milled properties was found, although the data did PMMA plate and a plane glass plate to study not support quantitative conclusions. the displacement process (see section 5.2.4). Schmittbuhl et al. 145 recorded the eleva­ tion of a granitic fault surface along one ­ 5.2.3 Earlier Percolation Models of Two- dimensional profiles and reported a self-affine Phase Fluid Flow in Single Fractures character of the profiles. Fresh brittle frac­ ture surfaces of granite and gneiss were ana­ Pruess and Tsang 131 and Pyrak-Nolte et al. 153 lyzed and found to exhibit self-affine proper ­ studied the relative permeability of fracture ties with a Hurst exponent H « 0.8. 146 models with spatially correlated aperture dis­

50 5.2 Invasion Percolation in Single Fractures

tributions. On a square lattice of sites repre­ more than one site. A variation of the algo ­ senting the fracture, each site with an aper­ rithm in which all "best" sites were invaded ture b less than a cut-off aperture was occu­ simultaneously was applied. pied with the wetting fluid, and the remain­ ing sites were occupied with the non-wetting 5.2.4 Invasion Percolation on Self-Affine fluid. Given the pattern of occupied sites, the Topographies steady-state flow of the fluids was simulated. Pyrak-Nolte et al. S5 studied the invasion Given the fractal character of fracture surfaces, of a wetting fluid in a fracture filled with a the simplest conceptual model of a fracture non-wetting fluid and modeled the process represents the fracture geometry defined by using a form of percolation with trapping 84 in the space between two self-affine fBm surfaces which the accessibility of different regions in that are parallel to each other. In some regions, the fracture was taken into account. The frac­ the surfaces may overlap; these regions repre­ ture model had a continuous spatially corre ­ sent the contacting regionis of the fracture in lated aperture distribution obtained from a re­ which no flow is possible. In the remaining cursive procedure. regions, the surfaces are separated from each Mendoza and Sudicky 154 reported simu­ other and form a void space with both wide lations of two-phase flow in a fracture using and narrow apertures. a fracture model with a spatially correlated, If the two statistically independent sur­ discrete aperture distribution. The aperture faces zi (z, y) and z2(x, y) are characterized by distribution was discretized in space and con ­ equal Hurst exponents H and equal ampli­ structed using spectral techniques. To com ­ tudes, then the aperture field b(x, y) given by pute the relative permeability of the model as a function of saturation with a non-wetting b(x,y) = zi{x,y) - z2(x,y) (5.5) fluid, a displacement process was simulated using a site IP model with trapping. Each is also a self-affine function characterized by the same exponent H.156’157 Statistically iden­ site in the model represented a fracture region with constant aperture. A pressure was im­ tical void space geometries with fractal prop ­ posed across the model and the effective per­ erties are obtained by considering the void spaces formed by a self-affine surface and a meability of the model was found by simu­ lating the steady-state laminar flow of the in­ planar surface z(x, y) = c, where c is a con ­ vader fluid in the regions that were invaded stant. In this case, the aperture field at breakthrough, and by simulating the flow b(x,y) = z1(x,y) -c (5.6) of the defender fluid in the regions that were not occupied at breakthrough. From the effec­ is, apart from the constant, identical to the tive permeabilities, the relative permeabilities surface zi (x, y), and also characterized by the could be determined.131 same Hurst exponent as z\{x, y). Figure 36 il­ Glass155 developed a modified invasion lustrates this model. percolation model to simulate the imbibition Amundsen132 used experiments and sim­ of wetting fluids into fractures, under the in­ ulations to study a fractal fracture model of fluence of gravity. At each stage in the simu­ this type. An array of 40 x 40 numbers rep­ lation, the local curvature of the fluid-fluid in­ resenting the elevations of a self-affine surface terface was used to define the invasion thresh­ z(x, y) with H = 0.8 was generated. The el­ olds of sites on a square lattice. The model was evations ranged from 0 to 0.3 mm in steps of effectively quite similar to the one described 0.01 mm. A PMMA plate was milled to ap­ in section 4.1.4. The aperture fields used in proximate an array of squares; the elevation the simulations were obtained from measure­ of each square was given by the surface eleva­ ments of optical absorption in fracture models tion in the corresponding region. The milled consisting of textured glass plates. plate served as the bottom surface in a frac­ Amundsen132 studied a modified inva­ ture model and was covered by a planar plate. sion percolation model using a discretized The model was filled with a wetting fluid (wa­ aperture field. At some stages during the ter), and a non-wetting fluid (air) was slowly simulation, the widest accessible region was injected through a hole in the center of the top not uniquely defined but was represented by plate.

51 5 INVASION PERCOLATION IN CORRELATED DISORDERED MEDIA

FIGURE 36: Illustration of a fractal fracture model. The fracture void space b(x, y) is formed by a self- affinefBm surface z(x,y) and a planar surface. The surface shown here is characterized by H = 0.8.

As the geometry of the model was known, fBm surfaces. In these simulations, the in­ the resulting flow patterns could be compared vasion threshold of a site was simply given with simulations based on the IP model, using by the elevation of the surface at the corre ­ the topography z(x, y) to represent the frac­ sponding position (Eq. (5.8)). It is apparent ture aperture field. Good agreement between that the cluster preferred to grow in regions experiments and simulations was found using that corresponded to regions of the surfaces invasion percolation with trapping. with low elevation, as expected. Compared A convenient procedure to simulate slow to ordinary IP clusters, the structures were two-phase flow in such a model is thus the fol ­ more compact and could be described as a lowing: collection of "blobs", connected to each other by fine "threads". Paper P7 is concerned with 1. Generate a self-affine fBm surface a quantitative study of the patterns generated z(z,y). by this model. Similar simulations were recently carried out by Paterson et al. 45,126 and 2. Assign to each site on a lattice with the by Du et a/.69,158 same dimensions as the surface an inva­ The main finding presented in paper P7 is sion threshold p 2- given by the inverse of that the fractal dimensionality D of the IP clus­ the surface elevation z(z,, #) at the ith ters increases systematically as H is increased site: from 0 to 1. Another issue discussed in pa­ per P7 is the structure of IP clusters that were grown in the absence of trapping. In this case D has a value close to, but less than the dimen­ where (x,, %) denotes the position of the sionality d = 2 of the underlying lattice. This ith site. A statistically equivalent pat­ result is in contrast to theoretical expectations tern is obtained by setting and earlier studies of percolation on self-affine substrates.130,159, 160 Pi — z(xi, yi) . (5.8) The interest in the fractal dimensionalities of cluster structures is related to the role of the 3. Carry out an invasion percolation simu­ trapping rule in IP. On both uncorrelated and lation using the thresholds p, . correlated substrates, the inclusion of trap­ ping reduces the cluster dimensionality to val­ Figure 37 shows IP clusters that were ues that are not understood theoretically (see grown on substrates representing self-affine section 3.1.2). To understand IP with trapping

52 5.2 Invasion Percolation in Single Fractures

FIGURE 37: IP clusters (with trapping) grown on self-affine topographies using different values of H. The left part of the figure shows a representation ofthefBm surface used in the simulation, and the right part of the lattice shows the cluster grown on the corresponding substrate. The grey scale indicates the elevation z(x,y) of the surfaces and the invasion thresholds p, of the ith site of the substrates, with dark shades representing regions with low elevation. In the simulations, the substrates were extended periodically. The boundary of the center tile is indicated by a box and the injection site is indicated by an arrow, (a) and (b) H = 0.1. (c) and (d) H = 0.5. (e) and (f) H — 0.9. it is worthwhile to study the process on corre ­ In the model by Wang et al., 156 the two lated topographies that may lead to "simpler" bounding surfaces of the fracture are repre­ cluster structures. sented by a self-affine fBm surface z(x,y) and its copy zc(x, y), oriented parallel to the origi ­ nal and displaced relative to the original by a 5.2.5 Invasion Percolation in a Fractal Frac­ distance b perpendicular to the fracture plane ture and by a distance vector r

53 5 INVASION PERCOLATION IN CORRELATED DISORDEREDMEDIA

Figure 38: Fracture model consisting of a self-affine fBm surface and its copy, displaced by a distance b in the direction perpendicular to the fracture plane and by a distance |rd | parallel to the fracture plane. ure 38 illustrates the procedure, using the sur­ The invasion threshold p, of the ith site on a face shown in Fig. 36 as an example. substrate representing an aperture field b(x , y) The properties of void spaces between was given the inverse of the aperture at the self-affine fBm surfaces were studied by point represented by the site, Wang eta.1 .156 and Roux et al. 162 and more recently by Plouraboue et a/.157,163 For a given fracture obtained by displacing a self-affine surface and its copy by a distance ]r 0.5, the self­ affinity implies that regions of the void space where the aperture is small are likely to be surrounded by other regions with small aper­ tures, and regions of the void space where the aperture is large are likely to be surrounded by other regions with large apertures. This behavior is in agreement with field studies (see section 5.2.1). On length scales greater than |rd|, the spa­ tial correlations are lost such that the apertures at two points that are separated by a distance larger than |rd| are independent from each other and Gaussian distributed. Figure 39 shows representations of the aperture fields obtained by displacing the surface shown in Fig. 36 relative to its copy. It is apparent that the size of correlated regions increases as the FlGURE 39: Representations of the aperture field magnitude |rd| of the displacement vector in­ b(x, y) obtained by displacing the self-affine fBm creases. Global properties like the fraction of surface shown in Fig. 36 relative to its copy using inaccessible regions and the percolation prob ­ the parameters b and |rd| defined in the text. The ability depend on the parameters |rd| and b. grey scale indicates the aperture value, with dark In paper P8 simulations of invasion perco ­ shades representing regions with narrow aper ­ lation (with trapping) are reported in which tures. (a) b = 0.7, rd = (20,20). (b) b =0.7, substrates obtained by displacing a self-affine rd = (100,100). The mean aperture b is measured fBm surface relative to its copy were used. in units of the variance of the surface elevation.

54 5.2 Invasion Percolation in Single Fractures

the amount of experimental data is limited. Based on the comparative studies by Amund­ sen,132 the IP algorithm appears to describe quite accurately slow drainage processes in fractures with narrow apertures. The frac­ tal fracture model used in paper P8 provided aperture fields with scaling properties similar to those of natural fractures (see section 5.2.1). It is thus possible that the modeling approach presented in paper P8 indeed captures the es­ sentials of slow drainage in fractures. The model certainly fails to correctly represent drainage processes in very wide fractures (with apertures on the millime­ ter scale). In such a scenario even slow flow is not dominated by capillary forces but by buoyancy forces and by thermal fluctuations. Interaction of the flowing fluids with the fracture matrix is another issue that is not captured by the present model. 164-166

Figure 40: IP cluster grown on a correlated sub­ 5.2.6 Applying the Site IP Model to Sim­ strate representing a fracture aperture field. The ulate Secondary Migration of Oil aperture field was obtained by displacing a self- Through Single Fractures affine fBm surface relative to its copy, using the pa ­ In two-phase flow through a natural frac­ rameters H = 0.8, b = 0.0, and |rd| = 32 (mea ­ ture, buoyancy forces will be present, as the sured in lattice units). Part (a) shows a representa ­ fracture is likely to be inclined. Migrating tion of the aperture field used in the simulation, and non-wetting fluid that is driven by gravity part (b) shows the cluster grown on the correspond ­ forces may invade narrow fracture regions ing substrate. The simulation was stopped when that could be impenetrable in a horizontal the substrate boundary was reached. The grey scale fracture with identical geometry. Buoyancy indicates the aperture, with dark shades represent­ forces are likely to change the displacement ing regions with narrow apertures or zero aper ­ pattern of flow through fractures, as in flow ture. The injection site is indicated by the arrow in through porous media. part (b). In the absence of reliable experimental data, the effect of buoyancy forces may be studied by using extended IP models capable ous ways by the structure of the IP cluster. of modeling fluid migration. Qualitative stud­ On length scales greater than |r

55 5 INVASION PERCOLATION IN CORRELATED DISORDERED MEDIA

Figure 41: Two stages in the simulation of the secondary migration of a oil (white) through a single fracture saturated with water, using the site IP model for fluid migration described in section 4.2.4 with the gradient parameter g = 0.02. The oil cluster is interspersed with trapped regions of water (black). The aperture field used in the simulation is shown in Fig. 39 (b). The grey scale indicates the aperture, with dark shades representing regions with narrow apertures or zero aperture. The oil entered at the bottom edge of the lattice, and the gravity forces were oriented in the positive vertical direction.

In this simulation, the invasion threshold 5.2.7 Applying the Site IP Model to Sim­ of the ith site was given by the inverse of the ulate Drainage in a Single Fracture aperture of the region represented by the site With Fluid Re-Distribution (Eq. (5.9)). The withdrawal threshold of the ith site was given by the aperture of the aper­ While the migration and fragmentation of ture of the region represented by the site, fluid in natural fractures is hard to investigate, fluid migration may be easy to observe in lab­ oratory experiments using artificial fractures. Fragmentation of invading non-wetting fluid Pi = b{xi,yi) . (5.10) in a fracture that is filled with a wetting fluid can occur even in the absence of buoyancy forces. Amundsen132 used an artificial frac­ ture model consisting of a textured glass plate As a result, regions with wide apertures were and a plane glass plate that were oriented hor ­ likely to be invaded by the migrating oil, and izontally. The aperture field was anisotropic these regions were likely to remain occupied and characterized by deep "valleys" and nar­ by the oil during the entire process. Regions row "ridges". with narrow apertures were unlikely to be­ The model was filled with a wetting fluid come invaded, and once such a region be­ (water), and a non-wetting fluid (air) was in­ came invaded it was abandoned by the oil jected slowly through a hole in the center of soon thereafter during a subsequent migra ­ the top plate. During the invasion process the tion step. air fragmented into numerous clusters, dis­ playing interesting dynamics. Fragmentation The relevance of this migration model to occurred rapidly on the time scale of the ex­ the physical migration process is not clear, due periment. The fragments migrated to regions to the lack of experimental data. Given the fact with wide apertures and remained there until that the withdrawal of the non-wetting fluid further fragments coalesced, forming a larger during the migration is poorly described by fragment. Eventually the larger fragments the site IP model (see section 4.2.5), deviations could re-distribute their fluid contents to an. from physical displacement patterns may be even wider region, undergoing further frag ­ expected. However, the model may still cap­ mentation on their way. the air did not form a ture essential features of the transport process connected path from the inlet to the boundary through correlated media. of the fracture model at any stage of the exper-

56 5.2 Invasion Percolation in Single Fractures

Figure 43: Two stages in the simulation of drainage in a singe fracture with re-distribution of the in­ vading non-wetting fluid (white). Dark spots indicate regions of trapped defender fluid. The process was simulated using a variant of the site IP model described in section 4.2.4. The grey scale indicates the aperture, with dark shades representing regions with wide apertures. The invader fluid was injected at the center of the lattice. The inset shows an enlarged portion of the displacement pattern shown in the right-hand figure.

iment. tion mechanism modeled the transport of air The process was simulated using the site through air bubbles observed in the experi ­ IP model described in section 4.2.4, with mi­ ment. nor modifications. As in standard IP, the sim­ During migration, invader fluid from a site ulation proceeded in steps and began by in­ at the perimeter of a cluster was withdrawn vading the injection site. After each invasion while an adjacent site was invaded, see Fig. 42. step, migration in which the invader fluid was A migration step was only possible if the aper­ re-distributed could take place. The migra- ture of the adjacent site was larger than that of the cluster site from which the air was with­ drawn. Transfer of invader fluid always took place between the combination of sites that had the largest difference in aperture. All pos ­ sible migration steps were carried out before a new invasion took place. Migration steps could lead to fragmenta ­ tion of an invader fluid cluster. A cluster frag ­ ment could migrate on its own and coalesce with other fragments. Invasion was restricted to the cluster fragment that was connected to the injection site. A trapping rule was used FlGURE 42: Illustration of the modified site IP during invasion steps and during migration model used to simulate displacement of water by air steps to conserve the volume of entrapped wa­ (white) in a fracture. Shaded sites indicate water- ter. Water-filled sites that were not connected filled regions, with dark shades indicating large to the lattice boundary by a path consisting of apertures. The occupied perimeter sites of the air nearest-neighbor water-filled sites could not cluster are marked with crosses. In a migration be invaded. step the perimeter sites with the smallest aperture To compare the simulation model with the (A) is filled with wetting liquid. If the adjacent experimental observations, the aperture field water-filled site with the largest aperture (B) has b(x, y) of the artificial fracture model used a larger aperture than A, air is withdrawn from A in the experiments was measured by optical and transported to B. The number of sites belong ­ absorption and used to assign the threshold ing to the cluster remains constant. pressures p according to Eq. (5.9). The quality

57 5 INVASION PERCOLATION IN CORRELATED DISORDEREDMEDIA of the simulation was surprisingly good. In cult to tell which one represented the experi- fact, from animated sequences of simulated ments. A detailed comparison of experiments displacement patterns and of experimental and simulations was presented in Ref. 132. displacement patterns it was at times diffi- Figure 43 shows two stages in the simulation.

58 6.1 Fragmentation of Invasion Percolation Clusters

6 Scaling Issues

In the laboratory experiments discussed in 6.1 Fragmentation of Invasion Per­ sections 2.2,2.3 and 3.2.1 and in papers PI, P2, colation Clusters and P3, the experimental cells had dimensions of the order of centimeters or one meter. Com ­ Papers PI and P2 are concerned with the frag ­ pared to the dimensions of a medium-sized oil mentation of IP clusters and IP cluster-like dis­ reservoir (several kilometers), the cells were placement structures (see section 4.1.3). The very small indeed. displacement structures were observed when clusters of non-wetting fluid in a porous cell In simulations based on the IP algorithm, saturated with a wetting fluid disintegrated the fluid displacement processes are modeled as the pressure of the wetting fluid increased on the length scale given by the size of a typ ­ (secondary imbibition). The original clusters ical pore of the represented porous medium. had a structure similar to an IP cluster and With today's computer power, simulations on were obtained by injecting the non-wetting lattices representing 106 pores are feasible, fluid slowly into the cell. When the pressure and simulations on lattices representing 107 of the wetting fluid was increased, part of the pores are just around the comer. Needless to non-wetting fluid was displaced out of the say, the number of pores involved in slow un­ cell while entire sections of the original clus­ derground flow in nature are typically many ter became disconnected from the outlet and orders of magnitude larger. remained immobile. The cluster fragments of If the work presented in this thesis is rele­ residual non-wetting fluid that were formed vant to slow two-phase flow on geologic scales during the secondary imbibition process re­ in spite of these limitations, it is due to the scal­ tained the IP cluster-like structure of the orig ­ ing properties of the displacement patterns. inal cluster. Given known scaling properties of the pat­ In the simulations, an IP cluster was terns, a model of the real-world flow processes formed initially. The fragmentation process may be obtained by upscaling the patterns. proceeded in an ordered manner, similar to Power laws are the simplest forms of math­ the experimental fragmentation process. The ematical relationships that are not character­ hierarchy of fragmentation steps was related ized by a typical size (length, duration, ...) of a to the volume conservation of trapped wet­ quantity. The absence of a finite characteristic ting fluid: At each stage, only those sections scale is the fingerprint of critical phenomena, of the non-wetting fluid cluster could be in particular the percolation transition in the withdrawn that were adjacent to untrapped percolation problem (see section 1.3). In the wetting fluid regions. absence of a finite characteristic scale, power- The process was analyzed quantitatively law dependencies of related quantities maybe by measuring the non-wetting fluid saturation expected. If a process is characterized by a T at the beginning and at the final stage of the typical size, power laws may be valid on the secondary imbibition process. At the begin ­ ranges below or above the typical size. ning of secondary imbibition, the dependence of the saturation on the size L of the lattice The characterization of the simulated pat­ could be described in terms of the scaling re­ terns is based on the analogy between dis­ lationship placement patterns and percolating systems pointed out in section 1.3. In section 6.1, re­ T ~ Ld ~2 (6.1) sults that were obtained in experiments and simulations on fragmentation of displacement where the value of the fractal dimensional ­ structures are discussed. Sections 6.2,6.3, and ity D % 1.82 (see section 3.1.2) was used, 6.4 present results on the properties of migrat ­ and the tilde denotes proportionality in the ing fragments. Scaling laws for IP clusters asymptotic limit of infinite lattice sizes. Since in correlated random media are given in sec­ D <2, Eq. (6.1) implies that the non-wetting tion 6.5. In the final section 6.6, the relevance fluid saturation goes to zero for L -» oo. At of the findings presented in the thesis is dis­ the final stage of the secondary imbibition pro ­ cussed. cess, the saturation of the system with resid­

59 6 SCALING ISSUES

ual non-wetting fluid depended on L through at a gradient value of a similar equation that included a correction term for finite lattice sizes. #4) The distribution A'{s, L)Ss of the number of fragments in a system of size L with sizes depending on the size of the IP cluster. The (numbers of sites occupied by fragments) in fragments retained the structure of an IP clus­ the range s to s + Ss (with Ss —»■ 0) was mea­ ter. At each stage during the migration, char­ sured. N(s,L) could be represented by the acterized by the gradient \g\, three classes of scaling form 167 fragments were studied in paper P2: The frag ­ ments that were immobile at gradients slightly lower than |g| and that became mobilized at , (6.2) \g\, the fragments that were in the process of migrating at \g\, and the fragments that were where the scaling function f(x) decayed faster newly formed at |g|. The typical (mean) sizes than any power of x for x 1, and sc(L) S'm) of the fragments at the onset of migration represented a characteristic cut-off fragment and of the typical sizes f") of the newly formed size. The cut-off fragment size was a measure fragments scaled with the gradient as for the size of the largest fragments of residual non-wetting fluid that a system of size L could gW ^ «(") _ |,,|-o . (6.5) support. The cut-off size was expected to scale with L as During a sequence of migration steps the extension lyP^ of the migrating frag ­ sc(^) - ^ , (6.3) ments with size gW in the direction of the gradient increased while the relative width as the secondary imbibition process was not w(p) — s(p) jl'A} decreased. In simulations characterized by any typical scale apart from of gravity IP the distortion of the threshold the lattice size L. map by a gradient contribution imposes a From the data, it was not possible to de­ characteristic length scale £ that scales with termine the values of the exponents t and % the magnitude of the gradient parameter g unambiguously. The difficulties were related as"* to the finite-size effects affecting the final non ­ wetting fluid saturation. If it is assumed that the cluster fragments have the same structure as the original IP cluster, y = D follows, lead­ where v is the ordinary percolation experi­ ing to t = 2. The measured cluster size distri­ ment with the value v = 4/3 in two dimen­ bution was consistent with this assumption. sions and v % 0.88 in three dimensions, re­ spectively. 73 In paper P2, the relative width of a migrating fragment was interpreted as a 6.2 Migration and Fragmentation of measure of the percolation correlation length £ Invasion Percolation Clusters and was found to scale with |y| in agreement with Eq. (6.6). Papers P3 and P2 discuss a scenario in which The distribution J\f(s)Ss of the size s of a cluster of non-wetting fluid in a porous cell fragments was measured and represented by saturated with a wetting fluid migrates and the form fragments under the influence of increasing buoyancy forces (see section 4.2.3). In the sim­ ulations, an IP cluster of size s0 with the fractal <67) dimensionality D was formed in the absence of a threshold gradient representing buoyancy where f(x) denoted a scaling function similar forces (see section 3.1.3). The cluster was not to f(x) introduced in Eq. (6.2), and sc a cut­ characterized by any finite length scale other off size dependent on the gradient parameter. than the size L of the lattice. The fragments with sizes comparable to the Under the influence of slowly increasing cut-off size had a typical extension lc in the di­ buoyancy forces, represented by a gradient rection of the gradient that scaled with g as \g\ >0, the cluster started to migrate and dis­ integrated into fragments. Migration started lc{g) ~ M"1 • (6.8)

60 6.4 Migration of Invasion Percolation Clusters at Constant Gradient

Assuming that the largest fragments had a to the form Eq. (6.7), as the total mass (num­ structure like an IP cluster and were not elon ­ ber of occupied sites) of scaling fragments was gated in the direction of the gradient, the cut­ not constant during these stages. The size dis­ off size is given by tributions were separated into two sets, cor ­ responding to the low gradient regime and to se{g) ~ \g\~ D - (6.9) the high gradient regime. The separation was reflected by the mean 6.3 Migration and Fragmentation of fragment size (s(g)) that did not follow a sim­ Clusters in One Dimension ple power law but became nearly constant at intermediate values of g. (s(g )) could be rep­ In paper P4, the scenario of fluid migration resented by the scaling form under the influence of a continuously increas­ ing gradient was further explored using the (s(g)) = sl0/2f{gsl /4) , (6.11) site IP model for fluid migration described where f(x) is a scaling function and .sr, is the in section 4.2.4 on a one-dimensional lattice. size of the initial cluster. Equation (6.11) could Needless to say, a one-dimensional model is be understood in terms of an even more sim­ a poor representation of physical reality. Yet, plified migration model. the study of a simple prototypical model for Using elementary probability theory, the migration through random media could pro ­ probability that non-wetting fluid is with­ vide a basic understanding of the dynamics of drawn from the ith site of a fragment includ­ fragmentation and coalescence events. ing j sites could be approximated. The overall A one-dimensional cluster of non-wetting probability V(s, g) with respect to a fragment fluid was obtained by occupying a string of of size s that any step was taken was shown to sites in the one-dimensional lattice that was scale as initially saturated with wetting fluid. The mi­ gration of the cluster started when a pressure ^(s, jr) = |fl|1/2/(srs2) , (6.12) gradient g was imposed. In each step, non ­ wetting fluid was withdrawn from one site where f(x) represented a scaling function. of the cluster and occupied a new site at the Based on the probabilities for migration and downstream end of the cluster. Practically ev­ fragmentation, the typical length As of newly ery withdrawal led to fragmentation of the mi­ formed trailing fragments stemming from mi­ grating cluster. grating fragments with an original size s was Most of the individual fragments were found to scale with g as separated by a single wetting fluid site. If As ~ M""1/2 (s oo,gs —> 1) . (6.13) one of the fragments could migrate, the in­ vasion of a new site at the downstream end of the fragment was likely to lead to coa ­ 6.4 Migration of Invasion Percola ­ lescence with another fragment. The newly tion Clusters at Constant Gradi ­ formed fragment tended to have a larger ex­ ent tension. Consequently, the driving buoyancy forces were more pronounced and migration In dimensions higher than one, the migration became more probable. The migration of a process is more complex and most of the scal­ single fragment could thus trigger a chain re­ ing relations applying to the one-dimensional action during which numerous fragments un­ case do not hold. Papers P5 and P6 were con ­ derwent coalescence and new fragments were cerned with a scenario in which a non-wetting formed. fluid migrates through a porous medium sat­ The distribution of fragment sizes urated with a wetting fluid. The simulations (lengths) could be described by the scal­ were carried out using the site IP model for ing form Eq. (6.7), with the characteristic fluid migration described in section 4.2.4. The fragment size sc scaling as migration process was driven by a constant hydrostatic pressure gradient g and could be ScW - . (6.10) regarded as a model for the secondary migra ­ tion of oil (see section 4.2.7). However, at very low values and at interme­ The resulting displacement structures diate values of \g\, the distributions did not fit could be characterized in terms of the fractal

61 6 SCALING ISSUES dimensionality D of IP clusters. The struc­ gradient (see section 6.2), the fragments were tures could be regarded as chains of fractal elongated in the direction of the constant gra ­ blobs, 100 each blob having a structure like an dient at stages when all migration had ceased. IP cluster. The typical diameter of a blob was The scaling forms Eq. (6.9) and Eq. (6.17) ex­ given by the correlation length £ according to pressing the dependence of the typical frag ­ Eq. (6.6). The blob chains were oriented along ment sizes in the two cases, respectively, are the direction of the gradient. different. In paper P5, the geometry of the struc­ In paper P6 slow two-phase flow was stud­ tures obtained in simulations using two- ied qualitatively using various forms of cor ­ dimensional lattices with uncorrelated related threshold maps (see section 5) that in­ threshold maps was characterized by mea­ fluenced the displacement patterns. The effect suring the linear mass density profile m(y) of the threshold correlations was an additional (the profile of the number of sites occupied factor that determined the scaling properties by the structure). The driving buoyancy of the simulated structures. No quantitative forces were oriented in the (/-direction. The characterization was attempted in this paper. quantity m(y)8y = s(y)/8y represented the number s(y) of non-wetting fluid sites with (/-coordinates in the range y to y + 8y. 6.5 Scaling of Invasion Percolation The linear mass density profile m(y) char­ Clusters on Correlated Thresh­ acterized the mass, or volume, of a structure per unit length and was defined by old Maps

m(y0) = N^°ySy , (6.14) As pointed out in section 3.1.2, IP clusters grown in the presence of a trapping rule on a two-dimensional substrate with an uncor ­ where N(y0)8y is the number of invaded sites related threshold map are characterized by a with the (/-coordinates in the range y0 toyo+8y fractal dimensionality D % 1.82. 51,59 In the (with 8y —> 0). The averaged linear mass den­ presence of correlations among the thresholds sity profile (m((/)) was related to the correla ­ assigned to different regions of the substrate, tion length £ characterizing the displacement the scaling properties of the cluster maybe af­ patterns through 62 fected on a global scale such that the fractal di­ mensionality changes. (m((/)> - ^ ^ ig) In paper P7, the fractal dimensionality Dh To obtain the right-hand side of Eq. (6.15), the of IP clusters grown on a substrate represent­ scaling relation Eq. (6.6) was inserted. ing a self-affine topography was measured The distribution J\f(s)Ss of the fragment (see section 5.2.4). The dimensionality was sizes s was studied using the scaling form measured in two ways, by counting the mass (the number of occupied sites) s(R) within a Af(s) ~ sqs 2 f ^ , (6.16) circle of given radius R, and by analyzing the two-point density function C(r). In an intermediate range 1< U < L, the where so and sc represented the total number number of sites s(f?) occupied by the IP clus­ of invaded sites and the cut-off size charac­ ters was found to scale with the radius of the terizing the fragment size distribution, respec­ circle as tively. Apart from the factor s0 correcting for the continuous increase of the total mass of the s(R) ~ RDh , (6.18) structure (the number of occupied sites), the scaling forms Eqs. (6.7) and (6.16) are equal. where the upper boundary for the scaling The cut-off size was also a measure for the regime was given by the system size L, and mean fragment size (s) and depended on the the center of the circle was located at the cen­ magnitude of the gradient as ter of gravity of the clusters analyzed. The sc ~ |5|-i-(0-iW(*'+i) _ (6.17) exponent Dh representing the fractal dimen­ sionality was found to vary with the Hurst In contrast to the scenario of fluid migration exponent H that characterized the threshold in the presence of a continuously increasing map. The two-point density correlation func-

62 6.5 Scaling of Invasion Percolation Clusters on Correlated Threshold Maps

tion was defined as size Sft(r) of a fractal blob that is characterized by the dimensionality Djj and that fills out a < 7r(r0)7r(r0 + r) >|r|=r C(r) = (6.19) region of linear extension r scales with r as < 7r(ro) 7r(ro) > s&(r) - r*" , (6.25) where 0 < ?r(r) < 1 denoted the number den­ sity of occupied sites and the averaging was in agreement with Eq. (6.22). The distribution taken over the occupied origins ro, orienta ­ M{sb) of blob sizes in Eq. (6.21) is obtained tions of the space vector r, and a large sample through of clusters. For small r, C(r) was found to de­ cay as Nr{r)8r — Af(sb)Ssb . (6.26)

C(r) ~ r2 Dh , (6.20) From Eq. (6.25), it follows

indicating a fractal dimensionality of Dh of (6.27) the IP clusters. The values Dh found by ei­ Ss b ther of the two methods were consistent and indicated a more dense structure (correspond ­ and thus Eq. (6.21), ing to a higher fractal dimensionality) of the IP clusters than observed for IP clusters that were H grown using uncorrelated threshold maps. (-2-Dh+H)/Dh sb (6.28) The IP clusters had a structure quite differ­ ent from the one of ordinary IP clusters and is obtained, as stated in Eq. (6.23). consisted of large blobs that were connected In paper P8, the structure of IP clusters by "threads" (see section 5.2.4; Fig. 37). The grown on a substrate representing a fractal distribution of blob sizes s* de­ fracture (see section 5.2.5) were studied us­ pended on the system size and on the thresh­ ing the same methods. On short length scales old map and was described by the scaling r (6.21) by the magnitude of the displacement vector Td used in the construction of the threshold where f(x) was a scaling form that decreased map. The value of exponent Dh was consis ­ for x 1 faster than any power of x, and tent with the values found in paper P7, using Sbc represented a cut-off blob size. The cut­ threshold maps characterized by comparable off blob size was assumed to be determined by values of H. the IP cluster size, On long length scales |r 0). The scribed the cross-over from correlated cluster

63 6 SCALING ISSUES growth at length scales R

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74 LIST OF PAPERS

List of Papers

Paper PI: Fragmentation of Invasion Percolation Clusters in Two- Dimensional Porous Media G. Wagner, A. Birovljev, P. Meakin, J. Feder, and T. Jossang Europhys. Lett. 31(3):139-144, 1995

Paper P2 : Experiments and Simulations on Migration and Fragmentation of Non-wetting Fluid in Two-dimensional Porous Media G. Wagner, A. Birovljev, P. Meakin, J. Feder, and T. Jossang submitted to Phys. Rev. E

Paper P3: Migration and Fragmentation of Invasion Percolation Clusters in Two-dimensional Porous Media A. Birovljev, G. Wagner, P. Meakin, J. Feder, and T. Jossang Phys. Rev. E 51(6)5911-5915,1995

Paper P4: Fragmentation and Coalescence in Simulations of Migration in a One-Dimensional Porous Medium G. Wagner, P. Meakin, J. Feder and T. Jossang Physica A 218:29-45,1995

Paper P5: Buoyancy-Driven Invasion Percolation With Migration and Frag ­ mentation G. Wagner, P. Meakin, J. Feder and T. Jossang in preparation

Paper P6: Fractals and Secondary Migration P. Meakin, G. Wagner, V. Frette, J. Feder and T. Jossang Fractals 3(4):799-806, 1995

Paper P7: Invasion Percolation on Self-Affine Topographies G. Wagner, P. Meakin, J. Feder and T. Jossang to appear in Phys. Rev. E

Paper P8 : Invasion Percolation in Fractal Fractures G. Wagner, P. Meakin, J. Feder and T. Jossang submitted to Water Resour. Res.

75 LIST OF PAPERS

76 EUROPHYSICS LETTERS 20 July 1995 Europhys. Lett., 31 (3), pp. 139-144 (1995)

Fragmentation of Invasion Percolation Cluster in Two-Dimensional Porous Media.

G. Wagner, A. Birovljev, P. Meakin, J. Feder and T. Jossang Department of Physics, University of Oslo Box 10A8, Blindem, 0316 Oslo 3, Norway

(received 27 March 1995; accepted in final form 26 June 1995)

PACS. 47.55MH - Flows through porous media. PACS. 47.55 Kf - Multiphase and particle-laden flows. PACS. 05.40 + j - Fluctuation phenomena, random processes, and Brownian motion.

Abstract - Two-phase fluid displacement in a two-dimensional porous medium has been studied experimentally and by simulations. A random porous medium was saturated with a wetting fluid (water/glycerol). A non-wetting fluid (air) was then slowly injected, forming a fractal invasion percolation-like structure. Shortly before breakthrough, the pressure of the wetting fluid was increased and the non-wetting fluid was forced to withdraw. The structure formed by the non-wetting fluid disintegrated into fragments. The displacement patterns were found to agree well with patterns obtained from simulations based on an invasion percolation model.

Introduction. - Much attention has been devoted to the problem of slow, immiscible displacement of one fluid by another one in porous media. While the intrinsic randomness of the media complicates exact analytical analysis, progress has been made by studying simple algorithmic models [1]. Quasi-static two-phase flow in porous media may be described using percolation concepts. When a non-wetting fluid (NWF) slowly invades a porous medium that is saturated with a wetting fluid (WF), the NWF forms a fractal structure that is similar to an invasion percolation (IP) cluster [2]. Here, we study the fragmentation of a two-dimen ­ sional IP-like NWF bubble formed in a random homogeneous porous medium. The fragmentation occurs during the displacement of the NWF by a WF, a process known as imbibition (see ref. [3]). In a recent study of the fragmentation of bond percolation clusters at the percolation threshold [4], the distribution bs „■ of sizes s' of fragments formed when one bond is removed from a cluster of size s was found and related to percolation exponents [5]. From this work, the fragment size distribution resulting from repeated removal of bonds from a parent cluster (without fragmenting any of the daughter clusters) may be derived. These results cannot be applied directly in the present study since the topology of the IP cluster changes during the displacement [6].

Experiment. - The rectangular experimental cell was constructed by randomly throwing 1 mm diameter glass beads onto a sheet of sticky contact paper surrounded by a rectangular

© Les Editions de Physique 140 F.riMI'HYSlCS LETTERS

Fig. 1. - The experimental set-up. a) Weight balance, b) wetting-fluid reservoir, c) supporting plates, d) inflated membrane, e) bead layer.

silicone border until no place for additional beads remained. After the excess beads were removed, another sheet of contact paper was applied on top to make the model air-tight. The bead model with approximately 100 x 200 pores was sandwiched between two 25 mm thick polymethylmethacrylate plates. Two inlet channels were drilled at two edges of one of the plates. The other plate had a transparent membrane attached to it which, when inflated, pressed the beads against the other plate. This ensured that the cell was only one-bead thick everywhere. The experimental cell was evacuated and saturated with a glycerol/water mixture with a viscosity of p = 6 ■ 10~2 P, density p = 1123 kg/m3 , and surface tension a- = 44 • 10-3 N/m, dyed with 1% black Nigrosine. This liquid is imperfectly wetting with respect to air at the contact paper/glass beads surfaces. The cell was positioned horizontally. One inlet was connected to an external WF reservoir. The other inlet at the opposite face of the cell was connected to the atmosphere. The pressure P of the WF was controlled by adjusting the height of the reservoir. Figure 1 shows the set-up. The experiment began by slowly reducing P from atmospheric pressure P0. Air invaded the cell through the inlet face, and WF was displaced to the reservoir. The invasion took place very slowly, so that the pores were invaded sequentially. Clusters of WF were enclosed by invading air («trapped») and were not displaced in the time scale of the experiment (days). The invading air formed an IP-like cluster that was interspersed with numerous WF regions («fjords») that connected rings of WF in the hollow spaces between the glass beads and the contact paper [7]. When P was reduced to approximately 200 Pa below P0, the air cluster extended over the entire cell. At breakthrough, the process was reversed by increasing P. The withdrawing air exited through the inlet. The air could withdraw only from pores and pore necks neighbouring the WF that was not trapped. When the air cluster receded, trapped clusters of WF became connected to the reservoir, as more and more air withdrew from pores and pore necks. Some of the withdrawals resulted in fragmentation of the receding air cluster. The fragments of the original IP-like cluster became disconnected from the air inlet. The disconnected fragments remained immobile when P was further increased. The fragments retained the IP-like structure of the original cluster. When P was approximately 250 Pa above P0, no more air could be displaced, and the experiment was terminated. Figure 2 shows a sequence of experimental patterns.

Displacement mechanisms. - Detailed studies [8] on square channel micromodels identified several pore scale mechanisms for the displacement of NWF by WF (see fig. 3). NWF may withdraw from pore necks either «piston-like» or as a «snap-off». Withdrawals from pores can occur by «I1» and «I2» mechanisms. Piston and II displacements require a G. WAGNER et al. : FRAGMENTATION of invasion percolation cluster etc. 141

-100 pores -100 pores -100 pores 100 lattice sites 100 lattice sites 100 lattice sites Fig. 2. - Patterns observed in the experiment and in the simulations, a) Non-wetting air (white) has invaded from top to bottom and displaced the wetting water/glycerin (black). The breakthrough saturation was F ~ 0.37. b) air is displaced by invading water/glycerol, c) At the final stage, the remaining air fragments were completely isolated (F ~ 0.24). d) Simulated system at breakthrough (F ~ 0.35). e) Simulated system in an intermediate stage. /) Simulated system when no more displacement was possible (F ~ 0.23).

lower pressure of the displacing WF than snap-off and 12. In the random media used here, the distinction between 12 and snap-off was not always clear. Nevertheless, the classification scheme appeared suitable for the simulation model described below.

Simulation. - The experiment was simulated using a site bond model based on invasion percolation [9]. The sites represented pores and were arranged in a square lattice and connected by bonds representing pore necks. Here, the main features of the simulation are described. One row of sites and bonds at one edge of the lattice was filled with non-wetting «fluid», and the remaining sites and bonds were filled with wetting «fluid». The row filled with NWF represented the air inlet of the experimental cell. The row at the opposite edge represented the edge connected to the reservoir of WF. Random numbers p and q were assigned to all bonds and sites, respectively. The numbers were used to compute invasion thresholds tt and withdrawal thresholds

a) b) c) d) Fig. 3. - Mechanisms of displacement of NWF (white) by WF (shaded) in porous media with square lattice geometry, a) Pore neck snap-off. 6) Piston-like pore-neck displacement, c) II pore displacement. d) 12 pore displacement. The figure is after ref. [8]. 142 EUROPHYSICS LETTERS invaded as well. By repeating this step, an IP cluster was formed. WF sites and bonds that became enclosed by NWF were trapped [10] and could not be invaded with volume conservation of the trapped WF. When the tip of the IP cluster had reached the reservoir edge, the displacement of NWF by WF was simulated by withdrawing NWF sites and bonds. The simulation of withdrawal of NWF was based on the concept of withdrawal thresholds 9 and included the four prevailing displacement mechanisms «piston», «snap-off», «I1» and «I2» mentioned above. At each stage, the withdrawal step with the lowest threshold 9 was carried out. The withdrawal threshold assigned to a given site or a given bond was varied according to the state of the neighbouring sites and bonds. NWF bonds that connected a WF site with a NWF site were withdrawn piston-like with a constant withdrawal threshold 9 = 0. Bonds that connected two colinear NWF sites were withdrawn by snap-off and had a high constant withdrawal threshold 9 = 1. NWF sites with three adiacent WF bonds withdrawn by II with a low withdrawal threshold 9 (g) = g/2. Finally, sites with only two adjacent WF bonds forming a right angle were withdrawn by 12 with a higher withdrawal threshold 9 (g) = = (g + l)/2. Withdrawal steps consisted either in the withdrawal of a NWF bond (by piston or by snap-off), or in the withdrawal of a NWF site (by II or 12). 12 withdrawals took place at later stages due to the high thresholds assigned. Snap-off was possible only at the final stage. Both 12 and snap-off could lead to fragmentation of the original IP cluster. Only those NWF sites that were connected to the inlet face by a path of nearest-neighbour NWF sites, linked by NWF bonds, could be withdrawn. Similar rules were used for bond withdrawals. Only sites (bonds) adjacent to untrapped WF sites (bonds) could be withdrawn with volume conservation of trapped WF. The random numbers q assigned to sites at the beginning of each simulation were distributed uniformly between 0 and 1. The distribution of the random numbers p used to simulate the growth of the initial IP cluster had no influence on the results. Figure 2 shows a sequence, of patterns obtained with the simulation model.

Comparison of experiment and simulation. - Figure 4 shows, on a log-log plot, the cumulative distribution of sizes of disconnected NWF fragments 2 Ms > s*) vs. s* (the number of fragments with size greater than a threshold size s*), measured in the experimental pattern shown in fig. 2c). The sizes of air fragments were determined by counting the number of pixels belonging to a fragment in the digitized image. The cumulative size distribution Ea t,(s > s*) of trapped WF clusters obtained from fig. 2a) and c) is also shown. Simulations were carried out on a lattice of size 100 x 200, corresponding to the cell size used in the experiments. Here, the size of a fragment was defined as the number of sites belonging to the fragment. The size distribution of NWF fragments was measured when no more withdrawal of NWF was possible. The size distribution of trapped WF clusters was measured at breakthrough and at the final stage. The simulation data fit the experimental data quite well. The cut-off at large s* seen in the simulated distributions is, for statistical reasons, not apparent in the experimental distributions. The saturation F has been measured in the simulations using different lattice sizes L. Here, L denotes the shorter, lateral length of the lattice. The breakthrough saturation is expected to scale as Fh~ LDl~2, where DY - 1.82 [2,10], and decreases with increasing system size. In the limit L —» 00, the saturation F{ at the final stage scales as Ft(L) ~ L°2~2 with an exponent D2 =S D, in order to fulfill F{ < Fh. The slower decay of Ft seen in the insert in fig. 4 is attributed to the fact that a given blob of NWF sites could be disconnected from the inlet in a small system whereas the blob could be part of a larger parent blob and still be connected to the inlet when embedded in a larger system [11]. The volatility of the blobs thus is expected to lead to enhanced withdrawal in large systems. In the absence of a known value G. WAGNER et ai: FRAGMENTATION of invasion percolation cluster etc. 143

4.0

2.0

0.0 %

-2.0 g

be -4.0 o

-6.0

log (s)

Fig. 4. Fig. 5. Fig. 4. - Cumulative size distributions of NWF fragments, measured at the final stage in the experiment (circles) and in simulations (solid lines) and plotted on a log-log scale. Also shown is the distribution of trapped WF clusters, measured at breakthrough (squares and dotted line) and at the final stage (diamonds and dashed line). In the insert is shown the NWF saturation r measured in simulations at breakthrough (circles) and at the final stage (squares) using different lattice sizes L. The standard deviations are indicated (not calculated for systems of size 800 x 1600). The dotted line shows a fit of the form eq. (1) for F{. F measured in fig. 2a) and c) is indicated by filled symbols. Fig. 5. - The size distributions of NWF fragments measured in simulations at the final stage. The simulations were carried out using lattices of size 25 x 25 (A), 50 x 100 (S), 100 x 200 (C), 200 x 400 CD), and 400 x 800 (E), respectively. The insert shows an attempt to scale the fragment size distributions using the scaling form given in eq. (2).

for D2, D1~ D2 may be assumed. In fig. 4, an attempt to approximate F{ with a fit of the form Ff (L) ahDl ~2 - bL~? (1) is shown, with the fit parameters a = 1.0, b - 1.3, and ,3 = 0.4. The second term on the right-hand side of eq. (1) is a correction to scaling. Figure 5 shows the size distribution of simulated NWF fragments at the final stage. The distributions were expected to scale with L as

W(s,D)-D*'s-YW8,:(Z')). (2)

The probability of finding a fragment of size s is proportional to the total amount Mf(L) = = 2L2Ff(L) — L°2 ~ LDl of NWF at the final stage. The cut-off size scales as sc(L) — Lx. The exponents % = 1.78 and r = 2.07 were determined from an analysis of the moments of the size distributions [4], If an IP-like structure of the fragments is assumed (x = Dx), r = 2 follows. The insert in fig. 5 shows an attempt to collapse the fragment size distributions onto a common curve, using the latter set of exponents r and %. The collapse is poor due to finite-size effects and to poor statistics for large system sizes. Using the other set of exponents did not improve the collapse. The quality of the data does not allow for a conclusive identification of the exponents D2, r and /. From fig. 5, cut-off function behaves asf(x) ~ x% for ,r «1 with x ~ 0.35, and decays rapidly for z»l. 144 EUROPHYSICS LETTERS

Conclusion. - We have presented an experiment in which a random fractal IP-like structure of NWF is formed in a porous medium and undergoes fragmentation. The fragmentation process proceeds in hierarchical manner due to volume conservation of trapped WF. The experimental patterns agree well with simulations based on a modified IP algorithm.

* * *

We thank A. Aharony, K. Christensen, V. Frette, L. Furuberg, P. King, R. Lenormand, and K. J. Mal0Y for helpful discussions. We gratefully acknowledge support by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (STATOIL) and by NFR, the Norwegian Research Council. The work presented has received support from NFR through a grant of computing time.

REFERENCES

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G. Wagner, A. Birovljev, P. Meakin, J. Feder, and T. J0ssang Department of Physics, University of Oslo Box 1048, Blindern, 0316 Oslo 3, Norway

Revised Version January 9, 1997

Abstract

We present experimental studies of two fluid displacement processes in porous media invoving extensive fragmentation of invasion percolation-like structures. In the first process, a two-dimensional porous cell saturated with a wetting fluid was slowly invaded by air. The air formed a fractal structure that fragmented when the pressure of the wetting fluid increased and the air was driven out of the system. In the second process, a fractal air structure migrated through a two-dimensional porous medium saturated with wetting fluid. The structure was driven by increasing buoyancy forces and fragmented. The fragments migrated, fragmented and coalesced with other fragmentes. The processes were simulated using new site-bond in­ vasion percolation models that captured various displacement mechanisms and reproduced the fragmentation events, and good agreement was found. In both processes, the fractal dimensionality of the fragments was equal to the dimensionality of the initial invasion percolation-like structures. The fragment size distributions measured in both processes are described by simple scaling forms.

1 Introduction

The slow displacement of one fluid by an immiscible second fluid in a porous medium is of practical interest in many fields of technology, including oil and gas recovery, chemical engineering, drying, and pollution control. The displacement of one fluid by an immiscible second fluid leads to the formation of complex, disorderly interfaces that often appear to have a fractal geometry [1]. For in­ stance, when a non-wetting fluid is slowly injected into a two-dimensional porous medium that is saturated with a wetting fluid, the non-wetting fluid forms a fractal structure with a dimensionality of D ~ 1.82 [2]. Concepts and models

1 related to percolation theory [3] have been used extensively to study this and other multiphase flow phenomena in the past few years [4, 5, 6, 7, 8, 9, 10]. The purpose here is to focus on two displacement processes in which the in­ vasion of non-wetting fluid is an initial step. In the first experiment, a fractal “cluster” of non-wetting fluid is formed by slowly displacing wetting fluid out of a two-dimensional porous medium. The injection of non-wetting fluid is stopped when the cluster has reached a given size, and the process is reversed by sucking the non-wetting fluid out of the medium [11]. In the second process, a fractal cluster of non-wetting fluid is formed in the same manner, by slowly displacing wetting fluid out of a porous medium. Then a hydrostatic pressure gradient is imposed on the system by slowly rotating the two-dimensional medium. Driven by increasing bouyancy forces, the cluster of non-wetting fluid starts to migrate through the medium [12]. In both processes, the initial fractal cluster of non ­ wetting fluid is found to fragment into smaller pieces. Here we show that the fragments can be described by the same fractal dimensionality as the initial clus­ ter, and the distribution of fragment sizes can be represented by simple scaling forms. The initial fractal clusters of non-wetting fluid can be described by a branch of percolation theory known as “invasion percolation ” [17]. Numerical studies of the fragmentation of percolation clusters have been carried out relatively recently [13, 14, 15]. These studies were motivated by the hope of advancing the understanding of the properties of percolation clusters. In this work, a similar hope has led to the development of numerical models based on the invasion percolation algorithm [5, 16, 17]. These models included mechanisms for the fragmentation of the non-wetting fluid, and were able to reproduce the experimental patterns in a satisfactory manner. The remainder of the text is organized as follows. In section 2, the experimen ­ tal procedures are described, and displacement patterns observed in the experi­ ments are presented. In section 3, the models used to simulate the displacement processes are described, and simulated displacement patterns are presented. A brief discussion of simplified and modified simulation models that were tested and discarded is included. In section 4, the distributions of fragment sizes measured in the experiments and in the simulations are compared. In section 5 attempts are presented to represent the fragment size distributions obtained from simulations by simple scaling forms. Conclusions from the work are presented in section 6.

2 Experiments

The experimental two-dimensional media consisted of a confined monolayer of 1 mm or 2 mm glass beads. The beads were randomly thrown onto a sheet of sticky contact paper bounded by a rectangular silicone border until no place for more beads was left. After removal of the excess beads, another sheet of contact paper was applied on top making the model air-tight. The two-dimensional bead model was sandwiched between two 25 mm polymethylmethacrylate sheets. The

2 Figure 1: The experimental set-up used in the experiments on IP cluster frag ­ mentation. (a) supporting plates, (b) inflated membrane, (c) bead layer, (d) inlet for the non-wetting fluid, (e) weight balance, (f) wetting fluid reservoir. The coordinate system is indicated. lower sheet had a transparent membrane attached to it which, when inflated, pressed the beads against the other sheet to ensure that the cell was only one bead thick everywhere. The pore space formed in this manner had a random geometry. The pores varied in diameter from approx. 0.5 - 2 mm and had 3-6 pore necks adjacent to them. In all experiments, air was used as the non-wetting fluid. A glycerin/water mixture with 1 % Nigrosin black dye was used as the wetting fluid with respect to air at the contact paper/glass beads surfaces. This fluid has a viscosity of = 6-10-2 P, a density of p — 1123 kg/m 3, and a surface tension of a = 44-10-3 N/m.

2.1 Experiments on IP cluster fragmentation The cell was positioned horizontally, evacuated and saturated with the wetting fluid. Along the shorter edge of the cell an open channel was prepared as an inlet. A corresponding channel on the opposite edge was used as an outlet. A reservoir with glycerine/water mixture was placed on a Mettler PE3600 weight balance. The weight balance rested on a stand with an adjustable height. The liquid in the reservoir was connected to the outlet of the porous medium. The pressure P of the wetting fluid was controlled by adjusting the height of the reservoir. The mass of liquid that entered or exited the reservoir was measured by the weight balance. Figure 1 shows the set-up. In the beginning of the experiment the pressure P of the wetting fluid was slightly higher than the atmospheric pressure P0. The experiment started by slowly lowering the reservoir with a constant speed v = 0.5 cm/h, to continuously reduce P. Wetting fluid was slowly withdrawing from the medium through the outlet while air entered at the inlet edge and formed an invasion percolation-like displacement pattern. Some regions of wetting fluid became engulfed and isolated from the reservoir by invading air. Shortly before breakthrough of the growing cluster, the pressure P of the wetting fluid that was connected to the reservoir had been reduced to a value of approx. 200 Pa below P0. At this stage, the process was reversed. The reservoir was lifted continuously at the same constant speed v. When P increased, the air was driven back. At first, the reservoir was still below the plane of the cell and P was less than fb, to counteract the capillary forces. Further pores were

3 Figure 2: Patterns observed in a IP cluster fragmentation experiment. Part (a) shows the system at breakthrough. The non-wetting fluid (air, white) had invaded along the top edge and displaced part of the wetting fluid (water/glycerol, black) out of the porous medium. The non-wetting fluid formed a connected IP-like cluster. The saturation was F % 0.37. (b) The system during the displacement of the non-wetting fluid by the wetting fluid, (c) The remaining cluster fragments that were isolated at the final stage at which no more displacement of non-wetting fluid was possible. The saturation was F % 0.24. abandoned by the air when the reservoir was lifted above the cell plane and P was increased to a final value of approx. 250 Pa above P0- The invading wetting fluid bypassed some of the receding air and occasionally caused fragmentation of the air cluster. Large fractions of the air could become isolated when air from a pore in the vicinity of the air inlet was withdrawn. These cluster fragments remained immobile and could not be displaced by the imbibing wetting fluid. The experiment was terminated when the remaining air was completely cut off from the edge of the model through which it had entered, and an increase of P did not lead to further displacement. Figure 2 shows a sequence of displacement patterns observed at breakthrough after air invasion, (Fig. 2 a), during the withdrawal of air (Fig. 2 b), and at the final stage when no more displacement of air was possible (Fig. 2 c). At this stage, the remaining air was disconnected from the outlet. The two-dimensional porous media used in the experiment shown in Fig. 2 had a size of approx. 100x200 pores. By the end of this experiment, about 35 % of the amount of air contained in the initial cluster was displaced from the cell. Figure 3 shows a close-up of a region of an air cluster observed in an experi ­ ment in which he air entered and left the cell through a hole in the center of the upper plate. Only a few experiments of this type were carried out, since the with-

4 Figure 3: Close-up images of patterns observed in a IP cluster fragmentation experiment using point-injection. The bright spots indicate the glass beads of the cell, (a) The cluster formed at the end of the invasion of non-wetting fluid, (b) The tips of the cluster receded, (c) Cluster fragmentation toook place, (d) Further receding and fragmenting of the tips of the cluster. drawal of non-wetting fluid could come to an early end when the center region became disconnected from the remaining non-wetting fluid cluster. The cluster of non-wetting fluid formed by the invading air is interspersed by thin “fjords ” of wetting fluid that separate pairs of adjacent invaded pores (Fig. 3 a). The fiords consisted of chains of pore necks in which the invading air did not manage to break through [25, 26]. Thus almost every invaded pore had at least one adjacent pore neck that was filled with wetting fluid but not necessarily connected to the reservoir other than through the wetting films along the edges and hollow spaces of the medium [19]. When the pressure in the wetting fluid increased, the branches of the air cluster became thinner, and the cluster tips retracted (Fig. 3 b). At higher pressures of the wetting fluid, fragmentation could occur, mostly at locations where a channel of air was strongly curved (Fig. 3 c). Once a fragment was disconnected, further retraction of tips occurred at other parts of the cluster (Fig. 3 d). The sequence of pores from which the air was displaced was different from the sequence of pores invaded during the preceding air invasion. Retraction of cluster tips often occurred simultaneously in different regions. The saturation F of the experimental cell with non-wetting fluid in a edge- injection increased steeply as the pressure P of the wetting fluid was reduced from atmospheric pressure P0. The increase is well known from pressure-saturation experiments [22, 27, 18, 21] and may be interpreted as a percolation phenomenon [4, 6, 7]. As the pressure difference was reduced and became negative, F decreased and reached a final value at about two thirds of the breakthrough value.

5 d

Figure 4: The experimental set-up used in the experiments on IP cluster migra ­ tion. (a) supporting plates, (b) inflated membrane, (c) bead layer, (d) rotation axis (e) inlet for the non-wetting fluid (f) valve. The coordinate system and the direction of gravity is indicated.

2.2 Experiments on IP cluster migration and fragmenta ­ tion

The cell was placed on a stand that allowed the inclination angle a between the plane of the cell and the horizontal plane to be controlled. The effective acceleration due to gravity acting on the cell was gsin(a). Figure 4 shows the set-up. The cell was evacuated and saturated with the wetting fluid. The cell was positioned horizontally (a = 0°), and air was injected slowly through an inlet in the middle of the cell. The displaced wetting fluid exited the cell through outlets at the edges of the cell. In this way an invasion percolation-like cluster of non-wetting fluid was formed. When the cluster covered sufficiently many pores (approx. 2000 to 10,000 pores), the injection of air was stopped and the cell was sealed. An increasing pressure gradient was imposed by rotating the cell slowly about its horizontal axis (z-axis). Since the air was less dense than the wetting fluid, the cluster experi­ enced buoyancy forces in the ^-direction that competed with the pinning capillary forces. As the inclination angle a was increased, the buoyancy forces surpassed the capillary forces and the cluster started to migrate. The migration occurred through a sequence of steps consisting of elementary displacements. The cluster migrated by withdrawing non-wetting fluid from pores at its “lower ” boundary and redistributing it in newly invaded pore spaces at the “upper ” boundary. An elongated branch-like structure was formed. The branches grew from the upper boundary of the original cluster and meandered in the redirection. As the buoyancy forces increased, the primary branch fragmented and the next generation of small percolation-like air fragments was formed. These frag ­ ments became mobilized and started to migrate at larger values of ct, later in the experiment. The experiment was terminated when the inclination angle reached its maximum value of a = 90°. The formation of branches increased the extension of the migrating cluster along the direction of the pressure gradient. The buoyancy forces that acted on the cluster were thus increased. At this stage, the cluster was elongated along

6 f > X

Figure 5: Three stages in the migration of a IP-like cluster of non-wetting fluid (air, white) covering approx. 4000 pores through a porous medium saturated with a wetting fluid (water/glycerol, black). The plane of the cell was rotated continuously. The original cluster (a) underwent fragmentation and distortion as the effective gravity force increased. The photographs were taken at a — 0° (a), 2.1° (b), and 2.4° (c), respectively. The coordinate system and the direction of gravity is indicated.

the direction of the pressure gradient. Eventually, withdrawal events occurred that led to fragmentation of the air cluster. Fragmentation reduced the buoy ­ ancy drive, and newly formed fragments were left “behind” the migrating cluster. When the capillary forces again surpassed the buoyancy forces, the migration of the cluster ended. When the inclination angle was further increased, the new fragments started to migrate. Migration was enhanced when fragments moved into contact and underwent coalescence as a result of an increase in the extension in the (/-direction. Migration and subsequent fragmentation occurred in bursts between long pe­ riods of inactivity. The inclination angle a was increased with a constant low rate of 4° per hour so that a can be considered to be constant during migration sequences. The time scale of migration was sufficiently slow to resolve elementary displacement events by eye. Figure 5 shows a sequence of displacement patterns observed in an experiment using a cell of approx. 100x200 pores. Beads with a diameter of 2 mm were used in this experiment. Figure 6 shows a similar sequence using a larger cell and 1 mm beads. In this case, the migration of the IP-like air cluster began at an even lower inclination angle a due to the higher buoyancy forces acting on large clusters.

7 Figure 6: Four stages in the migration of a IP-like cluster of non-wetting fluid (air, white) covering approx. 10,000 pores through a porous medium saturated with a wetting fluid (water/glycerol, black). The plane of the cell was rotated continuously. The photographs were taken at a = 0° [original cluster, (a)], 2.8° (b), 3.5° (c), and 85.4° (f), respectively. The coordinate system and the direction of gravity is indicated.

3 Simulations

The experiments were simulated using stochastic models, based on the invasion percolation algorithm [5, 16, 17]. In the simplest form of IP, an invasion threshold Pi is assigned to each site i on a lattice of sites. The invasion thresholds are random numbers uniformly distributed over the interval 0 to 1. The sites represent pores, and the invasion thresholds represent the sizes of channels that connect the pores. Initially, all but one site is occupied with “defender” fluid, and a seed site is filled with “invader” fluid. This site represents a growing cluster of invader fluid. At each step in the simulation, the site on the unoccupied perimeter of the invader fluid cluster with the lowest threshold is filled. The unoccupied perimeter includes all empty sites that are adjacent to filled sites. The filled sites form a fractal cluster embedded in the surrounding defender fluid reservoir.

3.1 Modeling displacement mechanisms The mechanisms of slow displacement of wetting fluids by non-wetting fluids in porous media have been studied extensively [18, 19, 20, 21]. The conclusion from these efforts is that the displacement of a wetting fluid out of a pore and the invasion of the pore by a non-wetting fluid is determined by the geometry of the pore neck that connects the invaded region to the pore. The pressure that the non-wetting fluid must overcome to invade a pore neck is inversely proportional to the size of the neck. The region invaded by the non-wetting fluid has a fractal

8 pftt r

Figure 7: Schematical illustration of imbibition mechanisms in a square net­ work porous medium. The wetting fluid (shaded) displaces the non-wetting fluid (white), (a) Snap-off invasion of a channel (b) Piston-like invasion of a channel (c) II invasion of a pore (d) 12 invasion of a pore. This figure is taken from Ref. [19]. geometry [2], and the invasion of pores occurs in bursts [22, 23]. The opposite process, slow displacement of non-wetting fluid by wetting fluid, is not only governed by the pore size but also by the local configuration of the fluid-fluid interface [18, 20, 24]. In quasistatic displacements of a non-wetting fluid by a wetting fluid, the channels that connect pores may be invaded by the wetting fluid in “piston-like ” processes, or by “snap-off ” events [19, 24] (see Fig. 7). Snap-off invasion occurs at channels that connect two pores filled with non-wetting fluid (Fig. 7 a). In narrow channels, the film of wetting fluid that covers the surface of the porous network may swell and eventually choke off the non-wetting fluid [19]. Piston-like invasion refers to the invasion of a channel at the interface, i.e., one that connects a pore filled with non-wetting fluid with a one that is already filled with wetting fluid (Fig. 7 b). The non-wetting fluid that occupies pores is preferentially displaced at pores that are connected to the rest of the non-wetting fluid by a single filled channel (“II imbibition ”, Fig. 7 c) [19, 24]. If no such pore is available along the interface and the pressure of the invading wetting fluid is sufficiently high, “12” displace­ ments occur (Fig. 7 d). 12 refers to the invasion of pores that are connected to the remaining non-wetting fluid by only two filled channels that are adjacent to each other. Other interface configurations have been observed to be of comparable stability. Although the random geometry of the porous media used in the present work prevented an unambiguous classification, the cluster tip retractions observed (Fig. 3 b) may be regarded as piston-like and II displacements, respectively. Fragmentation of the structures of air in pores where the fluid-fluid interface was strongly curved may be regarded as 12 and snap-off displacements, respectively. To simulate these processes, a site-bond IP model was used that incorporated the dominant displacement mechanisms II, 12, piston and snap-off. The pores in the two-dimensional porous medium were represented as sites on a two-dimensional square lattice and the channels connecting the pores were represented as bonds connecting the sites. A random number pi was assigned to each bond i, and a random number qj was assigned to each site j. The numbers {p} were used to compute the capillary threshold pressure 4>{q) that must be overcome to fill a region with non-wetting fluid. Similarly, the numbers {q} were used to compute the threshold pressure

9 rr' '"iis' 11 IT

Figure 8: The six nearest-neighbor bonds of the hatched bond are shaded.

were assigned to the bonds and sites at the border of the lattice (blocking boundaries). The distribution of the random numbers {p} aimed to represent the distribu­ tion of the inverse of the pore neck sizes identified in a digitized image of a section of the experimental cell. The random numbers {q} were distributed uniformly on the unit interval.

3.1.1 Simulation of the displacement of a wetting fluid by a non ­ wetting fluid

In the displacement of the simulated wetting fluid by the simulated non-wetting fluid, the invasion threshold 0 of a bond i was given by (p,) = p,, and the invasion threshold of a site was zero. In an invasion step, the non-wetting fluid displaced the wetting fluid either from a bond, or from a bond and a site. Each invasion step consisted of identifying all the bonds that were occupied by the wetting fluid and that were adjacent to the region occupied by the non-wetting fluid. The bond with the lowest threshold 4> was chosen and “invaded” by the non-wetting fluid. If the invaded bond led to a site that was still occupied by the wetting fluid, that site was invaded as well. To account for the very low compressibility of the displaced wetting fluid, a wetting fluid site was “trapped ” [16, 17] and could not be invaded by the simulated non-wetting fluid if it was not connected to the surrounding “infinite” reservoir of wetting fluid by a path consisting of steps between nearest-neighbor wetting fluid sites. Similarly, a wetting fluid bond was trapped if there was no path consisting of steps between nearest-neighbor wetting fluid sites or neighbor bonds leading to the reservoir. Wetting fluid transport by film flow along the hollow spaces of the cell [19] was included in the simulation in the following manner. The six nearest-neighbor bonds of a bond (see Fig. 8) could connect the wetting fluid in a given bond with the “infinite” reservoir of wetting fluid. Collinear bonds could form a connected path of nearest-neighbor wetting fluid bonds even if the sites adjacent to the bonds were filled with non-wetting fluid. In this way, “fjords ” consisting of bonds occupied with wetting fluid could penetrate the IP cluster of non-wetting fluid, as observed in the experiment (Fig. 3 a). Figure 9 illustrates the trapping rules.

10 t t t t t t

Figure 9: Illustration of the trapping rules used in the simulations. The simulated non-wetting fluid (white) entered and left the lattice through the inlet row (ar­ rows) and could trap clusters of wetting fluid (shaded). The wetting fluid site A and the bond a are trapped. The wetting fluid bonds b - e are not trapped since they are neighbors to the untrapped wetting fluid bonds /, g and h, respectively.

3.1.2 Simulation of the displacement of a non-wetting fluid by a wet­ ting fluid In the simulations of the displacement of the non-wetting fluid by the wetting fluid, the four dominant piston, snap-off, II and 12 displacement mechanisms were included in the model. In a displacement step, the wetting fluid displaced the non-wetting fluid either from a bond (by piston-like or snap-off processes), or from a site (by II or 12 processes). In each step, the bonds and sites from which the simulated non-wetting fluid could withdraw were identified. The bond or site with the lowest withdrawal threshold

■00—00 piston-like snap-off bond invasion bond invasion

C+0 C+1/2 C+1 Invasion Threshold

Figure 10: Schematical histogram of the distribution of (negative) withdrawal thresholds

11 simulated withdrawal mechanisms. Non-wetting fluid in bonds that connected a wetting fluid site with a non-wetting fluid site was withdrawn via a piston-like process. These bonds had a constant minimal withdrawal threshold of y = — 1. Fluid in bonds that connected two non-wetting fluid sites was withdrawn by snap-off. These bonds had a constant maximal withdrawal threshold of ip = 0. Non-wetting fluid at sites with a single adjacent non-wetting fluid bond was withdrawn via the II mechanism. These sites had a low withdrawal threshold of <~p{q) = —{q + l)/2. Finally, fluid at sites with two adjacent non-wetting fluid bonds forming a right angle was withdrawn by the 12 process, using a higher withdrawal threshold of tp(q) = —qj2. Non-wetting fluid at sites with all other configurations of adjacent bonds could not be withdrawn. To satisfy the condition of incompressibility of the simulated wetting fluid, only non-wetting fluid at sites that were adjacent to untrapped wetting fluid sites (connected to the surrounding reservoir of wetting fluid by a path consisting of steps between nearest-neighbor wetting fluid sites) could be withdrawn. Similarly, bond withdrawal was only possible for bonds with untrapped wetting fluid bonds among their nearest-neighbor bonds (see Fig. 8). When applying the trapping rules, the sites and bonds forming the blocking boundary were counted as non ­ wetting fluid sites and bonds.

3.2 Simulation of the experiments on IP cluster fragmen ­ tation To simulate the displacement experiments described in section 2.1, one of the shorter edges of a lattice of size L x 2L represented the injection edge for the non-wetting fluid, and the opposite edge represented the “infinite” reservoir of wetting fluid. The longer edges were impenetrable. The simulations began by labeling the row of sites and bonds to represent the injection edge at one end of the lattice with the non-wetting fluid. The remaining sites and bonds were labeled to represent the wetting fluid. An IP cluster of non ­ wetting fluid was formed by carrying out a series of invasion steps with trapping, as described in section 3.1.1. The invasion of non-wetting fluid was terminated at breakthrough when the IP cluster extended across the entire lattice and reached the edge representing the reservoir of wetting fluid. In the second part of the simulations, the withdrawal of the simulated non ­ wetting fluid and the invasion of the wetting fluid was simulated by withdrawing non-wetting fluid bonds and sites from the lattice, in the manner described in section 3.1.2. The withdrawals represented transport through the inlet (the for ­ mer injection edge at the edge of the lattice in edge-injection, or injection point in the center of the lattice in point-injection). Fluid from non-wetting fluid sites and bonds was withdrawn in the order of the withdrawal thresholds y assigned to the corresponding bonds and sites. In a typical sequence of withdrawals, bond and site withdrawals by piston and by II steps occurred until no more non-wetting fluid at sites with three adjacent wetting fluid bonds was present. Then a site withdrawal by an 12 step occurred.

12 Figure 11: Patterns obtained during the simulation of IP cluster fragmentation experiments using edge-injection. The non-wetting fluid is shown in white, (a) IP cluster of non-wetting fluid formed at the end of the displacement of wetting fluid by non-wetting fluid (breakthrough). The non-wetting fluid entered along the top, and 6995 sites had been invaded (P = 0.35). (b) The system during displacement of non-wetting fluid by wetting fluid at a wetting fluid “pressure ” of 1/2. At this stage, no 12 withdrawals had occurred. 6139 sites were invaded by non-wetting fluid at this stage (P = 0.31). (c) The system at the final stage when no more withdrawal was possible. 4898 sites remained invaded by non-wetting fluid (P = 0.24).

This always enabled a new series of II and piston withdrawal steps to take place. Snap-off withdrawals of fluid from bonds occurred only at the final stage of the simulation due to the high withdrawal threshold assigned to this process. Some of the withdrawals of non-wetting fluid sites and bonds caused fragmen ­ tation of the original IP cluster formed during the invasion of non-wetting fluid. Fluid at sites and bonds that were disconnected from the inlet (because no path consisting of steps between nearest-neighbor non-wetting fluid sites and bonds to the inlet existed) could not be withdrawn anymore. These bonds and sites formed immobile fragments of the original IP cluster. Figure 11 shows a typical sequence of displacement patterns obtained during a simulation on a lattice of size 100x200. The initial IP cluster of non-wetting fluid occupied 6995 sites. The simulation of the subsequent displacement of the non-wetting fluid by the wetting fluid lead to the withdrawal of about 30 % of the non-wetting fluid. Right after the beginning (Fig. 11a), all non-wetting fluid bonds that had untrapped wetting fluid bonds among their neighbor bonds and that connected a non-wetting fluid site with a wetting fluid site were withdrawn via piston-like processes. While the “pressure ” of the wetting fluid, represented by the maximum withdrawal threshold of the bonds and sites that had been

13 withdrawn at each stage, increased from —1 to —1/2, II withdrawal steps occurred (Fig. 11 b). When the “pressure ” of the invading wetting fluid increased further, 12 with­ drawal became possible. 12 withdrawal steps could lead to fragmentation of the IP cluster. Most of the 12 withdrawal steps turned one or more adjacent non-wetting fluid sites into “dangling ends” that had only a single adjacent bond connecting them to the IP cluster, and that then could be withdrawn in an II step. 12 withdrawal steps could also re-connect trapped clusters of wetting fluid to the wetting fluid reservoir. These clusters were then not trapped anymore, and the non-wetting fluid sites adjacent to these clusters became exposed to withdrawal too. When the wetting fluid “pressure ” reached the final value of 0, non-wetting fluid bonds that were still connected to the inlet row were withdrawn by snap- off. Each such withdrawal could enable further site withdrawals since it exposed additional sites to II or 12 withdrawal. In the final stage shown in Fig. 11 c, the remaining non-wetting fluid was split up in several clusters (some of them only separated by a bond occupied with wetting fluid). A small fraction of wetting fluid was still trapped.

3.3 Simulation of the experiments on IP cluster migration and fragmentation To represent the displacement experiments described in section 2.2, the injection of non-wetting fluid into the experimental cell at an inclination angle a = 0° was simulated, using a lattice of lattice of size L x 2L. An injection site in the center of the lattice was filled with non-wetting fluid. The remaining sites and bonds were occupied with simulated wetting fluid. An IP cluster was grown by filling bonds and sites adjacent to the non-wetting fluid region with non-wetting fluid, in the manner described above. The invasion was terminated when the cluster had reached a given size. At this stage, the migration of the IP cluster, under the influence of increas­ ing buoyancy forces, was simulated. The simulation consisted of a sequence of migration steps that involved either the migration of a bond, the migration of a site, or the simultaneous migration of a site and an adjacent bond. In each step, non-wetting fluid was withdrawn from a source site (bond), and migrated by invading a destination site (bond). The numbers of non-wetting fluid bonds and sites were conserved separately. The migration steps were driven by buoyancy, expressed as the product of a parameter / times the distance along the y-axis between the source and the des­ tination, and opposed by capillary forces. The gradient parameter / represented the effective buoyancy force per volume Apg sin a in the experiment, where Ap is the density difference between the wetting and the non-wetting fluid. At each stage, the pressure balance

II = + ip- fAy (1)

14 Figure 12: Patterns obtained during the simulation of IP cluster migration ex­ periments. The non-wetting fluid is shown in white, (a) The system at the end of the simulation of invasion of non-wetting fluid (white). An IP cluster covering 6000 sites was formed, (b) A meandering branch grew at / = 0.0036. (c) The non-wetting fluid cluster was highly fragmented at / = 0.0106. was evaluated for all possible migration steps, and the step yielding the mini­ mum balance IIm was determined. The first two terms on the right-hand side of Eq. (1) represent the capillary forces required for the non-wetting fluid to invade the destination site (bond) and to withdraw from the source site (bond). The framework of invasion thresholds and withdrawal thresholds <~p described above was used. Since the withdrawal thresholds

0. The first condition was met if the threshold assigned to a step was low or if the threshold cp and the height difference Ay between source and

15 destination were large. The second condition was necessary to avoid unphysical loops with non-wetting fluid migrating back and forth. Loops could arise since particular site migration steps and bond migration steps involving low thresholds could take place even in the absence of driving buoyancy force. If no migration step was possible, the gradient parameter / was increased in small steps. Some of the migration steps involving withdrawals by 12 steps or by snap- off processes led to fragmentation of the migrating IP cluster. The fragments migrated independently from each other and could coalesce. For a migration step to occur, the destination and the source had to be part of the same fragment of non-wetting fluid. In each migration step, incompressibility of the wetting fluid was taken into account. Clusters of wetting fluid that became engulfed by migrating non-wetting fluid were trapped. Migration steps were only possible if both the destination site (bond) and source site (bond) were adjacent to the same cluster of wetting fluid (including the surrounding “infinite” cluster). The withdrawal of non-wetting fluid frequently led to reconnection of trapped clusters of wetting fluid with the surrounding “infinite” cluster of wetting fluid. The migration of the non-wetting fluid fragments occurred in bursts, similarly to those observed in the experiments. After a fragment had started to migrate, a multitude of migration steps took place. The extension of the migrating fragment in the {/-direction was reduced by repeated fragmentation events. When the buoyancy drive became too weak to support further migration of the fragment, the migration ceased. When the buoyancy parameter had increased significantly, the fragment could migrate further. Coalescence with other fragments increased the {/-extension and prolonged migration. Figure 12 shows a sequence of displacement patterns obtained in a simulation in which / was increased in steps of size 10”4 from 0 to 0.01.

3.4 Modifications and simplifications of the simulation mod ­ els Figure 13 a shows the final stage of a simulation of the IP cluster fragmentation experiment (section 2.1) using edge-injection in which the distinction between II and 12 site withdrawal was dropped. For both cases, the withdrawal thresholds if were given by f(q) = q, rather than using the scheme described in section 3.1.2. The IP cluster at breakthrough was the same as shown in Fig. 11 a. Comparing Fig. 11 c with Fig. 13 a, the importance of the bias in favor of II site withdrawal becomes apparent. In the absence of such a mechanism, a smaller amount of non-wetting fluid was displaced. Non-wetting fluid was withdrawn from sites at all parts of the IP cluster since the enhanced withdrawal of sites with two adjacent neighbor bonds opened “channels” for the displacing wetting fluid. The regions of wetting fluid close to the outlet became untrapped at an early stage. This enabled withdrawal of defender fluid sites that were located near the outlet. Large fractions of the IP cluster became disconnected and could not be withdrawn.

16 Figure 13: The final stages of simulations of IP cluster fragmentation using mod ­ ified models. The original IP cluster is shown in Fig. 11 a. Part (a) show a model in which the distinction between II and 12 site withdrawal was dropped. 6128 sites remained invaded by non-wetting fluid at the final stage. Part (b) shows a model in which bond snap-off was enhanced. 5981 sites remained invaded by non-wetting fluid at the final stage.

In a similar simulation in which the thresholds for both piston bond with­ drawal and snap-off bond withdrawal was a random variable, defined by

17 using a set of invasion thresholds assigned to the sites. The only two withdrawal mechanisms included in the model were II and 12 withdrawal steps, the distinction between the two being based on the state of the adjacent sites rather than on the state of adjacent bonds. In another simplified version, a bond IP model was considered in which invasion thresholds and withdrawal thresholds were assigned to the bonds on a lattice of bonds. After the non-wetting fluid had formed an IP cluster consisting of occupied bonds, the non-wetting fluid was withdrawn using piston-like bond withdrawals and snap-off bond withdrawals. None of the simplified models was found to reproduce the experimental displacement patterns in a satisfactory manner. Versions of the site-bond IP model for cluster migration that were modified or simplified in manners analogous to the ones described above did not perform satisfactory. In the simulations discussed in the remainder of this text the site- bond IP models described in sections 3.2 and 3.3 were used.

4 Comparison of experiments and simulations

Quantitative comparisons of experiment and simulation were carried out by mea­ suring the size distribution and scaling properties of the fragments. In the ex­ periments, the size of a fluid fragment was determined by counting the number of pixels representing the fragment in the digitized image. One pore was repre­ sented by approx. 30 pixels. Fragments that occupied less than 30 pixels were ignored. Pixels belonging to a non-wetting fluid fragment were connected via nearest- or next-nearest neighbors. Wetting fluid fragments were defined via nearest-neighbor connection of pixels. In the simulations, the size of a fragment was defined as the number of sites occupied by the fragment. The sites belonging to a fragment were connected via nearest-neighbor sites, linked by bonds.

4.1 Comparison between experiment and simulation on IP cluster fragmentation Figure 15 shows, on a log-log plot, the cumulative distribution of non-wetting fluid fragments N(s > s*) vs. s*, measured in the experiment shown in Fig. 2 c. N(s > s*) denotes the number of fragments that have a size s equal to or greater than a size s*. In the same figure, the distribution of fluid fragment sizes observed in simulations is plotted. The distribution of non-wetting fluid fragments was measured at the end of the simulations, when no more withdrawal was possible, and was averaged over several hundred simulations. The simulations were carried out on a lattice of 100x200 sites, corresponding to the cell size used in the experiment. Also shown in in Fig. 15 is the cumulative size distribution of trapped wetting fluid clusters. The distribution was measured at the final stage in the experiment shown in Fig. 2 and in corresponding simulations. Only trapped wetting fluid clusters that were not adjacent to the edges of the experimental cell were taken

18 1.5 *

0.0 - gf 1.0 Slope = 1/1.82

-2.0 -

Figure 14 Figure 15

Figure 14: Cumulative fragment size distributions N(s > s*) measured in the experiment shown in Fig. 2 (circles) and simulations (solid line) on IP cluster fragmentation at the final stage, plotted as a function of s* on a log-log scale, s denotes the approximate number of occupied pores, and the number of occupied sites, respectively. Also shown are the cumulative size distributions N(s > s*) of trapped wetting fluid fragments, measured in the experiment (squares) and in simulations (dotted line).

Figure 15: The radius of gyration Rg of the non-wetting fluid fragments in ex­ periments on IP cluster fragmentation, measured at the final stage and plotted as a function of fragment mass s. The circles refer to fragments observed in the pattern Fig. 2 c, and the solid line refers to simulation data. The dotted line is a linear least squares fit to the simulation data with slope 0.53. The dashed line has a slope of 0.55 corresponding to the fractal dimensionality of IP clusters, with trapping. into account. The agreement between the fragment size distributions measured in the ex­ periments and in the simulations is fair. Deviations are found in the distribution of trapped wetting fluid clusters. These may be attributed to the fact that a substantial part of the wetting fluid belonging to a trapped cluster was located in pore necks and channels (see Fig. 3). Fluid in channels connected to a cluster contributed to the total cluster size (mass). In the simulations, only wetting fluid sites contributed to the cluster size, leading to a potential underestimation of the size. A kink at low s, observed in the simulated cluster fragment size distribu­ tions, is attributed to the geometry of the square lattice. The lattice structure makes the formation of non-wetting fluid fragments of 2 and 3 sites unlikely. Figure 15 shows, on a log-log scale, the radius of gyration Rg (s) of the non ­ wetting fluid fragments, plotted versus the fragment size s. Rg (s) was measured at the final stage in edge-injection experiments and simulations. Only fragments with size s > 10 were included. Rg($) was expected to scale with the fragment

19 size s as [3]

R,(,) ~ s''D (2) where D is the fractal dimensionality of the fragments. If the fragments retain the structure of the original IP cluster, Rg (s) ~ s1/1,82 % s0,55. Here, the fractal dimensionality D ~ 1.82 of IP clusters was inserted [2, 29]. Both the simulation data and the experimental data are consistent with this assumption. A least- squares fit of the simulation data yielded Rg (s) ~ s0,53.

4.2 Comparison between experiment and simulation on IP cluster migration and fragmentation

Each stage of a migration experiment was characterized by the ratio between the buoyancy and capillary forces acting, conveniently expressed by the modified Bond number Bo = (Apga 2 jcr) sin a, where a is a typical length scale describing the pore geometry of the medium. Using the values given in section 2 and setting a rj 1 mm, Bo was found to vary from 0 to % 0.04 when a was increased from 0° to about 10°. In the simulations, the quantity corresponding to Bo is given by the ratio f / < -\-tp > of the gradient parameter / to a typical value < + <£> >% 0.5 for the sum of thresholds. The range of buoyancy drive investigated in the experiments was covered in most of the simulations by letting / vary from 0 to 0.02 in steps of 10-4. Figure 16 shows a log-log plot of the cumulative size distribution N(s > s*) of fragments of the migrating IP cluster, measured in different experiments and in simulations. The distributions were measured at various inclination angles a (experiments) and values of the gradient parameter / (simulations). In the experiments, the initial IP-like cluster of non-wetting fluid covered about 4000 and 8000 pores, respectively. The simulations were carried out using IP clusters of size sq = 4000 and 8000 sites, respectively. Due to the computational effort required, only a few runs were carried out using the largest initial size, leading to poor statistics. At each stage, the size distribution was measured when all migration had ceased. The agreement between the cluster fragment distributions found ex­ perimentally and measured in simulations is satisfying. The steep decay of the distributions at large s is not seen in the experimental data for statistical reasons. As in Fig. 15, the radii of gyration Rg of the fragments observed in experiments and in simulations are plotted as a function of the fragment size s in Fig. 17. The experimental data was measured at inclination angles of a = 2.1° and 2.4°, respectively. The simulation data was measured at a gradient parameter of / = 0.0107, using initial IP cluster of size sq = 2000 sites. A least-squares fit of the simulation data yielded Rg (s) ~ -s0,57. Both the experimental data and the simulation data are consistent with the idea that the fragments form small IP-like clusters with the fractal dimensionality of D ~ 1.82.

20 f=0.0097

f=0.0012

1.5 -

1=0.0097

£1.0

1=0.0012

0.0 •

i°g,o(s*)

Figure 16 Figure 17

Figure 16: Cumulative size distributions of non-wetting fluid fragments N(s > s*) observed in migration experiments (dotted lines and symbols), plotted as a function of s* on a log-log scale. Different symbols refer to observations at different tilting angles a. The initial IP cluster covered about 4000 pores (a), and 8000 pores (b), respectively. The solid lines refer to the cumulative distributions observed in simulations.

Figure 17: The radius of gyration Rg of the non-wetting fluid fragments in exper ­ iments on IP cluster migration, measured at two slightly different tilting angles a (circles, squares). The solid line refers to measurements in simulations at a fixed value of the gradient parameter /. The dotted line is a linear least squares fit to the simulation data with slope 0.57. The dashed line has a slope of 0.55 corre ­ sponding to the fractal dimensionality of IP clusters, with trapping, observed in simulations were distributed.

5 Scaling properties of selected quantities

In the experiments, the system size could not be varied over a large range due to practical limitations, and only a handful of experiments could be conducted. The scaling behavior of a number of quantities was studied by means of simulations. In the simulations of IP cluster fragmentation, the analysis included the study of the fragment size distributions and of the saturation with non-wetting fluid at the final stage. In the simulations of IP cluster migration and fragmentation, the dynamics of the migration process was studied by measuring and comparing properties of fragments that were immobile and of fragments that were in the process of migrating.

21 TV -0.4 -

-1.0 • D E F 4.0 • 2 -2.0 1.5 2.0 2.5 3.0 -5.0-4.0-3.0-2.0-1.00.0 0.0 - 3.0 • -2.0 - ABC

•4.0 -

Figure 18 Figure 19

Figure 18: The size distributions N(s, L) of non-wetting fluid fragments measured in simulations of IP cluster fragmentation at the final stage, using lattices of size 25x50 (A), 50x100 (B), 100x200 (C), 200x400 (D), 400x800(E), and 800x1600 (F), respectively. The inset shows an attempt to collapse the distributions, using the scaling form given in Eq. (3). The exponents r and % were obtained from an analysis of the moments y^k\ assuming that D = D. The cluster sizes were shifted to As = s + 10. Figure 19: Ratio of moments y^ of the fragment size distributions measured in simulations of IP cluster fragmentation at the final stage using different system sizes L. The ratios y^/y^ (circles), y^> j y^ (squares), and yW/yW (dia­ monds), respectively, are shown as a function of L on a log-log scale. The dashed lines are linear least squares fits to the ratio y^ /y^ and to the ratio y^/y^2\ respectively. The inset shows the saturation P&at breakthrough (points) and at the final stage Pf (squares), plotted versus L on a log-log scale. The dashed lines indicate the scaling expected for IP clusters.

5.1 Results obtained in simulations on IP cluster frag ­ mentation Figure 18 shows the size distribution N(s, L) of the non-wetting fluid fragments, measured on a log-log scale for simulations at the final stage. The simulation data of Fig. 15 is included in this figure. The data sets corresponding to the largest systems studied (L — 800) appear noisy due to poor statistics. An attempt was made to represent the distributions by the scaling form [30]

N(s,L)~L*As-9(^L) . (3)

Here, D characterizes the dependence of the amount (mass) the of non-wetting fluid remaining at the final stage on the system size L. As = s + c is the fragment

22 size, shifted by a constant amount c. The scaling function g(x) decays faster than any power of r for x > 1. sc(L) represents the cut-off fragment size that was expected to scale with the system size as

5c(&) - I* . (4)

The exponents r and % may be extracted from the scaling behavior of the mo ­ ments of the fragment size distribution N(s, L). The k-th moment g^ k\L) of the distribution N{s, L) is defined as

OO /T = £s‘iv(5,i). (5) 5=0 For L 1, the ratios of the moments were expected to scale with the system size as [14]

~ const. (6)

£x (3-t)

LX .

Figure 19 shows a plot of these quantities as a function of the system size L. On a log-log scale, the ratios 1$ {l^i) and V/4^ appeared to obey power laws as expected. For the ratio of the second to the first moment and for the ratio of the third to the second moment, linear least-squares fites yielded , values of 1.65 ±0.02 and 1.78 ± 0.02, respectively. In the fits, the data point corresponding to the largest system size (L = 800) was ignored. Using Eq. (7), the exponents X = 1.78 ± 0.02 and r = 2.07 ± 0.03 were obtained. The inset in Fig. 19 shows a plot of the breakthrough saturation F& (the number of non-wetting fluid sites divided by the number of lattice sites at break­ through). For IP clusters in two dimensions, the saturation scales as F& ~ LD~2 since the cluster size (mass) scales with the system size as s ~ LD [17, 29]. The scaling of Ft, with the system size L apparent in Fig. 19 is consistent with this expectation. During the withdrawal of the non-wetting fluid, F decreases from the breakthrough value to the final value F/. The total amount of remaining non-wetting fluid may be expressed as the first moment of the fragment size distribution,

5=0 The first moment of the fragment size distribution depends on the system size L as [14] fj,^ ~ L® {Lx^2 t) ± const^j. Making use of r > 2, a power-law de­ pendence of T/ on the system size L is obtained for L 1, T/ ~ LD 2. Since

23 r/ < T(,, the final saturation cannot scale with an exponent D — 2 greater than the IP exponent D — 2. In the inset in Fig. 19, F/ is plotted versus L on a log-log scale. The final saturation had a roughly constant value F/ ~ 0.23 for all system sizes investigated. This finding indicates that the asymptotic regime has not been reached in the simulations. This is also indicated by the observation that the ra­ tio of the first moment to the zeroth moment, /g^\ was far from reaching a constant value in the range of system sizes studied, as shown in Fig. 19. In the absence of a known value for the saturation exponent, it was assumed that D = D. The inset in Fig. 18 shows the data collapse obtained using the scaling form given in Eq. (3) and the exponents r and % obtained from the analysis of the moments. The shift As = s + 10 was chosen judiciously. The data collapse is very good, and the scaling function in Eq. (3) is found to behave as g(x) ~ rr0'35 for i<1. Using a shift of zero (As = s), the collapse is not impaired for large values of the scaling variable, but considerable spread was found for low values. From Fig. 18, it is apparent that the distributions A(s, L) assume a power-law decay only for fragment sizes of the order of 10 and larger, up to the cut-off size sc(L). The transient behavior at very low s cannot be accounted for by the scaling function g(x) and is remedied by the shift applied in Eq. (3).

5.2 Results obtained in simulations on IP cluster migra ­ tion and fragmentation In the simulations of IP cluster migration, the fragmentation depended on the gra ­ dient parameter / (corresponding to the effective acceleration of gravity gs'm(a) in the experiment). Fragments that were formed could disintegrate into smaller fragments or coalesce with other fragments during migration. The total amount of non-wetting fluid was held constant at all stages. Figure 20 shows the fragment size distributions N(s,f) measured on a log-log scale at different values of the gradient parameter / in simulations using initial IP clusters with a size s0 = 2000. At low values of /, fragments of all sizes up to so were present. The distribution of fragment sizes is peaked at values slightly below s0, indicating that the IP cluster was essentially intact at this stage, in many simulation runs. When / was increased, smaller fragments were formed at the expense of the large fragments. The distributions measured at high / are characterized by cut-off sizes sc(/) that decrease with increasing /. A standard scaling form for fragment size distributions is [30]

JV(»,/)~s"29(^) , (8) with the cut-off size sc scaling with the parameter / as

(9) The insert in Fig. 20 shows the data collapse obtained by using the scaling form given in Eq. (8), using a value of 1.8 for the exponent z in Eq. (9). The distribu­ tions corresponding to the lowest values of / were omitted in the data collapse

24 4.0

3.0

-5.0 0.0 1.0 2.0 3.0 log !Q(s)

Figure 20: Plot of the fragment size distribution N(s, /) measured in simulations of IP cluster migration and fragmentation. The initial IP cluster had a size of So = 2000 sites, and the gradient parameter / was increased in steps of size 0.0001 from 0 to 0.02. At low values of /, a peak at s0 indicates incomplete fragmentation of the IP cluster. At high /, few large fragments remain. The insert shows an attempt to collapse the distributions onto a single scaling function g(s/s c), using the scaling form given in Eq. (8) with sc(f) ~ /_1'8 . The distributions corresponding to the five lowest values of / were removed. since fragmentation and migration had not began in all of the simulations at these stages. Equation (9) may be interpreted in terms of a simple picture in which the fragments are viewed as blobs with a IP cluster-like structure. In the simulations, the fragments that have a size close to the cut-off size sc(f) are characterized by an extension lc{f) in the {/-direction (the direction of the gradient imposed) that scales with / as (10) as will be shown below. Assuming that the radius of gyration Rg of the fragments is proportional to the extension lc(f) and making use of Eq. (2), the relationship

(11) is obtained. Inserting the IP exponent D % 1.82, Eq. (11) is consistent with the empirical data collapse observed in Fig. 20. Both the initial IP cluster and the fragments began to migrate when the buoyancy force acting on them became of order 1. For each value of /, a char­ acteristic extension l^(f) ~ / 1 of fragments in the {/-direction can be defined. Figure 21 a shows a scatter plot of the {/-extensions l^ of fragments that be­ gan a sequence of migration steps at a gradient /, on a log-log scale. The mean extension lym\f) — (s))2N(s)/ Y1 ^m^(s)Ar(s) scales as lym\f) ~ / 1 in the migration regime where / > /c(s0). At the onset of migration, most of the fragments were not elongated in the {/-direction. The size of these fragments

25 Slope=-1

£ 1-0

Slope=-0.57

Slope=-1

D I------■------1------:— ------«— -4.0 -3.0 -2.0 iog 10(f)

Figure 21: Scatter plots of fragment extensions vs. the gradient parameter / on a log-log scale. The plots include data from runs using IP clusters of size s0 = 2000. The solid lines indicate averages x(f) = J2S x2(s, f)N(s, /)/ J2s x(s, f)N(s, /). Part (a) shows the mean extension in the {/-direction l^n') of fragments that started to migrate. Part (b) shows the relative width of fragments that were in the process of migrating and underwent fragmentation. Part (c) shows the mean extension in the {/-direction of fragments that were newly formed. then may be expected to scale with the gradient parameter as >>(/) ~ rD, (i2) where D re 1.82 is the fractal dimensionality of IP clusters. During a typical sequence of migration steps, the {/-extension of the fragment increased up to the value l^yp\ leading to a further increase in the buoyancy force. Some of the steps led to the formation of fragments that were left behind. The migrating fragments were elongated in the {/-direction. The geometry of a migrating fragment of size s and {/-extension may be characterized by the relative width = s/l\pK The relative width wj.p) is a better measure for the width of an elongated fragment than the extension in the ^-direction /00 since the migrating fragments meandered along the {/-direction. Figure 21 b shows a scatter plot of as a function of the gradient /, on a log-log scale. The plot is consistent with the idea that the relative width is a measure for the percolation correlation length £ characterizing the structure of non-wetting fluid. In the presence of a gradient /, the correlation length is known to scale as [31] f _ l/l-VM (13) with vj(v + 1) ~ 0.57.

26 S!ope=-1.82

Figure 22: The mean fragment size s(f) (solid line) was bounded by the mean size sn(f) of fragments that were newly formed at a given / (dotted line), and by the mean size sm(f ) of fragments that started to migrate at a given / (dashed line), respectively. For low values of the gradient parameter /, little migration occurred and the statistics are poor. The initial IP clusters had a size of s0 = 2000 sites. The expected scaling behavior (Eqs. (12 and 14 ) is indicated (straight solid line).

The migration of a typical fragment of size ceased when the {/-extension of the fragment dropped below its original value after a sequence of frag ­ mentation processes. The remaining part of the migrating fragment had a size s' < s(m>(/), and was not elongated in the {/-direction. Similarly, the fragments formed during the sequence were not elongated in the {/-direction. The typical extension in the {/-direction of fragments formed at a given / was found to scale as l^n\f) ~ /-1, as shown by the scatter plot Fig. 21 c. The typical size of fragments formed during migration at / may then again be expected to scale as

«'">(/) ~ rD ■ (14)

Figure 22 shows the mean fragment size s(/) = J2 s2Ns(f)/^2 sNs(f) plotted as a function of the gradient parameter / on a log-log plot. Also shown is the mean size s(m)(/) of fragments that started to migrate at a given stage, and the mean size s(n)(/) of fragments that were newly form ed at a given stage. In view of the poor statistics leading to the plot for s(n)(/), the data is consistent with the scaling laws given in Eqs. (12) and (14).

6 Conclusions

In the experiments described in this paper, IP-like clusters of a non-wetting fluid embedded in a wetting fluid were fragmented in a porous medium. In the exper ­ iments on IP cluster fragmentation, the fragmentation of the cluster occurred by

27 removing parts of the cluster, in a way conceptually similar to numerical studies of cluster fragmentation [13, 14, 15]. In the experiments on IP cluster migration and fragmentation, the cluster fragments were deformed in the course of the ex­ periment. The fragments formed in these experiments not only depended on the sequence of invasion and withdrawal events imposed by the pore geometries but also on the migration dynamics. The main results emerging from the present study are: (i) The experiments were simulated using modified site-bond IP models, and good agreement was found. Emphasis was placed on modeling a hierarchy of possible displacement events based on experimental observations. Similar ordering of displacement mechanisms was used in simulation models introduced by Blunt et al. [32] and Glass [33]. The success of the simulation model indicates that IP models are well suited to model slow two-phase fluid flow with complex boundary conditions, despite the fact that the distinction between the various displacement mechanisms may not be not unequivocal. (ii) In both types of experiments and simulations, a fractal fluid cluster under­ went fragmentation. The fragmentation and migration processes led to constant changes of the displacement patterns at global and local scales, but did not change the internal arrangement of the fragmenting structure. The resulting irregular fragments could be described by the same fractal dimensionality D as the initial cluster, D % 1.82. This dimensionality characterizes invasion percolation clusters embedded in two dimensions and grown in the presence of a trapping rule. (iii) The distributions of fragment sizes measured in simulations could be represented by simple scaling forms (Eqs. (3) and (8)) and characterized by typical (cut-off) sizes.

Acknowledgements

We thank A. Aharony, K. Christensen, V. Frette, L. Furuberg, P. King, R. Lenor- mand, and K. J. Malpy for helpful discussions, and we gratefully acknowledge a valuable suggestion concerning Eq. (3) by one referee. We acknowledge support by VISTA, a research cooperation between The Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (STATOIL) and by The Re­ search Council of Norway (NFR). This research has received support from the NFR programme for supercomputing through a grant of computing time.

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[11] G. Wagner, A. Birovljev, P. Meakin, J. Feder, and T. Jpssang, Europhys. Lett. 31, 139 (1995).

[12] A. Birovljev, G. Wagner, P. Meakin, J. Feder, and T. Jpssang, Phys. Rev. E 51, 5911 (1995).

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29 [21] J. S. Buckley, in Interfacial Phenomena in Petroleum Recovery, edited by N. R. Morrow (Marcel Dekker, Inc., New York, 1991), pp. 157-189.

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30 PHYSICAL REVIEW E VOLUME 51, NUMBER 6 JUNE 1995

Migration and fragmentation of invasion percolation clusters in two-dimensional porous media

Aleksandar Birovljev, Geri Wagner, Paul Meakin, Jens Feder, and Torstein J0ssang Department of Physics, University of Oslo, P.O. Box 1048, Blindem, N-0316 Oslo, Norway (Received 7 June 1994) Experiments on and computer simulations of the migration of fractal, nonwetting fluid bubbles through a two-dimensional random porous medium saturated with wetting fluid are presented. A large invasion percolation bubble was initially formed by slow injection of a nonwetting fluid into a horizontal cell saturated with a denser wetting fluid. Slow, continuous tilting of the cell caused the bubbles to migrate through the medium. The interplay between local pinning forces and buoyancy led to fragmentation and coalescence of migrating bubbles. The process was simulated by a modified site-bond invasion percolation model.

PACS number(s): 47.55.Mh, 05.40,+j, 47.55.Kf, 64.60.Ak

Transport processes far from equilibrium generate tally, and through computer simulations in two and three complex spatial and temporal patterns. Familiar ex­ dimensions [7-9]. These studies were concerned with amples include convection, crack propagation, diffusion- gravity-stabilized displacements (less dense fluid on top). limited aggregation, erosion, river formation, and parti­ Gravity-destabilized drainage was recently studied ex­ cle deposition. In many cases, the transport process take perimentally [10], theoretically [11], and through simu­ place in disordered media and the dynamics are governed lations [11]. by global driving forces and local pinning forces. In our experiments, the two-dimensional medium con ­ In this study we are concerned with the migration sisted of a monolayer of 1 mm diameter glass beads. The of a nonwetting fluid (NWF) cluster in a random two- beads were randomly thrown onto a sheet of sticky con ­ dimensional porous medium that is saturated with a wet­ tact paper bounded by a rectangular silicone border until ting fluid (WF). The process is driven by an external no place for additional beads remained. After removal force (gravity) and opposed by local pinning forces (cap­ of the excess beads, another sheet of contact paper was illarity). The structure of the original NWF cluster was applied on top, to make the model air tight. The two- destroyed in the course of the experiment, as a result dimensional bead models with approximately 140x280 of a multitude of fragmentation, migration, and coales ­ or 280x560 pores were sandwiched between two 25 mm cence events. The experiment thus allowed us to observe thick polymethylmethacrylate plates. Inlet and outlet the fragmentation of a fractal object, a topic that has holes were drilled in one of the plates. The other plate recently become the subject of growing interest [1,2]. had a transparent membrane attached to it which, when Fluid-fluid displacement processes in porous media inflated, pressed the beads against the other plate. This have been studied intensively. The process of very slow ensured that the cell was only one bead thick, every ­ drainage, when a NWF displaces a WF in a porous where. The model was placed on a stand that allowed medium, produces a structure that is well described by the angle of inclination 6 between the plane of the model the invasion percolation (IP) model [3]. The growth of and the horizontal to be controlled. the IP cluster is governed by the local capillary forces The experimental cell was evacuated and saturated that are related to the sizes of the pore necks in the with a glycerol-water mixture with a viscosity of p = medium. The opposite process of imbibition occurs when 6 x 10~2 P, density p = 1123 kg/m 3, and surface tension a WF displaces a NWF. Detailed studies [4-6] using mi­ a = 64 x 10~3 N/m, dyed with 1% black Nigrosine. The cromodels of porous media and consisting of a square cell was positioned horizontally, while another, less dense network of channels led to the identification of four dom­ fluid (air) was slowly injected through a hole in the mid­ inant imbibition mechanisms, hereafter referred to as 11, dle of the cell until an IP cluster was formed that was 12, piston, and snap off. II denotes the imbibition of as large as possible, without touching the boundaries of pores that are connected to remaining NWF by only one the cell. The IP cluster was interspersed with numerous NWF pore neck. Analogously, 12 denotes the imbibition water-filled pores and pore necks that could be engulfed of pores connected to the remaining NWF by two NWF by air (“trapped ”) or could be parts of thin “fjords ” pene­ pore necks forming a right angle. The piston denotes the trating the interior of the cluster from its perimeter. The imbibition of pore necks at the NWF-WF interface, and WF connectivity in such chains of nearest neighbor pore snap off denotes the imbibition of pore necks that con ­ necks was maintained through WF “films” in the spaces nect two colhnear NWF pores. Both the 12 and piston between the glass beads and the contact paper. may lead to fragmentation of the structure formed by the When the IP cluster had reached a sufficiently large NWF. size (approximately 2000 to 7000 pores), the air injec­ The effects of external forces, such as gravity, on the tion was terminated and the cell was sealed. The cell IP process have been studied theoretically, experimen- was rotated about the horizontal axis with a constant low

1063-651X/95/51(6)/59 11 (5)/$06.00 51 5911 ©1995 The American Physical Society 5912 BIROVUEV, WAGNER, MEAKIN, FEDER, AND J0SSANG 51 velocity of 4° per hour. This corresponded to a continu ­ tion occurs. A rough estimate of the typical length of the ous increase of the Bond number, Bo = (Apga 2/cr) sin 9, newly formed fragments is provided by a similar scaling which is the ratio between buoyancy and capillary forces. relation, l'y ~ Bo -1, since the imbibition processes lead­ Here, Ap is the density difference between the two fluids, ing to fragmentation are also governed by a competition g the acceleration due to gravity, and a the grain size in between capillary and buoyancy forces. This suggests the the medium. Under the influence of the buoyancy the IP following simple picture: The set of NWF clusters that cluster migrated and fragmented, as shown in Figs. 1(a)- is present at a given moment consists of fragments that 1(c). The process was recorded using a Nikon camera. have been formed at different angles of inclination 9 (or The negatives were digitized [9] using a Nikon 35 mm values of Bo). Fragments that are formed at 9i remain film scanner connected to an Apollo 4000 workstation. immobile for a while and become mobilized at (?2, given The migration of the IP cluster started when a suffi­ by %(%) - Z,(%). ciently large inclination angle 9 was reached. The migra ­ From earlier theoretical and experimental work on in­ tion occurred through a sequence of steps consisting of vasion percolation in a gradient [7,10,11], it is known that drainage events at the upper front and imbibition events the correlation length £ scales as £ ~ 'Bo~ v^v+1\ where at the lower front of the migrating cluster. An elon ­ v — 4/3. For IP branches formed by destabilized dis­ gated branchlike structure was formed. As the angle of placement, £ measures the mean transverse width. In inclination 6 increased, the primary branch fragmented, the absence of fragmentation, a migrating cluster is ex­ forming the next generation of small IP-like fragments pected to match this scaling relation by dynamically ad­ that could become mobilized at some larger 9. Since the justing its shape. Fragmentation interrupts the process fragmentation reduced the typical size of the NWF clus­ of adjustment and prevents the direct observation of £. ters, the buoyancy drive became insufficient for further The experimental process was simulated by modeling migration and long periods of stagnation occurred. The the basic mechanisms of drainage and imbibition on the branches meandered along the direction of the compo­ pore level [12]. The model was based on the gradient- nent of gravity in the plane of the cell (y direction) . The invasion percolation algorithm [3,7,9], which was ex­ branches propagated at a rate faster than the very slow tended to include imbibition, fragmentation, and coales ­ rate used for IP cluster growth, but new pores were still cence [13]. A square lattice of sites and bonds was used. invaded sequentially. 9 was increased so slowly that the The sites represented pores and the bonds represented relaxation time (the time during which migration, frag ­ pore necks. The main features of the simulation model mentation and coalescence occur) was much smaller than are described below. A complete description will be given the duration of the experiment. The total volume of the in a forthcoming publication [14]. NWF was conserved. At the beginning of each simulation, every site was as­ A NWF cluster with a length ly in the y direction signed a random number p' uniformly distributed from 0 experiences a hydrostatic potential difference APg = to 1, and every bond was assigned a random number p Apgl y sin 9 from its upper to its lower front. The cluster (see Fig. 2). The distribution V{p) ranged from 0 to 1 starts to migrate when APg exceeds the capillary pres­ and was obtained using pore neck size distribution data sure p c ~ a/a at its upper front. Comparing these two from a section of the experimental cell. The sets of ran­ pressures leads to ly ~ Bo -1. During migration, the clus­ dom numbers {p} and {p'} were used to compute invasion ters become elongated in the y direction until fragmenta ­ thresholds H1 and “withdrawal thresholds ” H™, respec-

FIG. 1. (a)-(c) Pattern observed at 6 = 0°, 14°, and 53°, respectively. The original IP cluster shown in white covered approximately 2000 pores, (d)-(f) Simulated migrating IP clusters at / = 0, 0.005, and 0.013, respectively. 51 MIGRATION AND FRAGMENTATION OF INVASION ... 5913

FIG. 1 (Continued). lively. The results reported here were not expected to the thresholds. The tilting of the experimental cell was depend crucially on the exact shape of V(p) [11]. simulated by increasing / by small steps. After each The simulations began by invading the center site. The increment, the NWF bonds and sites were checked to de­ standard IP algorithm with trapping [3] was applied to termine if migration steps were possible. A migration grow an IP cluster of a desired size, using the invasion step included withdrawing a NWF source site (or-bond) threshold IP — p for WF bonds and IP — 0 for WF and invading a WF destination site (or bond). The source sites, respectively. A WF site was trapped and could not and the destination had to belong to the same NWF frag­ be invaded if there was no path of nearest neighbor WF ment. The invasion events were opposed by the invasion sites leading to the surrounding “infinite” WF cluster. A thresholds IP whereas the withdrawals were supported WF bond was trapped if there was no path of nearest by withdrawal contributions IP". The next step was de­ neighbor defender bonds or sites leading to the “infinite” termined by finding the global minimum of WF cluster. At this stage, the buoyancy force was simulated by n = IP - IP" - fAy . (1) imposing a uniform gradient /, in the y direction, on The last term in Eq. (1) accounted for the driving buoy­ ancy force given by the height difference Ay between source and destination, times the gradient /. II10 was a simple function of the random numbers p' and the local configuration of the interface that established a hierarchy between the four imbibition mechanisms listed above. II withdrawal was possible for NWF sites with only one ad­ jacent NWF bond (IP" = \p' + |). 12 withdrawal was possible for NWF sites with two adjacent NWF bonds FIG. 2. Illustration of the site-bond model. To the left: forming a right angle (IP" — |p'). NWF bonds that bonds are assigned random numbers p and sites are assigned connected a NWF site to a WF site could be withdrawn random numbers p'. To the right: the WF site A is trapped pistonlike (IP" = 1), and NWF bonds that connected two by the IP cluster (white). The WF bonds a, b, c, and d are not NWF sites could be withdrawn by snap-off (IP" =0). II trapped. The NWF site B can be withdrawn by II, and the and piston withdrawals were favored according to these site C can be withdrawn by 12 (two adjacent NWF bonds). rules. 12 site withdrawal and bond withdrawal by snap- The snap-off withdrawal of the NWF bond e would lead to off could cause fragmentation of the original IP cluster. fragmentation, in contrast to the pistonlike withdrawal of /. See Fig. 2 for an illustration. 5914 BIROVLJEV, WAGNER, MEAKIN, FEDER, AND J0SSANG 51

Physically, / corresponds to the Bond number Bo used in the experiment. A quantitative mapping between / and Bo was not attempted since the distribution of the imbibition threshold pressures in the experiment was not known. In each step, incompressibility of the WF was taken into account using the same trapping rule that was used during the formation of the original IP cluster. Only -4.0 r steps with Ay > 0 and II < 0 were allowed. If no possible step could be found, / was increased. Figure 1(a) shows patterns obtained in a simulation in which 2000 sites were invaded. The distributions of the fragment sizes in both the experiments and the simulations were measured. Fig ­ ure 3(a) shows the logarithmically binned fragment size distribution measured in two experiments (symbols) at several different Bond numbers and in the simulations (solid lines). About 2000 pores were invaded by NWF in one experiment (lower data set), and about 7000 in another experiment (upper data set). In the simulations IP clusters with corresponding sizes were generated on square lattices of size 200x600 and 400x2400. The gra ­ dient |/| was increased in steps of A|/| = 3 x 10-5 from 0 to 0.02 and in steps of A|/| — lx 10~5 from 0 to 0.01, respectively. The simulated distributions were averaged over 1400 runs for IP clusters of size 2000 and over 70 runs for clusters of size 7000. The kink at low fragment sizes is due to the geometry of the square lattice, which 0.0 1.0 2.0 3.0 4.0 5.0 made the formation of fragments of size 2 and 3 unlikely. r The simulation data in the lower set show a cutoff at high fragment sizes. In the experiments, and in the upper sim­ ulation data set, the cutoff is not apparent for statistical reasons. The immobile fragments observed in the experiment were found to have roughly the same extension in the: x and y direction, in agreement with the scenario described above. Figure 3(b) shows a plot of the probability density co 2.5 F(T), where F denotes the extension in the y direction divided by the extension in the x direction, and F(F)dF % 2.0 is the probability of finding a fragment with an aspect ratio in the range F, F+dF. The histogram is based on observations at several different values of Bo. The cor ­ responding histogram for immobile simulated fragments was measured at two values of /. The histogram for fragments that had migrated and were in the process of log(f) fragmenting [shaded in Fig. 3(b)], measured at different values of /, indicates adjustment of the fragment shape by elongation in the y direction. FIG. 3. (a) The number density of fragments Ns plotted Figure 3(c) shows that the simulated mean fragment as a function of size s on a log-log (base of log is 10) scale. size S(f) (measured when all migration had ceased) de­ The solid lines are simulated distributions for various values creases with increasing gradient. S is bounded from of the gradient / in which the IP clusters had a mass of 2000 (lower data set, 0 < / < 0.02) and 7000 (upper data set, below by the mean size Sf(f) of fragments that are 0 < / < 0.01). The symbols are distributions from the cor ­ newly formed, and from above by the mean size Sm(/) responding experiments, (b) The x/y ratio T of fragments of fragments that become mobilized and begin to mi­ observed in the experiment (dotted line) and of simulated grate. The mean values were defined as S(f) = fragments at / = 0.005 and / = 0.013 (solid line) is around 1. Y^s2N„(f)/'^2sNs(f), where N3(f) is the number den­ Just before they underwent fragmentation, fragments tended sity of fragments of size s at gradient /. At low /, Sf to have F > 1 (shaded area), (c) The simulated mean frag ­ fluctuates due to poor statistics. ment size S(f) (solid line) lies between the mean size S/(/) The experiment presented here deals with the migra­ of newly formed fragments (dotted line) and the mean size tion of a fractal fluid cluster of macroscopic size through a Sm(/) of fragments that begin to migrate (dashed line). The disordered medium. The migration is driven by a contin­ IP clusters had a mass of 2000. uously increasing buouyancy force. In the course of the 51 MIGRATION AND FRAGMENTATION OF INVASION ... 5915 migration process, the original cluster disintegrates and We thank A. Aharony, K. Christensen, V. Frette, P. fragments are formed. The fragment size distributions King, R. Lenormand, and K. J. Malpy for helpful dis­ appear curved on a log-log scale, giving no convincing cussions. We gratefully acknowledge support by VISTA, evidence for a power law. The process may be modeled a research cooperation between the Norwegian Academy with good qualitative and quantitative agreement using of Science and Letters and Den norske stats oljeselskap a modified site-bond IP model. The dynamics exhibited a.s. (STATOIL) and by NFR, the Norwegian Research by the migration process calls for further exploration. Council.

[1] M.F. Gyure and B.F. Edwards, Phys. Rev. Lett. 68, 2692 (Paris) 46, L1163 (1985); E. Clement, C. Baudet, E. (1992). Guyon, and J.P. Hulin, J. Phys. D 20, 608 (1987). [2] J.F. Gouyet, Phys. Rev. B 47, 5446 (1993). [9] A. Birovljev, L. Furuberg, J. Feder, T. J0ssang, K. J. [3] R. Lenormand and S. Bories, C. R. Acad. Sci. 291, 279 Mal0y, and A. Aharony, Phys. Rev. Lett. 67, 584 (1991). (1980); R. Chandler, J. Koplik, K. Lerman, and J.F. [10] V. Frette, J. Feder, T. Jossang, and Paul Meakin, Phys. Willemsen, J. Fluid Mech. 119, 249 (1982); D. Wilkinson Rev. Lett. 68, 3164 (1992). and J.F. Willemsen, J. Phys. A 16, 3365 (1983). [11] P. Meakin, J. Feder, V. Frette, and T. Jossang, Phys. [4] R. Lenormand, C. Zarcone, and A. Sarr, J. Fluid Mech. Rev. A 46, 3357 (1992). 135, 337 (1983). [12] M. Blunt, M.J. King, and H. Scher, Phys. Rev. A 46, [5] J.D. Chen and J. Koplik, J. Coll. lnt. Sci 108, 304 (1985). 7680 (1992). [6] N.C. Wardlaw and Y. Li, Trans. Porous Media 3, 17 [13] P. Meakin, G. Wagner, J. Feder, and T. Jyssang, Physica (1988). A 200, 241 (1993). [7] D. Wilkinson, Phys. Rev. A 30, 520 (1984); 34, 1380 [14] G. Wagner, A. Birovljev, P. Meakin, J. Feder, and T. (1986). J0ssang (unpublished). [8] E. Clement, C. Baudet, and J. P. Hulin. J. Phys. Lett. PHYSICA' ELSEVIER Physica A 218 (1995 ) 29-45

Fragmentation and coalescence in simulations of migration in a one-dimensional random medium G. Wagner, P. Meakin, J. Feder, T. J0ssang Department of Physics, University of Oslo, Box 1048, Blindem, 0316 Oslo 3, Norway

Received 12 April 1995

Abstract

A simple one-dimensional model of fluid migration through a disordered medium is presented. The model is based on invasion percolation and is motivated by two-phase flow experiments in porous media. A uniform pressure gradient g drives fluid clusters through a random medium. The clusters may both coalesce and fragment during migration. The leading fragment advances stepwise. The pressure gradient g is increased continuously. The evolution of the system is characterized by stagnation periods. Simulation results are described and analyzed using probability theory. The fragment length distribution is characterized by a crossover length s* (g) ~ g -1/2 and the length of the leading fragment scales as sp(g) ~ g -1. The mean fragment length is found to scale with the initial cluster length s0 and g as (s) = s'J12 f(gs 3J4).

1. Introduction

Dynamical processes in driven systems with many degrees of freedom frequently lead to the formation of complex spatio-temporal patterns. The dynamics may involve transport of mass, charge, or energy, and randomness may be involved at every step in the process. Typical phenomena of this type include flux pinning in high-T c superconductors [1], charge density waves in random media [2], stick-slip processes [3], and the propagation of fire fronts [4], General theoretical models for the motion of interfaces through random media are the subject of current research [5-7]. The computer model that is presented here is motivated by two-phase displacement experiments in random porous media [8], In these experiments a non-wetting fluid of low density was injected into a two-dimensional cell containing a porous medium that was saturated with a wetting fluid of higher density. The injection took place slowly, so that viscous forces could be neglected, and the cell was in the horizontal position.

Elsevier Science B.V. SSDI0378-4371(95)00130-1 30 G. Wagner et al./Physica A 218 (1995) 29-45

A “bubble” of non-wetting fluid embedded in the wetting fluid was formed. When the cell was tilted, the bubble was driven by buoyancy through the porous medium. The mass transport process was dominated by buoyancy and the local capillary forces required to drive the non-wetting fluid through the porous medium. During its migration the bubble fragmented, and the fluid fragments underwent additional fragmentation and coalescence events. The migration of the fragments occurred in bursts separated by periods of immobility. Fragmentation reduced the buoyancy force acting on the migrating fluid cluster, so that the capillary forces could inhibit further migration at a given tilting angle. Further migration took place when the tilting angle was increased. Experimental observation suggested that the migrating fluid clusters can be regarded as a string of fractal blobs forming a directed random walk along the direction of the buoyancy forces [9], The experiment was simulated using an invasion percolation (IP) model [10]. In this model the migration of non-wetting phase was modeled as a sequence of displacement processes on the pore level. The statistical properties of the simulated fragment size distribution agreed well with the fractal blob model. The dynamics observed in the two-dimensional simulations is complex. Since the migration is driven by buoyancy, the length of a fragment along the direction of the buoyancy determines the fragment ’s ability to migrate. Most of the migration steps either increase or reduce the length of a fragment along the direction of the driving force by coalescence or fragmentation. The interplay between fragmentation and coalescence is governed by the quenched randomness associated with capillary forces in the porous medium and the (fractal) geometry of the fluid bubble and its fragments. In this work a one-dimensional model of the experiment is discussed. A non-wetting fluid is injected into the middle of a one-dimensional “porous medium” (a string of pores of random size) that is initially horizontal and filled with a more dense, wetting fluid. The displaced wetting fluid can exit through outlets at both ends of the “cell”. The displacement of wetting fluid by non-wetting fluid in a pore is possible if the pressure of the invading phase exceeds the threshold pressure given by the size of the pore. Among all the invadable pores, the non-wetting fluid will therefore invade the one with the lowest threshold pressure. In a one-dimensional system, the non-wetting fluid cluster forms a string of invaded pores. The cluster covers the central region of the cell, and the number of empty invadable pores at the cluster front is always two. When the non-wetting fluid cluster has reached a given length, the injection is stopped, and the two “outlets ” are “sealed ”. The cell is increasingly tilted, imposing a buoyancy force on the cluster. The relaxation of the system is modeled as a sequence of migration events on the pore level. Each migration step consists of the withdrawal of non-wetting fluid from a pore that was invaded, and the subsequent invasion of an invadable pore. The initial non-wetting fluid cluster soon decays into a set of fragments, separated by gaps that arise after withdrawals. The numbers of pores filled with wetting and non-wetting phase are conserved separately. In a strictly one-dimensional setup, such a displacement experiment would be difficult to perform. If a pore is either filled with wetting fluid or with non-wetting fluid and the ends of the medium are sealed, the non-wetting fluid cluster is immobile. If the ends G. Wagner el al.JPhysica A 218 (1995) 29-45 31

Z

Fig. 1. The one-dimensional lattice with quenched disorder used in this work. Each site i is assigned two uncorrelated random numbers p, and p- to represent threshold pressures required for invasion and withdrawal of non-wetting fluid.

of the medium are connected to a reservoir of wetting fluid, the cluster may migrate, but fragmentation is excluded by incompressibility. In reality the wetting phase covers the surface of porous media by a film, so that all pores are constantly connected to reservoirs of wetting fluid. Both fragmentation and coalescence (transport of wetting fluid into and out of gaps between clusters of non-wetting phase) is then possible, but the flow rate of wetting fluid carried by film-flow is very low [11]. However, the model is not aimed to simulate a real-world experiment, but to provide a simple model to study the dynamics of transport processes in random media.

2. Description of the model

The simulation was carried out on a one-dimensional lattice of sites and is based on the IP algorithm [12-14], Each site i represented a pore and was assigned two random numbers p, and p' that were drawn randomly from a uniform distribution over the range 0 to 1 (see Fig. 1). Both the invasion of a site with non-wetting fluid and the withdrawal from a site required pressures that exceeded threshold values. The numbers {p} represented the threshold pressures that were required for the invading non-wetting fluid to fill the pores. Similarly, the numbers {p’} represented the threshold pressures required for the non-wetting fluid to withdraw from the pores. The random variation of the thresholds reflected the random geometry of the medium. A site could be either “filled” (invaded by non-wetting fluid) or “empty”. At the beginning all sites were empty. One site in the middle of the string of empty sites was chosen and filled. A one-dimensional cluster of filled sites was then grown from this filled site. At each stage in the growth process, among the two empty sites neighboring the filled part of the lattice the one with the lowest invasion threshold pressure p was filled. When the cluster had reached a pre-selected size %, a gradient g representing pressure was imposed on the random thresholds {p} and {p'}. The gradient could drive migration of the cluster of non-wetting fluid. Non-wetting fluid could withdraw from a filled site and invade an empty site adjacent to the end of the string of filled sites. The condition for a migration step from a source site i to a destination site j was given by

C =p'i+ pj - gdij < 0, (1) 32 G. Wagner et al./Physica A 218 {1995) 29-45 where d tj = |i — j| denotes the distance between the sites i and j. If this quantity is interpreted as a height difference, the gradient term in Eq. (1) becomes a buoyancy force. Since the thresholds {p} and {p'\ were always positive, migration was possible only in the direction of the pressure gradient. If the source site was not located at the end of the cluster, the withdrawal produced a gap and led to fragmentation. The fragments of non-wetting fluid could migrate individually. When several different migration steps were possible, the one leading to the lowest value of C in Eq. (1) was chosen. A step leading to a low value of C involved withdrawal from a site that had both a low withdrawal threshold />' and was situated far away from the destination site. The invasion threshold pj was the same for all possible migration steps within a fragment at a given time, since there was only one invadable destination site at the tip of the fragment . The migration stopped when a destination site with a high invasion threshold, which could not be overcome by the buoyancy term, was encountered. If the condition given in Eq. (1) could not be fulfilled for any of the fragments, the pressure gradient g was increased. Despite its simplicity, the model exhibited complex dynamics. Migration was pre­ vented by high invasion thresholds in the empty sites adjacent to filled sites in the direction of the pressure gradient. Once a cluster was driven by a sufficient buoyancy force to fill the destination site at its tip, a series of migration steps could follow if lower invasion thresholds were encountered. Since many of the migration steps led to frag ­ mentation, the driving force was steadily reduced and migration became less probable with each step. Short fragments were unlikely to migrate. They were however near to other fragments, and if one of them migrated, coalescence could take place. Migration then became more probable, and another cascade of migration events could take place. At the end of a cascade most of the fragments had a length that made migration unlikely if the buoyancy force was not significantly increased. Fig. 2 shows the spatio-temporal pattern obtained from a simulation in which 100 sites were filled at the start of the simulation. The pressure gradient g was increased in steps of 10~5. From left to right, the configuration of filled sites is shown at logarithmically increasing values of g. Fig. 3 shows the stepwise advancement Az of the tip of the leading string of filled sites as a function of g. The data was taken from one run with sq — 1000.

3. Simulation results

The fragment length distribution was measured as a function of the applied gradient g. Fig. 4(a) shows distributions obtained in simulations in which g was increased in steps of 10~6 from 0 to 0.1 on a log-log plot. Here Nsds is the number of clusters of filled sites (fragments) with lengths in the range s to s + ds. The length sq of the initial cluster was 1000. The curves shown represent distributions with g varying from 0.0005 to 0.1 with constant increment in log(g). The distributions are essentially constant for s less than a crossover length s*(g) and decrease sharply for s > s*(g). The decrease G. Wagner el al./Phvsica A 218 (1995) 29-45 33

Mill

▲

z

------► g Fig. 2. Spatio-temporal pattern obtained from a small scale simulation. 100 sites were filled, and the gradient g was increased from 0 to 0.1 in steps of 10~5. The configurations separated by a constant increment in log(g) are shown. At very low values of g (left), no migration is possible. As g increases, the tip advances step-wise while short fragments are left behind.

fog(g)

Fig. 3. The advancement of the tip of the leading fragment plotted as a function of the pressure gradient g, measured during one run in which the length % of the initial cluster was 1000. The tip advances stepwise, indicating the occurrence of a great number of migration and coalescence events at particular values of g. 34 G. Wagner el al./Physica A 218 (1995) 29-45

iog(g ,/2s)

log(gs)

Fig. 4. (a) A log-log plot of the number Ns of 5-site fragments vs. fragment length s. The distributions were averaged over 3000 runs. The length so of the initial cluster was 1000, and the gradient g was increased in steps of 10~6 from 0 to 0.1. The distributions correspond to 0.0005 < g < 0.1 with constant increment in log(g). (b) Apart from the curves at very low values of g, in which the fragmentation process had not yet started, the data can be scaled using the scaling form given in Eq. (2) and assuming that s*(g) ~ g™ 1''2. (c) The characteristic fragment lengths of the leading fragments sp(g) collapse when Ns(g) is plotted as a function of gs. has a roughly exponential form. At very low g the curves rise sharply at so, indicating that the decay has not yet started in some of the runs. At intermediate g the curves show a peak at a large length sp(g) indicating the characteristic length of the leading cluster. sp(g) shifts towards lower values with increasing g as further fragments break off the leading cluster. When sp(g) is close to s*(g) the distributions seem to change little as the gradient is increased. At slightly higher g the peak moves into the former crossover region and Ns increases significantly for s < 5* (g ). The crossover length is reduced and sp(g) decreases further. The fragment length distributions plotted in Fig. 4(a) can be represented quite well by the scaling form [15] G. Wagner et al./Physica A 218 (1995) 29-45 35

slope = -1

log(g) iog(gs 03/4)

Fig. 5. (a) The mean fragment length (s) as a function of the gradient g shown on a log-log plot for initial cluster lengths so of 100, 1000, 2000, 10000, and 20000, respectively. The dotted line shows that the onset of fragmentation lies on the curve cg~l with c « 0.9. (b) The data collapse obtained by plotting 1/2 (s) as a function of gs^4. The dotted line indicates the mean fragment length observed in the simplified model described in the text.

Ns(g) ~ S 2f(s/s*(g)) , (2) where fix) is a scaling function that decreases faster than any power of x for x )$> 1. Fig. 4(b) shows an attempt to scale the data using the scaling form given in Eq. (2) and assuming that the characteristic fragment length scales as s*(g) ~ g -1/2. The peak at large fragment lengths sp(g) disappears at large values of g and is not accounted for by the scaling form given in Eq. (2). The data collapse is poor for the distribution of smaller fragment lengths at very low values and at intermediate values of g. The total mass conservation on which Eq. (2) is based is not followed at these stages. Only fragments that broke off the leading cluster scale according to Eq. (2). At low and intermediate g, the overall mass of the disconnected fragments increases, as indicated by the shift of the characteristic length of the longest (leading) cluster sp(g) towards lower values. As a result, the distributions are separated in two groups, corresponding to roughly constant overall mass of scaling fragments. sp(g) scales with the gradient g as sp(g) ~ g~ a with a ~ 1, as can be shown by plotting Ns(g) as a function of gs (Fig. 4(c)). A convenient way to characterize the distribution of fragment lengths is provided by the mean fragment length

In Fig. 5(a) this quantity is plotted on a log-log scale as a function of g. The length sq of the initial cluster was 100, 1000, 2000, 10000, and 20000, respectively. The mean fragment length is constant for small values of g and decreases rapidly as g becomes large enough to cause fragmentation of the initial cluster. At intermediate values of g, 36 G. Wagner et al./Physica A 218 (1995) 29-45 the decrease of (s) is slowed down. The data collapse in Fig. 5(b) shows that the mean fragment length can be represented by the scaling form

(s) = sgf(gstf) , (4) with a = 1/2 and = 3/4. Using elementary probability theory, an expression can be obtained for the probability P(s,g, k) that site k in an s-site fragment is withdrawn and the destination site at the front end of the fragment is filled when a pressure gradient g is applied. Here the index k indicates the distance from the destination site. P(s,g,k) depends on the probability Pi(k) that the sum of the withdrawal threshold p'k and the buoyancy term —gdw = —gk for the site k is the smallest among all the s sites of the fragment, and the probability that the invasion threshold po of the destination site is sufficiently low to fulfill the migration condition given in Eq. (1). Pi(k) may be evaluated by defining the quantity qi = p\ — gdio (1 < i < s). The distribution of {<7 ,} is shown in Fig. 6. If qk > 0 for a source site k, withdrawal of the site is not possible since the migration condition cannot be fulfilled. For given g, the range of values of qk that does not exclude migration lies in the interval [ —kg, 0]. This interval is split into sub-intervals r\, r%, ..., of length g. The probability to find a value qk in a given sub-interval is equal to the length of the interval, since % is distributed uniformly on an interval of length 1. Hence the probability that % € r, is just g for 1 < j < k. If qk € rj for a given j, then the quantities q\, <72 , • ■qj-\ are all greater than <%. The probability that qj is greater than qk is (1 —g), to first order in g. The probability that the quantities qj, qj+\, ..., qk- 1 are all greater than % 6 rj, is the product (1 —g)(l — 2g) ■ ■ ■ (1 — [i— j]g). The probability that the quantities qk+\,qk+2, ...,

k-j s—k p{(k)=^n(i -«^ri(i - [t-y+i+iw (5) 77= j 71=!

The total probability Pi(k) that qk is the smallest of all the quantities <7,, 1 < i < s, is the sum

k-j s—k p^k) = Y,pi(v=sY, Y%(1 - ng) - [k - j + n + l]g) (6) >1 >1 77=1 The migration condition Eq. (1) is fulfilled if the invasion threshold po of the des­ tination site is sufficiently low, i.e., if qk + po < 0. If qk G rj then this requirement yields, to first order in g, po < (j — 1 )g. The probability that a random number x (0 < x < 1) will be in the interval 0 < po < (j — 1 )g is just (j - l)g. Hence the probability P(s,g, k) that migration is possible in an s-site fragment at gradient g and that site k is withdrawn, is given by the sum G. Wagner el al./Physica A 218 (1995) 29-45 37

-(M)g -

-(k-1)g -

-k -(k-1) -j -0-1) -1

Fig. 6. Illustration of a possible configuration of withdrawal thresholds p' in an s-site fragment. The figure shows the intervals [ —ig, 1 - ig] in which the quantities

k—j s—k P(s, g, k) O' !)%%(! -«£) - [k- j + n+ 1]#) (7) n=1 n=l which may be written as

k P(s,g,k) (8 ) 1 - [k-j + l]g

Writing the product as the exponential of a sum of logarithms and expanding ln( 1 — e) to first order in e, P(s,g,k) may be expressed as

y- 0 ~ O exp ~(s-j + \)(s-j + 2)g/2] P(s,g,k) (9) >1 1 - (k-j + l)g Fig. 7 shows the migration probability P(s,g,k) as a function of k, measured during a simulation and compared with Eq. (9). The agreement is good. 38 G. Wagner er al./Physica A 218 (1995) 29-45

0.0015 S = 1000 g = 0.0001

0.0010 -sc d> a D_ 0.0005

0.0000 700 800 900 1000 k

Fig. 7. The migration probability P(s,g, k) as a function of source site k, measured during a simulation with s = 1000 and g = 0.0001. 100000 runs have been averaged. The bold line shows the approximation given by Eq. (9). The dashed line indicates the characteristic fragment length s — As defined in the text.

The overall migration probability P(s,g) that any step is taken can be approximated by the sum

P{s,g) = ^P(s,g,k)

j k exp [~(s-j + l)(s-j +2)g/2] (10) 1 - (k-j + l)g

Fig. 8(a) shows a log-log plot of this quantity as a function of g. In the migration regime where P(s,g ) reaches a value of the order 1, the migration probability is proportional to the fragment length s, as shown in Fig. 8 (b). In the limit of very low gradients, P{s,g) may be further approximated by neglecting the g-dependent terms in the sum of Eq. (10):

k

4=1 f=l (g -» 0, S » 1) . (ID

For s —* oo, gs —> 1, the overall migration probability given by Eq. (10) may be further approximated by neglecting the g-dependent term in the denominator and introducing the summation index n = s — j:

s k P(s,g ) - 1) exp [-(s-j+ l)(s-j + 2)g/2] G. Wagner er ai/Physica A 218 (1995) 29-45 39

iog(gs 2) fig. 8. (a) The probability P(s, g) that any migration step is undertaken as a function of the pressure gradient g on a log-log plot. P(s, g) was evaluated using Eq. (10) for fragments of length 10000 (solid line), 1000 (dashed line), and 100 (dotted line) sites, respectively, (b) P(s,g) as a function of s for g = 0.0001 (solid line), 0.0005 (dashed line), and 0.001 (dotted line), respectively, (c) The data collapse obtained by using the scaling form given in Eq. (14).

5 ~ £ cx p H5 - V2] y=i

5—1 = g 2^2n(s-n)e\p[-n 2g/2] . (12)

Letting j strive towards infinity and going to the continuum limit, the result

P(s,g) ~£ I n(s — n) exp [-n 2g/2] An n=0 oo oo ■g 2s j nexp [-n 2gf 2] d« - g~ J n 2 exp \-n 2g/2]dn z?=0 n=0 40 G. Wagner el al./Physica A 218 (1995) 29-45

(s —> 00, gs -* 1) (13)

is obtained for this limiting case. Figs. 8 (a) and 8 (b) show that the approximation Eq. (13) is satisfactory of P(s,g) > 0.05. Fig. 8 (c) shows that P(s,g) scales with the fragment length s and the pressure gradient g as

P(s,g) =g l/2f(gs 2) . (14)

•The scaling function f(x) exhibits a crossover at x « 1 and must have the form f(x) = x1'2 for x » 1 in order to reproduce the linear behavior in the migration regime (large x). Migration of an s-site fragment must occur at the latest when P(s,g) approaches 1. Using Eq. (13), it may be concluded that g niax (s) = 1/s. A value g*(s) of g characteristic of the migration of an s-site fragment may be defined by

Neglecting the non-linearity of P(s,g) at low g and inserting Eq. (13) into (15), the simple result

8*(s) = ~ (16) s is obtained with c = 2/3. An alternative way to define g*(s) is to require that P(s,g*(s )) = c, where c is a constant in the range 0 < c < 1. Choosing c = 2/3 and using Eq. (13), Eq. (16) is recovered. Eq. (16) is consistent with the observation that the onset of fragmentation occurs roughly at g*(s) ~ 0.9/sq (see Fig. 5(a)), and the observation that the characteristic length of the leading fragment sp(g) identified in the fragment length distributions scales with the g as s,,(g) ~ g _1. A value for c of about 0.9 may be read off from Fig. 4(c). If coalescence events are excluded (for instance, the leading fragment cannot coa ­ lesce), P(s,g,k ) can be interpreted as the probability that an i-site cluster fragments into two clusters of length k and (s - k). Typically, k 3> (s - k), and the trailing fragment will be too short to migrate. A characteristic trailing fragment length As may be defined by taking the mean of the fragment lengths (s - k), weighted with the probabilities P(s,g,k) that fragmentation occurs:

. k)P(s,g,k) As =------—------

(17)

This length is indicated in Fig. 7. The sum in Eq. (17) may be calculated using the first-order approximations given above. As a result, it is found that the s-dependence is canceled and 4. probability that the trailing s*(g) if the independent In s g P(p) function distributed Fig. (a)

log(As)

and is

2.0 2.5 1.0 3.0 1.5 The Since

g

Figs.

Models

As 9. length invasion applied. - the Py(s,g,&) 6.0

is

(a)

as

plotted

results ~ ~

of

fragment

slowly

migration 9(a) probability

with The a s g py

of on The function with -1 that

characteristic of as .

an

thresholds a / presented the

and

2 For

curves a

log-log exponent the

increased the

function observed power-law

length

cut-off

and 9(b),

y original

site of

P(p) were log(g)

-5.0 plot, >

fragmentation s y

length

of

in k

and A

= for k As

— G.

fragment evaluated

(s

s using g in 0 from in the

p

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[(&-;

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(solid (g) distributed a is —

an of withdrawal

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observe > preceding fixed a

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s-cluster log-log = ~ zero.

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er first-order

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(a) g.

al./Physica

means of

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=

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> thresholds

sections thresholds

threshold is is

— the

with a 1 of

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~ 0.5 ) using

migrated -site

log-log length

o o> tempting s,

. Eqs.

approximation g regime

A

(dotted (&-;)'+?]

as 2.0 2.5 1.0 1.5 the

As

a

218 l/ cluster

(9) may ' 2.0 indicated

fragment 2

that

in initial

distributions

(1995)

are

scale

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becomes in

(b) breaks the be to

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which power-law

(20).

generalized length and identify 29-45 cluster as range 2.5

off by

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The

an

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1 s-site = p,p with

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length thresholds

(dashed log(s)

5000.

the (18). 3.0 distributed. in

cluster for +

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g*(s)

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s~\

s

as the 4.0

1//2 by 41 is a

42 G. Wagner et al./Physica A 218 (1995) 29-45

The generalized migration probability Py(s,g,k) is approximated quite well by

k = [(&- / + i)'+y_ (&-/)'+?]

(20)

For y — 1, Py(s, g, k) approaches a step function that vanishes for k =£ s. Similarly, for y —> oo, Pr(s,g,k) is expected to approach a step function since all thresholds become equal to 1. Using similar reasoning as the one leading to Eq. (13), the probability that any migration step is taken generalizes to

Py(s,g) ~ (g.$)1+7 (a -» oo, gs -» 1) . (21)

The characteristic length of the trailing fragment As that breaks off an s-site cluster when a gradient g is applied becomes

As~g-('+7)/(2+7> 00, gs 1). (22)

In Figs. 9(a) and 9(b), As is plotted for some values of y as a function of g and s on a log-log scale.

5. Analysis of a simplified model

Returning to the case of uniform threshold distributions, a simplified model has been constructed using the concept of characteristic pressure gradients and fragment lengths. A cluster of length so decays into a trailing fragment of length Asq and a leading fragment of length s, = s0 - Asq when the gradient reaches a value of g* (so). At g = g*(si), the leading cluster fragments further into clusters of length Asi and S2 = si — As,. The only cluster that fragments is the leading cluster. The length of the trailing fragments approaches zero at increasing g by virtue of Eq. (18). The leading cluster reaches a constant length at finite g since the length of a cut-off fragments must be rounded to an integer. Expressing g*(s„_i) by Eq. (16) and As„_i by Eq. (18), the following recursive equation for the length s„ = s„_t - As„_i of the leading cluster after n fragmentation events is obtained:

(23)

Here the brackets denote rounding to the nearest integer. When the leading cluster of length s„_] fragments into two clusters of length s„ and As„_i, the first moment sNs of the fragment length distribution is not changed whereas the second moment ^ s2Ns is changed by 2As„_i (As„_i — s„_i). At each step, the pressure gradient is incremented G. Wagner et al./Physica A 218 (1995) 29-45 43 by Ag* = g*(s„) - g*(s n-1). Expressing the change of the mean fragment length (s) by means of Eqs. (16) and (18) and expanding Ag* « cAs„-\/s2_x, the derivative of (s) with respect to g may be derived as

(24)

This simplified model may be refined by letting not only the leading fragment migrate but all fragments that exceed the length s*(g). At first the leading cluster is the only one that fragments and no difference occurs. After n migration steps at g*(Aso) the leading fragment is reduced to a length of s„ = Aso, equal to the length of the fragment that was cut off at the very first migration step. Both this last fragment and the leading fragment may now migrate. When this last fragment performs a migration step, its leading fragment coalesces with the fragment that was cut off at the second migration step. A chain reaction that leads to a leading fragment of length s„+i = A so — As„ and a number of fragments of equal length As„ is triggered. The mean fragment length drops abruptly to a lower value. Fig. 5(b) shows the scaled dependence of (s) on g obtained from this simplified model with the parameters c = 0.9 and so = 10000. The curve exhibits some of the features that characterize the mean fragment sizes for the full model. The decay sets in at g = cjso and (s) decreases as g~ 2, in good agreement with the curves obtained from the simulation model. The curve reaches a plateau with increasing g at which the decay of (s) slows down. In this simplified model, randomness is eliminated and only a subset of possible fragment lengths is permitted by the recursion formula Eq. (23). Before the first onset of the chain reaction, the system consists of a set of fragments of length A so > As, > ... > As„. The length distribution is approximately constant for the subset of realized fragment lengths and has a peak at the length of the leading fragment, in agreement with the fragment length distributions of the full simulation model (Fig. 4(a)). The moments of the length distribution scale as ^2sNs « N(Asn + As„_i + ... + A so) ~ (A so) 2 and Y] s2 Ns ~ (A so) 3. The mean fragment length is expected to scale as (s) ~ (Aso) ~ Sq , which accounts for the value a = 1/2 found empirically in the data collapse described by Eq. (4). The exponents a and appearing in Eq. (4) can be related to each other by the following argument: At very low g (Ag 1!2jc 1), the integer term in Eq. (24) may be neglected and integration yields (s) = c2/g 2so. Comparing this relationship with the scaling approach given in Eq. (4), it is found that

= ■$0/(^0) ’ (25)

with f(x) = c2x 2 and a - 2/8 - 1. This is consistent with the value /? = 3/4 found empirically. 44 G. Wagner et al/Physica A 218 (1995) 29-45

6. Conclusions

A simple one-dimensional model in which fluid strings are transported through a random medium has been discussed. The fluid transport is accompanied by fragmentation and coalescence while the driving force is steadily increased. The simulation results have been analyzed by elementary probability theory and some understanding of the fragment length distribution and the decrease of the mean fragment length has been gained. This model represents the gravity-stabilized, capillarity-limited migration, fragmenta ­ tion and coalescence of a non-wetting fluid in a porous medium that is saturated with a wetting fluid. For corresponding two-dimensional or three-dimensional experiments and simulation models, the simple scaling properties found in the one-dimensional model will not be present, and fractal concepts must be employed to analyze the patterns. Fragments may cover a great number of sites without necessarily having a great extension in the direction along the pressure gradient, and the number of invadable destination sites is proportional to the length of the perimeter of the fragment. The migration probability P depends not only on the fragment size s and the gradient g but also on the fragment geometry. Since the fragments extend in directions perpendicular to the gradient, there are are several “leading fragments ” that do not coalesce with any other fragment when migrating. Coalescence is, in general, less likely since the center of mass of fragments may be translated also in directions perpendicular to the gradient during a migration step. Consequently, the mean fragment size (s) decays as a simple power law of the gradient g ({s} ~ g~ x), unlike the more complicated function encountered here [10]. The mean fragment size may be identified with a crossover fragment size s*(g) and the size distribution scales as Ns ~ s~2f(s/s*(g )). The prominent peak well above the crossover size seen in Fig. 4(a) is absent in higher dimensions. It might be worthwhile to study the transition from d = 1 to d > 1 by investigating the presented model on a geometry of two, three, and more parallel strings of sites. Such a scenario may serve as a model for two-phase flow through a narrow porous column with a cross-section of only few pores. The model described in this paper may also be relevant to problems as the migration of bubbles in a “dirty” channel and the flow of granular media through a narrow pipe with rough walls.

Acknowledgements

We thank K. Christensen, T. Rage and M. Wagner for helpful discussions. We grate ­ fully acknowledge support by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (STATOIL) and by NFR, the Norwegian Research Council. The work presented here has received support from the NFR programme for supercomputing through a grant of computing time. G. Wagner el ai/Physica A 218 (1995) 29-45 45

References

I 11 E.H. Brandt, Int. J. Mod. Phys. B 5 11991) 751. 121 D.S. Fisher. Phys. Rev. Lett. 50 (1983) 1486-1489. I 31 H.J.S. Feder and J. Feder, Phys. Rev. Lett. 66 (1991) 2669-2672. ] 4] J. Zhang, Y.-C. Zhang, P. Alstrpm and M.T. Levinsen, Physica A 189 (1992) 383-389. 15 | M. Kardar, G. Parisi and Y.-C. Zhang, Phys. Rev. Lett. 56 (1986) 889-892. [6] G. Parisi, Europhys. Lett. 17 (1992) 673-678. I 71 K. Sneppen, Phys. Rev. Lett. 69 (1992) 3539-3542. 18 1 A. Birovljev, Experiments on Two-Phase Flow With a Gradient and Tracer Dispersion in Porous Media, Ph D. thesis. Physics Dept., University of Oslo (1994). 19 1 P. Meakin, A. Birovljev, V. Frette, J. Feder and T. Jessang, Physica A 191 (1992) 227-239. | 101 P Meakin, G. Wagner, J. Feder and T. Jpssang, Physica A 200 (1993) 241-249. [Ill R. Lenormand and C. Zarcone, Role of roughness and edges during imbibition in square capillaries, Paper presented at the 59th Annual Technical Conference of the Society of Petroleum Engeneers of AIME, held in Houston, Texas, Sept. 16-19 (1984). [ 121 R. Lenormand and S. Bones, C.R. Acad. Sc. Paris 291 (1980) 279-283. [ 131 R. Chandler, J. Koplik, K. Lerman and J.F. Willemsen, J. Fluid Mech. 119 (1982) 249-267. [ 141 D. Wilkinson and J.F. Willemsen, J. Phys. A 16 (1983) 3365-3376. 1151 T. Vicsek and F. Family, Phys. Rev. Lett. 52 (1984) 1669-1762. Buoyancy-Driven Invasion Percolation With Migration and Fragmentation

G. Wagner, P. Meakin, J. Feder, and T. J0ssang Dept, of Physics, University of Oslo, Blindern, PB 1048 N-0316 Oslo, Norway

January 17, 1997

Abstract A modified invasion percolation model is presented that includes migration and fragmentation processes. Migration of fluid clusters is driven by buoyancy forces. The model was studied in two dimensions. The structures obtained in simulations can be understood in terms of standard percolation and invasion percolation. The model may serve as a prototype of transport through random media.

1 Introduction

The slow immiscible displacement of one fluid by another fluid in a porous medium can be simulated by means of the invasion percolation (IP) al­ gorithm [1-3]. The IP algorithm models a dynamic growth process and produces a connected structure representing the distribution of the invader fluid that flows through the medium. IP clusters have fractal properties, in agreement with experimental observations of slow displacement processes in random heterogeneous media [4-9]. There is little doubt that IP is an ade­ quate model for the slow displacement of a wetting fluid by a non-wetting fluid in a homogeneous porous medium [10].

1 Frette et al. [11] carried out comprehensive displacement experiments of this type, using a three-dimensional transparent porous medium. Buoyancy- driven upward migration fluid was studied by slowly injecting a non-wetting fluid of low density into the medium that was saturated with a wetting fluid of high density. Similarly, the the downward penetration of a denser non-wetting fluid into the medium was studied. The dynamics of the dis­ placement process appeared to be complex. The invading non-wetting fluid formed a structure that consisted of many fractal “blobs ” that represented a directed random walk in the (upward) direction of the pressure gradient. The tip of the structure advanced via a series of bursts.

In connection with these steps, or bursts, occasional snap-off events oc ­ curred in which the regions invaded by the non-wetting fluid became discon ­ nected and wetting fluid re-occupied some of the pores. The disconnected fragments of non-wetting fluid could move independently some distance up­ wards. Further snap-offs diminished the extension of the moving fragments in the direction of the gravitational potential gradient. This reduced the buoyancy force acting on the fragments and the fragments were trapped temporarily in the matrix. These blocked fragments could move only after coalescening with other fragments, or after re-connection with the injection point at which non-wetting fluid was constantly injected.

The IP model may be extended in a simple manner to include some of the important effects of gravity [8,12-14], However, the simple gravity IP model still produces a single connected structure that cannot account for all the experimental observations. The purpose of this paper is to present a further extension to the gravity IP model that includes mechanisms to represent snap-off and the buoyancy-driven motion of disconnected fragments.

2 2 Description of the model

The model is based on the site invasion percolation model with trapping [3]. Unlike the ordinary IP model which simulates only the growth of the re­ gion occupied by the injected fluid, this model also represents the migration of non-wetting fluid through the porous medium. In this new model, a migration step consists of the invasion of a site with invader fluid and the si­ multaneous withdrawal of the invader fluid from another site. In the absence of these migration processes, the model reduces to the gravity IP model [8]. The sites on a lattice represent the pores in a porous medium. Random thresholds p z- and qt are assigned to each site i, and all sites are filled with the defender fluid (wetting). The thresholds are uncorrelated and distributed uniformly on the unit interval. The threshold p, represents the pressure at which the invading non-wetting fluid can invade the site i. Similarly, the threshold % represents the pressure at which the non-wetting fluid can withdraw from the site i. Initially, all sites are filled with the defender fluid. The (non-wetting) invader fluid is injected at a site in the center of the lattice. At each injection step, all perimeter sites adjacent to the invaded region that include the injection site are checked. Among the perimeter sites, the site i with the lowest effective invasion threshold tt,-, given by

Ki = Pi - 9Vi > (1) was invaded. Here, yi denotes the p-coordinate of site i. The gradient g > 0 represented the buoyancy force acting on the invading fluid, in the positive y- direction. This favored the invasion of sites with high ^-coordinates. When none of the invaded regions covered the injection site, the injection site was invaded. At each stage, trapping [3] was taken into account. If a defender site became surrounded by invader fluid and was disconnected from the “infinite” defender fluid cluster, the site was “trapped ” and could not

3 be invaded. — A trapping rule was also retained in migration steps. A defender site in a trapped region could become invaded in a migration step only if invader fluid was withdrawn simultaneously from a site adjacent to the trapped region. The invasion of a site by invader fluid represented the displacement of wetting (defending) fluid by non-wetting (invading) fluid. This displacement process is governed by the random geometry of the porous medium. The stochastic nature of the porous medium was captured by the random thresh­ olds used in the simulation. The trapping rule implies that the simulated fluids are incompressible. After each injection step, migration of the clusters of invader fluid was attempted. In each migration step invader fluid was withdrawn from an invaded site, and simultaneously a defender site was invaded. This required that the invasion threshold pi at the invaded site i as well as the withdrawal threshold qj at the abandoned site j must be overcome. The migration was driven by the buoyancy force. A migration step was possible if

nij = Pi qj ~ 9^ij <-- o j (2) and if the site i was on the unoccupied perimeter of the cluster of connected sites containing site j. The two first terms to the right in Eq. (2) represent the thresholds that must be overcome. The last term represents the driving buoyancy force, given by the product of the gradient parameter g and the difference A^ = yi — yj in the (/-coordinates between the sites i and j, respectively. Since the thresholds were positive, the site that was invaded in a step was required to have a larger (/-coordinate than the site that was abandoned, that is, A ij > 0. Figure 1 illustrates a migration step. An alternative way to write the migration condition Eq. (2) is to define the effective withdrawal threshold ip as

'-pj = 9j d™ 99j • (3)

4 The threshold balance II,y may then be expressed as

Hy = 7T,' + ipj < 0 . (4)

A cluster of invader fluid was defined as a set of sites that were connected to each other by a path consisting of steps between nearest neighbor invader fluid sites. In each migration step, one site of the cluster was abandoned, and another site at the perimeter of the cluster was invaded. In this way, the step-wise migration was used to simulate the movement of individual clusters of invader fluid. To account for trapping during migration, the invaded site was also required to be connected to the abandoned site by a path consisting of steps between nearest-neighbor defender sites. Most of the migration steps conserved the volume of the trapped defender fluid clusters. Some of the steps caused trapping of further regions of defender fluid, and some of the steps untrapped defender sites by opening up a gap between the enclosing invader fluid sites. At each stage, all possible combinations of sites that could be invaded and abandoned were checked, and the migration step yielding the lowest value Fly in Eq. (2) was carried out. If no step could be found fulfilling the migration condition II,y < 0, another injection step was carried out. The number of injection steps established a natural time scale for the simulations. A migration step often caused fragmentation of an invader fluid cluster as one part of the cluster became disconnected. On the other hand, migration could also lead to coalescence of two clusters. Figure 2 illustrates migration steps that lead to fragmentation and to coalescence. Figures 3 and 4 show examples of displacement patterns that were ob ­ tained using various values of the gradient parameter g. For small values of g, the structure is broad and large regions of defender fluid become trapped. For high values of

5 3 Formation of the structures

Figure 4 shows a sequence of patterns obtained in the simulations. At first, a branch protruded from the injection site. The branch meandered in the {/-direction along the direction of the gradient. At a later stage, several branches evolved, each of them growing at different stages of the simulation. The new branches sprung off at regions lower than the tip of the structure. The bifurcation observed in Fig. 4 is typical for the migration model, and is in contrast to the growth of a single branch observed in gradient IP. The branches in a structure consisted of numerous fragments that connected and disconnected to and from neighbor fragments many times during the growth process. The presence of the fragments and the “gaps ” between the fragments in the branches led to the occurrence of “growth cycles ” of various lengths. Among the sites that were invaded during a simulation, some sites were invaded and abandoned over and over again. Other sites were invaded only once and shortly thereafter abandoned. Yet another class of sites were in­ vaded once and never abandoned. A branch grew when injected invader fluid was transported from the injection site along the branch to the tip. Invaded sites that were not abandoned during the rest of a simulation (due to a high withdrawal threshold q) could be viewed as permanent parts of a “pipeline ” used in the transport process. These sites formed fluid fragments that remained frozen during the growth of the structure. The fragments were separated from each other by gaps and could be viewed as sections of the pipeline.

The gaps consisted of “active” sites that were invaded and abandoned over and over again. Active sites were characterized by both low invasion thresholds p and low withdrawal threshold q. When gap sites were invaded, fragments could coalescence and form a short-lived pipeline that extended over a fraction of a growing branch.

6 A growth burst occurred when a pipeline was formed that extended over a large distance in the direction of the gradient, causing a high buoyancy force. During the burst, active sites were abandoned one after another, while empty gap sites further up were filled, or new sites were invaded and the tip of the branch advanced. When a gap site was abandoned, an entire section of the pipeline became disconnected. Eventually, the length of the pipeline was reduced, and migration became less and less likely as the driving buoyancy forces diminished. The burst terminated when the tip reached a region in which all the empty sites had high invasion thresholds p, or when the supply of fluid from active sites was exhausted. During the following injection steps, the gaps were filled again, and another cycle could begin.

As the structure grew, the transport of fluid from the injection site to the growth region required more and more migration steps since the number of active sites belonging to the structure increased. At the end of a long series of migration steps, the tip advanced.

In a large structure, a growing branch contained many gaps of different sizes. The capacitative effects exerted by the gaps led to episodic advance­ ment of the tip. Strictly periodic advancement of the tip was, however, rarely observed over periods longer than a few cycle lengths. The changing environment experienced by the tip was reflected by subsequent modifica ­ tions of the structure. As the tip explored new regions, migration could be enhanced or hampered due to the random nature of the site thresholds. When the tip reached a region of low permeability (high invasion thresh­ olds), a growth burst could end prematurely. In this case, not all the gap sites became abandoned. Some of them could turn into “passive ” sites and retain their state for the rest of the simulation. In this way, gaps could become closed entirely as the growth proceeded.

Figure 5 shows a plot of the averaged migration step size (in the y- direction) Ay as a function of the total number of sites invaded. The step

7 sizes were measured at different stages in simulations at various values of g. As the plot shows, A,-j depended weakly on the total mass (the number of invaded sites) s of the structure, and possibly approached an asymptotic value. The inset in Fig. 5 shows, on a log-linear scale, an attempt to collapse the data from simulations with different values of g on a single curve. This suggests that the migration step sizes A,-j may be represented by the scaling form

(5) where J(s) is a universal scaling function that is equal for all values of the gradient parameter, and % % 0.5. The s-dependence indicates that most migration steps occurred in bursts across “pipelines ” in which numerous gaps were filled temporarily and long steps were possible. (Steps that occur within one fragment are limited by the extension of that fragment in the (/-direction; the mean (/-extension of the fragments does not depend on the entire structure mass, as shown below.) As the structure mass increased, some of the gap became closed permanently, and even longer pipelines could form. The exponent % can be related to the standard percolation exponent zz, as shown in section 5. The positive value of x reflects the migration condition Eq. (2) that requires that low values of g be compensated by long migration steps. For given g and s, A(j(s,g) has an approximately Gaussian distribution around the mean value A/j. Bifurcations, like that shown in Fig. 4, arose when a growing branch became disconnected from the injection site, and reconnection was less favourable than the formation of a new branch. Figure 6 shows a schematic example of a growing structure in which a bifurcation arises. Some invaded sites in a branch could become abandoned as that branch developed. A few of the sites could remain empty since they became trapped as the region was re-filled with migrating invader fluid. The trapped sites could prevent the re-filled region from becoming connected to the tip of the branch so

8 that growth was forced into a new direction. The use of a trapping rule is crucial for the formation of bifurcations. In three-dimensional simula­ tions the trapping rule is usually omitted, and no bifurcations should be expected. However, fragmentation will occur also in three dimensions. The scaling laws presented in the remaining parts of the paper can be expected to be valid, since they concern the strcture of the individual fragments. The fragment structure is not affected by the branching. The averaged velocity of the tip advancement depended on the gradient g imposed, as shown in Fig. 7. In this figure, yf^ is plotted as a function of the structure mass, using different gradients in the high-gradient regime.

The averaged velocity = yrJp/so, where sq denotes the structure mass (proportional to time if a constant injection rate is used), appears roughly constant. The tip velocity had the power-law dependence

VTip ~ 9X (6) on the gradient g, with x' ~ 0.46, as shown in the inset in Fig. 7. As indi­ cated by the notation, the exponent % appearing in Eq. (4) can be identified with x'i as shown in section 5.

4 Geometrical Characterization of the Structures

Figure 8 (a) shows the linear mass density profile m(y) of simulated struc­ tures, plotted as a function of the height y. Here m(y)Sy is the number of invaded sites with (/-coordinates lying in the range y to y + Sy. The curves were averaged over several dozen simulations using different gradi ­ ents. Structures consisting of 5000 sites were formed in each case, with the exception of simulations with g = 0.05 in which structures with 4000 sites were formed. The lattice size was 700 X 1400. The linear mass den­ sity decreases with increasing gradient, reflecting the fact that the branches meander to a lesser degree and the structures become thinner.

9 The width of the structures is a measure of the correlation length £w. The correlation length is expected to depend on the gradient g as [15]

, (7) where v is the ordinary percolation correlation length exponent with the value u = 4/3 in two dimensions [16]. The average m(y) = f m(y) dy/ f dy of the linear mass density over the entire length of the structure was found to exhibit a power-law dependence on the gradient g for high values of g (Fig. 8 (b)). The mass density m(y) is related to the correlation length £w of the displacement patterns through [12].

m(y) ~ £d_1 ~ g -{D-\)vl(v+\) ~ g -x" _ (g)

Here, the exponent x" — {D — \)v/{v +1) was defined. Inserting the fractal dimensionality D « 1.82 of IP clusters in two dimensions [3,17] and v = 4/3, a value of 0.47 is obtained for x"■ The data shown in Fig. 8 (b) is consistent with Eq. (7). In the case of low g, the average linear mass density could not be determined exactly since the structures did not reach a great height. This finite-size effect may cause the deviation from the scaling law Eq. (8) seen in Fig. 8 (b).

5 Geometrical Characterization of the Fragments

The structures consist of fragments of various sizes sy. At each stage when all migration has ceased, the number N(s/)dsf of fragments with size in the interval (sy,sy + dsy) is simply proportional to the structure mass s. Fig ­ ure 9 (a) shows a plot of normalized fragment size distributions s~1N(sf), measured in simulations using various values of the gradient parameter. For low fragment sizes sy, the distributions seem to follow a power-law. At

10 intermediate fragment sizes, a hump is apparent in the distributions corre ­ sponding to low values of the gradient g. The hump indicates that, at each stage, a constant fraction of the invader fluid forms a large, connected frag ­ ment. This large fragment is located at the tip of the growing structure. At the tip, withdrawal and migration is not possible anymore since the distance to the regions that can be invaded is short and the buoyancy force acting across the tip fragment is not sufficient to drive migration. For higher values of g, the typical length of a migration step, A,j, becomes shorter (Eq. (5)) and the mass density of the structure, m(y), decreases (Eq. (8)), such that the size of the tip fragment is comparable to the size of the remaining frag ­ ments. Part (b) of Fig. 9 shows an attempt to collapse all distributions onto a single curve according to the scaling assumption

AT(gf) ~ 83^2/ . (9)

Here, st denotes the size of a typical fragment, for a given value of g. The extension A in the (/-direction of a fragment tends to decrease with increas­ ing gradient. Migration and fragmentation of a fragment is possible if the driving buoyancy force overcomes the sum of two random thresholds, i.e.,

<7 A ~ 1. This suggests that A< ~ g~ l (10) where At denotes the extension in the (/-direction of a typical fragment. Assuming a width in the direction perpendicular to the gradient, an IP- like structure of the fragment, and an extension At in the direction of the gradient, the mass st of a typical fragment is found to be [18]

s, - . (ii)

Here, it was assumed that £w is given by Eq. (7), Eq. (10) was inserted, and the exponent x" ~ 0.47 appearing in Eq. (8) was used. In Fig. 9 (b),

11 the data collapse was attempted using st ~ y -1'5. The scaling assumption of a typical fragment size characterizing each distribution is justified. However, the collapse failed as f(s/st) depends non-trivially on y for Sf/st of the order of 1, due to the hump in the cluster size distribution mentioned above.

Figure 10 (a) shows the mean fragment mass S = 53(syjV(s/))/E s/^(s/), measured in simulations using different values of g. Here, N(sj)8sf denotes the number of fragments of mass in the range Sf,Sf + 5s/, measured when all migration had ceased. The mean fragment mass may be viewed as the mass of a typical structure fragment, 5 ~ st. This quantity is little dependent on the mass of the entire structure, except for very low g. In the low-gradient regime, larger structures must be formed in order to obtain a constant value for 5. An asymptotic mean fragment mass Sqq may be defined by simply evaluating the mean fragment mass of structures of the maximum mass sim­ ulated. When Soo is plotted as a function of g on a log-log scale, a power-law dependence is obtained that is consistent with Eq. (11) in the high-gradient regime (Fig. 10 (b)).

At this point, the value of the exponent % describing the dependency of the average migration step size, A,y , on the gradient g (Equation (5)) may be discussed. Ameasures the difference in the y-coordinates of the sites that are invaded and evacuated in a migration step, and hence the length of the “transport pipelines ” mentioned in section formation that are formed during a migration burst. In a simple picture, a pipeline is formed by re-distributing the fluid belonging to the lowest fragment of the structure and filling the “gaps ” separating the fragments further up. Initially, the pipeline increases in length as the gaps are gradually closed and a backbone of “passive ” sites remains at the lower end of the pipeline. If fluid from the next-lowest fragment is withdrawn, the pipeline cannot become longer since the gap between the lowest and the next-lowest fragment opens again.

12 During its remaining lifetime, the pipeline retains its length, independent of the total mass s of the structure. Assuming that the gaps in the structure are separated by the distance A< (Eq. (10)), and that the mass of the lowest fragment in a structure is given by Eq. (11), the typical length of a pipeline is found to be

~ tr" ~ 9~x ■ (12)

The numerical value x" ~ 0.47 (Eq. (8)) is in agreement with the estimated value x ~ 0.5 found in Fig. 5, such that the two exponents can be identified.

Finally, the scaling exponent x' in Eq. (6) can be identified with x", since the mass density m(y) given in Eq. (8) is inversely proportional to the tip advancement velocity vfl^ (at a constant injection rate). The dependency of the linear mass density of the structure, the advancement velocity, and the migration step size, are thus described by the single exponent x"•

6 Conclusion

The modified IP model presented in this work has complex fragmentation and migration dynamics. In the case of the two-dimensional model studied here, bifurcations were observed in the growing patterns. The patterns were fractal, and their geometry could be related to standard percolation theory and to standard IP clusters. The dynamics of the pattern formation is complex and merits further investigation.

The model represents an oversimplification of the actual migration pro ­ cess since as the displacement of the non-wetting fluid by the wetting fluid is a process that cannot successfully be described by a single withdrawal threshold. There exist several pore-scale mechanisms that can cause with­ drawal of the non-wetting fluid [19-21]. However, the model may serve as a prototypical model of transport in a random medium.

13 References

[1] Roland Lenormand and S. Bories. Description d’un mecanisme de con ­ nexion de liason destine a l’etude du drainage avec piegeage en milieu poreux. C.R. Acad. Sc. Paris, 291:279-283, December 1980.

[2] Richard Chandler, Joel Koplik, Kenneth Lerman, and Jorge F. Willem- sen. Capillary displacement and percolation in porous media. J. Fluid Mech., 119:249-267, 1982.

[3] David Wilkinson and Jorge F. Willemsen. Invasion percolation: A new form of percolation theory. J. Phys. A: Math. Gen., 16:3365-3376,1983.

[4] Roland Lenormand and Cesar Zarcone. Invasion percolation in an etched network: Measurement of a fractal dimension. Phys. Rev. Lett., 54(20) =2226-2229, May 1985.

[5] Christian Jacquin. Caractere fractal des interfaces fluide-fluide en mi­ lieu poreux. C.R. Acad. Sc. Paris, 300(Serie II) =721-725, 1985.

[6] T. M. Shaw. Drying as an immiscible displacement process with fluid counterflow. Phys. Rev. Lett., 59(15)=1671-1674, 1987.

[7] E. Clement, C. Baudet, E. Guyon, and J.P. Hulin. Invasion front struc­ ture in a 3d model porous medium under a hydrostatic pressure gradi ­ ent. J. Phys. D, 20:608-615, 1987.

[8] J. P. Hulin, E. Clement, C. Baudet, J. F. Gouyet, and M. Rosso. Quan­ titative analysis of an invading-fluid invasion front under gravity. Phys. Rev. Lett., 61 (3) =333-336, July 1988.

[9] Aleksander Birovljev, Liv Furuberg, Jens Feder, Torstein Jpssang, Knut J. Malpy, and Amnon Aharony. Gravity invasion percolation in two dimensions: Experiment and simulation. Phys. Rev. Lett., 67(5) =584-587, July 1991.

14 [10] Muhammad Sahimi. Flow and Transport in Porous Media and Frac ­ tured Rock. VCH Verlagsgesellschaft , Weinheim, FRG, 1995.

[11] Vidar Frette, Jens Feder, Torstein J0ssang, and Paul Meakin. Buoyancy driven fluid migration in porous media. Phys. Rev. Lett.., 68:3164-3167, June 1992.

[12] Paul Meakin, Jens Feder, Vidar Frette, and Torstein Jpssang. Invasion percolation in a destabilizing gradient. Phys. Rev. A, 46:3357-3368, 1992.

[13] Paul Meakin, Aleksander Birovljev, Vidar Frette, Jens Feder, and Torstein Jpssang. Gradient stabilized and destabilized invasion per­ colation. Physica A, 191:227-239, 1992.

[14] M. Chaouche, N. Rakotomalala, D. Salin, B. Xu, and Y.C. Yortsos. Invasion percolation in a hydrostatic or permeability gradient: Experi ­ ments and simulations. Phys. Rev. E, 49(4) =4133-4139, April 1994.

[15] David Wilkinson. Percolation model of immiscible displacement in the presence of buoyancy forces. Phys. Rev. A, 30(1) =520-531, July 1984.

[16] Dietrich Stauffer and Amnon Aharony. Introduction to Percolation The­ ory. Taylor & Francis, London, Washington D.C., 1992. 2nd edition.

[17] Liv Furuberg, Jens Feder, Amnon Aharony, and Torstein Jpssang. Dy ­ namics of invasion percolation. Phys. Rev. Lett., 61 (18) =2117-2120, 1988.

[18] Paul Meakin, Geri Wagner, Jens Feder, and Torstein Jpssang. Simula­ tions of migration, fragmentation and coalescence of non-wetting fluids in porous media. Physica A, 200:241-249, 1993.

15 [19] Roland Lenormand, Cesar Zarcone, and A. Sarr. Mechanisms of the displacement of one fluid by another in a network of capillary ducts. J. Fluid Mech., 135:337-353, 1983.

[20] Jing-Den Chen and Joel Koplik. Immiscible fluid displacement in small networks. J. Colloid Interface Sci., 108(2):304-330, 1985.

[21] Norman C. Wardlaw and Yu Li. Fluid topology, pore size and aspect ratio during imbibiton. Trans. Porous Media , 3:17-34, February 1988.

16 FIGURE 1: Illustration of the migration of a cluster of invaded sites (shaded sites).

In order to satisfy Eq. (2), site A is abandoned and site B is invaded only if qA +

Vb ~ g{VB - i/a) > o and if ys > Va-

17 D 1 J

A B H A B E C G F 0 >)

Figure 2: Illustration of migration steps, (a) Withdrawal of invader fluid (shaded sites) from C and invasion of D causes fragmentation of the cluster of invader fluid.

Withdrawal from E and invasion of D is not a legal step since the volume of trapped defender fluid (sites A and B) would not be conserved. Withdrawal from D and invasion of A or B is a legal step, (b) Withdrawal from F and invasion of G leads to coalescence of the two invader fluid clusters. Withdrawal of H and invasion of I traps the defender fluid site J and untraps the sites A and B.

18 Figure 3: Three displacement patterns using edge injection and values of the gradient parameter g of 0.0005 (a), 0.005 (b), and 0.05 (c), respectively. The sim­ ulations were terminated at breakthrough when the upper boundary was reached.

Periodic boundary conditions were used in the horizontal direction. Invaded sites are shown in black, and trapped defender sites are shaded.

19 L=256

Figure 4: A sequence of displacement patterns observed during a typical simula ­ tion using a value g — 0.007 for the gradient parameter. The number of invaded sites was 1000 (a), 2000 (b), 3000 (c), and 3394 at the final stage when the upper boundary was reached (d). Point injection was used at the site marked by an arrow.

Invaded sites are shown in black, and trapped defender sites are shaded. The inset shows a part of the fragmented structure at the final stage.

20 g=0.005 g=0.010 g=0.020

0.0 5.0 10.0 15.0 r 200 s (x10 )

10000 15000

Figure 5: Plot of the averaged migration step size Ay in the direction of the gradient, plotted as a function of the total structure mass (number of invaded sites) s. Ay was measured in bins of increasing size using three values of the gradient g. The inset shows, on a log-linear scale, an attempt to collapse the data on a single curve by plotting g°' 5Ay as a function of s. The curves were averaged over several runs, and lattices of 1200 x 4500 sites were used.

21 Figure 6: Schematic examples of a structure that forms a bifurcation of branches. Invaded sites that never become abandoned due to high withdrawal thresholds are shown in black. Invaded sites that become invaded and abandoned more than once are shaded. Abandoned sites are shown in white. Regions that never become in­ vaded due to high invasion thresholds are hatched, (a) The sites A, B, C, D, E, F, G, H, I and K have low invasion thesholds and are invaded one after another. The site L is not invaded at this stage due to a relatively high invasion theshold. (b) As the branch grows, the sites D and E are abandoned due to low withdrawal thresh­ olds. (c) More fluid is injected and D is re-filled. E becomes a trapped defender fluid site and cannot be re-filled, (d) The branch with the tip is disconnected from the injection site, and growth is stalled. Instead, the site L is invaded, forming the basis of a new branch.

22 0 1 2 3 4 5 s (x10 )

g=0.005 □ g=0.010 o g=0.015 a g=0.020

i ■______i______i______i______1------1------0 1000 2000 3000 4000 5000 s

Figure 7: Plots of the averaged maximum {/-coordinate grip of all the invaded sites belonging to simulated structures, plotted as a function of the structure mass s. Gradient parameters of g = 0.005 (circles), g = 0.01 (squares), g = 0.015 (diamonds), and g — 0.02 (triangles) were used, respectively. The inset shows an attempt to collapse the tip velocity vfi^{s) = Wip(s)/s on a single curve using Eq. (6) with % = 0.46.

23 e© g=0.001 oo g=0.005 « g=0.010 » g=0.015 g=0.020 g=0.050

slope=-0.45

Figure 8: Part (a) shows a plot of the linear mass density profile m(y) as a function of the structure height y, using various values of the gradient parameter g. Each of the curves was averaged over several dozen or hundred runs and measured when the total mass of the structure was 5000 (4000 for g — 0.05). The lattices used had a size of 700 x 1400 sites. Part (b) shows, on a log-log scale, a plot of m{y) as a function of g. The dotted line is a best linear fit through the data points for g > 0.001. The fitted line has a a slope of -0.45, consistent with Eq. (8).

24

CO

- -2.0 < g=0.010 g=0.020 g=0.050 s=3000

-2.0 -1.0 0.0

FIGURE 9: Part (a) shows, on a log-log scale, the logarithmically binned distribu­ tion N(sf) of fragment sizes s/, measured when all migration had ceased. N(sj) was normalized by dividing by the number of invaded sites s at each stage. Vari­ ous values of the gradient parameter were used. The distributions were measured for s = 1000, 2000, 3000, 4000, and 5000, respectively (1000, 2000, and 3000, re­ spectively, for g = 0.05), and averaged over several 100 runs using lattices of size 700 x 1400. Part (b) shows an attempt to collapse the data on a single curve by plotting s~1s'fN(sf) as a function of sj /g~ 15, according to Eq. (11). For clarity, only the distributions measured at s = 3000 are shown.

25 aa g =0.009 3.0 - ■o g=0.015 ee g=0.001 g=0.020 bq g=0.003 g=0.025 g=0.005 *-* g =0.050

1000 2000 3000 4000 5000

Slope=-1.51

FIGURE 10: Part (a) shows the logarithm of the mean fragment size S plotted as a function of the mass (number of invaded sites) s of the entire structure, using various values of g. Part (b) shows, on a log-log scale, the asymptotic mean fragment size Sas plotted as a function of g. The dotted line is a linear best fit with a slope -1.51.

26 Fractals, Vol. 3, No. 4 (1995) 799-806 © World Scientific Publishing Company

FRACTALS AND SECONDARY MIGRATION

PAUL MEAKIN, GERI WAGNER, VIDAR FRETTE, JENS FEDER and TORSTEIN J0SSANG Department of Physics, University of Oslo Box 1048, Oslo 0316, Norway

Abstract The process of secondary migration, in which oil and gas are transported from the source rocks, through water saturated sedimentary carrier rocks, to a trap or reser­ voir can be described in terms of the gravity driven penetration of a low-density non-wetting fluid through a porous medium saturated with a wetting fluid. This process has been modeled in the laboratory and by computer simulations using homogeneous porous media. Under these conditions, the pattern formed by the mi­ grating fluid can be described in terms of a string of fractal blobs. The low density internal structure of the fractal blobs and the concentration of the transport process onto the self-affine strings of blobs (migration channels) both contribute to the small effective hydrocarbon saturation in the carrier rocks. This allows the hydrocarbon fluids to penetrate the enormous volume of carrier rock without all of the hydro­ carbon being trapped in immobile isolated bubbles. In practice, heterogeneities in the carrier rocks play an important role. In some cases, these heterogeneities can be represented by fractal models and these fractal heterogeneity models provide a basis for more realistic simulations of secondary migration. Fractures may play a particularly important role and migration along open fractures was simulated using a self-affine fractal model for the fluctuating fracture aperture.

1. INTRODUCTION

In the process of secondary migration, hydrocarbon fluids (oil and gas) are transported from the source rocks in which they were generated, to a trap or reservoir in which they are

799 800 P. Meakin et al. found, many millions of years later. In some cases, the source rocks and the reservoir are separated by great distances. A separation of the order of 100 km is not uncommon 1 and migration over distances as great as 500 km has been suggested for some large accumulations such as the Athabasca tar sands in Alberta, Canada.2,3 A better understanding of the secondary migration process would be of value in oil exploration and in the assessment of probable reservoirs. In secondary migration, the hydrocarbon fluids are driven through the water-saturated porous sedimentary rocks between the source rocks and the reservoir by a combination of gravity acting on the water/hydrocarbon density difference and the hydraulic potential gradient associated with the flow of water. Water expelled during the compaction of sedimentary materials contributes to this flow, but the major contribution comes from the flow of meteoric water. Some of the basic aspects of secondary migration can be understood in terms of the displacement of a wetting fluid by a non-wetting fluid in a homogeneous porous medium. The secondary migration process is slow and our studies of the gradient destabilized fluid- fluid displacement process 4 indicate that this process can be represented quite well by a simple modification of the standard invasion percolation model. 5 Interesting dynamical phenomena are also associated with the gradient driven migration process. Some of these dynamical processes are quite rapid and viscous forces must play an important role in these events. However, the rapid events appear to be localized in both space and time. They do not influence the overall fractal scaling geometry of the displacement patterns. One of the important issues in the assessment of secondary migration is the residual sat­ uration of hydrocarbon in the enormous volume of porous rock between the hydrocarbon source and the reservoir. A residual saturation of the order of 1% would, in most cases, account for all of the hydrocarbon expelled by the source rocks and there would not be enough to reach the reservoir. Geological data1 as well as our laboratory experiments and computer simulations 4 indicate that the migration process is localized onto active channels with a characteristic width £. Within the migration channels, the hydrocarbon occupies a fractal region with a dimensionality of D ~ 2.5 (the fractal dimensionality of invasion per­ colation and ordinary percolation). A two dimensional cut, perpendicular to the migration direction, will have a fractal dimensionality of D\ — 1.5 and the residual saturation within the migration channel will be given by:

& = (W*- 2, (1) where s is an inner cut-off length scale. The overall saturation will then be given by:

3 = (2) where £2 is a length scale that characterizes the distance between the migration channels. We have not studied this length scale in detail, but laboratory experiments indicate that £2 £. Equation (2) indicates that the effective overall residual hydrocarbon saturations much smaller than 1% can be expected in typical cases. In practice, heterogeneities play an important role in secondary migration, and it is ap­ parent that heterogeneities must be taken into account if a satisfactory understanding of secondary migration is to be achieved. These heterogeneities range in scale from the micro ­ scopic roughness and diagenitic decorations on individual grains to length scales of the order of the length of the migration path. It does not seem probable that these heterogeneities can be described by a single simple fractal model but there is substantial evidence that Fractals and Secondary Migration 801 fractal models can describe the structure of sedimentary rocks on the pore scale and on the reservoir scale. Here we describe some of our recent work, in which a multifractal model for the distribution of rock properties is used to study the slow, gradient driven migration of a non-wetting fluid in a porous medium saturated with a wetting fluid. Fractures (faults and joints) are important components in the overall heterogeneity of rocks and may play a particularly important role in secondary migration. Depending on how they are filled, as a consequence of mineralization, diagenitic, mechanical and other processes, fractures may act either as channels for fluid flow or as barriers. We have recently begun to study several aspects of the geometry of fractures and the way in which it controls both single and multi-phase fluid transport processes.

2. STABILIZED AND DESTABILIZED MIGRATION The buoyancy-driven upward migration of a less dense, non-wetting fluid through a porous medium is an example of a gradient destabilized migration process. The propagation of the fluid-fluid interface through the porous carrier bed rocks, during secondary migration, can be thought of in terms of a gradient destabilized migration process. However, the downward propagation of the oil/water interface during the filling of the reservoir or in the vicinity of a temporary trap can be described in terms of gradient stabilized displacement. The patterns formed by both stabilized and destabilized displacement can be described in terms of a fractal blob model. This structure is a result of the “competition ” between capillary forces and the effects of the external gradient g, acting on the migrating fluid. On short length scales l < £, the capillary forces are dominant and the pattern has the same self-similar fractal structure as the invasion percolation patterns generated by slow displacement, in the absence of a gradient. On long length scales l > £, the effects of the gradient are dominant and the shape of the displacement depends on the nature of the gradient. In the case of stabilized displacement, the pattern can be described as a compact packing of fractal blobs of size £. If the less dense non-wetting fluid is injected all over the base of the porous medium, the displacement pattern consists of a layer of blobs covering the base from which a self-affine string of blobs emerges. The relative magnitudes of the capillary forces and the forces associated with the gradient, on the pore scale, can be characterized by the Bond number B0, given by:

Bo = = GApe2/T = ge 2/ T, (3) where G is the acceleration due to gravity, T is the interfacial tension, Ap is the density difference between the non-wetting fluid and the wetting fluid, and e is a length scale, which can be taken to be the grain size for a homogeneous packing. It is useful to think of the migration process in terms of the propagation of an active zone through the porous medium that leaves behind a pattern that does not change, as a record of its passage. In the case of stabilized displacement, the active zone consists of a rough surface with a width £ that can be described as a planar array of fractal blobs of size £.6 In the case of destabilized displacement, only the tip of the propagating finger is active and the active zone can be described as a single fractal blob. In order for the active zone to propagate through the porous medium, the leading edge must fill the porous medium up to the percolation threshold pc (a fraction pc of the pore 802 P. Meakin et al. space is filled). In the case of stabilized displacement, the pressure across the trailing part of the interface will exceed that required to reach the percolation threshold, by an amount given by AP ~ g£, where g is the gradient. As a result the percolation threshold will be exceeded by an amount 6p given by:

6p = (p- pc) ~ g£ ■ (4) A second relationship between f and (p — pc) is obtained from percolation theory,

£ ~ \{P-Pc)\~v (5) where v is the correlation length exponent. It follows from Eqs. (4) and (5) that7:

£~ g? ~ ~ . (6)

Similar arguments can be used to estimate the blob size in destabilized invasion percolation. In this case, the pressure across the interface, behind the leading edge of the invading finger, is smaller than the critical value. This reduced pressure prevents the finger from spreading too far to the sides and controls the finger width. Equations (4) and (5) can be used to estimate the correlation length £, leading again to Eq. (6). In this case p < pc, but Eq. (4) still describes the dependence of £ on \p — pc|. Equation (6) has been confirmed using both laboratory models 8 and computer simulations. 9 According to Eq. (6), the mass per unit length in the invading finger in destabilized invasion percolation should be given by:

s ~ ^Dx ~ ~ ^ _ (7)

Using the best available values for D and v from computer simulations Eq. (7) indicates that S ~ ga , with a ~ 0.70. Figure 1 shows some results from experiments in which the mass per unit length S was measured for the penetration of water/sucrose solutions into a porous medium consisting of a packing of transparent polymethylmethacrylate particles with a size of £ ~ 2 mm, saturated with a refractive index matching fluid (n-butylphthalate) . In these experiments the Bond number was controlled via the density difference between the two liquids, which could be varied via the sucrose concentration. In this manner, both the penetration on a denser fluid from the top of the porous medium and the buoyancy-driven migration of a less dense fluid injected at the bottom were studied. Figure 1 indicates that the experiments were in good agreement with Eq. (6), over about an order of magnitude in the Bond number B0.

3. HETEROGENEITIES

The effects of heterogeneities on the slow migration of a non-wetting fluid have been studied using invasion percolation models. In the standard invasion percolation model, random numbers are placed on the sites or bonds of a lattice to represent the capillary forces associated with the pores in a porous medium. Since only the order of the random numbers is important, the random numbers R are usually selected from a uniform distribution over the range 0 < R < 1. In the invasion percolation model, it is assumed that the non ­ wetting fluid always invades the widest pore, with the smallest capillary force or threshold. Fractals and Secondary Migration 803

o o V

log i0 IBol

Fig. 1 The mass per unit length S' as a function of the Bond number Ba for different displacement experiments in three dimensions. The straight line has a slope of 0.72. The open symbols correspond to downward migration, solid to upward. Injections rates were 0.2 (A), 2 (e, o), 5 (V) and 20 (□) ml/h, respectively.

Consequently, the unoccupied perimeter site with the smallest threshold is filled at each stage in the simulation. In the two-dimensional case, trapping of incompressible wetting fluid surrounded by the invading non-wetting fluid is usually represented by excluding the trapped perimeter sites from the set of unoccupied perimeter sites that can be filled. To represent the effects of a gradient, the random thresholds {ti} are replaced by:

U = Ri + gzi, (8) where z; is the projection of the position of the ith threshold or pore neck in the direction of the gradient. In Eq. (8), g is positive for a stabilizing gradient and negative for a destabi­ lizing gradient. Figure 2 compares the results obtained from three dimensional simulations carried out using this model and experiments similar to those used to obtain the results shown in Fig. 1.

3.1 Migration in Fractures

There is a considerable body of empirical evidence in favor of the idea that fractures in brittle materials can be described using a self-affine model. 10,11 In most cases Hurst exponents in the range 0.7 < 0.9 have been reported but the idea that the Hurst exponent is universal remains controversial. We have modeled the gradient destabilized migration of a non ­ wetting fluid in an open fracture, assuming that the capillary thresholds are given by:

U = Fa”1 + gzi, (9) where at is the fracture aperture at the ?’th site. The fracture apertures were calculated assuming that the rough, parallel walls of the fractures could be described as self-affine fractal surfaces with a Hurst exponent of 0.8. 804 P. Meakin et al.

Fig. 2 Gradient destabilized invasion percolation patterns, (a) and (b) show structures obtained during buoyancy-driven migration through a three-dimensional porous medium at Bond numbers of about —0.4 and —0.046, respectively, (c) and (d) show projections of simulated clusters obtained with gradients (g) of —2.7 x 10-2 and —3.0 x 10-3, respectively. (The ratio of the Bond numbers in the experiment to the gradients in the simulations is the same (about 9).) The experiments were carried out in a container of size 18.3 x 18.3 x 28.3 cm3 and the simulations were carried out on lattices of size 128 x 128 x 360, so that a lattice unit in the simulations corresponds approximately to a typical “pore ” volume in the experiments.

Fig. 3 Two stages in the simulation of the buoyancy driven migration of a non-wetting fluid (yellow) through a fracture filled with a wetting fluid (blue). The blue color scale represents the fracture aperture or depth of wetting fluid. The self-affine fracture surfaces partially overlap, occluding parts of the fracture. These occluded regions are represented by the deepest shade of blue. The shade becomes lighter with increasing aperture. The earlier stage is shown on the left hand side. The gradient acts from bottom to top.

Figure 3 illustrates two stages in a simulation of gradient driven migration in a model fracture. In this simulation, both invasion and withdrawal processes were included.12 The inclusion of withdrawal steps leads to a complex fragmentation/coalescence dynamics, sim­ ilar to that observed in experiments. Fractals and Secondary Migration 805 4. MULTIFRACTAL POROUS MEDIA

A variety of fractals models have been used to represent heterogeneities in oil reservoirs. In this work, we used a random multiplicative algorithm to generate a multifractal distribution 13 of capillary thresholds. Since cuts in different directions will have differ­ ent multifractal scaling properties, these distributions may be considered to be multi-affine. Similar models have been used previously, to represent the heterogeneous distribution of properties in sedimentary rocks. 14 Figure 4 shows some of the results from a simulation carried out using an inclined multi-affine porous medium. This simulation is intended to illustrate the lateral migration of the non-wetting fluid under the combined effects of buoyancy through an inclined, stratified carrier formation with long range Bond number correlations. Figure 5 shows two stages in a similar three-dimensional simulation.

Fig. 4 Two stages in a two-dimensional simulation of the migration of a non-wetting fluid. The multi-affine porous medium was tilted by 20° and the fluid is driven through the porous medium by a vertical buoyancy gradient. The combination of the inclination and the vertical gradient leads to a large lateral component to the migration. The dome-like structures near the top of the figure represent formations that cannot be penetrated; they trap the migration fluid. The shade becomes darker with increasing Bond number of the medium.

Fig. 5 Two stages in a three-dimensional simulation of the migration of a non-wetting fluid through an inclined, stratified, multi-affine porous medium. The multi-affine porous medium was tilted by 55° and the fluid is driven through the porous medium by a vertical buoyancy gradient. The fluid was injected near the bottom of the medium. Only the boundaries of the medium are indicated. 806 P. Meakin et al. 5. SUMMARY

Computer simulations and laboratory model studies suggest that the secondary migration of hydrocarbon through the carrier rocks is confined to well separated fractal channels. This results in a very low effective saturation of hydrocarbon in the enormous volume of rock between the source and trap. Our work lends support to the idea that hydrocarbon fluids can be transported through very large distances during secondary migration without loss of all of the hydrocarbon. Simple variations of the standard invasion percolation model provide a surprisingly real­ istic representation of the laboratory experiments. The invasion percolation model that we have used includes both withdrawal and invasion processes. This enables us to study the aggregation/fragmentation dynamics associated with gradient destabilized migration phe­ nomena. This aspect of the models cannot be appreciated from Figs. 3-5. It was illustrated using computer generated videos during the oral presentation.

ACKNOWLEDGMENTS

We gratefully acknowledge support by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (STATOIL) and by NFR, the Norwegian Research Council.

REFERENCES 1. H. Dembicki Jr. and M. J. Anderson, “Secondary migration of oil: Experiments supporting efficient movement of separate, buoyant oil phase along limited conduits, ” AAPG Bulletin 73(8), 1018-1021 (1989). 2. G. J. Demaison, “Tar sands and supergiant oil fields,” AAPG Bulletin 61, 1950-1961 (1977). 3. S. O. Moshier and D. W. Waples, “Quantitative evaluation of Lower Cretaceous Mannville Group as source rock for Alberta’s Oil Sands,” AAPG Bulletin 69, 161-172 (1985). 4. P. Meakin, J. Feder, V. Frette and T. J0ssang, “Invasion percolation in a destabilizing gradient, ” Phys. Rev. A46, 3357 (1992). 5. R. Lenormand and S. Bories, “Description d’un mecanisme de connexion de liason destine a 1'etude du drainage avec piegeage en milieu poreux, ” C.R. Acad. Sc. Paris 291, 279-283 (1980). 6. A. Birovljev, L. Furuberg, J. Feder, T. Jpssang, K. J. Malpy, and A. Aharony, “Gravity invasion percolation in two dimensions: Experiment and simulation, ” Phys. Rev. Lett. 67(5), 584-587 (1991). 7. D. Wilkinson, “Percolation model of immiscible displacement in the presence of buoyancy forces, ” Phys. Rev. A30(l), 520-531 (1984). 8. V. Frette, J. Feder, T. Jpssang and P. Meakin, “Buoyancy driven fluid migration in porous media,” Phys. Rev. Lett. 68, 3164-3167 (1992). 9. P. Meakin, A. Birovljev, V. Frette, J. Feder and T. Jpssang, “Gradient stabilized and destabilized invasion percolation, ” Physica A191, 227-239 (1992). 10. B. B. Mandelbrot, D. Passoja and A. J. Paullay, “Fractal character of fracture surfaces of metals,” Nature 308, 721-722 (1984). 11. P. Meakin, “The growth of rough surfaces and interfaces,” Physics Reports 235, 191-299 (1993). 12. P. Meakin, G. Wagner, J. Feder and T. Jpssang, “Simulations of migration, fragmentation and coalescence of non-wetting fluids in porous media,” Physica A200, 241-249 (1993). 13. B. B. Mandelbrot, “Intermittent turbulence in self-similar cascades: Divergences of high moments and dimension of the carrier,” J. Fluid Mech. 29, 331-358 (1974). 14. P. Meakin, “Fractal aggregates in geophysics, ” Rev. Geophys. 29, 317-354 (1991). Invasion Percolation on Self-Affine Topographies

G. Wagner, P. Meakin, J. Feder, and T. J0ssang Department of Physics University of Oslo, Box 1048, Blindern, 0316 Oslo 3, Norway (Revised version: 14 October 1996)

Invasion percolation (IP) with trapping was studied on two-dimensional sub­ strates with a correlated distribution of invasion thresholds. The correlations were induced by using the heights of (2+l)-dimensional self-affine rough sur­ faces with Hurst exponents in the range 0 < H < 1 to assign the threshold values. The resulting IP clusters consist of “blobs ” with sizes up to the entire cluster size that are connected by fine “threads”. The fractal dimensionality Dh of the IP clusters is dominated by the blobs. The blob size distribution is is related to H and Dh .

I. INTRODUCTION rule” [3]. When the invader fluid has surrounded a region of defender fluid, the defender fluid cannot become dis­ placed and is trapped. Growth of the IP cluster must take The invasion percolation (IP) algorithm [1-3] is re­ place at unoccupied perimeter sites that are not trapped, markably successful in describing the slow immiscible i.e., a path consisting of steps between nearest-neighbor displacement of a wetting fluid by a non-wetting fluid unoccupied sites to the outside of the cluster must ex­ in a porous media [4-10]. In slow displacements, viscous ist. The trapping rule appears to change the universality forces can be neglected, and the process is governed by class of the model. the capillary forces. In equilibrium, the pressure of the In standard IP, the thresholds are distributed uni­ non-wetting fluid must exceed the pressure of the wetting formly on the unit interval without any spatial correla ­ fluid by an amount pc, the capillary pressure, to sustain tion in their values, representing a homogeneous porous the curvature of the interface. When a non-wetting fluid medium. However, the pore sizes in geological fields is injected into a pore filled with wetting fluid, the capil­ clearly are strongly correlated on large scales, and there lary pressure must be overcome. For a circular throat of are indications that correlations also exist on scales down radius R and for the interfacial tension P acting between to the pore scale [11]. Motivated by this observation, the the two fluids, pc = —2P cos 9/R, where 6 is the contact IP model with trapping and without trapping was stud­ angle. The contact angle denotes the angle at which the ied using a multifractal distribution of thresholds [12,13]. interface between the two phases meets the solid surface In these studies the fractal dimensionality of the IP clus­ of the matrix. Thus the non-wetting fluid preferentially ter was reported to be little or not at all affected by the invades the pores with the largest throats. spatial correlations. In contrast, changes in the IP cluster The displacement process can be mapped on the IP growth pattern were observed in simulations using site- model in a straightforward manner. In site IP, the porous bond lattices in which correlations were induced by the medium is represented by a lattice of sites. Each site i constraint that a bond size be less or equal to the size of is assigned a random number r,- and represents a pore the smaller site to which it was connected [14]. with the diameter 1/r-j. Initially, all sites are occupied In the present work, IP with trapping in two dimen­ by the wetting “defender” fluid. An injection site is cho ­ sions was studied using substrates with a different kind sen and filled with the non-wetting “invader” fluid. The of correlated disorder. For each site i at the position algorithm consists of repeating the following three steps: (xi,yi) in a two-dimensional lattice of L x L sites, the 1. Identify all defender sites that are adjacent to the in­ threshold value assigned to the site was given by the z- vaded sites. 2. Among these perimeter sites, find the one value of a rough surface z(z, , %) with the same exten­ with the lowest number r2-. 3. Fill this site with invader sion L x L in the x — y-plane. The substrate obtained fluid. The process is terminated when the edge of the in this way may be used to model the slow displacement lattice is reached by the cluster formed by the invader of a wetting fluid in a fracture by a non-wetting fluid. fluid. The three-dimensional fracture is represented by the two- In two dimensions, the incompressibility of the two flu­ dimensional lattice such that each lattice site corresponds ids must be taken into account by using a “trapping to a region of the fracture plane, and the threshold r,

1 assigned to the site corresponds to the aperture of the fracture. For a perfectly non-wetting fluid [6 — 180°) to advance and displace a wetting fluid in a fracture region of infinite extension and of aperture a, the capillary pressure

must be overcome. The non-wetting fluid tends to in­ vade fracture regions with wide apertures and does not displace the wetting fluid from narrow aperture regions. In the quasistatic case, the displacement process is gov ­ erned entirely by the fracture geometry. Field measurements of natural rock surfaces indicate a fractal character [15-18]. Fresh brittle fractures of differ­ ent types of rock were shown to generate self-affine rough surfaces [19]. An isotropic self-affine surface z(x, y) re­ mains statistically invariant to the scaling transformation x —> Aar, y —¥ Ay, and z —» AHz. The roughness expo ­ nent or Hurst exponent H lies in the range 0 < H < 1. For H = 0.5, a “vertical” cross-section of the surface has the same statistical properties as a Brownian process. For H > 0.5, the cross-section is persistent (segments leading in the positive or the negative z-direction are likely to be followed by segments leading into the same z- direction), and for H < 0.5, the surface is anti-persistent. For H 0.5, the cross-section is characterized by a pro ­ cess called fractional Brownian motion [20]. A simple model of a fracture aperture field a(x,y) is provided by taking the difference of two self-affine surfaces z\(x, y) FIG. 1. Clusters of non-wetting fluid (black) obtained from and Z2{x, y) (with the same amplitude and roughness ex­ simulations of IP with trapping using self-affine substrates of ponent) representing the fracture boundaries, size L = 256, at the stage when the edge of the substrate was reached. The grey shade indicates the invasion thresholds a(x,y) = z2{x,y) - zx(x,y) . (2) used, with bright shades corresponding to high tliresholds. As long as the surfaces do not overlap (a(x,y ) > 0), For comparison, the random number generator seed was kept the invasion thresholds r,- are well defined after discretiz­ constant and the Hurst exponent was varied from H = 0.13 ing the aperture field a(x,y) on a lattice of sites. The (a), H = 0.23 (b), H = 0.39 (c), H = 0.55 (d), H = 0.73 aperture field also forms a self-affine surface with the (e) to H = 0.89 (f), respectively. The arrow indicates the injection site. same roughness exponent as the two surfaces [21]. In the present work, aperture fields were modeled by the z-value fields z(x,y) of single self-affine surfaces: algorithm with random successive addition [26.27], The Hurst exponent characterizing the surfaces was measured a{x,y) - z{x,y) . (3) using the height difference correlation function Invasion percolation using this type of substrates was A(r) =< \z{xi,yi) - z(z, + r=,% +ry )|2 >r2+r2=r2 . studied recently also by Paterson et al. [22] and by Du et al. [23]. Percolation on self-affine topographies was (4) applied by Sahimi [24] to model transport phenomena in heterogeneous media. Invasion percolation in three di­ For self-affine surfaces with positive Hurst exponent H, mensions, using heterogeneous substrates characterized the correlation function scales as A(r) ~ r2fl [26]. The by fractional Brownian motion, was studied by Paterson invasion threshold fields r(x,-,y,) = z(z,, y,) were ob ­ et al. [25]. tained from the substrates. All sites on the lattice were filled with the wetting fluid, and the central site of the lat­ tice was filled with the non-wetting fluid. The IP Simula II. SIMULATION tion was carried out by letting the cluster of non-wettin fluid grow stepwise and invade perimeter sites with min Periodic self-affine surfaces z(z,-, y,) were generated on imal threshold. a square lattice using a random midpoint displacement Figure 1 shows a series of IP clusters obtained in thi

2 manner. For low values of H(0 < H < 0.5), the clus­ ters were reminiscent of ordinary IP clusters grown on a lattice with uncorrelated invasion thresholds. Numer­ ous regions of defender fluids became trapped, and the distribution of the sizes of the regions followed a power law. For higher values of Ff(0.5 < H < 1), the clusters had a disordered shape and could be described in terms of ™ 0.0 “blobs ” of different sizes, connected by thin “threads ”. Compared to the case of low H, a lesser amount of de­ fender fluid became trapped. The IP cluster size dis­ tributions were skew and extended over a large interval. Using H as 0.85, the largest clusters covered more than 50% of the substrate while the smallest ones covered less than 0.005% when the edge of the substrate was reached. FIG. 3. The dependence of the mean number of sites S(R) In contrast, the size distribution of IP clusters grown on in a circle of radius R around the center of gravity of IP uncorrelated substrates is approximately Gaussian. clusters, on a log-log scale. R~2S(R) was measured for IP with trapping on uncorrelated substrates (circles) and on cor ­ related substrates with H « 0.31 (squares), H w 0.47 (dia­ monds), and H ~ 0.85 (triangles), respectively. R~2S(R) is also plotted for IP without trapping on correlated substrates with H a 0.47 (filled diamonds). The solid lines represent linear least-squares fits for distances in the range 2 < R < 64. The substrate size was 512 x 512 sites.

size (mass) S is defined by the ratio of the second mo ­ ment of the size distribution N(s, R) to the first moment, S — s2N(s, R)/J2 sN(s, R)- The clusters grew on FIG. 2. Illustration of the IP cluster growth model. The the correlated substrates using periodic boundary condi ­ invasion threshold of each site i is given by the height z(x,, y, ) tions. If a cluster reached the edge of the substrate, it of a rough surface. A blob is formed as the cluster (shaded could re-enter the substrate from the opposite edge. Each sites) fills out a region around the local threshold minimum simulation was terminated when a cluster attempted to Mi. When the cluster reaches the site S, a thread leading intersect with itself. For a self-similar fractal of fractal to a second local minimum M2 is formed. The region around dimensionality D, M2 is then filled. S(R) ~ RP . (5) A compact blob was formed when the growing IP clus­ In an intermediate range 1 < R |r|=r ' ^ Figure 3 shows on a log-log scale the mean number S{R) of cluster sites s counted in a circle of radius where 0 < p(r) < 1 is the cluster mass density and the R around the center of mass of an IP cluster. The mean averaging is over the occupied origins ro, orientations of

3 -4.0 •

Si 0.8

-10.0 •

i°g,„(b)

FIG. 4. Log-log plot of the density-density correlation func­ FIG. 5. Log-log plot of the number n(b) of “blobs ” per lat­ tion C(r) versus r, on a log-log scale. C(r) was measured for tice site of size b obtained after removing single-connecting IP with trapping on uncorrelated substrates (circles) and on cluster sites from IP clusters (with trapping), using a log-log correlated substrates with H x 0.31 (squares), H x 0.47 (di­ scale. The distributions were averaged over a sample of clus­ amonds), and H x 0.85 (triangles), respectively. Also plotted ters in bins of logarithmically increasing size. The IP clus­ is C(r) for IP without trapping on correlated substrates with ters were grown on uncorrelated substrates (circles) and on H x 0.47 (filled diamonds). The solid lines represent lin­ correlated substates with H x 0.31 (squares), H x 0.47 (dia­ ear least-square fits to the curves for those values of r where monds), and H X 0.85 (triangles), respectively. The substrate log(C(r)) appears to be a linear function of log(r). The sub­ size was 256x256 sites. Each simulation was terminated when strate size was 512 x 512 sites. the IP cluster reached the substrate edge.

the space vector r, and a large sample of clusters. For a At this stage, cluster sites were removed if the removal self-similar fractal of dimension Dona two-dimensional implied fragmentation of the IP cluster. The remaining substrate, C(r) ~ r2~D f(r/rc) is expected, where rc is a sites defined the blobs. The distributions n%,,#(&) of the distance characteristic of the overall cluster extension. number of blobs of with sizes in the range b to b + Sb The cut-off function f(x) has the form f(x) = 1 for (8b —1 0) in on a substrate of size L with a Hurst exponent z 1. Figure 4 shows a plot of C(r). using different values of the Hurst exponent H and the (?) same sample of clusters that was used to measure S(R) (Fig. 3). The correlation function obtained from sim­ where f(x) is a scaling function that decreases faster than ulations on uncorrelated substrates shows a decay con ­ any power of r for r > 1. Here, the cut-off blob size sistent with C(r) ~ r2~Duc for small r. A linear least- bc ~ LDh was assumed to be equal to the IP cluster square fit to C(r) yielded Duc = 1.80 ± 0.01. Turning size. The exponent r is given by the size distribution o to correlated substrates, linear least-squares fits to C(r) regions in the self-affine surfaces with a height z(x, y) less (for r -C L) yielded dimensions of Do .31 = 1.88 ± 0.01, than a “horizontal ” cut parallel to the the x — y plane D0.47 = 1.89±0.01, and D q .ss = 1.95±0.01, respectively, at an arbitrary height zq. For self-affine surfaces with for H % 0.31, H « 0.47, and H x 0.85, respectively. 0 < H < 1, the linear extension r of such regions scales These values are consistent with the results presented in as N(r) ~ rH~3 [20,30]. Making use of Eq. (5), the Fig. 3 and confirm the systematic increase of the frac­ size distribution of blobs with fractal dimensionality Djj tal dimensionality of the IP clusters with the degree of filling out a random sample of regions with z(x, y) < zq is spatial correlations. found to be N(b) ~ b~T with r = (2 + Dh — H)/Dh- The The IP cluster structure was studied by measuring the L-dependence in the scaling form Eq. (7) is obtained fror number (per lattice site) n(b) of blobs of size (mass) b the requirement that the first moment pW = J bn(b)d obtained by removing the connecting “threads”. For each of the blob size distribution scale with the system size a site of IP cluster, it was tested if the removal of the site pW ~ LDh~2. Figure 6 shows an attempt to collaps fragmented the cluster. In this case, the site was marked several blob size distributions n.L,.y (&) corresponding t as a thread site. When all sites had been tested, the different values of L and H on a single curve f (x), usin thread sites were removed. The remaining sites defined Eq. (7) and the fractal dimensionalities Dh found fro the blobs. Figure 5 shows the distribution n(b) measured the scaling behavior of the mean cluster size. From th on substrates of 256 x 256 sites counted at the stage when figure, the scaling function f(x) is found to be linear wit the growing clusters reached the edge of the substrate. a slope of approx. 0.33 for z

4 attempts to determine the fractal dimensionality for IP without trapping on correlated substrates with H to 0.47. From the measurement of S{R) in Fig. 3 and of C(r) in Fig. 4, D%A7 = 1.95 ± 0.01 and D$ A7 = 1.95 ± 0.01 was obtained, respectively. These results do not permit a definite conclusion since finite-size effects may reduce the effective fractal dimensionality from 2.0 to the mea­ sured values of DJj to 1.95. The IP clusters studied in

slope = 0.33 these measurements had a size between approx. 50.000 and 150.000 sites. This finding may be compared with the fractal codimensionality a = 2 — = 0.03 ± 0.01 measured for H = 0.5 in IP without trapping by Paterson et al. [22]. Isichenko [30] analyzed the percolation problem on self- affine topographies and predicted that the perimeter of FIG. 6. Attempt to collapse the blob size distributions n(b) measured for IP with trapping onto a single curve by using the infinite percolation cluster is fractal, with a fractal the scaling form given in Eq.(7). For low values of the argu ­ dimensionality D# = (10 — 3ff)/7 (0 < H < 1). The ment b/L°H, the scaling function f(bjL°H) is linear with a trapping rule does only apply to sites in the interior of the slope of approx. 0.33 (straight solid line). The IP clusters IP clusters such that the structure of the cluster perime­ were grown on correlated substrates with H to 0.31 (squares), ters is not affected by the rule. IP with trapping may H to 0.47 (diamonds), and H to 0.85 (triangles), respectively. thus lead to IP cluster perimeters that are equivalent to The substrates were of 64 x64 sites (solid lines), 128 x 128 sites the perimeters of infinite percolation clusters. was (dotted lines), 256 x 256 sites (dashed lines), and 512 x 512 measured by box-counting the perimeters of some of the sites (dot-dashed lines), respectively. Each simulation was IP clusters and was found to be consistent with the the­ terminated when the IP cluster reached the substrate edge. oretical prediction. For IP with trapping, the results presented here indi­ IV. DISCUSSION cate that the cluster dimensionality Dh depends on the Hurst exponent H characterizing the threshold correla ­ tions of the underlying substrate. Trapping is not a local It was recently shown that the percolation transition rule but affects the growth of the cluster on a global scale, [31] on a correlated substrate of the type used here is leading to the deviation of Dh from Djj . A similar dif­ never critical for H > 0 [32,33]. The percolation expo ­ ference is well known for IP on uncorrelated substrates in nent vh (characterizing the divergence of the correlation two dimensions [3]. In three dimensions, trapping occurs length at the percolation threshold) becomes infinite for rarely and no such global effect of the trapping rule on substrates with topographies given by the z-value field the cluster dimensionality should be expected. of self-affine surfaces with positive Hurst exponent. On The effect of the trapping rule on two-dimensional sub­ such a substrate, the percolation threshold may be inter­ strates is related to the dynamics of the IP process. The preted as the minimal threshold height zc up to which the dynamics may be expected to be quite different from the underlying self-affine surface must be “flooded ” (repre­ dynamics of standard IP [29]. For example, the pair senting the invasion of the corresponding substrate sites) correlation function P(r) giving the probability that IP in order to obtain a spanning cluster of flooded regions cluster sites that are invaded in two subsequent steps that are connected to each other. If the relations between are separated by a distance r, decays approximately as critical exponents known from percolation theory carry P(r) ~ r~2 on uncorrelated substrates, for r below a over, the fractal dimensionality of the infinite percolation cut-off length. On correlated substrates using H = 0.5, a cluster at the percolation threshold must be equal to the much slower decay was found (P(r) ~ r~~< with 7 to 0.9). substrate dimension d = 2. Du et al. [23] measured the The cluster growth proceeded “smoother ”, with promi ­ density of percolation clusters at the critical threshold nent “bursts” occurring less frequently than in standard on self-affine topographies with if = 0.5 and reported no IP. For large values of H, trapping of large regions was dependence on the size of the lattice, implying a cluster rare and the deviation of Dh from D* H became small. dimension of 2. The same result was found earlier by For low values of H, the fractal dimensionality Dh of Schmittbuhl et al. [32]. the IP clusters appears to be close to Duc characterizing There are strong indications that IP without trapping IP with trapping on uncorrelated substrates. This result generates a percolation cluster that is equivalent to the is consistent with the findings of Meakin et al. [13] men­ infinite percolation cluster [34,35]. IP without trapping tioned in the introduction. The multifractal substrates thus may be expected to lead to IP clusters with the studied in the cited work were generated recursively and fractal dimensionality D* H = 2. Figures 3 and 4 show are equivalent to the self-affine “hierarchical” substrates

5 used by Schmittbuhl et al. [32] in the H —» 0 limit. [12] P. Meakin, J. Phys. A: Math. Gen. 21, 3501 (1988). In summary, IP with trapping on substrates with a [13] P. Meakin, Physica A 173, 305 (1991). spatially correlated threshold distribution resulting from [14] A. M. Vidales et al., Invasion Percolation in Correlated the mapping of a self-affine surface has been studied. The Porous Media, submitted to europhysics letters, 1996. [15] S. R. Brown and C. H. Scholz, J. Geophys. Res. 90, 12575 fractal dimensionality of the IP clusters appears to be a (1985). tunable parameter, depending on the Hurst exponent H [16] W. L. Power et al., Geophys. Res. Lett. 14, 29 (1987). characterizing the threshold correlations. [17] W. L. Power and T. E. Tullis, J. Geophys. Res. 96, 415 (1991). [18] J. Schmittbuhl, S. Gentler, and S. Roux, Geophys. Res. ACKNOWLEDGEMENTS Lett. 20, 639 (1993). [19] J. Schmittbuhl, F. Schmitt, and C. H. Scholz, J. Geophys. We gratefully acknowledge support by VISTA, a re­ Res. 100, 5953 (1995). search cooperation between the Norwegian Academy of [20] J. Feder, Fractals (Plenum Press, New York, 1988). Science and Letters and Den norske stats oljeselskap a.s. [21] F. Plouraboue, S. Roux, J. Schmittbuhl, and J.-P. (STATOIL) and by NFR, the Norwegian Research Coun ­ Vilotte, Geometry of Contact Between Self-Affine Sur­ cil. We thank an unknown referee for bringing refs. [24] faces, to appear in Fractals, 1995. [22] L. Paterson, S. Painter, M. A. Knackstedt, and W. V. and [23] to our attention. The work presented has re­ Pinczewski, Patterns of Fluid Flow in Naturally Hetero ­ ceived support from NFR and from the Institute for Com ­ geneous Rocks, 1996. puter Applications at the University of Stuttgart through [23] C. Du, C. Satik, and Y. C. Yortsos, AIChE Journal 42, a grant of computing time. 1 2 3 4 5 6 7 8 9 10 11 2392 (1996). [24] M. Sahimi, AIChE Journal 41, 229 (1995). [25] L. Paterson and S. Painter, Simulating Residual Satu­ ration and Relative Permeability in Heterogeneous For­ mations, 1996, paper presented at the Annual Technical Conference and Exhibition of the Society of Petroleum Engenieers, held in Denver, CO., Oct. 6-9, 1996. [1] R. Lenormand and S. Bories, C.R. Acad. Sc. Paris 291, [26] R. F. Voss, in Fundamental Algorithms for Computer 279 (1980). Graphics, edited by R. A. Eamshaw (Springer-Verlag, [2] R. Chandler, J. Koplik, K. Lerman, and J. F. Willemsen, Berlin, 1985), pp. 805-835. J. Fluid Mech. 119, 249 (1982). [27] D. Saupe, in The Science of Fractal Images, edited by H.- [3] D. Wilkinson and J. F. Willemsen, J. Phys. A: Math. O. Peitgen and D. Saupe (Springer-Verlag, Berlin, 1988), Gen. 16, 3365 (1983). pp. 71-136. [4] N. C. Wardlaw and R. P. Taylor, Bull. Can. Petroleum [28] J. F. Willemsen, Phys. Rev. Lett. 52, 2197 (1984). Technology 24, 225 (1976). [29] L. Furuberg, J. Feder, A. Aharony, and T. Jpsszmg, Phys. [5] J. P. Hulin et al., Phys. Rev. Lett. 61, 333 (1988). Rev. Lett. 61, 2117 (1988). [6] A. Birovljev et al., Phys. Rev. Lett. 67, 584 (1991). [30] M. B. Isichenko, Rev. Modem Physics 64, 961 (1992). [7] M. Chaouche et al., Phys. Rev. E 49, 4133 (1994). [31] D. Stauffer and A. Aharony, Introduction to Fercolatio [8] K. J. Malpy, L. Furuberg, J. Feder, and T. Jpssang, Phys. Theory (Taylor & Francis, London, Washington D.C., Rev. Lett. 68, 2161 (1992). 1992), 2nd edition. [9] P. Meakin, J. Feder, V. Frette, and T. Jpssang, Phys. [32] J. Schmittbuhl, J.-P. Vilotte, and S. Roux, J Phys. A: Rev. A 46, 3357 (1992). Math. Gen. 26, 6115 (1993). [10] P. Meakin et al., Physica A 191, 227 (1992). [33] Z. Olami and R. Zeitak, Phys. Rev. Lett. 76, 247 (1996). [11] T. A. Hewett, Fractal Distributions of Reservoir Hetero ­ [34] B. Nickel and D. Wilkinson, Phys. Rev. Lett. 51, 71 geneity and Their Influence on Fluid Transport, 1986, (1983). paper presented at the 61st Annual Technical Conference [35] D. Wilkinson and M. Barsony, J. Phys. A: Math. Gen. and Exhibition of the Society of Petroleum Engenieers, 17, L129 (1984). held in New Orleans, La., Oct. 5-8, 1986.

6 Invasion Percolation in Fractal Fractures

G. Wagner, P. Meakin, J. Feder, and T. J0ssang Department of Physics University of Oslo, Box 1048, Blindern, 0316 Oslo 3, Norway January 9, 1997

Abstract A model for the slow flow of a non-wetting fluid through a fracture filled with a wetting fluid is presented. The fracture was formed by a fractional Brownian noise surface with a Hurst exponent of H = 0.8 and its replica, displaced relatively to each other in the fracture plane. The aperture field generated in this manner is self-affine on length scales less than the horizontal displacement vector, and uncorrelated on larger length scales. Depending on the displacement parameters and on H, a fraction of the fracture is occluded. Slow two-phase flow was simulated using a modified invasion percolation algorithm. The properties of the correlated substrates are reflected in the structure of the resulting invasion percolation clusters.

1 Introduction

Slow flow of fluids through a network of fractures that are saturated with a second, immiscible fluid is of importance to geological applications in both the petroleum and environmental fields. In the process of secondary migration, oil is transported from a source rock through porous or fractured rock to a trap in which it accumulates in a reservoir [1, 2]. The void spaces in the carrier rock are filled with water that is displaced as the oil passes. In spills, the flow of organic chemicals through fissured media saturated with groundwater is studied to assess damage and design remediation strategies [3]. In this paper, the simpler problem of slow two-phase flow through a single fracture is addressed. This problem is a necessary first step towards the understanding of more complex flow systems and has been the subject of renewed interest in recent years [4, 5, 6, 7, 8, 9,10, 11, 12]. A number of studies support the idea that the fracture surfaces and profiles of a large number of synthetic and natural materials can be characterized using fractal geometry (see [13] and references therein). In particular, brittle fracture of different types of rocks was shown to generate self-affine rough surfaces [14]. The fractal-like character of natural rock surfaces has also been demonstrated [15, 16, 17, 18].

1 However, the simplest conceptual model envisages a fracture as a single pair of parallel plates [19, 20, 21], in the form of a Hele-Shaw cell [22]. Experiments on two-phase flow through such a system revealed interesting pattern formation processes that take place in different flow regimes, depending on the flow rates, the wetting properties, and the interfacial tension of the two fluids [23, 24, 3, 25]. Single-phase tracer experiments indicate that the flow of a carrier fluid is unevenly distributed over natural Assures [26, 27], in contradiction to the as­ sumption made in the parallel-plate model [28, 29, 30]. The experiments were interpreted in terms of channel models in which a single fracture is viewed as an array of non-intersecting channels or pipes with apertures that are either con ­ stant or varying in space [31, 32, 33]. While channel models are successful in reproducing experimental breakthrough curves in single-phase flow [26, 33], the complex, hetereogeneous displacement patterns characteristic for two-phase flow in a natural fracture [9] are represented poorly. A better understanding of multiphase flow through fractured media may be obtained by studying variable-aperture models. In this class of models the rough surfaces of a fracture is approximated using a square lattice of sites. Each site represents a region of the fracture with a more or less constant aperture, and the aperture is varied from site to site [34, 35, 30, 36]. The aperture field ) determining the aperture at the site (z*, y 2) may be generated using geostatistical methods [36, 37, 4, 38, 10], or by means of stochastic approaches that introduce correlations [39, 35, 30, 40]. Piggott and Elsworth [41] constructed variable- aperture models by measuring the hydraulic pressure in a rock sample at selected nodes, and converting the data to an aperture field using an iterative procedure. Simulations of two-phase flow in disordered media (like a variable aperture fracture) are mostly based on the mass continuity equation and on an extension of Darcy ’s law [42] leading to the notion of relative permeabilities. In this ap­ proach, the flow of a fluid from one node to a neighbor node is computed as a function of the pressure difference between the two nodes, the fluid saturation, the viscosity, and the relative permeability assigned to the phase. One of the fluids is customarily assumed to be perfectly wetting, and the other fluid is as­ sumed to be perfectly non-wetting. In the steady-state flow simulator developed by Pruess and Tsang [4], the void space regions occupied by the two fluids were determined through the aperture field by defining a cut-off aperture vc. All nodes with apertures less than vc are occupied by the wetting fluid, and the remaining nodes with apertures greater or equal than vc are occupied by the non-wetting fluid. This rule accounted for the capillary pressure difference

7 cos 9 A Pc = (1) v/2 acting at the interface between the non-wetting fluid and the wetting fluid in a parallel-plate fracture with aperture v. In Eq. (1), 7 and 9 denote the interfacial tension between the two fluids and the contact angle, respectively. Equation (1)

2 is derived from the more general Young-Laplace equation

-L + J-) APC = 7 (2) Pi P2/' where Ri and R2 denote the principal radii of curvature of the interface. Assuming that the interface is nearly flat in the direction parallel to the fracture plane

(Pi 1), Eq. (1) is obtained from Eq. (2) by inserting R2 = v/2. Physically, the equations express the fact that the non-wetting fluid flows in regions with wide aperture while it cannot displace the wetting fluid from regions with narrow apertures. Based on the general equation Eq. (2), Murphy and Thomson [10] simulated dynamic two-phase flow in a variable-aperture fracture using a set of different interface configurations describing the flow pattern at each node. Common to these models is the need to solve large systems of linear equations expressing the relationships between local saturation and pressure. Pyrak-Nolte et al. [9] carried out a simulation of flow of a wetting fluid in a fracture that was initially filled with a non-wetting fluid. Regions with an aperture less than a cut-off aperture vc were filled with the invading wetting fluid if they were connected to the inlet edge by a path of wetting fluid. A trapping condition was used to prevent non-wetting fluid sites with v > vc that were surrounded by the wetting fluid at a given value of vc from being invaded at a later stage, when vc was increased. The non-wetting fluid in the surrounded regions did not participate in the flow and blocked the wetting fluid. Similar models were used by various workers to represent quasistatic displacement processes in porous media (see [43] and references therein). The approach is closely related to a branch of percolation theory known as “percolation with trapping ” [44]. In percolation with trapping, a lattice of sites is used to represent the medium. Quenched disorder is introduced by assigning a random number rt- (corresponding to the aperture height v(xi, yi)) to each site i. One species called the defender is defined which initially occupies a fraction p = 1 of the sites, and which is gradually displaced by the other species, called the invader. When the occupation fraction is decreased from p to p — Sp the invader displaces the defender from sites that are part of the “infinite” defender cluster and have their random number r in the interval \p — Sp,p\. The “infinite” defender cluster consists of all defender sites that are not completely surrounded by invader sites. In standard percolation with trapping, no attention is paid to the inlet accessibility of the sites that become invaded at each stage. The invader sites form many separated clusters that coalesce as the defender occupation fraction p is reduced. A different representation of fluid-fluid displacement processes in disordered media is provided by an algorithm known as “invasion percolation ” (IP) [45, 46, 47, 48]. In IP models, a lattice of sites with quenched disorder, expressed by random numbers r, that are assigned to the sites, is considered. Initially, all sites are occupied by the defender. One site is chosen as an inlet site and becomes occupied by the invader. The algorithm proceeds stepwise. In each step, among all the defender sites that are nearest-neighbors to invaded sites, the one with the lowest random number r is chosen and becomes occupied by the invader. The

3 invader sites form a single cluster of increasing size that is connected to the inlet. A trapping rule may be introduced by excluding all defender sites from becoming invaded that have been completely surrounded by invader sites, so that there is no path of nearest-neighbor defender sites from the trapped region to the outlet [48]. In the absence of a trapping rule, the invader cluster formed in IP shares the properties of the infinite cluster at the percolation threshold in the percolation problem [49]. Trapping rules affect the cluster structure on all scales, up to the global scale, since the trapped regions may be of any size within the size of the cluster [48]. IP models have been used successfully to model the quasistatic displacement of a wetting fluid by a non-wetting fluid in a homogeneous porous medium [50]. The effects of gravity [51] and of pore size variations [52] on the displacement process have been studied by including simple modifications of the model. Two-phase flow in a random aperture field representing a fracture was simu­ lated based on IP by Mendoza and Sudicky [6]. The aperture field was character­ ized by an exponential correlation function describing the spatial persistence of the apertures. The IP model was used to find a flow path of least resistance for a non-wetting fluid that propagated through the fracture and displaced a wetting fluid. Based on the flow path, the transport properties of the fracture model were studied as a function of the non-wetting fluid saturation. In the present work, the IP model was used in a similar manner to study two-phase flow in a variable-aperture fracture. The fracture was assumed to be filled initially with a wetting fluid that is displaced as a non-wetting fluid enters into the fracture. Such a scenario may model the slow migration of oil as it moves through fractured rock that is saturated with water. The displacement was modeled in the quasistatic limit in which the process is dominated entirely by the capillary forces acting at the fluid-fluid interface. The simulation was carried out using “invasion thresholds ” r assigned to the sites on a lattice. The threshold rt- assigned to the site (xt-,y 4) was given by r,- = l/u(zi,2A), (3) where u(x,-, y,-) is the local aperture of the fracture. Thus, the invasion threshold corresponds to the capillary pressure APc (Eq. (1)) required for the non-wetting fluid to displace the wetting fluid at a given site. The aperture fields v(x,y) used in the simulations were generated using a method suggested by Wang et al. [53] and Piggott and Elsworth [54], and more recently by Roux et al. [55] and Plourabone et al. [12]. A rough surface with fractal properties was generated using an iterative algorithm. A copy of the surface was the displaced relative to the original in the directions normal and parallel to the surface plane. Each pair of surfaces represented the walls of a fracture, and the aperture field was given by taking the difference between the two surfaces. Depending on the model parameters, the asperities of a surface and its copy could overlap in some regions, leading to negative differences. The aperture field v(x, y) was taken to be zero in these regions and represented occluded regions.

4 The simulated fracture void spaces obtained in this manner are studied in section

2. In section 3, the simulation model is defined, and results are presented. Section 4 discusses the model and ends with some concluding remarks.

2 Characterization of the Fracture Model

In the Brownian process, the distance B(t) moved by a Brownian particle (a random walker) in the time t follows the scaling law

(4)

Fractional Brownian motion (fBm) is a generalization of Brownian motion in which the motion is either persistent or anti-persistent (steps in a given direction are more likely to be followed by another step in the same direction, or more likely to be followed by a step in the opposite direction) [56]. A fBm trace Bu{t) follows the scaling law

Bu{t) is a self-affine fractal and is characterized by its power spectral density St(k), which decays as [57]:

&(&) - &-C+2") . (6)

Here, k denotes a wave vector in Fourier space, and the tilde denotes propor ­ tionality in a statistical sense in the asymptotic limit. The Hurst exponent or roughness exponent H varies from 0 to 1, with H = 0.5 corresponding to ordinary Brownian motion. fBm may be extended to two or more dimensions. An isotropic fBm surface z(x,y) is characterized by a power spectral density Ss(k) that decays as

(7) with k — {k2 + k2)1?2. On a local scale, the surface appears fractal and has the dimensionality Ds = 3 — H; on a global scale the surface is two-dimensional. A convenient method to measure the Hurst exponent H is provided by the height difference correlation function Cs(r) or variogram js(r) that is defined by

= q,(r) = ^ < |z(ro) - z(ro - r)|^ . (8 )

In Eq. (8 ), r and r0 denote vectors in x — y space, and r = |r|. The height difference correlation function scales as [57]

Cs(r) ~ rH (9)

Thus the expectation value for the height difference between two points on the surface increases with the distance r separating the points.

5 Figure 1: Illustration of the process by which the void space aperture fields are generated. Part (a) shows a fBm surface with H = 0.8, mean value < z(r) >r= 0 and variance a 2 = 1 that is translated in the negative z-direction by an amount —6/2. A copy of the surface is translated in the positive z-direction by 6/2. If the top surface is not translated in the lateral direction, the two surfaces bound a void space of irregular shape but constant aperture (b), and the aperture field shown in (c) corresponds to the vertical displacement. In part (d) the top surface was translated laterally by an amount rg in a random direction. For some values of the parameters 6 and rj, the bounding surfaces overlap and the void space (e) has zero aperture (dark regions in the figure). Part (f) shows the aperture field resulting from a simulation with 6 = 0.25 and rj = 32.

Several different methods have been developed to generate fBm surfaces on the computer. In the present work, the fBm surfaces were generated on a square lattice of size LxL using random midpoint displacement with successive addition [57]. Periodic boundary conditions were used, and a value of 0.8 was chosen for H in all cases. The surface variance cr2 = %(T) was 1. Figure 1 shows a typical surface generated in this manner. A fracture void space was generated following the approach used by Wang et al. [53]. The surface with mean value < z(x,y) >(x,y) — 0 was displaced by an amount 6/2 in the negative z-direction, where the parameter 6 is the mean fracture aperture. A copy of the surface was displaced by an equal amount in the positive z-direction. The copy of the surface was then displaced by a displacement vector ra parallel to the plane of the fracture formed by the original surface and its copy. The magnitude of the displacement vector was a second model parameter that could be chosen freely. The direction of the displacement vector was chosen at random.

6 The lateral displacement was done using “periodic wrapping ” so that the top surface covered the bottom surface exactly. The top surface zt(x,y) was related to the bottom surface Zb(x7 y) by Zt(z,2/) = Z(,(z-Zd,2/-2/d)-l-&, (10) where the displacement vector was given by r z&(%, 3/) %(z,3/) = (11) 0 otherwise. Figure 1 illustrates the procedure. For ru = 0, the bottom surface and its copy match perfectly, and the void space generated has a constant aperture b (Fig. 1 c). For a finite displacement vector, a mismatch occurs that leads to a rough aperture field (Fig. 1 f). Wang et al. [53] have derived the equation „2H ,2 H [C„(r)r 1 + ,2 H r2H lFA-H,-H, 1; (12) for the aperture difference correlation function Cv(r) of the aperture field v(x. y) generated in this manner. Here, r< = r and r> = if r < r^; r< = rd and r> = r if r > rd, Cv(r) approaches the asymptotic limit crv, defined by =< (y(r) - y)2 >r , (13) where v is the mean aperture. The aperture field is not self-affine at length scales of the order of rd and beyond. Figure 2 shows the aperture difference correlation functions measured in simulated fracture void spaces, using b — 1 and rd < Lj 16. For these parameter values, the two bounding surfaces did not overlap in the simulations. For parameter values leading to significant overlap of the two bounding surfaces, qualitatively similar aperture correlation functions were measured if the occuled regions were excluded. In Fig. 2, the expressions predicted by Wang et al. [53] (Eq. (12)) are plotted as well, using the coefficients <7 y that were measured in the simulations. The insert in Fig. 2 shows an attempt to collapse the measured correlation functions on a single curve. The aperture correlation function Cv(r) is related to the power spectral density Sv(k). Wang et al. [53] found that the spectral density for the aperture field has the form &(&) - [1 - Jo(W]&-2-2" ' (14)

7 -4.0 - -5.0 - 0.0 - -6.0 •

-3.0 -

-4.0 -

log(r) log(k)

Figure 2 Figure 3

Figure 2: Plot of the aperture difference correlation functions Cv(r) measured in simulations using 6=1 with lattice size L = 1024. The lateral displacement vectors used in the aperture held generation process had a magnitude of 1 (A), 2 (B), 4 (C), 8 (D), 16 (E), 32 (F), and 64 (G), respectively. The dashed lines represent the theoretical solutions given in Eq. (12), multiplied with the aperture variances a 2 measured in the simulations. The insert shows an attempt to collapse the correlation functions on a common curve. The dashed line has a slope of 2H.

Figure 3: Plot of the power spectral densities Sv(k) measured in simulations using 6=1. The densities were measured for wave vectors k < tt/8 L with the lattice size L — 1024. The lateral displacement vectors used in the aperture held generation process had magnitudes of 1 (A), 2 (B), 4 (C), 8 (D), 16 (E), 32 (F), 64 (G), 128 (H), 256 (I), and 512 (K), respectively. The insert shows an attempt to collapse the power spectra onto a common curve using Eq. (14).

Figure 3 shows the spectral densities measured in simulations using different values for with 6=1. For k rj1, the spectral densities decay as Sv(k) ~ k~2~2H, as expected from Eq. (14). The decrease of the power spectrum at long wavelengths reflects the correlation of the bounding surfaces at length scales greater than the correlation length rj introduced by construction [58]. If the two bounding surfaces do not overlap, the aperture distribution V{v) is Gaussian [12], with a mean of v — 6 and a variance of a „(rj). For r 0, and with a 5-peak at v = 0, as shown in Fig. 4. Figure 5 shows how the fraction p of the void space with zero aperture (the occluded area) increases as 6 is reduced. From Fig. 4, p may be expressed as

8 0.4 •

0 2 4 6 8 10

Figure 4 Figure 5

Figure 4: Void space aperture distribution V{v) measured in simulations with a lattice size of L — 1024 using lateral displacements rd = 32. The vertical displace­ ments were b = 1.0 (A), 0.5 (B), 0.25 (C), 0.125 (D), and 0.0 (E), respectively. For v > 0, the distributions have a Gaussian shape. Figure 5: The fraction p of zero aperture void space, measured in simulations with the lattice size L = 1024 using various values of b and rd = 32 (A), 64 (B), 128 (C), 256 (D), and 512 (E), respectively. The insert shows an attempt to collapse the data on a single curve by plotting p as a function of the ratio b/(rd /L)H. The squares correspond to L = 256 and the circles to L = 1024, respectively. Each data point is based on 100 runs.

1 fO = —f= - / e 2°t> du y27r

= erf(—6/a v) , (15) where the definition of the error function,

erf(z) = --L= f e~t2/2dt , (16) ^ y/2W-co was used. The fraction p depends on via the the variance cr^(rj) ~ The insert in Fig. 5 shows that the measured fractions p(rj, b) of impenetrable void space fall on a single curve when plotted as a function of b/Rf, where Rj = r^/T is the magnitude of the lateral displacement vector, expressed as a fraction of the lattice size L. For 5 = 0, half of the void space is blocked (zero aperture), independent of rd . As the fraction p of zero aperture void space increases, the probability that a fluid can percolate through the fracture is reduced. Percolation in the ^-direction or y-direction is possible if there is a path of nearest-neighbor sites with positive

9 0.4 -

0.4

Figure 6 Figure 7

Figure 6: Probability P of finding a path leading from one edge of the lattice to the opposite edge in the ^-direction (solid lines) or in the {/-direction (dotted lines), measured in simulations with a system size of L = 1024 with b = 0.5 (A, A’), 0.25 (B, B’) and 0.0 (C, C’), plotted as a function of rj on a log-scale. The error bars indicate the statistical uncertainty. Each data point is based on 100 runs.

Figure 7: Probability P of finding a path leading from one edge of the lattice to the opposite edge, plotted as a function of the ratio bj(rd/L) H. P was measured in simulations with a system size of L = 256 (squares) and L = 1024 (circles). Each data point is based on 100 runs.

aperture that leads from one lattice edge to the opposite edge in the x-direction or in the {/-direction, respectively. The path must not cross the lattice boundaries. Figure 6 shows a plot of the percolation probability P measured in simulations using various parameter values b and r^. Since the aperture field is isotropic, the percolation probabilities in the x-direction and in the {/-direction are equal within the statistical uncertainties. P is always finite for b > 0 since the overlap of the

two bounding surfaces is never complete (for > 0). Figure 7 shows an attempt to plot P as a function of the ratio b/(rd/L) H using different values for b, rj and L, respectively. In standard percolation, the percolation probability P approaches a step function as L —)■ oo [49]. For perco ­ lation through self-affine surfaces with H > 0, P has been shown not to approach a step function even for L —y oo [59]. Percolation through the aperture field

studied here is determined by the ratio 6/(rj/L)H, corresponding to the occu ­ pation probability in standard percolation. For any value of this ratio, can be of the order of the system size L, and the aperture correlations are dominant on all length scales. Hence the percolation probability does not approach a step function but is a smooth function of the occupation probability, even in the limit of infinite system size.

10 3 Simulation of Capillary-dominated Two-phase Flow

Quasistatic two-phase flow through the fracture was simulated using an IP model [46, 47, 48]. The IP algorithm is suitable to model dynamic displacement pro ­ cesses on geologic time scales where capillary forces are dominant and viscous forces may be neglected. In the work described in this paper, two scenarios were studied in which a non-wetting fluid (oil) migrated through a fissured medium and encountered a fracture filled with a wetting fluid (water). In the first scenario (A), the non-wetting fluid entered into the fracture through one edge and displaced the wetting fluid until the opposite fracture edge was reached. Periodic boundary conditions were used in the direction perpendicular to the flow direction. In the second scenario (B), the non-wetting fluid entered into the fracture through a hole in the bottom boundary of the fracture, and the wetting fluid was displaced through the edges of the fracture. Periodic boundary conditions were used in the lateral directions. To simulate this scenario with as large IP clusters as possible, the cluster front was allowed to cross the lattice boundaries. Each simulation was terminated when the IP cluster spanned the lattice. At the beginning of each simulation, a “fracture” was generated, and all sites (aq, yi) on the lattice were filled with the wetting fluid (defender). To study scenario A, the sites at one edge of the lattice were then filled with the non ­ wetting fluid (invader). To study scenario B, one site in the center of the lattice was filled with the non-wetting fluid. The edge or the center site represented the inlet through which the non-wetting fluid entered the fracture. The displacement of the wetting fluid by the non-wetting fluid was controlled by capillary forces and driven by an external “pressure”. In each step, the non ­ wetting fluid displaced the wetting fluid from a new site (aq,y,-), and occupied that site. The invaded site was the site among all the wetting fluid sites adjacent to non-wetting fluid sites that had the lowest invasion threshold r4- = 1 /v(xi,yi). This rule expressed the fact that the non-wetting fluid displaces the wetting fluid from regions where that aperture is widest (Eq. (3)). A trapping rule [48] was used so that the wetting fluid sites that had become surrounded by the non ­ wetting fluid could not be invaded. This rule accounted for the incompressibility of the fluids.

Figure 8 shows aperture fields and displacement patterns (the structure of non-wetting fluid immersed in the wetting fluid that is being displaced) observed in typical simulations of scenario A. Depending on the magnitude of the lateral displacement vector (equal to the aperture field correlation length), pronounced differences in the structure of the cluster of invading non-wetting fluid are visible. For rj

11 Figure 8 : Aperture fields and displacement patterns observed at the final stage in two typical simulations of scenario A using b = 0.25 on a 256x256 sites lattice.

Part (a) shows the aperture field generated in a simulation using r,j = 8 , and part (b) shows the same picture with the non-wetting fluid drawn in white. The non-wetting fluid entered into the fracture through the bottom edge and the simulation was terminated when the opposite edge was reached. Parts (c) and (d) show similar pictures obtained using the same fracture bounding surfaces and Td = 128, using the same direction of relative surface displacement. The grey shade indicates the aperture height v from zero (black) to the maximum height (light grey). holes are quite large. In a large range of sizes, the hole size distribution follows a power law, similar to IP on uncorrelated substrates (see below). In contrast, for of the order of the lattice size L, the IP clusters consist of large, more compact “blobs ”. The blobs may be connected to each other by fine “threads” and do not include large trapped wetting-fluid regions. The non ­ wetting fluid forms threads when the advancing cluster tip passes a region with a small aperture and enters a region in which there is an overall aperture gradient (with apertures increasing with increasing distance from the penetration point). If the invasion thresholds rt- assigned to the sites decrease rapidly in a given direction (the apertures increase rapidly), the tip grows in the direction of the aperture gradient, without growing in the direction perpendicular to the gradient. If the region becomes more narrow towards its periphery, the non-wetting fluid forms a compact blob. In this case, growth in the region may cease completely and continue elsewhere at the cluster front.

12 Figure 9: Aperture fields and displacement patterns observed at the final stage in two simulations of scenario A using = 32 on a 256x256 sites lattice. The same bounding surfaces and the same direction of the displacement vector as in Fig. (8 ) were used. Part (a) shows the aperture field and displacement pattern generated in a simulation using b = 1.0, Part (b) shows a similar pattern obtained using b = 0.0. The non-wetting fluid cluster (white) obtained in the two cases are identical. The grey shade indicates the aperture height v from zero (black) to the maximum height (light grey).

In the IP algorithm, the site with the lowest threshold among all other sites at the cluster perimeter is invaded. The alogorithm is not sensitive to the absolute magnitude of the thresholds, nor to the shape of the threshold distribution [51]. When the parameter b is reduced to simulate flow through a more narrow fracture, the capillary pressures required for the non-wetting fluid to occupy sites in the fracture increase. The increase preserves the order of the thresholds such that the order of invasion is not changed. This is demonstrated in Fig. 9 which shows the aperture fields and displacement patterns obtained in two simulations (scenario A) in which b was varied and otherwise identical parameters were used. The IP clusters obtained in both cases are identical. This is not true in cases in which the overlap of the two bounding surfaces inhibits percolation across the fracture, or in a tilted fracture where gravity imposes a linear hydrostatic pressure gradient. These cases have not been included in the present study. The two-point density correlation function C(r) is frequently used to charac­ terize fractal structures. This quantity is defined as

< p(r0)p(r0 + r) >|r|=r < pWXro) > where 0 < p(r) < 1 is the cluster mass density and the averaging is over the occupied origins r0, orientations of the space vector r, and a large sample of clusters. Figure 10 shows plots of C(r), measured in simulations of displacement with point injection (scenario B), using different displacement vector magnitudes rd- The simulations were carried out on lattices of size 512 x 512. The correlations built into the substrates are reflected by the density corre ­ lation function. On short length scales (r

13 log(r) log(s)

Figure 10 Figure 11

Figure 10: Log-log plots of the density correlation functions C(r) measured in IP simulations with trapping, using rd = 4 (circles), rd — 16 (squares), rd = 64 (diamonds), and rd = 256 (triangles) on lattices of 512 x 512 sites. Point injection and periodic boundary conditions were used. For comparison C(r) is also shown for IP with trapping on uncorrelated substrates (dotted line), and a straight line with slope -0.08 (dashed line). For small rd , the values of rd are indicated in the plot.

Figure 11: The dependence of the number Nt(s) of the trapped clusters of wetting fluid on the size s using rd = 2 (circles), rd — 4 (squares), rd = 8 (diamonds), rd = 16 (triangles), and rd = 32 (plus) on lattices of size 512 x 512. Point injection and periodic boundary conditions were used. For comparison Nt(s) is also shown for IP with trapping on uncorrelated substrates (dotted line). To improve readability, the quantity s1’91Nt(s) is plotted on a log-log scale. The values of rd are indicated in the plot.

self-affine, the IP clusters appear compact and C(r) decreases with a slope of approx. -0.08, indicating a fractal codimension of 2 — D Rd —0.08 and a fractal dimension of D Rd 1.92. This result is in qualitative agreement with studies of IP with trapping on correlated substrates [60] in which D = 1.89 and D = 1.95 was found for the fractal dimensionality of clusters grown on self-affine substrates characterized by H = 0.47 and H = 0.85, respectively. On longer length scales (r >• rd ), a cross-over occurs and C(r) decreases with a larger slope of approx. - 0.2, comparable with the corresponding correlation function for IP with trapping on uncorrelated substrates. On uncorrelated substrates, IP clusters are charac­ terized by a fractal codimension of 2 — Duc % —0.18 (Duc Rd 1.82) [48, 61]. - The density correlation function decays faster than any power law for distances r of the order of the lattice size. Consequently, the cross-over is not visible for values rd that are not much smaller than L. A similar reflection of the properties of the substrates is found in the depen-

14 slope=-0.1

0.2 -

0.1 -

Figure 12: Log-log plot of the number of sites ra(A) contained in circles of radius A around the center of gravity of IP clusters. Point injection and periodic boundary conditions were used on lattices of Lx L sites, with L = 32, 64, 128, 256, and 512. The ratio r^/L was varied from 1/32 (circles) to 1/8 (squares) to 1/2 (diamonds), respectively. To test the scaling form Eq. (19), AL92M(A) was plotted against A/rj. dence of the number Nt(s) of holes (trapped wetting fluid clusters) on the size s of the holes, Fig. 11. For IP with trapping on uncorrelated substrates, the hole size distribution can be represented by the form [51]

, (18) where the exponent r — (Duc + 2)/2 ~ 1.91 and s* is a cross-over size given by the lattice size L. The function f(x) approaches a constant for z 4C 1 and f(x) decreases more rapidly than any power of x for x Z$> 1, consistent with Fig. 11. For IP clusters growing on correlated substrates, a second characteristic size Sd = r\ comes into play and Nt(s) cannot be described by a simple power law. Compared to IP with trapping on uncorrelated substrates, holes with small sizes s Sd appear less frequently. Large holes with sizes s > extend over the correlation length and appear with approximately equal frequency. Figure 12 shows, on a log-log scale, the averaged number of invaded sites M(A) that were counted in a circle of radius A around the center of gravity of simulated clusters in simulations (scenario B). The center of gravity was determined at the final stage when the IP cluster intersected with itself. M(A) was expected to follow the scaling form M{A) ~ ADmRd j , (19) where the exponent D describes the increase of the cluster mass with increasing circle radius A, and Rd = r^/T. The function rriRd {x) describes the cross-over from correlated cluster growth (at A

15 length rj. In Fig. 12, the scaling hypothesis Eq. (19) was tested by plotting \~D M(X) as a function of A/r^, using different lattice sizes L and various values of rd- The value D — 1.92 obtained in connection with Fig. 10 was used. The data sets cluster into groups depending on the parameter r^/T. For rd/L = 1/32, growth is similar to IP on uncorrelated substrates with trapping (M(A) ~ \Duc) [48, 61]. For rd /L = 1/8, m(A) initially decays slower, and a cross-over to IP-like growth is apparent for larger A. For r^/L = 1/2, the data is consistent with the law m(A) ~ 1.92 for the range A

4 Conclusion

In the present work, a model for slow two-phase flow of a non-wetting fluid through a fracture saturated with a wetting fluid was studied. The model is a simple IP model applied on substrates with built-in correlations. The correlations correspond to a fractal fracturing process, followed by a shear offset of the two fracture surfaces relative to each other. Such a fracture formation scenario is not realistic if the mean aperture is small and the two surfaces touch each other. Lateral displacement of the surfaces is then not possible without damage and loss of the fractal properties. The fracture model may be considered as an idealized approximation that is simple and that agrees with field observations [15, 58, 62, 16, 63, 18] in terms of power spectra and other measures of correlation. The applicability of the IP model to simulations of two-phase flow through narrow void space was recently demonstrated [64]. However, the interface between the wetting and the non-wetting fluid may be tortuous and characterized by both negative and positive local curvatures. The capillary pressure APc (Eq. 2) is then not a constant but varies dynamically, in contrast to the assumptions made in the IP model. Without doubt, the model fails for very wide fractures in which even slow flow is not dominated by capillary forces. Bearing in mind these reservations, the following results emerge from the study:

1. The fracture model used in this work is defined by three parameters related to the geometry and positioning of the two surfaces that bound the fracture void space: The mean separation b of the surfaces in the direction perpen ­ dicular to the fracture plane, the relative displacement of the surfaces in the direction of the fracture plane, and the Hurst exponent H characterizing the surface roughness. Global properties such as the fraction of inaccessible regions and the percolation probability (Figs. 5 and 7) of the fracture void space depend on the single parameter combination % = b/(rd/L )H, where the lattice size L gives the fracture size (or the spatial resolution) in the fracture plane.

2. The displacement pattern has fractal properties. For instance, the total volume V of non-wetting fluid contained in a fracture region of size L scales with the size of the region in a non-trivial way, V ~ LD with D < 2. The fractal character of the pattern is conserved even in wide fractures without

16 occluded regions, as the displacement pattern is not dependent on the mean aperture (Fig. 8 ). The fractal properties are related to the topography of the fracture surfaces.

3. If the geometry of the void space is characterized by correlations up to a typical length scale r^, the displacement pattern may be expected to reflect the length scale in numerous ways (Figs. (10 - 12). On length scales far beyond r^, the correlations do not influence the displacement process, and the displacement proceeds as if the void space geometry was of uncorrelated randomness.

The effect of buoyancy forces on the displacement process was not studied in the present work. For IP in homogeneous random porous media, it is well known that a hydrostatic pressure gradient due to buoyancy imposes a length scale £ on the displacement patterns [65, 51]. On length scales below £, the displacement patterns show the characteristics of standard IP (without buoyancy). On length scales far beyond £, the patterns loose their fractal properties. - Displacement process in tilted fractures (in the presence of buoyancy) may be expected to be characterized by two length scales and £. If the bouyancy forces are low, £ > rd and the displacement pattern is affected only at length scales of the order of the system size L. For high buoyance forces, £ may be less than the void space correlation length. The displacement process is then characterized only by £ and rd looses its significance.

Acknowledgements

We thank A. Aharony, K. J. Malpy and J. Schmitt buhl for helpful discussions. We gratefully acknowledge support by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den Norske Stats Oljeselskap A.S. (STATOIL) and by NFR, the Norwegian Research Council. The work presented has received support from NFR through a grant of computing time.

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21