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Studies of Tracer Dispersion and Fluid Flow in Porous Media
A Thesis in Partial Fulfillment of the Requirements for the Degree of Doctor Scientiarum
MASTER Thomas Rage Department of Physics, University of Oslo P.O.BOX 1048 Blindern N-0316 Oslo, Norway 1996
WSTHBWIOM OF THIS DOCUMENT IS tMJMTH) DISCLAIMER
Portions of this document may be illegible in electronic image products. Images are produced from the best available original Preface This thesis presents main parts of the project work I performed in the period 1993 to 1996 as a doctor student in the Condensed Matter Group at the Department of Physics, University of Oslo. The project was part of the Cooperative Phenomena Program, initiated by Torstein Jpssang and Jens Feder. My work during that period included studies of tracer dispersion and fluid flow in porous media as well as numerical studies of diffusion limited aggregation (DLA) and related growth models. In order to create a consistent and comprehensive thesis, only work related to tracer dispersion and fluid flow is presented here.
Financial support was provided by the Royal Norwegian Research Council (NFR) through a DEMINEX-scholarship, Grant No. 101874/410, and is gratefully acknowl edged. I am indebted to Jens Feder, Paul Meakin, and Torstein Jossang who super vised my work and taught me the physics of condensed matter on the scientific, administrative, and social level. When I joined the group as a student with a ger man Master’s degree on the subject of chaos (and relatively little knowledge about modern condensed matter physics), they took a chance in hiring me. I have hope to believe that they didn ’t regret. I would like to thank Unni Oxaal for her endless will to prepare experimental results such that they could be compared to the results of my computational studies. To work with Eirik Flekkpy was a challenging pleasure. His numerical approach towards dispersion being different from mine, the friendly competition between us had inspiring influence on my work. Finn Roger, Thomas Walmann and Anders Malthe-Sprenssen were most helpful in my efforts to make computers do the kind of things that I wanted them to do. I acknowledge valuable suggestions of Geri Wagner, Ragnhild Halvorsrud, and Thomas Walmann who read an early version of the thesis. Finally, I mention that this thesis could not have been written without the support from my family and friends, especially Beate, Johannes, and my parents.
Oslo, November 1996
(Thomas Rage) ii Abstract
This thesis is based on computational studies of tracer dispersion and fluid flow in porous media. Its objective is to explore the connection between the topology of a porous medium and its macroscopic transport properties. In porous media, both diffusion and convection contribute to the dispersion of a tracer. The investigation of the combined action of diffusion and convection is therefore an important ingredient in the work presented here. The differential equations that govern fluid flow in porous media are solved numerically by finite-difference schemes. To model tracer transport, a finite-difference scheme and a Monte Carlo method were employed.
The thesis is divided into two parts. The first part (Section 1-7) provides an introduction to the subject of tracer dispersion and fluid flow in porous media, dis cusses its scientific and industrial relevance, establishes the required mathematical framework, and presents the numerical methods that were used. Some specific stud ies serve as illustrations in the otherwise relatively brief presentation. The reader will be referred to standard textbooks and reviews for deeper discussions.
The second part of the thesis consists of reproductions of five scientific papers, listed on page 39. All five publications are based on original research that was carried out as part of the doctorate work presented in this thesis. In the following, these five papers are referred to as PI - P5. Paper 1 (PI) discusses a so-called echo-experiment on tracer dispersion. In such experiments a tracer is first convected into a porous media, and then - by flow reversal - echoed back towards its initial position. Echo-experiments have become an important tool to probe a number of aspects of flow and dispersion in porous media. The paper investigates the influence of finite Reynolds number on the outcome of echo-experiments. By comparing results of experiments and simulations, it is shown that at surprisingly low values of the Reynolds number effects of non linear inertial forces lead to a visible deformation of the returned tracer. Apart from its scientific relevance, this observation may be of significant importance in the interpretation of results from echo-experiments in natural porous media. Paper 2 (P2) is a study on tracer dispersion and fluid flow in periodic arrays of discs. The accuracy of the numerical methods is demonstrated by comparing results from simulations with some well-established analytical predictions. The connection between microscopic and macroscopic dispersion mechanisms is investigated by cal culating macroscopic dispersion coefficients for samples of different porosity and a large range of Peclet number. Since relatively little is known about macroscopic dispersion coefficients, the detailed study of the dispersive properties of such simple geometries adds valuable information to the subject of tracer dispersion in porous media. In detail, it is demonstrated that the mechanisms of mechanical disper sion in periodic media and in natural (non-periodic) porous media are substantially different. The results of P2 illustrate the close connection between the topology of a porous medium and its macroscopic transport properties. To quantify geometric charac teristics of a porous medium (such as homogeneity, connectedness, and tortuosity) and to rigorously relate them to macroscopic transport coefficients is, however, still an unfinished enterprise. In this context, Paper 3 (P3) presents first measurements on the percolation probability distribution of a sandstone sample. Local porosity theory predicts that this simple geometric function of a porous medium is of domi nant importance for its macroscopic transport properties. It is demonstrated in P3 that the measurement of percolation probability distributions on digitized samples presents only little difficulty. Results are consistent with theoretical expectations
m on the shape of the obtained curves. The subject of Paper 4 (P4) is essentially iden tical to the one of P3, but discusses our results from a somewhat different point of view. Paper 5 (P5) presents a number of qualitative results on tracer dispersion and two-phase flow in rough fractures. Faults and fractures play a dominant role for geo logical phenomena (such as metamorphism and mineral deposition) , waste disposal by deep-well injection, and the transport of hydrocarbon fluids from the source rock to a reservoir. Due to the small length scales and large time scales that the trans port in a natural fracture involves, experiments are difficult to perform. Moreover - since the void space is usually not accessible - they yield only limited information. Consequently, relatively few quantitative results on flow and transport in fractures exist. It is demonstrated in P5 that, using simple but realistic models and readily available computer resources, many aspects of transport through fractures can be studied numerically on scales large enough to allow quantitative questions to be addressed.
IV Zusammenfassung
Diese Abhandlung basiert auf computergestiitzten Studien fiber die Ausbreitung (Dispersion) eines Indikators und Fliissigkeitsstromungen in porosen Median. Ziel ist, die Verbindung zwischen der Topologie eines porosen Mediums und seinen makroskopischen Transporteigenschaften zu erforschen. In porosen Median tra- gen sowohl Diffusion als auch Konvektion zur Dispersion eines Indikators bei. Die Untersuchung der kombinierten Wirkung von Diffusion und Konvektion ist da- her wesentlicher Bestandteil der Abhandlung. Die Differentialgleichungen , denen Stromungsprozesse in porosen Medien unterliegen, warden numerisch durch Diffe- renz-Verfahren gelost. Simulationen des Transports eines Indikators beruhen auf einem Differenz-Verfahren , sowie einer Monte Carlo Methode.
Die Abhandlung gliedert sich in zwei Teile. Der erste Teil (Abschnitt 1-7) gibt eine Einfiihrung in die Physik der Dispersion und der Stromung in porosen Medien, diskutiert deren wissenschaftliche und wirtschaftliche Relevanz, prasentiert den notwendigen mathematischen Apparat, und erlautert die benutzten numerischen Methoden. Einige einfache Studien dienen als Illustrationen in dieser ansonsten recht knapp gehaltenen Einfiihrung. Der Leser wird auf Lehrbiicher und Ubersichts- artikel zu den jeweiligen Themen verwiesen.
Der zweite Teil dieser Abhandlung umfafit Reproduktionen von fiinf wissenschaft- lichen Arbeiten, die auf Seite 39 aufgelistet sind. Alia fiinf Publikationen basieren auf originaren Ergebnissen der Doktorarbeit, die diese Abhandlung reprasentiert. Die Publikationen sind im folgenden mit P1-P5 bezeichnet. Die erste Publikation (PI) diskutiert ein sogenanntes Echo-Experiment. In solchen Experimenten wird ein Indikator mittels einer stromenden Fliissigkeit zuerst in ein porosen Medium transportiert und anschliebend - durch Umkehr der Stro- mung - zuriick zu seiner Ausgangsposition getrieben. Echo-Experimente haben sich zu einem bedeutenden Werkzeug bei der Untersuchung zahlreicher Aspekte der Stromung und Dispersion in porosen Medien entwickelt. Die Publikation untersucht den Einflufi endlicher Reynolds-Zahl auf das Ergebnis von Echo-Experimenten. Durch Vergleich der Resultate von Experimenten und Simulationen wird gezeigt, dafi bei erstaunlich niedrigen Werten der Reynolds-Zahl Effekte von nicht-linearen Tragheitskraften zu einer sichtbaren Verformung des returnierten Indikators fuhren. Neben seiner grundsatzlichen wissenschaftlichen Relevanz kommt diesem Refund unter Umstanden grofie Bedeutung bei der Auswertung der Ergebnisse von Echo- Experimenten in naturlichen porosen Medien zu. Die zweite Publikation (P2) ist eine Studie fiber Dispersion in periodischen Anordnungen von Kreisscheiben. Die Genauigkeit der verwendeten numerischen Methoden wird durch einen Vergleich der numerischen Ergebnisse mit wohlbekan- nten analytischen Resultaten aufgezeigt. Zur Untersuchung der Verbindung zwi schen mikroskopischen und makroskopischen Dispersionsmechanismen werden Sim ulationen bei verschiedener Porositat und fiber ein breites Spektrum der Peclet-Zahl diskutiert. Da relativ wenig fiber makroskopische Dispersionskoeffizienten bekannt ist, tragt eine solche detaillierte Studie der dispersiven Eigenschaften dieser ein- facher Geometrien wesentlich zum Verstandnis der Dispersion in porosen Medien bei. So wird gezeigt, dafi der Einflufi mechanischer Dispersion in periodischen porosen Medien und in naturlichen (nicht periodischen) porosen Medien grundsatz- lich verschieden ist. Die in P2 dargestellten Ergebnisse illustrieren die enge Verbindung zwischen der Topologie eines porosen Mediums und seinen makroskopischen Transporteigen schaften. Die Quantifizierung geometrischer Konzepte (wie Homogenitat, Verbun- denheit und Tortuositat) und die rigorose Herleitung ihrer Relation zu makroskopi- schen Transportkoeffizienten sind auch heute noch aktiver Forschung unterworfen. In diesem Zusammenhang werden in der dritten Publikation (P3) erste Messun- gen der Verteilung der Perkolationswahrscheinlichkeit eines natiirlichen Sandsteines prasentiert. Im Rahmen der lokalen Porositatstheorie wird dieser einfachen geome- trischen Funktion eines porosen Mediums groBe Bedeutung fur dessen makrosko- pische Transporteigenschaften zugeschrieben. In P3 wird gezeigt, daB die Messung der Perkolationswahrscheinlichkeit von digitalisierten Proben eines porosen Medi ums nur geringe Anforderungen stellt. Die Form so erhaltener Verteilungsfunk- tionen ist konsistent mit theoretischen Er wart ungen. Die vierte Publikation (P4) umhandelt im wesentlichen das gleiche Them a, diskutiert unsere Ergebnisse aber aus einem anderem Blickwinkel. Die fxinfte Publikation (P5) prasentiert einige qualitative Ergebnisse iiber Dis persion und Zwei-Phasen FluB in Rissen. Verwerfungen und Gesteinsrisse haben entscheidenden EinfluB auf geologische Transportprozesse (wie z.B. Metamorphis- mus und Mineralablagerung), die Tiefbrunnen-Entsorgung chemischer und radioak- tiver Abfalle, und den Transport von Kohlenwasserstofverbindungen (01 und Gas) vom Quellgestein zum Reservoir. Aufgrund der kleinen Langenskalen und groBen Zeitskalen, die solche Prozesse involvieren, sind Experimente nur schwer durchfiihr- bar. Dariiberhinaus liefern sie nur begrenzte Information, da der Leerraum eines Risses der experimentellen Beobachtung iiblicherweise nicht zuganglich ist. Aus diesem Grunde existieren nur wenige quantitative Aussagen iiber Transport und Dispersion in Rissen. In P5 wird gezeigt, daB numerische Simulationen (mittels einfacher aber realistischer Modelle und leicht zuganglichen Rechnerkapazitaten) die Untersuchung zahlreiche Aspekte des Transports in Rissen auf Langenskalen ermoglichen, die groB genug sind, um quantitative Analysen zu erlauben. Contents Preface i Abstract iii Zusammenfassung v
1 Background and Motivation 1 1.1 Organization of the Introductory Sections ...... 5
2 Basic Concepts of Porous Media Theory 7 2.1 Percolation Properties ...... 8
3 Tracer Dispersion 10 3.1 The Convection-Diffusion Equation ...... 10 3.2 Macroscopic Equations ...... 11 3.3 Dispersion Mechanisms ...... 11
4 Fluid Flow 16 4.1 The Navier-Stokes Equation ...... 16 4.2 The Stokes Equation ...... 17 4.3 The Darcy Equation ...... 18
5 Modeling Fluid Flow 19 5.1 Time-Dependent Flow ...... 19 5.2 Steady Flow...... 23 5.3 Alternative Methods ...... 25
6 Modeling Tracer Transport 27 6.1 A Finite-Difference Scheme ...... 27 6.2 A Monte Carlo Scheme ...... 28
7 Concluding Remarks 33
References 34 List of Symbols 38 List of Research Papers 39 1 Background and Motivation 1
1 Background and Motivation Consider the situation that a drop of drawing ink is suspended in a glass of water, such that the ink initially colors only a small amount of the water, as sketched in Fig. 1 a). Daily experience tells us that the ink will spread relative quickly, Fig. 1 b), and eventually get evenly distributed, Fig. 1 c). This spreading is due to molecular diffusion of the ink particles, discussed further in Sec. 3.
Figure 1: Sketch of the dispersion of a drop of ink in a glass of water.
Physicists use the term dispersion as a synonym for the spreading of a distri bution of mass (or energy) in a carrier medium. In the above example, the ink is dispersed in the water by diffusion. Dispersion is thus a fundamental mechanism of transport. Other examples of dispersion phenomena include the spread of sound waves and the spread of heat.
Molecular diffusion is not the only mechanism that leads to dispersion. Transport by convection, for example, often leads to dispersion, too. To see this, imagine that an ink is poured into a river, and convected downstream along with the flowing water. The local velocity of the carrier will usually not be constant throughout the river, but vary with position. Such velocity fluctuations introduce a number of different dispersion mechanisms, some of which are discussed in more detail in Sec. 3. The interplay between dispersion and transport can be illustrated in the following way:
Diffusion -> Dispersion # T ransport <- Convection
Here, arrows should be read as ‘leads to ’, and diffusion and convection are mentioned as examples of a dispersion- and a transport process, respectively.
It is obvious that dispersion is a phenomenon of fundamental importance in daily life as well as in engineering. Consequently, there is a profound need to quanti tatively understand and describe dispersion processes. If, for example, the ink in the above mentioned example was a poison (as was the case in the Sandoz-accident in November 1986, when firemen in-voluntarily polluted the river Rhine with 30 tons of highly toxic waste), one would very much like to know where (or when) one should not drink from the river.
Dispersion phenomena occur in a large variety of situations, and the underlying dispersive mechanisms can be of quite different nature. Only one specific class of dispersion processes is considered in this thesis, namely the dispersion of a passive tracer by diffusion and convection in fixed beds. The inherent assumptions are as follows: 2 1 Background and Motivation
• The tracer is assumed to be perfectly miscible in the carrier fluid. If in the above example the ink was replaced by a drop of oil (which is immiscible with water), there would be a sharp interface between the oil and the water. Such an interface has drastic influence on both transport and dispersion. This is especially true for transport in porous media and fractures, where capillary effects become important. • Only passive tracers are considered, that is, density- or viscosity contrasts between the tracer and the carrier are neglected. Consequently, passive tracers do not affect the flow of the carrier fluid and the flow of the carrier plays the role of an ‘input parameter ’ to the dispersion process, compare Fig. 2. • The carrier is assumed to be an incompressible, Newtonian fluid, with constant density and viscosity. This restriction is not as severe as they might seem, since many fluids of practical importance satisfy these conditions to a high degree. Examples include water, glycerol, and many hydrocarbon fluids. It will be seen that the flow of such fluids is described by a relatively simple differential equation, the Navier-Stokes equation. • The fluid geometry is assumed to be fixed, that is, not changed by the flow of the carrier. Although this restriction is employed mostly for computational ease, this situation is frequently encountered in packed beds, porous soils, and porous or fractured rocks. Flow through fixed beds is usually steady, such that the velocity of the carrier fluid at a given point does not change in time. In this case, studies on tracer dispersion can be divided into three steps, as sketched in Fig. 2: First, the geometry of the porous medium is defined. Next, the steady flow of the carrier in the porous matrix is calculated. Finally, the transport of a tracer under the combined action of molecular diffusion and convection by the steady flow of the carrier is studied.
Figure 2: Sketch of the typical three-step procedure in studies of tracer dispersion by a steady carrier flow: a) A porous medium is created. The solid matrix is shown in grey. The sample shown here is periodic in order to mimic an infinitely extended porous medium, b) A steady flow field of the carrier in the porous matrix, indicated by arrows, is calculated, c) The process of tracer transport is modeled. Dots denote individual tracer particles. The solid matrix and the flow field are not shown. The unit cell of the porous medium is indicated by a square. The arrow inside the unit cell indicates the direction of the average carrier velocity.
In addition to diffusion and convection, a passive tracer might be submitted to processes like adsorption, radioactive decay, or chemical reactions. In this thesis, however, main focus is put on the influence of the geometry in which the carrier resides to the dispersion process. More specifically, we take interest in the case of tracer dispersion in porous media (like soil or sandstone), and fractures. Since molecular diffusion and convection are usually the most relevant dispersion mecha nisms, only these two microscopic processes will be considered. 1 Background and Motivation 3
Despite of the above assumptions the results presented in this thesis have practical implications to • the transport of minerals in soils, • the disposal of radioactive or chemical waste in air or water, • chromatography, • different types of oil recovery, • sewage waste disposal into aquifers, and • aquifer tracer tests. Moreover, transport by diffusion and convection is an interesting phenomenon from a scientific point of view: Diffusion is a stochastic process, that is, a diffusing particle has no memory about its position at earlier times. Consequently, the process of diffusion is irreversible in time. In contrast, convection is a deterministic, and (under certain restrictions) completely reversible process. The interplay of two processes so different in nature leads to remarkable phenomena, some of which are discussed in this thesis. Finally, it will be seen that tracer dispersion is relatively simple to access theoretically. Consequently, a number of analytical results exist, and can be checked against the results of experiments and simulations. For these reasons, tracer dispersion has been a subject of study in the Cooperative Phenomena Program for many years [1-10]. The work presented in this thesis contributes to this activity by numerical studies of tracer dispersion and fluid flow in porous media. Such computational studies commonly provide a deeper understanding of experimental and theoretical results. Especially for the purpose of testing theories, numerical studies are often preferred to real experiments. The reason for this is, that computer simulations provide complete knowledge of both, the modeled process and the results of the simulations. This is usually not the case in experiments. In detail, natural porous media (like soils, sandstones, or fractured rocks) are not transparent, prohibiting the observation of the tracer inside the porous medium.
The echo-experiments by Oxaal nicely illustrates many of the above arguments: For these experiments, a cylindrical obstacle was placed in the middle of a flow channel (Hele-Shaw cell) to create a simple porous medium with a well-defined geometry. The top- and bottom plate of the cell are transparent in order to allow inspection from above. The cell and the attached pumping devices were filled with a mixture of glycerol and water as a carrier fluid. The experimental setup is shown schematically in Fig. 3. Detailed discussions of the experiment can be found in Refs. 9-11, and PI. In the beginning of an experiment, a thin line of tracer fluid - a mixture of glycerol, water and nigrosine - is injected with a syringe into the cell. With the help of a syringe pump, a fixed amount of carrier fluid is pumped through the cell to convect the tracer line towards and past the cylinder. By reversing the syringe pump, the same amount of carrier fluid is then withdrawn to ‘echo ’ the tracer line back towards its initial position. A typical experimental observation is reproduced in Fig. 4. Note the region of enhanced dispersion , visible in Fig. 4 b) as a smearing of the central part of the returned tracer line. This enhanced dispersion can qualitatively be understood in the following way: During forward convection, the tracer line partially folds around the cylinder. At the same time - since the fluid velocity decreases towards the cylinder - the tracer line is decelerated when approaching the cylinder and thereby compressed. In a similar fashion, the acceleration of the tracer line during backward convection yields an expansion of the line. Due to the relatively low velocity of the fluid these two processes can be considered to be completely reversible, such that - in the absence of molecular diffusion of the tracer 4 1 Background and Motivation
syringe pump
valve peristaltic pump tracer line
Hele-Shaw cell wii obstacle^^^
fluid reservoir syringe for^ tracer injection
Figure 3: Sketch of the experimental setup by Oxaal [9,10].
- the returned tracer line is expected to have the same shape as the initial tracer line. Molecular diffusion, however, tends to smooth the sharp concentration gradi ents that the compression of the tracer line during forward convection introduces. Consequently, the compression of the line by convection is opposed by a diffusive spread. At flow reversal the tracer line is therefore ‘thicker ’ than it would be in the absence of diffusion. The expansion of the tracer line during backward convection leads then to the enhanced smearing of the mid-channel part of the tracer line, which was subject to the largest compression.
Figure 4: a) Superposition of four snapshots of a tracer line during return flow in an echo experiment. Time increases going from right to left. The cylinder does not show, since this is a difference image. The grid mark spacing is 1 cm. b) Enlargement of the characteristic mid-channel dispersion of the returned tracer line. The grid mark spacing is 0.1 cm. The figure is reproduced from [10] with permission of the authors.
Loosely speaking, the ’echo ’ of this tracer experiment is the result of the in terplay between convection as a transport mechanism and diffusion as a dispersion mechanism. The role of the obstacle is to introduce velocity gradients into the flow field, by which convection and diffusion interact. When such an echo-technique is 1.1 Organization of the Introductory Sections 5
applied to a natural porous medium, one may thus extract information about the topology of the porous medium from the tracer echo [12-15].
The above explanation of the experimental result is based on the assumption that diffusion and convection are the only relevant processes. However, several other mechanisms might be relevant for the origin of the experimental observation: The density of the tracer and the carrier - although carefully matched - will in practice be slightly different. Such density contrasts introduce buoyancy forces and may severely affect the results. Additionally, inertial effects during start-up, reversal, and stop-down of the carrier might be important, especially if the carrier is compressible. Yet, Monte Carlo simulations of the echo-experiment that rely on steady state flow of an incompressible fluid and a completely passive tracer reproduce the ex perimental results to a high degree. This becomes evident from comparing Fig. 5 to Fig. 4, and was further confirmed by a quantitative analysis of the results of experiment and simulation. From this agreement between experiments and simula tions, one concludes that the experiment is indeed accurately described as a passive tracer experiment in incompressible flow. Moreover, the dispersion of the tracer can be studied in detail in the simulations, leading to the qualitative picture presented above. In summary, the echo-experiment illustrates how numerical simulations can be used to verify and explain experimental results.
Figure 5: a) Superposition of six snapshots of a tracer line in a two-dimensional Monte Carlo simulation of the echo experiment with N = 103 particles. Only half of the computational domain is shown. The side walls of the cell and the disc that represents the cylinder are shown in grey. Arrows indicate the local fluid velocity, as calculated with a finite-difference scheme. Each tracer particle is shown by a black dot. The leftmost ‘line ’ corresponds to the initial tracer, and the rightmost ‘line ’ shows the tracer at flow reversal. Molecular diffusion of the tracer was here excluded, such that the initial line and the returned line were identical, b) Enlargement of a returned tracer line in a simulation with molecular diffusion of the tracer. Each of the dots marks the position of a tracer particle at the end of the simulation. The grid mark spacing is 0.1 cm.
1.1 Organization of the Introductory Sections The notion of a porous medium is of fundamental importance in this thesis. Some basic concepts in the theory of porous media are therefore discussed in Sec. 2. 6 1 Background and Motivation
Section 3 gives a brief introduction to the mathematical framework in which tracer dispersion can be described, and discusses a collection of dispersion mecha nisms and their contribution to the macroscopic dispersion coefficients. Section 4 presents three differential equations that are relevant for flow in porous media. These are the Navier-Stokes equation, the Stokes equation, and the Darcy equation. Section 5 presents finite-difference schemes by which the Navier-Stokes equation (or the Stokes equation) can be solved numerically. Results of tests that demon strate the accuracy of the schemes (and the correct implementation) are included. In a similar fashion, Sec. 6 discusses two different models to simulate tracer transport; a finite-difference scheme and a Monte Carlo method. Most attention is put on the discussion of the Monte Carlo method that render the tracer as an ensemble of independent particles. As a test of this scheme, numerical results on Taylor dispersion in a channel are compared with analytical predictions. Section 7 sums up the introductory part of this thesis through a short conclusion. 2 Basic Concepts of Porous Media Theory 7
2 Basic Concepts of Porous Media Theory
For our purpose, a porous medium is a partition of space into a solid matrix, M., and a porous matrix, V, such that any point in space, x, is either part of the solid (x £ M) or the porous matrix (x £ V) [16]. The boundary between porous matrix and solid matrix is denoted as F. Throughout this thesis, the porous matrix, or pore space, is assumed to be saturated with a carrier fluid, while the solid matrix is occupied by a solid substance. The carrier fluid is required not to penetrate into the solid, and the solubility of the tracer in the solid is assumed to be zero. In other words, both carrier fluid and tracer will be found in the porous matrix only. Since the properties of both the carrier fluid and the tracer differ so strongly in the solid and the porous matrix, the geometric characteristics of the porous medium, like its symmetry and connectivity will have a large impact on the transport properties of the porous medium.
Figure 6: Simple models of two-dimensional porous media. The solid matrix is shown in grey, a) Tube model: The porous matrix consists of isolated channels, b) Grain model: The solid matrix consists of grains, c) Network model: The porous matrix consists of connected channels.
A number of simple geometric models that are frequently used in the theory of porous media are sketched in Fig. 6. Other models of porous media include reconstruction models and process models. In reconstruction models, a porous media with a-priori chosen geometric characteristics is reconstructed through, for example, Fourier methods. Such a method was employed in P5 in order to generate fractures with rough surfaces. It is described in some detail in Ref. 17. Process models subject a porous medium (one of the above models, for example) to a process like consolidation, fragmentation, erosion, or porosity reduction. In the introductory part of this thesis only tube models, compare Fig. 6 a), and grain models, compare Fig. 6 b), are considered.
In order to quantify the geometric properties of a given porous medium, it is customary to introduce the characteristic function, %, such that
(1)
Note that the assumption of a fixed bed (see Sec. 1) amounts to the requirement that the characteristic function be time-independent. In a similar fashion, periodicity and anisotropy of a porous medium are directly related to translational invariance and rotational invariances of the characteristic function, respectively. Since such invariances introduce strong correlations in the characteristic function, correlation functions of % are important quantities in the geometric characterization of a porous media. 8 2 Basic Concepts of Porous Media Theory
One of the most important quantities in the theory of porous media is, however, the porosity ,
(2) where V denotes a sample of volume V, centered at position x. The porosity of a sample is thus the ratio of pore volume to sample volume. When evaluated on samples of small volume, the porosity necessarily fluctuates strongly with position. This observation suggests the introduction of the local porosity distribution as a function that measures the probability that a sample of given volume (or length) has a certain porosity. For samples of large volume, the fluctuations of the local porosity may decay, and a large-scale limit be obtained. In that case, the porous medium is said to be homogenous. Homogeneity of a porous medium involves a characteristic length , l, on which fluctuations of the properties of a sample of the porous medium (like porosity) become negligible. This length can usually be identified as a typical size of the grains of the solid matrix, a typical pore size, or the correlation length of the two- point correlation function.
The specific surface, 5, defined as the ratio of the area of the boundary F inside a sample to the volume of the sample, is mentioned here as another purely geometric quantity of a porous medium. For homogenous porous media, the specific surface is related to the slope of the two-point correlation function at the origin [18]. Since the specific surface has the dimension of an inverse length, it provides yet another possibility to define the characteristic length l. Fundamental macroscopic transport coefficients, such as the effective diffusion coefficient (compare Sec. 3) or the spe cific permeability (compare Sec. 4) of a homogenous porous medium are strongly correlated to its porosity and its specific surface.
2.1 Percolation Properties As far as tracer transport and fluid flow through porous media is concerned, the connectivity of the porous matrix is of dominant importance, since macroscopic transport will only occur if the porous matrix is percolating. For our purpose, a porous sample is said to percolate if it is possible to push the carrier fluid through it. To illustrate the concept of percolation, we note that the model porous me dia shown in Fig. 6 b) and Fig. 6 c) are percolating in both the vertical and the horizontal direction, while the channel model, Fig. 6 a) percolates only in the hor izontal direction. A more general introduction to percolation theory can be found in Ref. 19. The percolation properties of porous media are discussed in P3 and P4 in the context of percolation probability distributions of a natural sandstone sample. The percolation probability distribution connects the local percolation probability, P, to the local porosity,
three-dimensional case is straightforward . We identify sites of the porous matrix (x = 1) as ‘white ’, while sites that belong to the solid matrix are ‘black ’. The algorithm to test if the sample percolates in the horizontal direction consists of the following steps:
1------1------1------1—^ l 12 3 4
Figure 7: Two-dimensional, digitized (4 x 4)-sample of a porous media. Sites that belong to the solid matrix are colored ’black ’. The sample percolates in the hori zontal direction, since it is possible to connect the first column (i = 1) and the last column (i = 4) by a next-neighbour path of white sites. Sites that belong to this connecting path are marked by grey circles. Sites of the porous matrix that belong to a ‘dead end ’ are marked by black circles.
Step 0: Take all white sites in the leftmost column of the sample into an ordered list and mark them ‘grey’. Continue with the next step. Step 1: If there are no sites in the list, say ‘Sample does not percolate ’ and stop the algorithm. Continue otherwise with the next step. Step 2: Choose the last site in the list and continue with the next step. Step 3: If the selected site is in the last column (i = l), say ’Sample percolates ’ and stop the algorithm. Continue otherwise with the next step. Step 4: If the selected site has only black next-neighbours , paint it black, remove it from the list, and continue with step 1. Continue otherwise with the next step. Step 5: Add all white next-neighbour sites of the selected site to the list, paint them grey, and continue with step 2. Step 3 of the algorithm is the appropriate criterion for percolation of the sample. Step 1 is an exit-criterion in case that there are no more sites left to construct a connecting path. Step 4 serves to ‘burn ’ the dead-end sites of the pore space by marking them as solid. Once a dead-end site is filled it will thus no longer be considered as a candidate for a next-neighbour path of white sites that connects the first and the last column. Step 5 guarantees that all possible candidates for the connecting path are added to the list. Since only white sites are added and then colored ‘grey’, no site is added to the list more than once. 10 3 Tracer Dispersion
3 Tracer Dispersion In this section, the differential equations that capture the process of tracer transport and tracer dispersion are derived. The discussion is kept relatively brief, and the reader is referred to textbooks and reviews on this subject, Refs. 20-23.
3.1 The Convection-Diffusion Equation We start the discussion on microscopic scales, that is, length scales smaller than the characteristic length l of the porous medium. These length scales are, however, assumed to be large compared to molecular scales, so that the carrier fluid and the tracer can be discussed in the frame of continuum mechanics. The derivation of the transport equation relies on the principle of conservation of tracer particles.
Let y denote a sample of volume V, centered at position x, and let A(x, t) denote the number of tracer particles inside the sample at time t. The local molecular concentration, C(x,t), is then defined by [20] dN C = limr —N y-»o V dV' (3) This quantity is often directly measurable as it locally affects the color, opacity, or a similar property of the carrier. Consider now the case that the carrier fluid is in motion, such that there is a single valued vector function u(x, t) that measures the local fluid velocity at position x and time t. Associated to this flow is a convective flux of concentration, given by jc = u C. (4) For a tracer that undergoes molecular diffusion, the diffusive flux is given by Pick’s law 3D = -DmVC. (5) The Nabla operator V = (d/dx,d/dy,d/dz)T denotes partial derivation with re spect to position, and Dm is the molecular diffusion coefficient. Here and through out the thesis, the superscript T denotes transposition. Since the creation or annihilation of tracer particles by, for example, radioactive decay or chemical reactions, is excluded (compare Sec. 1), the rate of change of number of tracer particles in V is given by the sum of the concentration fluxes across its surface, denoted by dV: ^Cdy = -^jdA = -^(Vj)dK (6)
Here d/dt denotes partial derivation with respect to time, dA is the normal vector of an infinitesimal surface element, and the last equality follows from Gauss’ divergence theorem. Inserting j = jc +jz> into the continuity equation, Eq. (6), and taking the limit y —> 0, one obtains the partial differential equation ^C=-V.(uC) + D,»V:C. (7)
Equation (7) is called convection-diffusion equation since it describes the transport of a tracer under the influence of convection and molecular diffusion. It is instructive to cast Eq. (7) into dimensionless form by non-dimensionalizing all involved quantities according to
u -A u/tZ, x —>x/l, t-*tU/l, C-+C13, (8) 3.2 Macroscopic Equations 11
where U is a characteristic flow velocity. In these dimensionless variables Eq. (7) takes the form
The quantity
(10)
is the Peclet number and compares a typical time scale of diffusion (l2/Dm) to a typical time scale of convection (l/U). At large Pe, diffusion is thus much ‘slower ’ than convection, and the transport process is dominated by convection. If, on the other hand, Pe is very small, convective transport is negligible compared to diffusive transport.
3.2 Macroscopic Equations Equation (7) gives a valid description of tracer transport on length scales that are small compared to the characteristic length of the porous medium. To obtain a macroscopic description of the dispersion process, one may average Eq. (7) over macroscopic samples to obtain [21,23]
i-C = -V - (uC) + V • D • VC, (11) where now C denotes a macroscopic concentration, u a macroscopic velocity, and D is the symmetric dispersion tensor. It is assumed here that Pick’s law holds, such that the diffusive flux is given by jc — —DVC. The dispersion tensor contains in general contributions from molecular diffusion as well as from convection. It is therefore dependent on Dm, u and the geometric characteristics of the porous medium in the samples over which the average was performed. Consequently, D depends on position. If, however, the porous medium is homogenous on the length scales of the samples, fluctuations of the macroscopic quantities can be neglected. In this case, u equals the overall (average) flow velocity, U, the dispersion tensor becomes independent on position, and one obtains [21]
C, (12) dy2 dz2 where U is the magnitude of U, and the coordinate system is chosen such that U points into the ^-direction. Note that this approach requires the flow to be unidi rectional. Moreover, Eq. (12) is derived under the assumption that the dispersion tensor is diagonal in the chosen coordinate system. The longitudinal and transversal dispersion coefficients, D|| and D±, characterize the dispersion parallel and perpen dicular to the average flow. These coefficients are expected to depend on Dm, U, and geometric characteristics of the porous medium, such as its porosity
3.3 Dispersion Mechanisms In the remainder of this section, some ‘prominent ’ macroscopic dispersion mecha nisms and their contribution to the dispersion coefficients D\\ and D± are discussed.
Molecular Diffusion In 1827, the Scottish botanist Robert Brown observed in a microscope the ’’rapid oscillatory motion ” of tiny particles suspended in a stagnant fluid. This Brownian motion is a thermodynamic effect, and due to the bombardment of the particles 12 3 Tracer Dispersion by the thermally excitated molecules of the fluid. If interactions between the dif fusing particles can be neglected, and the particles are suspended in a uniform carrier, each particle can be assumed to trace out a random walk. Since individ ual steps of a random walk are random events, the central limit theorem applies, and the statistical properties of random walks are easily calculated. The mean displacement is (x(t) - x(t 0)) = 0, while the mean square displacement is given by {(x(t) — x(Z 0))2) = 2 Dm(t - to)- Here, x(f) denotes the position of the diffusing par ticle at time f, Dm is the molecular diffusion coefficient, and () denotes the average over many individual steps. The diffusion coefficient is a measure of the amplitude of molecular diffusion: Increasing Dm increases the mean square displacement per unit time. The geometric properties of random walks are well understood. They were among the first natural geometric objects to be recognized as being self-similar fractals [24,25]. The concept of self-similarity is illustrated in Fig. 8, where a finite segment of a two-dimensional random walk is displayed. The pictures in Fig. 8 a) and Fig. 8 b) are self-similar since their similarity becomes apparent only after an appropriate rescaling of time (t —> at) and space (x -4- a H x), where the Hurst exponent has the value H = 1/2.
Figure 8: On the self-similarity of random walks: a) 200 successive observations of a two-dimensional random walk. Each step between two observations has mean square length r2 oc DmAt, where At is the time interval between two successive observations. Initial position of the walker and its position after 200 steps are marked by grey circles, b) 40.000 steps of a two-dimensional random walk of mean square length r2 each. Initial position of the walker, its position after 200 steps and its position after 40000 steps are marked by grey circles. Only each 200th step is displayed. When averaged over many realizations, the pictures in a) and b) are indistinguishable . In other words, b) is statistically identical to the result of 200 steps of a random walk with mean square step length R2 = 200 r2.
If an ensemble of diffusing particles is considered, it is customary to consider the time evolution of the molecular particle concentration C(x, t), defined in Eq. (3). It can be shown that the associated flux obeys Pick’s law, Eq. (5). By combining Pick’s law with the continuity equation, Eq. (6), one obtains the diffusion equation
^C=V (DmVC) (13) 3.3 Dispersion Mechanisms 13 that is, Eq. (7) without the convective term. Note, that - in d-dimensional space - the Gaussian distribution
C'(x.t) = (27tct2) d/2 exp (—x 2/(2a 2)) (14) with variance a 2 = 2Dmt is a fundamental solution to Eq. (13). Thus, Gaussian distributions play a major role in the theory of diffusion.
The simple picture of diffusion as a sequence of random steps changes if the carrier resides in a porous medium of porosity
Deff~
Mechanical Dispersion Mechanical dispersion is due to fluctuations of the local velocity in the pores of the porous matrix, compare Fig. 9. If these fluctuations can be assumed to be uncorrelated on macroscopic scales, they lead - in complete analogy to molecular diffusion - to dispersion.
Figure 9: On the origin of mechanical dispersion by flow through a porous media: The solid matrix is shown in grey. A collection of tracer particles is shown at time to (o) and a later time t > to (•). Each particle follows a tortuous streamline of the flow, indicated by the dashed lines (- -). As a result, the distribution of the particles at time t is wider in both the horizontal and the vertical direction than it was at time to, as indicated by the arrows.
Since mechanical dispersion is an effect of convection only, and occurs also in the absence of diffusion, its contribution to the macroscopic dispersion coefficients must be independent of Dm. By dimensional analysis, the dispersion due to convection 14 3 Tracer Dispersion is therefore proportional to Ul. A detailed discussion led Koch and Brady [23] to the result (D,|)M = |[T, and = (16) expected to hold at large porosity. In periodic porous media the fluctuations of the local fluid velocity are strongly correlated, altering the mechanism and the contribution of mechanical dispersion [29,30]. This subject is discussed in detail in P2 for the case of periodic arrays of discs. Another transport coefficient of such periodic arrays of discs, the specific permeability (compare Sec. 4) is discussed in Sec. 5.
Taylor Dispersion Taylor dispersion is the result of a combination of convection and molecular dif fusion. It was first studied by Taylor [31] for flow in long tubes of circular cross section. He concluded that at large Pe, the coefficient of longitudinal dispersion is given by where U denotes the average fluid velocity in the tube, and a the radius of the tube. There is of course no transversal dispersion, since the walls of the tube limit the square-width <72(f) of any tracer distribution to be less or equal to a2. The phenomenon of Taylor dispersion is further discussed in Section 6 in the context of dispersion in two-dimensional channels.
Other Mechanisms A number of other, non-mechanical dispersion mechanisms exist, and - depending on the geometry of the porous medium - may be relevant. These non-mechanical mechanisms have in common that they contribute to the longitudinal dispersion, but have no effect on transversal dispersion [23]. Boundary layer dispersion stems from the small fluid velocity close to the fluid- solid boundary (compare Sec. 4). Tracer particles can enter these boundary layers by both diffusion and convection, but - since the fluid velocity is small - diffusion is the only mechanism, by which they can escape from these regions. This dispersion effect was first investigated by Saffmann [32] in networks of capillary tubes. A rigorous treatment shows that the contribution of boundary layer dispersion to the longitudinal dispersion coefficient is [23]
(18)
The "hydrodynamic dispersion at stagnation points ”, [9,10] is thus essentially an effect of boundary layer dispersion.
Holdup dispersion is a consequence of stagnant fluid regions, introduced by dead ends in the porous matrix and pore channels that are perpendicular to the flow direction. Tracer particles can both enter and leave such ‘pockets ’ by diffusion only. A tracer slug that passes such a pocket will by diffusion loose some concentration into the pocket, which will only slowly escape from it. Thus pockets tend to create small tracer reservoirs that are emptied when the tracer slug has passed them. It can be shown that the contribution of this dispersion mechanism is of order [D\\)h oc Dm(j>2Pe2, that is, quadratically in Pe, like Taylor-dispersion [23]. Note that the experimentally observed dispersion will in general be the sum of the contributions of each of the discussed dispersion mechanisms, such that 3.3 Dispersion Mechanisms 15
D|| = Deff + [D\\)m + (D||)t + (D\\)b + {D\i)h - Consequently, the dependence of D\\ on the Peclet number will be quite complex, and it will be difficult to separate the contribution of a single dispersion mechanism. In order to test the theoretical predictions, Eqs.(15) to (18), it is therefore important to study situations in which one or more dispersion mechanisms can a-priori be ruled out. In flow in long chan nels, for example, only molecular diffusion and Taylor-dispersion will contribute to the longitudinal dispersion coefficient. 16 4 Fluid Flow
4 Fluid Flow
We now turn the attention to equations that describe the motion of a fluid in a porous medium. We give a brief discussion only and refer to standard textbooks on fluid dynamics, Refs. 33,34
4.1 The Navier-Stokes Equation The principles of conservation of mass and momentum lead to the continuity equa tions
ftP + V • (pu) 0, (19)
P ^■u + (u ■ V)u ) - V ■ E f. (20)
Here, p is the density of the fluid, E denotes the stress tensor, and f the externally applied force per unit volume, or body force. In order to process these equations further, a constitutive relation for the stress tensor must be supplied. In Newtonian fluids, for example, the stress tensor is a linear function of the velocity gradient, and Newton ’s law is valid:
E = -pl + A(V -u)l + 2pR, (21) where p is the pressure, I the identity tensor, X and p are the two coefficients of viscosity, and R = ({Vu} + {Vu}r)/2 denotes the tensor of rate of strain. Here {Vu} denotes the dyadic product of V and u. In cartesian coordinates, using the notation x = (xi, x%, x3 )T and u = («i, u2, uz)T, one finds {Vu}ij = dui/dxj, such that Rjj = (dui/dxj + duj fdxi)/2. Introducing Newton ’s law into Eq. (20) yields
JU = - -Vp - (u • V)u + -V(A (V -u)) + -V-(p R) + -. (22) ut p p p p
Next, the assumption of constant density of the carrier is utilized. Inserting dp/dt = 0 and Vp = 0 into Eq. (19) leads to the incompressibility equation
V • u = 0. (23)
Inserting this result into Eq. (22), and using that the viscosity is constant, one finally obtains -Iju = — -Vp — (u • V)u + —V2u-f -, (24) ut p p p where the relation 2VR = V2u + V(V ■ u) was used. Equation (24) is called the Navier-Stokes equation. Together with Eq. (23) it provides a complete description of the dynamics of a Newtonian, incompressible fluid. The unknowns are the velocity, u, and the pressure, p. Note, that Eqs. (23) and (24) do not explicitly define the dynamics of the pressure. Instead, by taking the partial derivative of Eq. (23) with respect to time, one finds the Poisson equation
V2p = —V • ((u • V)u + f), (25) where the relation V - (V2u) = V2(V • u) = 0 was used. This corresponds to the observation, that the incompressibility of the fluid demands the dynamics of the pressure to be infinitely fast. 4.2 The Stokes Equation 17
Equation (24) is readily cast into dimensionless form by non-dimensionalizing all involved quantities with the characteristic length / and the characteristic velocity U according to
x -> x//, u-mi/17, t-ttU/l, p ->• p/{U 2p), f-*f l/(U2p). (26)
In dimensionless variables Eq. (24) takes the form
^■u = -Vp - (u • V)u + ^V2u + f, (27) where the quantity pUl Re = (28) is called the Reynolds number. Similar to the Peclet number, the Reynolds number is an important dimensionless quantity that allows one to distinguish between different flow regimes, as discussed below. When the left hand side of Eq. (27) vanishes, the flow is referred to as steady. In this case, the velocity at any given position is constant in time. Steady state flows are often encountered in flow in porous media, since the forces that drive the flow change very slowly in time.
4.2 The Stokes Equation Since u in Eq. (27) is a dimensionless quantity, the inertial term (u • V)u is of the order 0((u • V)u) = 1. Similar, the viscous term V2u/Re is of the order order 0(V2u/Re) = 1/Re. Consequently, the Reynolds number is a measure of the ratio of inertial to viscous forces. If the Reynolds number is small, one may therefore neglect the inertial term (u ■ V)u in Eq. (27) to obtain the Stokes equation [34]
|u=_Vp+ J_V'u + f. (29)
The Stokes equation is linear in u and therefore much easier to access theoretically than the Navier-Stokes equation. To mention an example, the steady flow of a Newtonian, incompressible fluid around a spherical obstacle can be solved analyt ically in the Stokes approximation, while only approximate solutions exist for the Navier-Stokes equation [35,36]. Additionally, a steady flow which is governed by the Stokes equation (steady Stokes flow), is completely time reversible. In other words, reversal of the average fluid velocity (U —> —U) in steady Stokes flow is simply obtained by reversing the local fluid velocity (u —» —u, p —» —p). This is not true for the steady Navier-Stokes equation. This important difference between the steady Stokes equation and the steady Navier-Stokes equation is discussed in detail in in the paper PI. Generally, the Stokes approximation is considered to be valid for Re < 1 — 5 [37]. The relevance of the Stokes equation for flow in porous media, becomes obvious when noting that typical Reynolds numbers of water flow under reservoir conditions are of the order Re ~ 3 x 10-4 [16]. To complete the discussion of the influence of the Reynolds number, it is men tioned here that turbulent flows are characterized by large values of Re, typically Re > 10. In a turbulent flow, eddies and vortices introduce an inherent time- dependence of the flow field. In other words, even is the flow is driven by a time- independent force, a turbulent system will not reach a steady state. 18 4 Fluid Flow
4.3 The Darcy Equation While the Navier-Stokes equation and the Stokes equation both are microscopic equations, the Darcy equation [38] gives a macroscopic description of flow through porous media. Similar to the macroscopic convection diffusion equation, it can be obtained by volume-averaging the steady Stokes equation. The resulting equation reads [16,21] VD = -k-{-Vp + f), (30) where k is the specific permeability tensor and U# =
5 Modeling Fluid Flow Partial differential equations can almost never be solved analytically when applied to a specific situation. Fortunately, modern computers can be used as ‘number crunchers ’, that is, as fast and accurate devices to perform arithmetic manipula tions. They provide a convenient tool for the approximate, numerical solution of differential equations. In the following, algorithms that allow the numerical solution of the two-dimensional Navier-Stokes equation are discussed. The generalization to the three-dimensional case is straightforward .
5.1 Time-Dependent Flow We first present an explicit finite-difference (FD) scheme for the solution of Eqs. (23) and (27). The method was first proposed by Fortin [40]. A complete discussion can be found in [41].
Time Discretization For our purpose, the Navier-Stokes equation should be read as an initial value problem: Given an ’initial state’ (u(Zo),p(Zo)) = (u°,p°), we want to construct a sequence of states (u”,p n ) = (u(tn),p(tn)) at predescribed times t„ that fulfills Eqs. (23) and (27). To this extent, the time-derivative dg/dt is replaced by the forward difference where At = tn+ i — tn is the increment between successive time steps. Such a ‘slicing ’ technique is typical for FD-schemes: Derivatives are approximated by finite differences of constant step length, compare Fig. 10.
r1 £
At At
tn tn+l t n+2
Figure 10: Finite difference approach to the solution of differential equations: In stead of calculating the complete curve, g(t), only its values g n at certain points of time tn is evaluated, and the differential dg/dt is replaced by the finite difference
The time-discretized versions of Eqs. (27) and (23) read now
(33) At 20 5 Modeling Fluid Flow
0 = V • un+1 (34)
Note that the difference between the Navier-Stokes equation and the Stokes equation amounts to a slight difference in the precise form of the quantity kn . Thus, the scheme can easily be implemented as a solver for both the Navier-Stokes and the Stokes equation. Equations (33) and (34) are solved by a fractional step method: First one calculates the (physically unimportant) quantity
u* = u” + At k”. (35)
Next, the new pressure field p ra+1 is calculated by solving the Poisson equation
V2p" +1 = v-V-u*. (36) At Taking the space-derivative of Eq. (37), one finds that Eq. (34) indeed is fulfilled with this choice of pn+1. Finally, un+1 is calculated from
un+i = u*_AtVp n+1 . (37)
Inserting Eq. (35) into Eq. (37) shows that un+1 satisfies Eq. (33). In practice, Eq. (36) is solved by the method of successive overrelaxation (SOR). Introducing the notation p(n’ml such that p(n<°l = pn, = pn+1 , the SOR- scheme reads p(n,m+ 1) _ p (n,m) _ V2p {n,m) _ J_V -u*, — (38) where £ corresponds to the time step of the relaxation procedure. This iteration is truncated after a finite number of iterations, m, when sufficient convergence towards the asymptotic solution is obtained. A useful convergence criterion is
jp(n,m+l) _ p(n,m)j max - < €, (39) where c is a small number. Note the completely explicit nature of the scheme: The state of the fluid at time f„+i is calculated from the state at time tn by a simple ‘updating ’ scheme.
Space Discretization In addition to the time-discretization, one needs to perform discretization of space. Here, we employ the centered finite-difference operators n _ g(x + Ax/2,y)-g(x-Ax/2, y) (40) As n S(x, V + Ay/2) - g(x, y - Ay/2) ^ - Ay (41) XX9 = T-^XX \j^xx9): (42) Dyy9 — 'Dyy{'Dyy9)i (43) T) g = (/Dxx 4" ,)g. (44) Note that this procedure imposes a lattice structure on space, where Ax and Ay are the lattice constants in the x- and y-direction, respectively. The values of p and u = (u, v)T are defined on a so-called staggered marker-and- cell (MAC) mesh [41,42], compare Fig. 11. This mesh is especially useful for our purpose, since all finite differences are automatically placed at the desired positions, such that no interpolations are required. 5.1 Time-Dependent Flow 21
3.) ■<>► • ■<>►
Figure 11: The two-dimensional staggered MAC mesh: a) Distribution of velocity and pressure nodes on the mesh, b) The mesh near a fluid-solid boundary. Grey sites represent solid. Some velocity and pressure nodes are indicated as in a). The values of uo inside the solid can be interpolated by the first order formula «o = —«! in order to mimic zero velocity at up (non-slip boundary condition). The Neumann condition for the pressure on the fluid-solid boundary, pr, is employed by the interpolation po = Pi-
Boundary Conditions We demand the fluid velocity to vanish on the fluid-solid boundary (Non-slip bound ary conditions). In practice, this is done by initializing and keeping the velocity at all nodes at the fluid-solid boundary F to zero. Additional interpolations are re quired for some velocity nodes inside the solid, compare Fig. 11. In the pressure relaxation, Eq. (38), a Neumann boundary condition must be employed, which is derived from projecting Eq. (37) on the outward normal unit vector, A, of the fluid-solid boundary F:
(un+1 — u*)r • A, (45) At
where (u"+1 + u*)r denote the values of u”+1 + u* on the boundary. Due to the non-slip boundary conditions, u”+1 and u* are both zero on the boundary, and the Neumann condition reduces to an interpolation of some pressure nodes inside the solid, compare Fig. 11. The resulting discretized version of the Navier- Stokes equation is readily cast into an explicit time marching scheme that allows the calculation of (un+1 ,p ”+1) from (un ,p n ). In this thesis only unidirectional flows are considered, and a body force is applied to drive the flow. Radial flows and other more complex flows can be modeled by initializing and keeping the value of the pressure at selected sites of the lattice to a predescribed value.
Stability and Convergence The question of the stability of an explicit FD-scheme is very important since such schemes are only conditionally stable, and convergence of the scheme towards the physical solution is only achieved with a suitable choice of the parameters Ax, Ay, At and £. The size of the spatial discretizations Az and Ay will usually be determined from the requirement of a sufficient spatial resolution of the geometry. Thus, the stability criterion for the above FD-scheme reduces to the question how the involved time steps At and $ must be chosen. In general, stability criteria for explicit FD-schemes are obtained from the requirement that the successive cor rections to the iterated quantity must decay during the iteration. This amounts to 22 5 Modeling Fluid Flow the condition that the spectral radius of the amplification matrix of the scheme be less than unity (strict Von Neumann condition) [41]. We do not go into the details of such a stability analysis here, but only state the results of its application to the above scheme: In the case of the SOR-scheme, Eq. (38), the strict Van Neumann condition leads to the requirement [41]
o;(Az)2(Ay)2 2((Az)2 + (Ay)2)' where 1 < w < 2. The optimal value of u (the one that leads to fastest convergence) depends slightly on the size of the complete lattice, but a value of w = 1.85 is generally a good choice. It is assumed here that a Gauss-Seidel iteration for the pressure relaxation is implemented, such that updated pressure values are used as soon as they are computed. In a similar fashion, it can be shown that - when neglecting the non-linear term (u • V)u (Stokes flow) - the time step At must satisfy
Re(Az)2 At < (47) 4 where it was assumed that Az = Ay. In the case of Navier-Stokes flow and relatively large values of Re, it is recommended to determine a good value of At from some test runs of the scheme.
Starting Flow in a Channel As a simple test of the scheme, we consider the two-dimensional problem of starting Stokes flow in an infinite channel, compare Fig. 12 a). The fluid is stagnant at time t0 = 0, and a body force f = (/, 0)T, pointing in the z-direction, acts for t > to- The pressure is constant throughout the channel. The analytical solution to this problem can be written as [33]
„ _ f us(y)-w(y,t) u~ V o where us — tto (1 — (y/°)2)> such that us = (us, 0)T denotes the steady state solution (Hagen-Poiseuille flow), and uq = fa 2/(2jj,). The deviation from the steady state solution, w, can be expressed as a Fourier-series
00 w ~ ^2 exp(-n 2t/t) sin( 7„ (y + a)) , (49) n—1 where r = 4a 2p/(n 2y), 7„ = im/(2a), and
^ + ")) <% = { 32^/(^u=) ebT^' (%>)
Note that w decays exponentially in time, and that the coefficients qn decay as n~3 , such that the series converges quickly. For sufficiently large times, the mid-channel velocity u(y = 0, t) obeys
u(y = 0,t) ~ u0 (1 - exp(-t/r)) . (51)
Thus, Mo is the maximum velocity in the channel, obtained at y = 0 in the limit of sufficiently large times. 5.2 Steady Flow 23
The geometry that was used in the simulation is sketched in Fig. 12 a). The spatial resolution was Ax = (2a)/25, while the increment between successive time steps was At = rj158. The excellent agreement between numerical and analyti cal solution, documented in Fig. 12 b) and Fig. 12 c), shows that the FD-scheme reproduces the analytical solution despite of the relatively coarse resolution of the geometry. The complete calculation took only a few seconds on a HP-workstation.
2a r > 1 r X
L y % -
tfz Figure 12: Starting flow in an infinite channel: a) Geometry of the problem as used in the simulation. Grey sites denote solid, b) Velocity in z-direction, u{y,t), as a function of position, y, shown at times U = (r/2) i, i = 1,2,..., 16. Bullets (•) are results from the simulation, and lines (—) denote the exact result, c) Mid channel velocity, u(y = 0,t), as a function of time, t. Bullets (•) are results from the simulation at times ti — (r/2) i, i = 1,2,..., 16. The line (—) denotes the exact result, Eq. 51. The excellent agreement between the simulation and the exact solution indicates that even with a relatively coarse resolution of the geometry, the FD-scheme yields high-quality results.
5.2 Steady Flow In order to solve the steady Navier-Stokes equation one can use the discussed FD- scheme and iterate the initially stagnant fluid into a steady state. However, faster convergence is obtained with the artificial compressibility method [41,43], derived from the following modification of the time-dependent Navier-Stokes equation ra+l u 11 -Vp" -(un ■ V)u” + ^-V2un + r, (52) At '------' k”
pn +1 -pn V • un+1 . (53) c2At If now the sequence (un ,p n ) converges towards a steady state (u°°,p°°), then (u°°,p 00) is the desired solution of the steady state problem. A suitable initial 24 5 Modeling Fluid Flow
condition is (u°,p°) = (0,0), and in practice, the iteration is truncated when suffi cient convergence is achieved. A useful convergence criterion is
max < €. (54) At ’ At c2At Here, the maximum is taken over all nodes, and e is a small number. Convergence of the scheme must again be ensured by a suitable choice of the free parameters c2 and At, discussed below. The name ‘artificial compressibility scheme ’ stems from the observation that Eqs. (52) and (53) describe the time-evolution of pressure and velocity for a com pressible fluid with p = c2p. This compressibility is only present during the non steady state, while the steady state is characterized by the incompressibility of the fluid. The interpretation of the artificial compressibility scheme as a SOR-method is obtained from the following consideration: If in the non-steady scheme only one cycle of the SOR-iteration, Eq. (38), is performed, one finds
(55)
On the other hand, using Eq. (52) one can eliminate ura+1 in Eq. (53), yielding
(56)
Thus, the artificial compressibility scheme is identical to the time-dependent scheme if £ = c2 At and only one SOR-cycle is performed. This observation emphasizes that the ‘time step ’ At in the artificial compressibility scheme corresponds to a relaxation parameter rather than a physical time step. The scheme has been used in a variety of studies on steady fluid flow in porous media, Refs. 44-47.
Stability and Convergence The free parameters At and c2 in the artificial compressibility scheme must again be chosen such as to ensure the stability of the scheme. Due to the close relation between the artificial compressibility scheme and the time-dependent scheme, one finds for the time step At the criterion Eq. (47) to hold. When the inertial term (u-V)uis neglected (Stokes flow), one can show that c2 must satisfy the criterion [41]
(57)
In practice, Eq. (57) was found to yield good convergence properties up to Reynolds numbers of at least order unity.
Specific Permeability of Periodic Arrays of Discs To demonstrate the accuracy of the steady-state scheme, we present results on the calculation of the specific permeability of periodic arrays of discs, compare Fig. 13 a). The problem of tracer dispersion in this type of porous media is discussed in P2. In the simulations the two-dimensional steady Stokes equation were solved. The fluid was driven by a body force. It was found that the average fluid velocity in the porous matrix, U, is always parallel to the applied body force, f. Thus, the specific permeability tensor reduces to a scalar for this type of porous media. This result 5.3 Alternative Methods 25
may be somewhat surprising, since a periodic array of discs is not isotropic. Indeed, the observation that the permeability is a scalar is here connected to the rotational symmetry of the characteristic function: Since the lattice constants in the x- and ^-direction are chosen to be equal, the resulting invariance of the geometry under 90-degree rotations leads to a degeneracy of the eigenvalues of the permeability tensor. The scalar specific permeability is calculated directly from Darcy’s equation as k = n V8 (f - 2r):/: 4A 5/2 -1/4 -1 (58) 9 tt /!/2 7T \\J 4(1 - Figure 13: Flow through periodic arrays of discs: a) Geometry of porosity * ~ 0.5 as used in the simulations. The unit cell of the porous medium is enclosed by the square. The finite spatial resolution used in the simulation is visible through the corners in the discretized discs, b) The dimensionless specific permeability k/A, where A = nr2 denotes the area of a disc, is shown as a function of porosity, 5.3 Alternative Methods Explicit FD-schemes provide, of course, only one way to solve the Navier-Stokes equation. Alternatively, one could employ an implicit FD-scheme, explicit or im plicit finite-element (FE) schemes, or a lattice Boltzmann (LB) scheme. We have, however, implemented only the above mentioned schemes for the following reasons: • Implicit schemes require the solution of (sparse) linear systems of equations and therefore a relatively large amount of memory, when compared to explicit schemes. Especially for large systems, the required memory easily exceeds available resources. • When compared to FE-methods, FD-schemes require in general less book keeping, due to the simple ’slicing ’ of space. Consequently, FD-schemes are 26 5 Modeling Fluid Flow much easier to implement than FE-methods. The meshing process requires only a simple digitalization of the true geometry, while FE-methods require a more complex meshing. Finally, the evaluation of the fluid velocity at a given position is relatively fast in the case that lattices are used for the spatial dis cretization. This aspect is especially important here, since the generated flow fields are used as the input to a model for tracer transport, that - as will be seen - requires many such evaluations. • • LB-methods were implemented and successfully applied by Flekkpy [4,5] and thus at hand in the group. Their connection to the Navier-Stokes equation is, however, less obvious. Moreover, the incompressibility of the fluid is only approximately satisfied in LB-methods. Finally, we wanted to create an in dependent tool that both experiments and LB-methods could be compared to. 6 Modeling Tracer Transport 27 6 Modeling Tracer Transport This section addresses the question of how to model tracer transport, described by the microscopic convection-diffusion equation, Eq. (9). We only discuss our own implementations, and refer to Ref. 49 for a deeper discussion of the subject. 6.1 A Finite-Difference Scheme An explicit FD-scheme for the approximate solution of Eq. (9) is readily derived in the fashion described in Sec. 5 for the solution of the Navier-Stokes equation. Here, only the scheme to solve the two-dimensional convection-diffusion equation in case of steady-state flow is discussed. The generalization to the three-dimensional case, or to the case of time-dependent flow, is straightforward . When the finite-difference operator Eq. (32) is employed for time-discretization, Eq. (9) reads Cn+1 _ cn - At V • (uCn - j^VC"). (59) In practice, one evaluates first the flux jn according to the flow field u and the concentration Cn before updating the concentration field according to the computed flux. For space-discretization , the finite-difference operators Eqs. (40) and (41) were employed, and the staggered MAC mesh, Fig. 11, was used, where now concentration nodes replace the pressure nodes, and the flux nodes are placed at the same positions as the velocity nodes. The boundary condition that the tracer shall not enter the solid amounts to a non-slip boundary condition of the fluxes on the fluid-solid boundary, implemented by initializing and keeping the flux at all nodes at the fluid-solid boundary to zero. Additional interpolations are required for some flux nodes inside the solid, compare Fig. 11. The time step At must again be chosen such as to guarantee the stability of the scheme. When neglecting the convective term uC” it can be shown that the stability criterion amounts to the restriction [41] wPe(Az)2(Ay)2 (60) - 2((Az)2 + (Ay)2)' where the relaxation parameter u must again satisfy 0 < w < 2. A major drawback of the resulting explicit FD-scheme is, however, that it exhibits numerical dispersion: The dispersion observed in the results of such simulations has not only physical contributions from diffusion and convection, but contains also a contribution from an artifact of the numerical method [50]. At large Pe, this artifact usually spoils the numerical results completely. To a certain extent numerical dispersion can be reduced by sophisticated models for the evaluation of the flux j” [51,52], Such an improved algorithm was used in P5 for qualitative studies on tracer dispersion in fractures with rough surfaces. But even the improved algorithm shows numerical dispersion, which becomes the dominant dispersion mechanism at large Pe. Before turning the attention to a model that allows the investigation of tracer dispersion at large Pe, it seems appropriate to shortly discuss the significance of numerical dispersion in the FD-schemes discussed in Sec. 5: 28 6 Modeling Tracer Transport • The artificial compressibility scheme and the SOR-scheme, Eq. (38), are not affected by numerical dispersion, since velocity and pressure are iterated into steady state, whereas numerical dispersion affects only time-dependent quan tities. ♦ The time marching scheme Eq. (33) is in principal subject to numerical disper sion. However, since relatively small Reynolds numbers are considered here, the contribution of numerical dispersion can be neglected. 6.2 A Monte Carlo Scheme The idea of this scheme is to compute the individual trajectories of a large number of non-interacting tracer particles, and to reconstruct concentration profiles from the spatial distribution of the tracer ensemble. Similar methods were employed in Refs. 15,53-55 Convective Steps The purely convective motion of a single tracer particle is governed by the equation of motion (61) where x denotes the position of the particle, and u the local velocity of the carrier. A large number of numerical integration schemes for Eq. (61) exist, and the interested reader is referred to standard textbooks on numerical integration routines, Refs. 56- 58. Generally speaking, numerical integration relies on a more or less sophisticated approximation of the time derivative d/dt by a finite-difference operator. In the following, the notation x(t + At) = x(t) + xc(u, At) (62) is used, where xc denotes the convective step, as obtained from a suitable numerical integration routine. The notation xc(u, At) shall emphasize that the length and direction of xc depend on the time step At, and the velocity u of the carrier fluid. Since the flow fields as computed from the FD-schemes are defined on the nodes of a lattice only, whereas the tracer particles shall move continuously, a bilinear interpolation scheme [59] was employed to obtain off-lattice velocities (compare Fig. 14). Random Steps In order to introduce molecular diffusion, a random step is added to each convective step of a tracer particle. Equation (62) is thus replaced by x(t + At) = x(t) +xc +xD(Dm,At), (63) where X£> denotes a random step of mean square length that depends on the molec ular diffusion coefficient Dm and the time step At. In our implementation, each component of the d-dimensional vector X£> is taken from a Gaussian distribution with zero mean and variance 2DmAt. The mean square length of xp is thus (xp) = 2dDmAt, where () denotes the average over many individual steps. To generate Gaussian distributed random numbers, the Box-Muller transfor mation [56] was implemented. It transforms a pair of random numbers that are uniformly distributed in the interval [—1,1] into a pair of Gaussian distributed ran dom numbers with zero mean and unit variance. Uniformly distributed random 6.2 A Monte Carlo Scheme 29 Figure 14: Bilinear interpolation in two dimensions: Circles (o) denote velocity nodes of the MAC mesh. The black dot (•) denotes a tracer particle. The interpo lated velocity of the tracer particle is u = A\Ui + A2U2 + A3U3 + A4M4, where A; denotes the shaded areas. numbers are obtained with the pseudo-random number generator R.250 [60], initial ized with 250 random numbers, created with the pseudo-random number generator CONG [61]. This procedure guarantees a relatively fast production of high-quality random numbers [61]. Boundary Conditions Since the tracer is not allowed to enter the solid matrix, any step that would lead a tracer particle from its present position in the porous matrix into the solid is rejected. Instead, a new diffusive step is taken. After several unsuccessful trails of this kind, the time step At is decreased, and the convective step as well as the diffusive step are recalculated. This rejection-adjustment method is preferred to a reflection technique [15], since individual steps have both a convective and a diffusive contribution, making the implementation of a reflection technique rather cumbersome. To implement periodic boundary conditions, a counter keeps for each particle the number of crossings of the boundaries of the unit cell. Accuracy and Time Step Control Numerical integration schemes to generate convective steps are usually uncondition ally stable, and yield results for all values of the time step At. The accuracy of this result, however, is strongly dependent on the choice of At and the order of the nu merical integration scheme. Here, we consider the first order scheme Eq. (32) (RK1), a second-order Runge-Kutta scheme (RK2), a fourth-order Runge-Kutta scheme (RK4), and a fourth-order Runge-Kutta scheme with automatic adjustment of the time step At (RKQC) [56]. Since flow fields are relatively slowly varying functions, higher order schemes were not considered. The choice of the time step At relies on the following considerations: It seems ap propriate to require each diffusive step to be at most of length r = 2 min(Ax, Ay, Az) in order to ensure that no single diffusive steps lets a tracer particle jump over a 30 6 Modeling Tracer Transport part of the solid matrix. This leads to the requirement At < Atmax = r2/(2Dm). The same constraint applies to convective steps, yielding At = min(At max ,r/|u(x, t)|). (64) In the case of the RKQC-routine, this value of At serves as a first trial time step, and will automatically be adjusted during the integration process. In practice, one finds that this implementation of the RKQC-routine yields good results for all Pe, when the accuracy parameter of the routine, e, is taken in the range 10~2 to 10-6. Note that decreasing e increases the accuracy of the scheme and results in both smaller time steps, and an increased number of evaluations of the velocity field per time step. When e is chosen in the above mentioned range, the scheme requires typically 10 — 20 velocity evaluations per time step, compare Table 1. At low Pe, the schemes RK1, RK2, and RK4, when implemented in the described way, yield good results, too. Since the RK1-, the RK2-, and the RK4-scheme require only one, two, and four evaluations of the particle velocity in each time step, respectively, these schemes are preferred to the RKQC-routine at low Pe. At large Pe these low order integration routines are not accurate enough, such that computed average tracer velocities do no longer coincide with average fluid velocities. More important, the low accuracy of the integration scheme leads to the artifact that convective steps may take the tracer into the solid matrix from which is can not escape by a diffusive step. In effect, the routine goes into an infinite loop. This defect can be avoided by decreasing the allowed time step according to the Peclet number. Such an adjustment still leaves the low-order schemes to be rather unefficient when compared to the RKQC-routine . Average time Average number of Routine step in units velocity evaluations of r/[f per time step RKQC (e = 10-*) 0.19 21 RKQC (e = 10-4) 0.83 15 RKQC (e = 10-2) 1.35 11 RK4 0.76 4 RK2 0.76 2 RK1 0.76 1 Table 1: Comparison of the integration routines RK1, RK2, RK4, and RKQC in the case of steady Stokes flow through a periodic array of discs of porosity 4> ~ 0.5 at Pe ~ 200. N = 1000 particles were integrated from t = 0 to t ~ 2500 r/U, where r = 2 Ax is the maximum size of a single step. Taylor Dispersion in Infinite Channels As a test for the Monte Carlo scheme, we modeled Taylor dispersion in a two- dimensional channel. The steady flow field is given by the Hagen-Poiseuille law 3f7 (1 - (y/a) 2) u = (65) 2 0 where the x-direction is chosen along the tube, a denotes the half width of the channel, and U the average fluid velocity in the channel, compare Fig. 16. 6.2 A Monte Carlo Scheme 31 An analytic treatment of this problem due to Aris [62] yields the result (66) for the longitudinal dispersion coefficient, where l = 2a. Note that for large Pe = Ul/Dm the dispersion coefficient D\\ is inverse proportional to the molec ular diffusion coefficient Dm. 10“ 10" 10' 102 10> 10* 10s Pe Figure 15: a) Dimensionless variance, tr2/2/2, of the tracer distribution plotted as a function of dimensionless time, tDm/l2, from a simulation with N = 104 particles at Pe = 200 (solid line) . Note the initial quadratic increase of the variance with time, corresponding to the stretching of a rubber band {or2 oc t2) rather than a diffusive spread {a 2 oc t). A linear fit of the curve for tDm/l2 > 1/3 yields a slope D\\/Dm = 187±3, in good agreement with the theoretical value 1 + Pe2/210 ~ 191.5 (dashed line), b) Dimensionless longitudinal dispersion coefficient, D^jDm, plotted as a function of Peclet number, Pe = Ul/Dm. Results from simulations are indicated by bullets (•). The prediction Eq. (66) is shown as a solid line (—). Dashed lines (- -) are the results D\\ — Dm at small Pe, and Dy = DmPe2/210 at large Pe. Since the velocity along the streamlines is constant, the first order integration routine RK1 can be employed in our simulations, yielding exact results for any time step At. Thus, we choose here the time step according to At = min(At max , 4(a - y)2/(2Dm)), (67) such that a single diffusive step will be of the order of the distance of the particle to the channel wall. The maximum step size Afmax is set such as to guarantee that each particle performs a large number of steps during the simulation. In practice, one finds that Atmax must be chosen according to the Peclet number in order to obtain consistent results: The smaller Pe, the smaller values of Atmax must be chosen. Fig. 15 displays results of simulations with N = 104 particles at different Pe. Fig. 15 b) demonstrates the good agreement between the computed values of D± and Eq. (66). The problem of Taylor dispersion in a channel was studied by Gut- fraind et. al. [63] to verify the accuracy of a lattice-gas model for fluid flow and tracer dispersion in self-affine fractures. Their computed values of the longitudinal dispersion coefficient were about 10% off the prediction Eq. (66). Here, we obtained errors of the order 1%. Results of a single simulation with N = 5 x 10s particles are shown in Fig. 16. At to = 0, the tracer particles were uniformly distributed in the channel with 32 6 Modeling Tracer Transport Figure 16: Taylor dispersion in an infinite channel at Pe = 200: a) Snapshot of distribution of N = 104 particles. The line corresponds to the mean tracer position, {x(y)}. b) Reconstructed concentration profile from a simulation with N = 5 x 106 tracer particles. The concentration field was evaluated on a 30x30 lattice by counting the number of particles inside each lattice site. Each lattice site is grey-shaded according to the concentration, c) Diffusive flux, —Dm V2C, reconstructed from the concentration profile in b). d) Flux in a frame of reference that moves with the mean fluid velocity, (u — U)C — Z?m V2C, reconstructed from the concentration profile in b). Note the vertical magnification of the four pictures: The true pictures would have the aspect ratio of the horizontal lines that indicate the channel walls. —0.2 < x/a < 0.2. The snapshot is taken at time tDm/l2 = 1.25. Note the curvature of the mean tracer position, {x(y}}, across the channel, visible in Fig. 16 a). Here (} denotes an average over x. Our simulations indicate that the mean tracer position is well fitted by the form {x(y)} oc cos(m//a). (68 ) At present we are not aware of any theory that allows the prediction of this result: A variation of tracer concentration across the channel was completely neglected by Taylor [31]. Aris [62] seems aware of such effects, but does not include them in his quantitative analysis. We therefore find our results very interesting, and prepare currently a manuscript for publication [64]. 7 Concluding Remarks 33 7 Concluding Remarks This thesis presents selected studies of tracer dispersion and fluid flow in porous media. Finite-difference schemes were implemented to solve the (Navier-)Stokes equation for an incompressible, Newtonian fluid. To model tracer dispersion, a finite-difference scheme for the solution of the microscopic convection-diffusion equa tion, and a Monte Carlo method were implemented. The accuracy of the different numerical methods was demonstrated in the introductory part of this thesis by a number of applications of the schemes to situations where an analytical solution exists. The programs are easily applied to a specific situation. Finite-difference schemes are well known and have been employed in a variety of contexts. The Monte Carlo scheme, however, is relatively little used, partly due to the observation that the reconstruction of concentration profiles requires a large number of independent tracer particles (of the order N = 106 to N = 107), making the algorithm rather unefficient for this purpose. We found, that the Monte Carlo scheme works well for studies of tracer dispersion, where only the first few moments of the tracer distribution are required. The mean position of the tracer distribution and its variance are accurately evaluated with particle numbers of the order N = 104 to .V = 105, yielding tolerable requirements as far as memory and time consumption are concerned. We therefore consider the Monte Carlo scheme to be a powerful tools for the numerical investigation of tracer dispersion in porous media. The following papers discuss some aspects of tracer dispersion and fluid flow in porous media of current scientific interest. Three of these papers (PI, P2 and P5) are based on simulations of fluid flow and tracer dispersion. In PI, the effect of finite Reynolds numbers on the outcome of echo-experiments is discussed. The other two papers are concerned with the influence of the geometric characteristics of a porous medium to its macroscopic transport properties. 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On the Taylor dispersion in channels and tubes, (in preparation). 38 List of Symbols The following symbols and notations were used the preceeding sections: d dimension of space t, d/dt time, operator of time differentiation x = (%i, xz)T = (z, y, z)T position vector d/dt, d/dx, d/dy, d/dz operators of partial differentiation V = (d/dx, d/dy, d/dz)T Nabla operator V2 = V • V Laplace operator M solid matrix of a porous medium V porous matrix of a porous medium r boundary between solid- and porous matrix X characteristic function V sample of a porous medium y,d/dy volume of V, operator of volume differentiation surface of V A, A normal vector (and area) of a surface element porosity P percolation probability l characteristic length of porous medium a, r, R other lengths S specific surface N number of tracer particles in a sample C molecular tracer concentration a 2 variance of a distribution Dm molecular diffusion coefficient Pe = Ul/D„ Peclet number J vector of tracer flux H Hurst exponent D macroscopic dispersion tensor Deff, D\\, Dx macroscopic dispersion coefficients F = Dm/{Defi) formation factor U = (ui,u2,u3)t fluid velocity vector U, U average fluid velocity (vector and magnitude) f,/ external forces (vector and magnitude) P local pressure P density P, A coefficients of viscosity Re = pUl/y Reynolds number 1 identity tensor H stress tensor R rate of strain tensor k, k specific permeability (tensor and scalar) I vector of current $ electrostatic potential A conductivity tensor T>t, Vx, Vy, Vxx, Vyy, p 2 finite difference operators At, Ax, Ay finite differences (time step and lattice constants) 4u, £, c2 parameter of relaxation schemes e accuracy parameter (a small number) a, i, j, n, m, k, u*, w, g, q, 7, r other quantities (explained in the text) 39 List of Research Papers This thesis is based on the following publications: PI : E. G. Flekkpy, T. Rage, U. Oxaal, and J. Feder. Hydrodynamic irreversibility in creeping flow. Phys. Rev. Lett. 77, 4170-4174 (1996). P2 : T. Rage and J. Feder. Simulations of tracer dispersion and fluid flow in periodic arrays of discs. In preparation. P3 : R. Hilfer, T. Rage, and B. Virgin. Local Percolation probabilities for a natural sandstone. Accepted for publication in Physica A (1996). P4 : R. Hilfer, B. Virgin, and T. Rage. Local Porosity theory for the transition from microscales to macroscales in Porous Media. ER- COFTACBulletin 26, 6-9 (1996). P5 : P. Meakin, T. Rage, G. Wagner, J. Feder, and T. Jpssang. Simula tions of one- and two-phase flow in fractures. In: B. Jamtveit and B. W. D. Yardley, editors, pp. 251-261. Fluid flow and transport in rocks: Mechanisms and effects. Chapman and Hall, London, 1996. Hydrodynamic irreversibility in creeping flow E. G. Flekk0y, T.Rage, U.Oxaal and J. Feder Department of Physics, University of Oslo P.O. Box 1048 Blindern, 0316 Oslo 3, Norway Abstract Using experiments as well as lattice Boltzmann and Finite Difference simula tions we study creeping flow in a Hele-Shaw cell at Reynolds numbers below unity. We describe an ‘echo ’-technique that is very sensitive to effects of hy drodynamic irreversibility. By combining experimental and numerical studies we show that the irreversibility observed in the experiments is due to iner tial forces. As a byproduct we validate the numerical techniques used—in particular the recently introduced lattice Boltzmann model. 1 I. INTRODUCTION The flow of an incompressible, Newtonian fluid is governed by the non-linear Navi Stokes equations, Eqs. (1) and (2), which can be solved exactly only in simple geometri In flow, which is slow in the sense that the ratio of inertial to viscous forces (the Reynol number Re) is sufficiently small, the non-linear inertial term can be neglected to obta the linear Stokes equation [1]. In contrast to the non-linear equations, the steady sta Stokes equation is invariant under flow reversal and yields exact results in simple geometri so it has been a matter of fundamental discussion below which value of Re the Stok approximation to Eq. (2) is justified. Numerous authors [2-4] have calculated correction factors to the Stokes drag at sma non-zero Reynolds numbers. The corrections are linear in Re and have prefactors of ord unity. Correspondingly, streamlines around a cylinder look symmetric in the fore-and- direction until Re is of order unity [5]. Generally, for single phase flow the Stokes a proximation is considered to be valid if Re < 1-5 [6]. However, when there are finite si particles suspended in the fluid, inertial effects can be observed, in principle, at arbitra small Reynolds numbers. This is the case in the so called tubular pinch effect, which cans particles suspended in a tube flow to migrate to a stationary off-center position in the tub The effect, which was first recorded by Poiseuille [7] in blood streams and later studi systematically by Segre and Silberberg [8] cannot be explained without the non-linear ter in the Navier-Stokes equation. But only the transient state, i.e.: the speed with which t’ particles migrates to their steady state position, depends on Re—the final radial position the particles is Re-independent, even in echo experiments [9]. In the present study we observe non-linear effects at low Re in single phase flow, contrast to the tubular pinch of suspended particles the ‘echo ’ signal in our single pha experiment with passive tracer molecules depends continuously on Re and thus serves quantify the role of the non-linear term, also when its relative magnitude is very small, immediate conclusion from our measurements is that the proper condition on Re may highly problem dependent, even in simple flow geometries. We present ‘echo ’-experiments on creeping flow in a Hele-Shaw channel with an obstac (see Fig. 1), where a passive tracer is convected forwards and backwards with a carri fluid such that (ignoring molecular diffusion) each tracer molecule would exactly retu to its initial position if the flow were perfectly reversible. Since the flow as described the Stokes equation is time-reversible [10], the echo-technique effectively integrates out t reversible part of the velocity field, making the experiment sensitive to any deviation fro reversibility. In the experiments we detect visible departure from reversibility at Re ~ 0.02 throu an ‘M’-shaped deformation of an otherwise perfectly returned tracer line (see Fig. 2), whe the size of the deformation increases linearly with increasing Re, see Fig. 3. Comparing t experiments to two independent numerical models—where the deformation is visible alrea at Re ~ 0.0006—we demonstrate that the observed irreversibility is due to the effect of t 2 non-linear inertial force, he., of hydrodynamic nature. II. EXPERIMENTS The experimental setup was previously used to study enhanced dispersion in creeping flow [11,12]. In each experiment, a thin, straight line (radius a ~ 0.3 mm) of tracer fluid was first placed across the filled cell at a distance of 3 cylinder-radii, r = d/2 = 0.5 cm, in front of the cylinder center (see Fig. 1). Next, a fixed amount of carrier fluid (glycerol-water mixture) was pumped into the cell at constant volume flux Q. The pump was then abruptly reversed, withdrawing the same amount of fluid. At flow reversal, the maximum extension of the tracer line was approximately xmax = 10 r. and the smallest distance to the cylinder was xmin = 0.08 ± 0.01 r. Buoyancy forces on the tracer were minimized by carefully matching the density (p = 1.2318 ± 0.0001 g/cm3) of the two fluids. The viscosity of the carrier fluid was p ~ 220 cP, while the viscosity of the tracer was 2-3 % lower. The effect of this viscosity contrast appeared to be negligible. Average flow velocities U between U = 0.031 cm/s and U = 0.28 cm/s produced Re in the range 0.016-0.17. The molecular diffusivity of the tracer was Dm = (2.8 ± 0.2) x 10-8 cm2/s, giving Peclet numbers Pe = Ud/Dm of the order 106-107. Using a photometric CCD camera with a spatial resolution of (20 pm) 2/pixel, we identi fied the positions x(y) of maximum concentration for all y across the channel (see Fig. 2). III. SIMULATIONS Mathematically the flow is described by the Navier-Stokes equations: V*« = 0, (1) d*tjb p-^ + pu'Vu = —Vp + pV2u + / . (2) Here u(r,t) denotes the local fluid velocity at position r and time t, p the pressure and / the external forces [1]. For flow with average velocity U in a geometry with a characteristic length, d, the Reynolds number is Re = Udjv, with v = pfp. In the steady state (dufdt = 0) the reversibility of the flow can be expressed by the invariance of Stokes equation under the transformation u —> —u. f —> —and Vp —> — Vp. This simple symmetry [10] is broken by the presence of the non-linear term, and in general the flow that results from reversal of the external forces, must be recomputed. However, in the present case where there is a mirror symmetry of the flow geometry, the reversed flow field is given by ur(r) = —u(—r). The simulations can be performed in a 2-D plane corresponding to the central fluid layer if the modification of the forcing /—>•/ — Spu/h2, (where h is the plate separation) is made in Eq. (2) to account for the viscous drag of the top and bottom plates of the cell [12]. This correction factor assumes that the ^-dependence of u is parabolic. For the Reynolds number 3 of these quasi 3-D simulations the velocity U must be replaced by 217/3, where U is t average value of u in the central layer. Steady state solutions to this quasi 3-D Navier-Stokes equation were obtained using lattice Boltzmann (LB) method as well as a finite difference (FD) scheme. To verify t validity of the quasi 3-D method, a full 3-D FD simulation at Re = 0.01 was computed. T relative difference between the central layer of this 3-D field and a quasi 3-D calculation the same average velocity U was about 1%. In the FD-scheme a space-discretized version of the steady state Navier-Stokes equati was solved approximately through a relaxation process discussed in Ref. [13] (‘artifici compressibility method on a MAC mesh ’). The LB-method [14] models the fluid as a large number of particles [15] that move fro site to site on a triangular lattice, where they interact in mass and momentum conservi collisions. We use the BGK model [16], which differs from other Boltzmann models by simplified collision operator. The basic variables of the LB models are the probabiliti Ni(r,t) of finding a particle on a site at position r at time t moving with unit velocity one of the six lattice directions C;, with i = 1,..., 6. The precise scheme according to whi these probabilities are updated is discussed in Refs. [11,17]. The conserved site densities mass and momentum are defined as p = M and pu = respectively where p is t total averaged density. This definition of u differs from the usual one by the replaceme p p. This minimizes the slight effects of compressibility [15]. The present applicati is very sensitive to effects of compressibility and p has spatial variations of the order 1 Provided the Mach number is small the velocity u satisfies the two-dimensional version Eqs. (l)-(2) with the modification that pu-Vu —$■ Gpu*Vu. This extra G-factor is a fr parameter of the model [11,18]. For steady flows Eqs. (l)-(2) can be recovered by absorb! the factor IfG in z/, P and f. This causes the pertinent Reynolds number to take t (generalized) form Re = GUdjv [12]. For computational efficiency Re was tuned by varyi G alone. Although the FD and LB methods are based on completely different perceptions of t fluid, calculated flow fields were almost identical. The two methods will be compared detail elsewhere [19]. In both methods, tracer particles were convected by integrating t equation of motion r = u with a fourth order Runge Kutta scheme [20], and off-latti velocities were obtained using linear interpolation schemes. IV. RESULTS AND DISCUSSION Qualitatively Fig. 2 shows agreement between LB-, FD-simulations and experiment Quantitatively, the main result of this letter is shown in Fig. 3 where the ‘peak-to-pe distance ’, A = A+ + A- of the ‘M’-shaped deformation is plotted as a function of Re f experiments and simulations (see Fig. 2d) as well as for the analytic solution of the line Oseen equation [2] for the flow around a cylinder in the absence of boundary walls. T data give that A fr = a Re. For experiments, we find aexp = 1.37 ± 0.03, while LB- an 4 FD-simuIations lead to the values a# ~ 1.47 ±0.1, and ajd — 1.33 ± 0.03, respectively. In experiments the £M’ is blurred for Re < 0.02 (see Fig. 2a) since enhanced diffusive spreading [11,12] becomes important, and at Re < 0.004 this effect dominated. The slope of the Oseen result is larger than the slope of the LB/FD results by almost a factor 9, thus demonstrating the Oseen solution captures the rough qualitative- but not the quantitative aspects of the present inertial effects. Solving the Oseen equations for the experimental geometry we find aos — 12 ± 0.03. The shape of the ‘M’ may be understood by inspecting the ‘difference field’ u + ur, see Fig. 4. The net velocity towards the cylinder along the central flow-line produces the central kink of the ‘M’. It expresses the fact that the flow is slightly slower behind the cylinder. The arches of the ‘M’ develop through the net transport transverse to the flow since the return stream-line is not identical to the forward stream-line as illustrated in Fig. 4. This causes the tracer to travel faster during return, making it overreach the starting position. Note, the passive tracers follow the stream lines, and do not move across them. In spite of the good quantitative agreement between experiments and simulations shown in Fig. 3, the exact shape of the returned tracer lines is different in the two cases (see Fig. 2). We find the ratios A+/A~ ~ 1 and ~ 4.7 for experiments and simulations respectively. In order to understand this discrepancy a number of simulations were carried out. By varying the channel geometry we found that end effects are insignificant. On the other hand, by including transient effects, modeling the start and reversal of the flow, described by the dufdt term in the quasi 3-D version of Eq. (2), we found A+/A- ~ 3.9 instead of 4.7. Preliminary investigations indicate that while the ratio A+/A“ depends on the time dependent characteristics of the full three dimensional flow field, the sum A depends mainly on the non-linear term [19]. The present problem strongly challenges the numerical methods, since the main signal, the reversible component of the flow field, is integrated out. The agreement between LB- and FD-simulations, and the agreement with experiments, therefore provide solid validation of the two numerical models. This is particularly non-trivial for the LB-model where the connection with the Navier-Stokes equations is less than obvious [14,15]. However, we have been unable, with our computer resources and algorithms, to obtain the full 3-D time- dependent solutions to Eqs. (1) and (2), required to get the exact experimental echo profile. Fig. 5 shows an experiment where one tracer line was reversed and returned four times. The linear increase of the amplitude A with the number of flow reversals can be used as a means to magnify the signal at very small Re. Earlier studies of oscillatory flows [21] have focused on small amplitude oscillations compared to our experiments. The inherent irreversibility of low Reynolds number flows that we demonstrate is also of interest when discussing experiments on hydrodynamic dispersion where the ‘echo ’-technique is used [11,12,22,23]. Rigord et al. [23] performed experiments with a porous medium made of packed glass beads and observed that the spread of the returned tracer profile increased with the Reynolds number, which was in the range 0.2-1.1 in their case. Our results suggest that some of their observations may be rationalized by considering the effects of the present 5 hydrodynamic, in addition to diffusive, mechanism of irreversibility. V. CONCLUSION We have discussed a particular case of creeping flow in a Hele-Shaw cell, where inert! effects are experimentally observable for Re > 0.02. Numerical simulations of the stea state Navier-Stokes equation reproduced this effect. The sensitive echo-technique could also be used to detect mechanisms other than inert that destroy reversibility. In particular it might be used to investigate non-Newtonian b havior of complex fluids like paints, pastes and polymer solutions. Finally, our results rela to hydrodynamic problems that involve particles in oscillatory flows (see Fig. 5) and to t" study of dispersion in porous media. ACKNOWLEDGMENTS This work has been supported by NFR (Norwegian Research Council) through Gran 100339/431, 101874/410, 100666/410 and 100198/410 and by Grant no. 6311 from VIST (a research cooperation between the Norwegian Academy of Science and Letters and D norske stats oljeselskap a.s. (Statoil)), and by Norsk Hydro a.s. T. R. acknowledges a gra of computing time from the Norwegian Super-computing Committee. 6 REFERENCES [1] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon Press, New York, 1987). [2] H. Lamb, Phil. Mag. 21, 112 (1911). [3] I. Proudman and J. R. A. Pearson, J. Fluid. Mech. 2, 237 (1957). [4] W. Chester and D. Breach, J. Fluid Mech. 37, 751 (1969). [5] M. V. Dyke, An Album of Fluid Motion (The Parabolic Press, Stanford California, 1982). [6] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Prentice Hall Inc, Englewood Cliffs, N.J., 1965). [7] J. L. M. Poiseuille, Ann. Sci. Nat. 5, 111 (1836). [8] G. Segre and A. Silberberg, J. Fluid Mech. 14, 136 (1962). [9] L. I. Berge, J. Feder, and T. Jpssang, J. Coll. Interface Sci. 138, 480 (1990) [10] F. P. Bretherton, J. Fluid Mech. 14, 284 (1962). [11] U. Oxaal, E. G. Flekkpy, and J. Feder, Phys. Rev. Lett. 72, 3514 (1994). [12] E. G. Flekk0y, U. Oxaal, J. Feder, and T. Jpssang, Phys. Rev. E 52, 4952 (1995). [13] R. Peyret and T. D. Taylor, Computational Methods for Fluid Flow (Springer-Verlag, New York, 1983). [14] G. McNamara and G. Zanetti, Phys. Rev. Lett. 61, 2332 (1988). [15] U. Frisch et al.7 Complex Systems 1, 648 (1987). [16] Y. Qian, D. D’Humieres, and P. Lallemand, Europhys. Lett. 17, 479 (1992). [17] E. G. Flekkpy, Phys. Rev. E 47, 4247 (1993). [18] E. G. Flekkpy, Ph.D. thesis, University of Oslo, 1993. [19] T. Rage, E. G. Flekkpy, U. Oxaal, and J. Feder, (to be published) . [20] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986). [21] E. J. Chang and M. R. Maxey, J. Fluid Mech. 277, 347 (1994). [22] J. Hulin and T. J. Fiona, Phys. Fluids A 1, 1341 (1989). [23] P. Rigord, A. Calvo, and J. Hulin, Phys. Fluids A 2, 681 (1990). 7 FIGURES FIG. 1. Sketch of the experiment: During forward convection, a line of passive tracer w convected towards (full line) the cylinder of radius r = 0.5 cm in a channel of width 5 cm su that it folded around the cylinder. Upon flow reversal, the tracer line was ‘echoed ’ back towar its initial position (dashed line). 8 FIG. 2. Enlargements of the characteristic deformation of the returned tracer (a) Re = 0.016, (b) Re = 0.069 and (c) Re = 0.17. White pixels show the calculated coordinates that are used to measure distances, (d) FD-simulation at Re ~ 0.062. Data are compared by measuring A+, A~ and their sum A. 9 simulations; number ( — FIG. ) - A/r analytic Re 3. for: Peak-to-peak ( — Oseen ) (•) - finite - solution experiments; difference distance [2] for A simulations (A) infinite of the - lattice returned channel of 10 the Boltzmann Oseen Re width. tracer equations lines simulations; as a for function experimental (o) - of finite the geomet differe Reyno f t t ' t t \ \ \ ' / / / ' - t ; FIG. 4. The difference between the forward and reversed velocity field at Re = 1.58. In case of complete reversibility the field should be zero everywhere. The grey lines show the tracer line in the initial straight position, fully stretched at the point of reversal and deformed at the return. The black line represents the path of diffusionless tracer molecules that start on the initial, are converted along with the forward flow and convected back, not retracing its path, to over-reach its starting point. 11 K- FIG. 5. A tracer line after one, two, three and four convection cycles at Re = 0.05. 12 Simulations of Tracer Dispersion and Fluid Flow in Periodic Arrays of Discs T. Rage and J. Feder Department of Physics, University of Oslo P.0. Box 10f8 Blindem, 0316 Oslo 3, Norway Abstract We study numerically the dispersion of a passive tracer in periodic arrays of discs. Flow fields are calculated by a finite-difference method, while a Monte Carlo method is used to model the process of dispersion by molecular diffusion and convection. For small Peclet number, we find that effective diffusion coefficients agree very well with a result from conductivity theory. At intermediate values of the Peclet number non-diagonal dispersion is observed. At large Peclet number, the transversal dispersion coefficient saturates at a value of the order of the molecular diffusion coefficient. We argue that this result is due to the periodicity of the porous media, in agreement with the theory by Koch et. al.. We also present results on the permeability of the porous media and show that they agree with a prediction of lubrication theory. 1 I. INTRODUCTION The dispersion of a passive tracer in porous media is a phenomenon of fundamen importance in aquifer tests, waste disposal, and oil recovery. It has been studied extensiv in the past by experiments, theory and numerical simulations, see Refs. 1-3 and referen therein. In simulations, two-dimensional (2-D) porous media have often been considerd, si they require less computational efforts than 3-D simulations. The problem of dispersion periodic media was studied in, for example, Refs. 4-8. We discuss here several aspects tracer dispersion and fluid flow in 2-D periodic arrays of discs and show that this cl of porous media exhibits pecularities that will not be present in natural porous med In detail, we study the formation factor, F, the scalar specific permeability, k, and t macroscopic dispersion coefficients D\\ and D±. Results are compared to correspond! theoretical predicitions, specifically the results of Koch et. al. [8-10]. Experimental measurements of dispersion coefficients usually rely on the evaluation tracer breakthrough curves [11]. By such methods, longitudinal dispersion coefficients ha been measured for a large number of porous media. As far as transversal dispersion coe cients are concerned, relatively little is known, since transversal dispersion is more diffic to observe experimentally. The practical relevance of transversal dispersion is pointed o in Ref. 11. In the present study, we obtain results on the longitudinal and the transver dispersion simultaneously by modeling the dispersion of a tracer spot in a porous mediu We extract the variance, cr2, of the tracer distribution both parallel and perpendicular the average tracer flow and - since cr2 is found to increase linear with time, t - compute t corresponding dispersion coefficient, D, from the Einstein relation cr2(t) oc Dt. II. THEORY The transport of a passive tracer can be described by the convection-diffusion equati [1.2] + V • (uC'j = DmV2C. ( where C is the local tracer concentration, u the local velocity of the carrier fluid, Dm the molecular diffusion coefficient of the tracer. Equation 1 relies on the assumpti that the tracer is submit to molecular diffusion and convection by the carrier fluid on Additionally, the diffusion coefficient is assumed to be constant, specifically independent C. In its above form, Eq. 1 is valid on length scales smaller than the characteristic lengt l, of the porous matrix that the carrier resides in, such that molecular diffusion is the on dispersive mechanism. Note that the existence of such a characteristic length requires t porous medium to be homogeneous [12]. Due to the presence of the porous matrix, streamlines of the velocity field u are usual tortuos. The resulting fluctuations of the local fluid velocity may then introduce a numb 2 of different macroscopic dispersion mechanisms that act ‘on top ’ of molecular diffusion, such that the macroscopically observed dispersion is larger than what one expects from molecular diffusion alone. On macroscopic scales Eq. 1 must thus be replaced by [1] ^C + V(uC) = VDVC, (2) where now C is a macroscopic concentration, u a macroscopic tracer velocity, and D the dispersion tensor. In general, D will depend on Dm, u, and the fluctuations of the local fluid velocity. If the porous medium is homogeneous one can assume the existence of a large scale limit on which the local fluctuations of D are neglegible, such that D becomes independent on position. If additionally the porous medium is isotropic, the off-diagonal elements of D vanish. Furthermore, the macroscopic velocity, u, will be identical to the mean tracer velocity U. In this case, the dispersion parallel and orthogonal to the mean flow of the tracer is characterized by two scalar dispersion coefficients, D\\ and Dj_, such that [1] Jtd Cu + vYdx Cm = ( D"d?d2 + DlWd22 \) Cm' (3) where U = ||U|| denotes the magnitude of the macroscopic tracer velocity, taken here to point in the ^-direction. In the present study, the tracer is excluded from the solid matrix, such that the macroscopic tracer velocity equals the average fluid velocity in the porous matrix. The macroscopic length scale on which this approach is valid is typically much larger than the characteristic length l but still smaller than a typical sample size [9]. Additionally, the system must have had time enough to ’equilibrate ’ the inital concentration distribution. The Peclet number, Pe = U ljDm, compares the typical time scale of molecular diffusion (l2/Dm) to the time scale of convection (l/U). For small Pe, convection is ‘slower ’ than molecular diffusion, such that its contribution to the dispersion coefficients can be neglected. In this regime, one finds that D\\ ~ D± ~ 1 [{(f>F) = Deff, where 4> is the porosity of the porous medium, and Des the effective diffusion constant, as measured in a stagnant fluid [2]. Here and through the rest of the paper, dispersion constants are non-dimensionalized with Dm. Due to the analogy between diffusion and electrical conduction, the formation factor is experimentally accessible as the ratio of conductivity of the pure fluid to the conductivity of the porous media saturated with the fluid [2]. The formation factor is usually strongly correlated with the porosity of the porous medium. In homogenous, isotropic, unconsolidated porous media, F is believed to be a function of the porosity only [13]. At large Pe, dispersion by convection becomes important, and the contribution of the convection process to the dispersion coefficients must be taken into account. Several dis persion mechanisms must be distinguished [9,10]: Taylor dispersion, for example, results from velocity gradients perpendicular to the local fluid velocity and contributes a term of the order 0(Pe2) to the longitudinal dispersion coefficient, but has no contribution to the transversal dispersion coefficient. This is also true for holdup dispersion, due to stagnant fluid zones that the tracer can both enter and leave by molecular diffusion only. Boundary- layer dispersion is an effect of the non-slip boundary conditions on the fluid-solid boundary, 3 resulting in regions of very small fluid velocity close to the solid. A tracer can enter th boundary layers by both diffusion and convection, but can leave them by diffusion only, can be shown that this effect contributes a term of the order 0(Peln(Pe)) to D\\. Final mechanical dispersion is due to the tortuosity of the streamlines and the associated fluct tions of the fluid velocity. Usually, this dispersion mechanism has a contribution of the or (9(Pe) to both, longitudinal and transversal dispersion. Moreover, the relation D oc Ul s gests, that mechanical dispersion is independent of molecular diffusion. In periodic poro media, however, the effect of mechanical dispersion is altered, since fluctuations of the lo fluid velocity are strongly correlated. We return to this point, specifically the theory Koch et. al. [8], when discussing our results. III. SIMULATIONS Two-dimensional, ordered model porous media were constructed by flagging each site a square lattice of dimension l x l to represent either porous matrix or solid such that t solid matrix constitutes a digitized disc of diameter d, compare Fig. 1. Periodic bounda conditions were imposed on the borders of the lattice in order to mimic a periodic array discs. Since the solid matrix occupies A of the l2 pixels of the lattice, the porosity of t so-defined porous media is 0 = —Vp + pV2u + f, ( 0 = V • u, ( where p is the fluid viscosity, p is local pressure, and f the external force. We solve this s of equations by a finite difference method, the so-called ‘artificial compressibility schei [16,17]. Pressure and velocity nodes are placed on a MAC mesh [17,18]. Non-slip bounda conditions are implemented on the fluid/solid boundary. The fluid is driven by an extern force, such that the pressure drop over any length L > l was zero. To simulate the dispersion process, we implemented a Monte Carlo method: At to = a fixed number, N, of tracer particles were placed inside the porous matrix. The motion each particle under the combined action of molecular diffusion and convection was model by iteratively updating the position, xn = x(tn), of each tracer particle at time tn accord! to the scheme X^+1 — Xn A X(7 [Atn, u) + xD(Atn,Dm), ( 4 where Atn = in+1 — tn is the time step, Xc denotes a convective step, and xp a diffusive step. For a given flow field, the Peclet number is varied by changing the diffusion constant. At large Pe, convective steps were calculated from a fourth order Runge-Kutta scheme with automatic time step control [19]. At low Pe, a second order Runge-Kutta scheme was employed [19]. Diffusive steps were constructed by taking each component of the random vector from a Gaussian distribution of zero mean and variance <72(At) = 2DmAt. Any step, (xc +X£>), that would have led a particle from its present position into the solid matrix was rejected, and the random vector replaced by a new one. After several unsuccessful! trials of this kind, the time step Atn was decreased, and both x IV. RESULTS AND DISCUSSION Formation factor First, we discuss the case Pe = 0, that is, a stagnant fluid. Effective diffusivities, Des, were calculated from the Monte Carlo scheme with N = 104 particles, initially placed all at the same point in the porous matrix. The second moment of the particle distribution, cr2{U) = <7jj(t2) + cr\{ti), was evaluated at times ti = i l2/(2Dm), i = 1,2,..., 100. Effective diffusion coefficients were obtained from fitting this data to a straight line of slope 4Deff- We considered discs of diameter d = 10 {A = 316), 20(1264), 30(2828), and 70(15380), respectively, and varied porosity by changing l. Results of these simulations are shown in Fig. 2, where the inverse formation factor is plotted as a function of porosity. Fig. 2 shows that there are no systematic deviations of the data for different d, indicating that the discretization error is low. As a theoretical prediction of the formation factor, we mention here the result [21] 1 0.305827c 4 = 1 - 2c / 1 + c - - [0.013362c8 ] (7) 1 - [1.402958c 8 ] 5 where c = 1 — Specific permeability We discuss now the case Pe > 0, that is, steady motion of the fluid. From our simulatio we found that the mean velocity of the carrier in the porous matrix, U, was always parall to the force, f, that drives the fluid. Thus, the specific permeability tensor of a period array of discs reduces to a scalar, k. This result is connected to a rotational symmet of the geometry: Since the distance of the discs is the same in both lattice directions, t" eigenvalues of the specific permeability tensor are degenerate. The value of k was calculat directly from Darcy’s law [1] where / = flf|| is the magnitude of the force that drives the fluid, and 4>U denotes t magnitude of the Darcy velocity. Again, we considered discs of diameter d — 10,20,3 and 70, respectively, and show in Fig. 3 our results together with a result of lubricati theory [5,23] V8(f-d):/2 k = ( 9?r ' Although Eq. 9 was derived under the assumption of small porosity, its agreement wi experimental result is excellent over a large range of porosity, as noted previously by Mart' et. al. [5]. The simple relation k oc F~2 was observed to hold for many three-dimensional poro media [24,25]. In this spirit, we correlate Fig. 4 our results on the specific permeabilit k, and the formation factor, F. We conclude that at large values of the formation fact (low porosity), the relation k oc F~4 seems to hold. At large porosity, a relation of t form k oc F~9 is more likely. We put little significance to these ‘powerlaws ’, but find crossover-behaviour (at Macroscopic dispersion coefficients Next, we report simulations of tracer dispersion at Pe > 0 in which the Monte Car scheme was employed to transport a large number of tracer particles by convection and di fusion through the porous matrix. Deviations of the mean tracer velocities from the averaj flow velocity were always within 5% of the mean flow velocity, confirming the accuracy 6 the Monte Carlo method. Four snapshots of a Monte Carlos simulation in a disordered geometry of porosity 4> ~ 0.5 are shown in Fig. 5. Generally speaking, we found that the square width of the tracer distributions parallel to the flow increases linearly with time only after a sufficient ‘relaxation time’, r, that depends on porosity and Peclet number. For t > r, reconstructed concentration profiles attained a nearly Gaussian shape, compare Fig. 6. The square width of the tracer distributions both parallel and orthogonal to the averge flow direction increased then sufficiently linearly with time to allow the extraction of the corresponding dispersion coefficients D\\ and Dj_, respectively, compare Fig. 7. For Pe > 104, the relaxation time was usually so large, that simulations became very time consuming. We therefore consider here the case Pe < 104 only, and display in Fig. 8 so obtained dispersion coefficients in geometries of porosity D± = £>eff + (Dj_)m (10) Longitudinal dispersion, on the other hand, is influenced not only by effective diffusion and mechanical dispersion, but also by non-mechanical dispersion mechanisms, such as boundary-layer dispersion, Taylor dispersion, and holdup dispersion. Here, we include the contribution from mechanical dispersion only, such that D\\ = £>eff + (11) Finally, Koch et.al. give an explicit formula for the mechanical dispersion tensor in periodic arrays of discs [8] [(i - kk/p) • v/uy (D)m = a 53 (12) ^ IG^F (4Ft 4 /Pe^ + (U/(7. k)2)' Here, I is the identity tensor, k denotes a dimensionless reciprocal lattice vector of magnitude k = ||k||, and A = (4n/(ln(l/r) — 1.3015))2 is an amplitude factor, related to the drag force on the discs. Equation 12 is expected to hold at large values of 7 (relatively little amount of) disorder in our disorderd geometries does not affect the d persion properties of the medium, which are dominated by its periodicity. Next, we no that - at sufficiently large Pe - the transversal dispersion coefficients are independent Pe, as predicted by Koch et. al.. This phenomenon can nicely be illustrated by not! the ‘band structure’ of the streamlines in periodic porous media for ‘rational ’ flow dir tions, as sketched in Fig. 9. Here, a (unit) vector in 2-D space, x = (x, y)T is call rational if x/y is a rational number. A ‘prominent ’ irrational direction is the golden me x/y = (1 + ^/(5))/2 ~ 1.618 (a ~ 53.8°). For such irrational flow directions, any streamli in a 2-D periodic porous medium is space-filling, that is, it fills the entire unit cell of t porous medium. Again, a detailed discussion led Koch et. al. to the observation that t contribution of mechanical dispersion to the transversal dispersion becomes independent Pe in this case. Finally, we point out that the dependence of the longitudinal dispersion coefficient rather poor predicted by Eq. 11. In detail, it is evident from Fig. 8 b) and Fig. 8 c) that at large Pe - Eq. 11 predicts too small longitudinal dispersion coefficients. One might arg that this is due to the fact, that the contribution of boundary-layer dispersion and hold dispersion was neglected. However, Fig. 8 a) shows that including only the effect mechanic dispersion already overestimates the calculated longitudinal dispersion coefficient. We would also like to point out that the dispersion tensor is not necessariliy diagon For reference, we include the tracer distribution from a simulation in an ordered array porosity 8 V. CONCLUSION We performed numerical studies on fluid flow and tracer dispersion in periodic arrays of discs. The dependence of the formation factor and the specific permeability on porosity was discussed and good agreement between theory and simulations obtained. By correlating the results on F and k, we found a non-trivial relation between these two quantities that is inconsistent with the heuristic relation k oc 1 /F2. observed to hold for many three- dimensional porous media. The crossover-like character of the observed relation between k and F excludes any simple algebraic relation between k and F to be valid over the entire range of porosity. Tracer dispersion in periodic arrays of discs at Pe > 0 was modelled and transversal and longitudinal dispersion coefficients calculated as a function of Peclet number in ordered and disordered periodic arrays of discs of various porosity. Results were discussed in the framework of the theory by Koch et. al. [8-10]. To our knowledge, such a test of the theory has never been carried out before. We find that some aspects of the Pe-dependence of the macroscopic dispersion coefficients from simulations are qualitatively predicted by the theory. However, quantitative agreement was not obtained. It seems therefore appropriate to study the problem of dispersion in periodic arrays of discs in the case that the tracer is allowed to enter the solid matrix. We are currently investigating this problem numerically, and results will be published elsewhere [26]. Our work demonstrates once more that the ‘artificial compressibility scheme ’, a rela tively simple finite-difference method for solving the steady (Navier-) Stokes equation is able to meet the needs of scientific investigations as far as accuracy and computer resource con sumption (time and memory) are concerned. Finally, it is worth noting that the Monte Carlo method for solving the miscroscopic convection-diffusion equation is generally believed to be doomed to fail [27]. From our experience, concentration profiles that are reconstruced from tracer distributions show - when the number of independent particles is of the order N — 104 - indeed fluctuations that may be intolerable in many situations. However, the evaluation of the first two moments of the tracer distribution yielded reliable results already for N = 104. A major drawback of the scheme is the relatively large amount of computing time required. ACKNOWLEDGMENTS The authors acknowledge valuable discussions with E. Flekkpy on the subject of this pa per. T.R. acknowledges financial support by the Royal Norwegian Research Council (NFR) through Grant No. 101874/410 and a grant of computing time from the Norwegian Super computing Committee. 9 REFERENCES [1] A. E. Scheidegger, The Physics of Flow through Porous Media, 2nd ed. (University Toronto Press, Toronto, 1974). [2] J. J. Fried and M. A. Combarnous, Adv. Hydrosci. 7, 169 (1971). [3] K. W. Morton, Numerical Solution of Convection-Diffusion Problems (Chapman a Hall, London, 1996). [4] N. S. Martys, Phys. Rev. E 50, 335 (1994). [5] N. S. Martys and E. J. Garboczi, Phys. Rev. B 46, 6080 (1992). [6] A. E. Saez, J. C. Perfetti, and I. Rusinek, Transport in Porous Media 6, 143 (1990). [7] H. Brenner, Phil. Trans. R. Soc. London A 297, 81 (1980). [8] D. L. Koch, R. G. Cox, J. F. Brady, and H. Brenner, J. Fluid Mech. 200, 173 (1989) [9] J. F. Brady and D. L. Koch, in Disorder and Mixing, edited by E. Gy non, J. P. Nad and Y. Pomeau (Kluwer Academic, Dordrecht, The Netherlands, 1988), Chap. 6. [10] D. L. Koch and J. Brady, J. Fluid Mech. 154, 399 (1985). [11] F. I. J. Stalkup, Miscible Displacement, Monograph, Volume 8 (Soc. of Petroleum E gineers of AIME, Henry L. Doherty Series, New York, 1983). [12] R. Hilfer, Advances in Chemical Physics 92, 299 (1996). [13] M. Quintard, Transport in Porous Media 11, 187 (1993). [14] N. S. Martys, S. Torquato, and D. P. Bentz, Phys. Rev. E 50, 403 (1994). [15] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon Press, New Yor 1987). [16] A. J. Chorin, J. Comput. Phys. 2, 12 (1967). [17] R. Peyret and T. D. Taylor, Computational Methods for Fluid Flow (Springer-Verla New York, 1983). [18] F. H. Harlow and J. E. Welsh, Phys. Fluids 8, 2182 (1965). [19] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recip (Cambridge University Press, Cambridge, 1986). [20] T. Rage, Ph.D. thesis, University of Oslo, 1996. [21] W. T. Perrins, D. R. McKenzie, and R. C. McPhedran, Proc. R. Soc. London A 36 207 (1973). 10 [22] Lord Rayleigh, Phil. Mag. 34, 481 (1892). [23] S. Kostek, L. M. Schwartz, and D. L. Johnson, Phys. Rev. B 45, 186 (1992). [24] L. M. Schwartz, N. S. Martys, D. P. Bentz, and E. J. Garboczi, Phys. Rev. E 48, 4584 (1993). [25] P. Wong, J. Koplik, and J. P. Tomaniac, Phys. Rev. B 30, 6606 (1984). [26] T. Rage and J. Feder, (in preparation) . [27] A. L. Fogelson and R. H. Dillon, J. Comput. Phys. 109, 155 (1993). 11 FIGURES FIG. 1. Unit cell of an ordered, 2-D, periodic array of digitized discs of porosity 12 FIG. 2. Inverse formation factor, 1/F, as a function of porosity, 13 FIG. 3. Dimensionless specific permeability, k/A, as a function of porosity, 14 log 10 (1 / F) FIG. 4. Specific permeability, k, as a function of formation factor, F. Bullets (•) are results from simulations. The solid line has slope 4 and the dashed line has slope 9. 15 FIG. 5. Four snapshots of a dispersion experiment with N = 104 particles at Pe ~ 2 x 103 in disordered geometry of porosity 16 • parallel ° perpendicular ) <—i—i—' —i—>—i—i——1—'—yE>—i—i—'—i—i— i1 ######### i—i—i—) -20 -15 -10 -5 0 5 10 15 20 Position FIG. 6. Reconstructed concentration profiles parallel and perpendicular to the flow direction for the tracer distribution shown in Fig. 5 d). Position is measured from the center of the distribution and non-dimensionalized by l. Concentration is normalized such that the area under both curves is one. Lines show Gaussian distributions with the same square width as the particle distribution. 17 • parallel ° perpendicular -3.1 -2.9 -2.7 -2.5 -2.3 -2.1 -1.7 -1.5 -1.3 FIG. 7. Dimensionless variance, cr2//2, of tracer distribution parallel and perpendicular to t flow direction as a function of time for the simulation shown in Fig. 5. Data at the snapshots a) d) of Fig. 5 is indicated. From linear fits of the data we found dispersion coefficients Dy ~ 159 and Dx. — 4.4 ± 0.1, respectively. 18 FIG. 8. Dimensionless dispersion coefficients as a function of Peclet number for periodic arrays of discs of porosity 19 FIG. 9. Sketch of the ‘band structure’ of streamlines in a period porous medium with rat nal flow direction: The unit cell of a periodic array of discs of porosity 4> = 0.5 is shown, a a = tan -1(l/3) ~ 18.4°. The line that separates the black ‘band ’ from the white ‘band ’ is streamline that connects the stagnation points of the fluid. However, any streamline of the fl field yields a similar tilation. A tracer particle that is started inside the white ‘band ’ can leave t band only by diffusion. The underlying spatial correlations in the flow field lead Koch et. al. the observation that the effect of mechanical dispersion in periodic porous media and in natu (non-periodic) porous media is different. 20 FIG. 10. Tracer distribution from a Monte Carlo simulation in an ordered array of porosity 4> ~ 0.35 with a ~ 53.8 at Pe ~ 12. Dots denote tracer particles. The thick line denotes the flow direction. The thin line is orthogonal to the flow direction. The two lines meet in the center of the tracer distribution. The ellipse denotes the variance of the tracer distribution parallel and perpendicular to the flow direction. Clearly, the tracer ‘spot ’ is not aligned with the flow, that is, the dispersion tensor contains off-diagonal elements. Moreover, it can be seen that transversal dispersion is larger than longitudinal dispersion. 21 0.3 0.4 0.5 0.6 0.7 0.8 0.9 4> FIG. 11. Transversal dispersion coefficients at Pe = 104, Dj_(Pe = 104), as a function porosity, 22 Local Percolation Probabilities for a Natural Sandstone R. Hilfer1,2, T. Rage3 and B. Virgin 3 lICA-l, Universitat Stuttgart, Pfaffenwaldring 27, 70569 Stuttgart 2Institut fur Physik, Universitat Mainz, 55099 Mainz, Germany 3 Institute of Physics, University of Oslo, 0316 Oslo, Norway Abstract Local percolation probabilities are used to characterize the connectivity in porous and heterogeneous media. Together with local porosity distributions they allow to predict transport properties [1]. While local porosity distribu tions are readily obtained, measurements of the local percolation probabili ties are more difficult and have not been attempted previously. First mea surements of three dimensional local porosity distributions and percolation probabilities from a pore space reconstruction for a natural sandstone show that theoretical expectations and experimental results are consistent. According to the mixing law of local porosity theory [1-10] the effective frequency pendent dielectric function £e(u) of a heterogeneous mixture may be calculated by solvi the integral equation L1£c{oj; 1 - £C{u-,4>) = £\ (cv) (1 - (1 — £2(uj)/£i(u>)) 1 — £B(cu;0) = e2(u) (1 (1 — £i(u)/£2(u>))~1 ~ (1 — 0)/3) In (1) the local porosity distribution fi(4>) and the local percolation probability A(^) geometrical input functions which will be defined shortly. The functions £i(u;), and £2{ are the frequency dependent dielectric functions of the pure materials. If £i(u>),i = 1,2 a A(0) are known then (1) gives a prediction for the effective dielectric function £e{uo Measurements of the local porosity distribution /z(<^) are readily obtained from t sections [6,8,11]. Measurements of the local percolation probabilities A(^), on the ot hand, are more difficult, and have not been reported. The main purpose in this paper is present preliminary measurements of local percolation probabilities for a natural sandsto specimen. Define P to be the pore space of a two component porous medium. The compleme of P in R3 is called the matrix space M. More generally P and M may represent t two components in a heterogeneous medium. The local porosity <£(K) measured within measuring region K is defined by y(pnK) 1 cubic lattice, and let K(r, L) = {r' E Hf* : |r< - < 1/2, i = 1,2,3} (5) denote a cube of sidelength L centered at r. Given the notation introduced above the local porosity distribution may be defined as [10] i M = ^im^ —51^(0-^(K(Lgi,l))) (6) if the limit exists. Here 5(a;) is the Dirac 5-function. In practice the sample is finite, and hence the limiting process terminates after a finite number M of measurement cells. The resulting histogram is used as an approximation for y,. Local porosity distributions quantify the fluctuations in volume fractions. To characterize the transport properties,however, it is necessary to quantify the degree of connectedness of the porous medium. Two points inside the pore space P are called connected if there exists a path entirely within the pore space that connects the two points. A cubic measurement cell Kj in a cubic partitioning is called percolating in the ^-direction if there exists a path within P fl Kj connecting those two opposite faces of EC, that are perpendicular to the ^-direction. Percolation in the y- or ^-direction is defined analogously. The local percolation probability Xx( To measure n{4>, l) and X((f>, L) in practice it is necessary to reconstruct the three dimen sional pore space P for digital processing. This was done by the method of serial sectioning for a Savonnier oolithic sandstone specimen [12,13]. This type of sandstone exhibits oomoldic porosity with ellipsoidal pores and grains of sizes between 200/um and 300/mi [12]. After 2 filling the stone with a coloured epoxy resin the specimen was cut and polished. The ished surface was photographed before removing another layer of material parallel to surface. The second surface was then polished and photographed before iterating the p cedure. A total of 99 sections was prepared. The distance between consecutive planes about 10p.m. Each photograph represented an area of roughly 2cm x 2cm. The photogra were scanned and thresholded into a binary image with 1904 x 1904 pixels. This correspo to a resolution within each image of roughly 10pm. Figure 1 shows a typical binary im from within one of the 100 planes measuring roughly 1cm along each side. The family of three dimensional local porosity distributions p( obtained straightforwardly. The results are displayed in Figure 2 for various L. Ref. [9] investigated the question to what extent the three dimensional local porosity distribut' can be measured from two dimensional sections. While this is not generally possible, it found that a two dimensional measurement combined with an appropriate rescaling of length L gives good results. A preliminary measurement of the local percolation probabilities has also been carried for the oolithic sandstone specimen. The results are shown in Figure 3 for cubic measurem cells of size L = 32. The curve 32) is shown in black, the curve Xy(4>; 32) is shown medium gray, and the curve Az( While the data show much scatter these preliminary estimates for the local percolati probabilities indicate that the sandstone has anisotropic connectivity. The same trend observed for smaller as well as for larger values of L, although the strength of the anisotro appears to change slightly. The data are in agreement with the theoretical expectation that the local percolati probabilities should increase monotonically from 0 to 1. The shape of the curves confor 3 to the one expected for systems with a percolation threshold. An example is the grain consolidation model with random packings [1]. For small values of L, however, the shape of the local percolation probabilities resembles that for the central pore model [1], All of these conclusions are very preliminary. More work is necessary to corroborate them, and to validate the measurement. ACKNOWLEDGEMENT: The authors are grateful to Chr. Ostertag-Henning, E. Haslund, Dr. U. Mann, Prof.Dr. B. Npst, Prof.Dr. D.H. Welte and Prof.Dr. R. Koch for discussions, and to Norges Forskningsrad and the Forschungszentrum Jtilich for partial financial support. 4 FIG. 1. Digitized thin section image of Savonnier oolithic sandstone [13]. The pore space P coloured black, the matrix space is rendered white. 5 -- L=24 — L=32 - - L=64 3 4.0 — L=80 FIG. 2. Local porosity distributions p(<£; L) for a cubic measurement cells of sidelengths L. The values of L for the different curves are indicated in the legend. 6 porosity FIG. 3. Local percolation probability functions Ax(4>, L) (black), Xy( Xz( 7 REFERENCES [1] R. Hilfer, “Geometric and dielectric characterization of porous media,” Phys. Rev. B, vol. 44, p. 60, 1991. [2] R. Hilfer, “Local porosity theory for flow in porous media,” Phys. Rev. B, vol. 45, p. 7115, 1992. [3] R. Hilfer, “Geometry, dielectric response and scaling in porous media,” Physica Scripta, vol. T44, p. 51, 1992. [4] F. Roger, J. Feder, R. Hilfer, and T. Jpssang, “Microstructural sensitivity of local porosity distributions, ” Physica A, vol. 187, p. 55, 1992. [5] R. Hilfer, “Local porosity theory for electrical and hydrodynamical transport through porous media,” Physica A, vol. 194, p. 406, 1993. [6] B. Hansen, E. Haslund, R. Hilfer, and B. Npst, “Dielectric dispersion measure ments of salt water saturated porous glass compared with local porosity theory, ” Mater.Res.Soc.Proc., vol. 290, p. 185, 1993. [7] R. Hilfer, B.Npst, E.Haslund, Th.Kautzsch, B.Virgin, and B.D.Hansen, “Local porosity theory for the frequency dependent dielectric function of porous rocks and polymer blends, ” Physica A, vol. 207, p. 19, 1994. [8] E. Haslund, B. Hansen, R. Hilfer, and B. Npst, “Measurement of local porosities and dielectric dispersion for a water saturated porous medium,” J. Appl. Phys., vol. 76, p. 5473, 1994. [9] R. Hilfer, “Probabilistic methods, upscaling and fractal statistics in porous media,” Z. Blatt f. Geol. Palaont., p. in print, 1996. [10] R. Hilfer, “Transport and relaxation phenomena in porous media,” Advances in Chem ical Physics, vol. XCII, p. 299, 1996. 8 [11] B. N0st, B. Hansen, and E. Haslund, “Dielectric dispersion of composite material Physica Scripta, vol. T44, p. 67, 1992. [12] C. Ostertag-Henning , “Modellierung von Porositat und Permeabilitat in Realgestein anhand der lokalen Porositatstheorie, ” tech, rep., Institut fur Palaontologie, Universi Erlangen, Germany, 1995. [13] C. Ostertag-Henning, B. Virgin, T. Rage, R. Hilfer, R. Koch, and U. Mann, “Messu dreidimensionaler lokaler Porositatsverteilungen ,” 1995. to be published. 9 LOCAL POROSITY THEORY FOR THE TRANSITION FROM MICROSCALES TO MACROSCALES IN POROUS MEDIA R. Hilfer1-2^, B. Virgin 1 and T. Rage1 1 Institute of Physics, University of Oslo, 0389 Oslo, Norway 2Institut fur Physik, Universitat Mainz, 55099 Mainz, Germany ^ICA-1, Universitat Stuttgart, 70569 Stuttgart, Germany Introduction diffusion where C corresponds to the diffusivity and in A quantitative understanding of fluid flow and other trans dispersion where C is the dispersivity. port processes in porous media remains a prerequisite for Distributions of local geometric observables form the progress in many disciplines such as hydrology, petroleum basis of local porosity theory [Hil91, Hil96], The simplest technology, chemical engineering, environmental protec and most important geometric quantity of porous media is tion, nuclear waste storage, drug transport in biological the porosity defined as 4>(S) = U(P)/Vr(S) where U(P) tissues, catalysis, paleontology, filtration and separation denotes the volume of the pore space P and U(S) is the technology to name but a few. While the microscopic volume of the sample S. The basic idea underlying lo equations governing flow and transport in porous media cal porosity theory is to consider the fluctuations of “local are often well known, the macroscopic laws are usually porosities ” or local volume fractions inside mesoscopic different and much less understood. regions (measurement cells). The size of these regions Most approaches in computational fluid dynamics for becomes a parameter controlling the transition from mi porous media avoid to discuss or control the problems croscales to macroscales. The length scale dependent lo arising in the transition from a microscale (pores) to the cal porosity distributions are used to calculate length scale macroscale (field or laboratory). As a consequence the dependent effective transport coefficients. pscaling of transport processes, particularly for immis- ible fluid-fluid displacement [Hil96], has remained diffi- Local Porosities ult. A new and general upscaling methodology was in re- Given a porous medium the sample space S is partitioned ent years introduced under the name local porosity theory into M mutually disjoint subsets, called measurement Hil96, Hil9I, HiI92b, Hil92a, BFHJ92, Hil93, HNH+94, HHN94], Local porosity theory provides a general up- cells Kj, such that |J>Li Kj = S and K, A K, =0 if caling methodology that is applicable to all transport pro- i j. The partitioning is denoted as K = {Kj,..., Km}. esses involving the disordered Laplacian V • C(r)V. A simple partitioning is illustrated in Figure 1 showing a Tere V denotes the Nabla operator, and C(r) is a sec- quadratic lattice of measurement cells superimposed on nd rank tensor of fluctuating local transport coefficients, the digitized image of a thin section of an oolithic sand xamples arise in Darcy flow where C is related to the stone [OHVH+95], This partitioning has the convenient uid flow permeability, in electrical conduction where C feature that the K, are translated copies of the same set, s the conductivity or dielectric constant, in mass or heat and hence have all the same shape. Figure 1: Measurement lattice of squares Kj superimposed on the discretized thin section image of a sandstone. The pore space P is coloured black, the matrix space is rendered white [OHVH+95]. FFCOFTAC nuRptin Local porosities within each measurement cell K; distribution changes. Maximizing the Gibbs-Shan [j = 1,M) of the partitioning K are defined as entropy of p allows to determine an entropic len scale for each porous sample [BFHJ92]. More im 7(PnKj) tantly, however, the dependence of the local poro ^(K;) = (1) distributions on the length scale of the measurem cell provides a parameter for controlling the transit where V(Kj) is the volume of the measurement cell from microscales to macroscales in porous media. Kj and F(P fl K;) is the pore volume inside K;. Let The local porosity distributions characterize 5(z) denotes the Dirac ^-distribution concentrated at porosity fluctuations in a porous medium. T zero. The discrete probability density function can be observed experimentally from digitized ages [OHVH+95]. Figure 1 shows part of a di 1 A tized cross section from a sample of Savonnier oolit (2) sandstone. This type of sandstone exhibits oomol i=i porosity with ellipsoidal pores and grains of sizes 1 tween 200pm and 300pm [OH951. The cross sectio is called one cell local porosity distribution. The local taken from a series of 100 parallel cross sections e porosity distribution p gives a quantitative geometric measuring 2cm x 2cm. The distance between conse characterization of the pore space which can be mea tive planes was about 10pm, and the resolution wit sured directly using methods of image analysis. It de each image is of similar magnitude. The family of pends on the partitioning K, and in particular on the cal porosity distributions p{ Figure 2: Local porosity distributions p( Local Percolation Probabilities black, the curve Ay( F.RCOFTA r Bulletin 0.4 0.6 porosity figure 3: Local percolation probability functions Ax(4) (black), Xy(4) (medium gray), and Az(4) (light gray) lor the Savonnier oolithic sandstone shown in Figure 1 using a simple cubic lattice of measurement cells with Iside lengths (lattice constant) L = 32. The local porosity distribution fi(4) for L = 32 is shown as the inset. Macroscopic Limit semble limit [Hil94] which can be defined as M —* oo definition the local porosity distribution fi(4; K) and Kj —» R3 for all j. This limit models the situa spends on the partitioning K. Intuitively one ex tion where the individual cells are much larger than acts that this dependence becomes unimportant the typical pore size, but still much smaller than the |hen the individual measurement cells K; are much total sample so that the concept of “local ” porosities rger than the typical pore size but still much smaller remains useful. The limiting local porosity distribu ban the sample S. This expectation is supported by tions were found to characterize universality classes le fact that for K — {S}, i.e. M = 1, one expects of porous media with macroscopically heterogeneous W; {S}) = 6(4 - <£(S)). For mixing and ergodic porosity. Analytical expressions for the limiting lo cal porosity distributions fjt(4\ to, C, D) were found ledia the macroscopic limit is limg_jj 3 4( S) = 4 which depend only on three macroscopic parameters iere 4 is the average macroscopic porosity. Hence w,C, D, where 0 < td < 1, but not on the partition | Figure 4: Universal limiting local porosity density fi(4; C, D) for w = 0.5, C — 2 and 0.05,0.5,1.0,2.0,5.0,10.0. For D —* 0 the density function is concentrated at 4 — 1/C. It vanishes for all D whenever 4 > 1/C. FvrnFTAr The divergence at small ERCOFTAC Bulletin A Simulations of one- and two-phase flow in fractures P. Meakin, T. Rage, G. Wagner, J. Feder, and T. J0ssang Department of Physics, University of Oslo Box 1048 Blindern, Oslo 0316, Norway November 16, 1996 Abstract We present numerical studies on one- and two-phase flow in fractures. The fractures were modeled as self-aiBne fractal surfaces with a Hurst exponent of H=0.8. In the simulations of one-phase flow, a finite-difference scheme was used to solve the steady Stokes equations. Tracer dispersion was simulated by solving the Convection-Diffusion equation. In the simulations of immiscible two-phase flow, a modified invasion percolation algorithm was used to model quasistatic displacement processes that are dominated by capillary forces and by buoyancy forces. 1 Introduction Faults and fractures play an important, if not dominant, role in the transport of fluids through rocks. Under a wide range of circumstances the fluid flow process is concentrated onto a network of interconnected fractures. However the flow of fluids through highly permeable rocks may be seriously impeded if fractures have become filled by impermeable materials resulting from a combination of mechanical phenomena and chemical processes, generally involving the flow of 1 water carrying dissolved or colloidal minerals, into, along and out of the fracture. In either case the geometry of individual fractures, the transport within or across individual fractures, their interactions with the surrounding rocks, and the manner in which they are connected, is essential to a good overall understanding of fluid transport phenomena. Transport in fractures is important in many processes of fundamental importance in geology such as metamorphism (Manning, 1994)- Flow in fractures also has important practical and economic consequences in processes such as the accumulation of economically valuable mineral deposits, the disposal of chemical waste by deep-well injection and the containment of radioactive waste. In particular, faults and fractures play an important role in the transport of hydrocarbon fluids from source rocks to a trap or reservoir and in the economic recovery of oil and gas from the reservoir. This class of applications has provided the major motivation for the work described here. We describe simulations of miscible displacement (dispersion) of a fluid by a second fluid with identical properties as well as the slow immiscible displacement of a wetting fluid by a non-wetting fluid in individual fractures. The Hele-Shaw cell (Hele-Shaw, 1898) (two transparent plates separated by a distance b) provides the most simple model for a fracture and important results have been obtained from Hele-Shaw cell experiments and associated simulations (particularly in the case of the unstable displacement of a viscous fluid by a much less viscous fluid). However the variable aperture, and other heterogeneities, are important in most fluid flow processes in fractures. The variable aperture concentrates the flow onto preferred regions and, in the case of immiscible flows, the aperture is the most important quantity that controls the capillary pressure that must be overcome to drive the fluid-fluid interface through the fracture. The Hele-Shaw cell can be improved, as a model for fluid flow, by randomly placing obstacles in the channel (D0vle, 1993). The computer modelling studies described here are based on a self-affine fractal model for the roughness of fracture surfaces. A self-affine surface h(x,y) is characterized by its Hurst exponent H and looks “similar to itself” when the length scales in the x- and in the y-direction are rescaled by a factor 6, and the length scale in the ^-direction is rescaled simultaneously by the factor bH. In contrast, a self-similar fractal looks “similar to itself” when all the length scales in all directions are rescaled simultaneously by the same factor b. Following the pioneering work of Mandelbrot et ai, 1984, the rough surfaces of 2 a large number of synthetic and natural materials have been analyzed (see Meakin, 1993, for a review) and the results of most of these studies support the idea that the surfaces generated by brittle fracture have a self-affine fractal geometry, with a purportedly universal Hurst exponent of H ~ 0.8. In particular, experiments on both igneous (Schmittbuehl et al, 1995) and sedimentary rocks indicate that fresh fracture surfaces can be described in terms of self-affine fractal geometry, with H ~ 0.8. Studies carried out on anisotropic rock materials (Brown and Scholz, 1985; Scott et al., 1992) and wood (Eng0y et al., 1994) indicate that the Hurst exponent is not sensitive to the orientation of the fracture with respect to the anisotropy. An unambiguous demonstration of self-affine scaling and the accurate determination of Hurst exponents requires data covering a wide range of length scales (Boger, 1993)} Unfortunately the “scaling regime” found in most fracture experiments extends over only 2 or fewer decades in the plane of the fracture. For this reason, and the absence of a convincing theoretical model, the idea of self-affine fracture roughness with a universal Hurst exponent should not be accepted without severe reservations. However the broad based experimental studies do provide strong motivation for the self-affine model for fracture geometry used in this work. The self-affine fracture model does not take into account the effects of fracture debris within the aperture, partial closure and deformation due to ambient stress fields, the evolution of the fracture aperture due to a wide range of physico-chemical processes such as diagenesis, interactions with the surrounding porous rock or the spatial organization of fractures with a wide range of sizes into a fracture network. We plan to include some of these effects in future modelling work and to carry out experiments to evaluate our simulation results and motivate future simulation studies. 2 Two-Phase flow The slow displacement of a wetting fluid from a fracture aperture by an invading non-wetting fluid, under the influence of gravity, was simulated using a modified site invasion percolation model (Lenormand and Bories, 1980; Meakin et al., 1993). In the model, the aperture was represented by the two-dimensional aperture field /i(x) = h(x, y) and the non-wetting phase was 3 assumed to enter the fracture along one of its edges, at y = 0. The fracture was represented as a square lattice of sites. To each site i with the coordinates thresholds tf and were assigned. The thresholds represented capillary forces that impede the non-wetting phase from invading a site in the fracture that is filled with the wetting fluid or from withdrawing from a site in the fracture that it occupies. The thresholds were given by = K1 - gxji ; tf = hi + gy { . (1) Here, hi = h(xi,yi) is the local aperture of the fracture, and g denotes a gradient parameter representing buoyancy. Little is known about withdrawal tresholds. In our simulations the withdrawal thresholds are smaller than the invasion thresholds and play only a minor role. The simulations were carried out on square lattices with sizes of Lx x Ly lattice sites. Initially all of the sites were labelled to indicate wetting fluid, except for a row of Lx sites at y = 0, which were given a different label to indicate non-wetting, invading fluid. The simulation proceeded in steps in which the non-wetting phase either invaded a site i, or withdrew from a site j and invaded another site j'. The invasion of sites without withdrawing represented the filling of the fracture void space with non-wetting phase that emerged from a reservoir connected to the entrance at y = 0. The wetting phase was displaced from the void space during these steps. Only sites that were connected via a path consisting of steps between nearest neighbors, within the non-wetting phase, to the reservoir could be invaded by this mechanism. Incompressibility of the phases was taken into account by using a trapping rule (Wilkinson and Willemsen, 1983). Wetting phase sites that were surrounded by the non-wetting phase and no longer connected to the opposite face of the fracture via a path of wetting phase sites were trapped and could not be invaded. Among all the sites that could be invaded, the site with the lowest invasion threshold tf was chosen in each step. This rule modeled the slow invasion of non-wetting phase into the fracture aperture. In steps with simultaneous invasion and withdrawal events, the non-wetting phase migrated along the fracture without occupying additional sites. In these steps, non-wetting phase withdrew from a source site and invaded a destination site that was connected to the source site by a path of non-wetting phase sites. A migration step involving the withdrawal from the site j and the invasion of the site j' was possible if the condition c — tj + ^ — hjl + hji — g(yj> — < 0 (2) was fulfilled. Eq. (2) modeled buoyancy-driven migration in the quasi-static regime in which viscous forces may be neglected and the capillary forces are 4 dominant. At each stage, the migration step yielding the lowest negative value for c was carried out. If no migration step with c < 0 was possible, the site on the untrapped external perimeter of the cluster of sites connected to the entrance at y = 0 with the lowest invasion threshold was filled to represent invasion by the non-wetting phase. After each invasion event, a new search for pairs of sites j and j' satisfying equation Eq. (2) began. The simulations were terminated when the non-wetting phase reached the exit at y — Ly. This model was motivated by an interest in secondary migration in which less dense hydrocarbon fluids are driven through fractured rock by buoyancy forces 1. The gradient g imposed a tendency for the non-wetting phase to rise towards the upper exit at y = Ly. For low values of g, Eq. (2) could not be fulfilled, and the non-wetting phase formed a single large cluster. For high g , a multitude of migration steps took place. Some of the migration steps led to fragmentation of a cluster of non-wetting phase, and some led to coalescence of two clusters formed by a combination of earlier fragmentation and coalescence events. The local apertures h(x,y) used to assign thresholds (Eq. (1)) were obtained by generating rough, periodic surfaces using a Fourier Transform filtering algorithm (Voss, 1985). A random fractal self-affine surface is characterized by its two-dimensional spectral density S behaving as S(fk,fl) ~ _|_ J2^H+1 ’ (3) where /&, /; denote frequency variables in directions corresponding to the x- and the ^-direction in real space, and H is the Hurst exponent (Feder, 1988). A cross-section through a surface with H = 0.5 corresponds to a random walk in which steps in the positive and in the negative direction occur with equal probability and without any correlation with previous steps. A cross-section through a surface with 0.5 < H < 1 corresponds to a generalized random walk (fractional Brownian noise) with long range, persistent correlations. For such a surface, a step in a particular direction is likely to be followed by a step in the same direction. In the simulations reported in this work a Hurst exponent of H = 0.8, corresponding to a quite strongly persistent process, was used. hydraulic potential potential gradients resulting from the flow of water through the same formation can also play an important role. We believe that hydraulic potential gradients can produce effects similar, but not identical, to the effects of gravity acting on the density difference between the two fluids and that migration driven by hydraulic potential gradients can also be represented by gradient invasion percolation models. 5 Two-dimensional Fourier spectra that fulfilled Eq. (3) were obtained by assigning values of a*,; = (&: + (4) to the complex, two-dimensional array of Fourier coefficients (Saupe, 1988). Here, $ is a random variable selected from a Gaussian distribution with mean 0 and variance 1, and ip is a random phase uniformly distributed between 0 and 2tt. After transformation into real space, the surface was normalized to have a mean height of zero and a maximum extension d/2 in the positive and in the negative directions. A fracture void space model was created by copying the surface and translating the copy by an amount a in the ^-direction and by a vector (iq,, vy) in the x — y plane. The translation in the x — y plane was carried out using periodic “folding ” such that both surfaces overlapped completely. The void space aperture h(x,y ) was given by y) - z/) if y) > y) (5 ) 0 otherwise, where zt denotes the top surface and the bottom surface. The displacement patterns obtained depended strongly on the parameters o, d, and (vx, vy) used to generate the void space. Figure 1 a shows a surface with H = 0.8 and d = 0.7 on a lattice of size 256 x 256, and Fig. 1 b shows the void space created by translating a copy of the surface using a = 0.15 and (vx, vy) = (20, 20). This parameter set led to a slight overlap of the two surfaces such that the aperture was zero at some of the sites. Figure 1 c shows a void space created in a similar manner using a larger lateral translation vector {vx,vy) = (100,100). In this case, large regions of the fracture were inaccessible with zero aperture. Figures 2 a and b show two stages in a simulation using the void space geometry shown in Fig. 1 b. The non-wetting fluid entered the fracture through one of its faces and migrated in the direction of the gradient. The displacement pattern formed by the non-wetting phase consists of large blobs that are connected by thin “strings ” of invading fluid. The strings indicate that the non-wetting phase passed regions with large apertures (low thresholds) in which propagation was easy. In contrast, when narrow regions were encountered, the non-wetting phase explored and invaded neighboring sites. Most of them had similar invasion thresholds due to the long-ranged aperture correlations. Small amounts of wetting phase became surrounded by non-wetting phase at these stages. The connecting strings could break when non-wetting phase withdrew 6 during migration steps The strings were re-formed when more non-wetting phase was supplied from the reservoir. Figures 2 c and d show two stages in a simulation using the void space geometry shown in Fig. 1 c. The blobs formed by the non-wetting phase tended to be larger in this case, indicating pronounced long-ranged aperture correlations. 3 Single-phase flow The dispersion of a passive tracer is a consequence of the combined effects of convection and diffusion. To simulate dispersion in a fracture the flow of an incompressible, Newtonian fluid was studied. It was assumed that the fracture is saturated with fluid, and that the convective motion of the fluid is described by a velocity field v(x) which fulfills the steady Stokes equations (Batchelor, 1967; Landau and Lifshitz, 1987) 0 = —Vp + — V2v, (6) P 0 = (V-v), (7) where p(x) is the fluid pressure, p is the fluid density, and p is the viscosity. Eq. (6) is a valid approximation to the general Navier-Stokes equations if the Reynolds number Re = uclcp/p is small. Here, uc is a typical flow velocity and lc a typical length of the flow geometry. Non-slip boundary conditions were assumed at the boundary between the fluid and the fracture. To calculate approximate solutions to Eq. (6) and Eq. (7), the fracture aperture was divided into equally sized, rectangular cells of dimension (Az x Ay x Az). Each of the cells was labelled to represent either fluid or solid, such that a cell k with midpoint %/&,%&) represented solid if zk < zb(xk,yk) or zk > zt(xk,yk), where zb(xk,yk) and zt(xk,yk) are the top and bottom surfaces of the horizontal fracture. The fracture surfaces were obtained using the same self-affine fracture model that was used to simulate the two-phase fluid-fluid displacement processes, described above. Nodes of the pressure field were defined on the centers of the cells, while nodes of the velocity field were defined on cell boundaries (MAC-mesh). After assigning initial values, pressure- and velocity-nodes were iteratively updated according to the “Artificial Compressibility ” scheme (Peyret and Taylor, 1983). This explicit 7 finite-difference scheme guarantees the convergence of the lattice-based fields towards the solution of the (spatially) discretized version of Eq. (6) and Eq. (7). The dispersion of the tracer under the combined influence of convection and molecular diffusion, i.e. the time evolution of the local tracer concentration C, was assumed to follow the Convection-Diffusion equation (Scheidegger, 1974) Ac = -(v.V)C + DmV2C, (8) where Dm is the molecular diffusivity of the tracer. Eq. (8) must be supplied with appropriate boundary conditions in order to guarantee the uniqueness and existence of the solution. The Peclet number Pe = uclcjDm compares a typical time scale of diffusion over the length lc {tdi// — l2c{Dm) to a typical time scale of convection (tconv = lc/uc). For large Pe, diffusion is much “slower” than convection and the dispersion process will be dominated by convective mass transport. If on the other hand Pe is small, convective transport will be negligible compared to diffusive transport. To model Eq. (8) numerically, we implemented a finite-difference scheme. The evolution of the tracer concentration field was calculated using the forward difference approximation d C(x,t + At) - C(x, t) Jt ~ At ’ so that an explicit updating scheme was obtained. To improve numerical accuracy, the Barton scheme (Centrella and Wilson, 1984; Hawley et al., 1984) was used in the calculation of the convective flux (v • V)C. Fig. 3 shows a typical dispersion front at Pe ~ 1. The (continuous valued) height functions and zt of the fracture were here defined on a (128 x 128)-lattice. The fracture was translated onto a 3D-lattice of size as explained above. Some “free space ” was added at the inlet and outlet of the fracture. The resulting 3D-lattice covered 148 x 128 x 21 sites. When solving for the velocity field, the pressures at the inlet and outlet of the fracture were kept at different constant values pi and pQ. Periodic boundary conditions were used in the ^-direction. Fig. 4 shows results from three different dispersion simulations on the same fracture, all at Re ~ 6 • 10-5. In the first simulation, the Peclet number was 8 approximately Pe ~ 0.1, while values of Pe ~ 1 and Pe = 10 were used in the two other studies, respectively. In all three cases, the tracer concentration was kept constant on the left hand side of the fracture (x = 0, inlet) in order to simulate a continuous supply of tracer. For each value of Pe, Fig. 4 shows three snapshots of the concentration profiles at different times. The fracture is seen from above, and the color codes represent average tracer concentrations (averaged over the ^-direction) from black (high concentration) to grey (low concentration), and white (zero concentration). It is evident that smaller Peclet numbers lead to smooth concentration profiles, while large Peclet numbers create relatively complex dispersion fronts with a sharp interface. Finally, Fig. 5 shows a dispersion front in an “obstacle-model ” of a fracture. Here, the fracture was modeled as a Hele-Shaw cell of height h in which cylindrical obstacles of diameter d were placed at random positions such that the porosity was $ = 0.715. The resulting geometry was translated on a two-dimensional lattice and the 2D steady Stokes equations (including a linear term that captures the viscous drag from the bottom- and top-plate of the Hele-Shaw cell (Oxaal et al., 1994) were solved with the Artificial Compressibility scheme. Similar models have been used by Martys, 1994, in numerical studies on dispersion in porous media. 4 Conclusion This chapter describes the relatively early stages in a program that is being carried out with the objective of developing a better understanding of flow in fractures. This work is part of a broader study of flow in disordered media, the fracture of complex materials and pattern formation far from equilibrium that is being carried out in the Cooperative Phenomena program at the University of Oslo Physics Department. We are planning to compliment the simulations described above with laboratory experiments. The results presented above are qualitative in nature. There are good reason to believe that the patterns (iso-concentration lines, fluid-fluid interfaces etc.) generated by these simulations can be described in terms of fractal geometry (Mandelbrot, 1982; Feder, 1988) and related scaling concepts. However the interpretation of the patterns generated in the self-affine fracture aperture models will be complicated by crossovers related to the characteristic length 9 scales introduced by the lateral and vertical displacement of the fracture surface. In the gradient driven migration process the scaling will be further complicated by length scales related to the “competition ” between capillary and buoyancy (gradient) forces. Despite these anticipated difficulties we believe that a quantitative analysis of our simulation results will lead to rewarding results and a better fundamental understanding of single and multi-phase flow in fractures. Another interesting aspect of the self-affine fracture model is the existence of a percolation threshold. Because of the self-affine correlations associated with surface roughness, this is not an ordinary percolation phenomenon. We expect that a quantitative study of the behavior of our models near to the percolation threshold will take us into areas such as statistical topography (Isichenko, Perhaps the most important result of this work is the demonstration that it is possible to carry out simulations, using simple but realistic models on readily available computer resources (a workstation), on a scale that is large enough to allow quantitative questions to be addressed. Acknowledgements We thank K.J. Malpy and J. Schmittbuehl 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 here has received support from the NFR programme for supercomputing through a grant of computing time. References Batchelor, G. K., An Introduction to Fluid Dynamics, (Cambridge University Press, Cambridge, 1967), pp. 216-228 Boger, F., Rough surfaces, synthesis, analysis, visualization and applications. 10 PhD thesis, Department of Physics, University of Oslo, (1993). Brown, S. R. and Scholz, C. H., Broad bandwidth study of the topography of natural rock surfaces. Journal of Geophysical Research B90, 12575-12582 (1985) Centrella, J. and Wilson, J. R., Planar Numerical Cosmology II: The difference equations and numerical tests. Astrophys. J. Suppl. Ser. 54, 229-249 (1984). Dpvle, M., Dispersion in 2-d homogeneous and inhomogeneous porous media. Master thesis, Department of Physics, University of Oslo, (1993). Engpy, T., Malpy, K. J., Hansen, A. and Roux, S., Roughness of two-dimensional cracks in wood. Physical Review Letters, 73, 834-837 (1994) Feder, J., Fractals (Plenum Press, New York, 1988). Hawley, J. F., Smarr, L. and Wilson, J. R., Astrophys. J. Suppl. Ser. 55, 211-246 (1984). Hele-Shaw, H. S., Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions. Transactions of the Institution of Naval Architects, 40, 21-46 (1898). Isichenko, M. B., Percolation, statistical topography, and transport in random media Reviews of Modern Physics, 64. 961-1043, (1992). Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, 2nd ed. (Pergamon Press, Oxford, 1987), pp. 58-68. Lenormand, R. and Bories, S., Description d’un mecanismede connexion de liason destine a l’etude du drainage avec piegeage en milieu poreux. C.R. Acad. Sc. Paris 291, 279-283 (1980). 11 Mandelbrot, B. B., The fractal geometry of Nature. Freeman, New York, (1982). Mandelbrot, B. B., Passoja, D. E. and Paullay, A. J., Fractal character of fracture surfaces om metals. Nature, 308, 721-722 (1984). Manning, C., Fractal clustering in metamorphic veins. Geology, 22, 335-338 (1994). Martys, N. S., Fractal growth in hydrodynamic dispersion through random porous media. Phys. Rev. E 50, 335-342 (1994). Meakin, P., The growth of rough surfaces and interfaces. Physics Reports, 235, 189-289 (1993). Meakin, P., Wagner, G., Feder, J. and Jpssang, T., Simulations of migration, fragmentation and coalescence of non-wetting fluids in porous media. Physica A 200, 241-249 (1993). Oxaal, U., Flekkpy, E. G. and Feder, J., Irreversible dispersion at a stagnation point: Experiments and Lattice Boltzmann Simulations. Phys. Rev. Lett. 72, 3514-3517 (1994). Peyret, R. and Taylor, T. D., Computational Methods for Fluid Flow, (Springer-Verlag, New York, 1983). Saupe, D., Random fractal algorithms. In The Science of Fractal Images, edited by Heinz-Otto Peitgen and Dietma Saupe (Springer-Verlag, Berlin, 1988), pp. 71-136. Scheidegger, A. E., The Physics of Flow Through Porous Media, 3rd ed. (Uni versity of Toronto Press, Toronto, 1974). Schmittbuehl, J., Schmitt, F. and Scholz, C., Scaling invariance of crack surfaces. Journal of Geophysical Research, 100, 5953-5973 (1995). 12 Scott, P. A., Engelder T. and Mecholsky Jr., J. J., The correlation between fracture-toughness anisotropy and crack surface morphology of siltstones in the Ithaca formation, Appalachian basin, in Fault mechanics and transport properties of rocks edited by Brain Evans and Teng-Fong Wong (Cambridge University Press, Cambridge, 1992), pp. 341-370. Voss, R. F., Random fractal forgeries. In Fundamental Algorithms for Computer Graphics, edited by R.A. Earnshaw (Springer-Verlag , Berlin, 1985), pp. 805-835. Wilkinson, D. and Willemsen, J. F., Invasion percolation: A new form of perco lation theory. J. Phys. A: Math. Gen. 16, 3365-3376 (1983). 13 Figure 1: (a) A surface with a Hurst exponent H = 0.8 generated using a Fourier filtering algorithm on a lattice of size 256 x 256. (b) Fracture void space generated by translating a copy of the surface in the ^-direction. The void space is shown such that the height of the structure at a given site i corresponds to the aperture h{. The surfaces overlap partly, leading to zero aperture at some of the sites. The parameters used were d = 0.7, a = 0.15, (vx,vy) = (20,20). (c) Fracture void space generated in a similar manner using d = 0.7, a = 0.15, (vx, vy) = (100,100). 14 Figure 2: Stages in two simulations of quasi-static two-phase flow. The shade of the wetting phase (grey) indicates the local aperture. In the regions where the two fracture surfaces overlap, the aperture is zero (black regions). The non wetting phase (white) enters the fracture at the lower face and migrates upwards. A small fraction of the wetting phase becomes completely surrounded by non wetting phase (black spots). (a),(b): Simulation using the void space geometry shown in Fig. 1 b. The gradient parameter was g = 0.05. (c),(d): Simulation using the void space geometry shown in Fig. 1 c. The gradient parameter was g 0.02. 15 Figure 3: Iso-concentration front of a dispersion pattern in a fracture at Pe ~ 10. The front corresponds to the (quite low) concentration C = 0.004, and follows mainly the two fracture surfaces. The inlet of the fracture is clearly visible. 16 Pe ~ 0.1 Figure 4: Dispersion patterns in a fracture for three different values of the Peclet number Pe. For each Pe. three snapshots of the dispersion pattern (average tracer concentration) at different times are shown. Between the snapshots, a constant amount of time goes by. 17 Figure o: Dispersion pattern in a quasi three-dimensional obstacle model of fracture at He ~ 4 • 1CT2 and Pe ~ 10.