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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
In this standard reference of the field, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible displacements in porous media and fractured rock are considered. Two different approaches are discussed and contrasted with each other. The first approach is based on the classical equations of flow and transport, called 'continuum models'. The second approach is based on modern methods of statistical physics of disordered media; that is, on 'discrete models', which have become increasingly popular over the past 15 years. The book is unique in its scope, since (1) there is currently no book that compares the two approaches, and covers all important aspects of porous media problems; and (2) includes discussion of fractured rocks, which so far has been treated as a separate subject.
Portions of the book would be suitable for an advanced undergraduate course. The book will be ideal for graduate courses on the subject, and can be used by chemical, petroleum, civil, environmental engineers, and geologists, as well as physicists, applied physicist and allied scientists that deal with various porous media problems.
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Seitenzahl: 1520
Veröffentlichungsjahr: 2011
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Contents
Cover
Half Title page
Related Titles
Title page
Copyright page
Dedication
Preface to the Second Edition
Preface to the First Edition
Chapter 1: Continuum versus Discrete Models
Introduction
1.1 A Hierarchy of Heterogeneities and Length Scales
1.2 Long-Range Correlations and Connectivity
1.3 Continuum versus Discrete Models
Chapter 2: The Equations of Change
Introduction
2.1 The Mass Conservation Equation
2.2 The Momentum Equation
2.3 The Diffusion and Convective-Diffusion Equations
2.4 Fluid Flow in Porous Media
Chapter 3: Characterization of Pore Space Connectivity: Percolation Theory
Introduction
3.1 Network Model of a Porous Medium
3.2 Percolation Theory
3.3 Connectivity and Clustering Properties
3.4 Flow and Transport Properties
3.5 The Sample-Spanning Cluster and Its Backbone
3.6 Universal Properties
3.7 The Significance of Power Laws
3.8 Dependence of Network Properties on Length Scale
3.9 Finite-Size Effects
3.10 Random Networks and Continuum Models
3.11 Differences between Network and Continuum Models
3.12 Porous Materials with Low Percolation Thresholds
3.13 Network Models with Correlations
3.14 A Glance at History
Chapter 4: Characterization of the Morphology of Porous Media
Introduction
4.1 Porosity
4.2 Fluid Saturation
4.3 Specific Surface Area
4.4 The Tortuosity Factor
4.5 Correlations in Porosity and Pore Sizes
4.6 Surface Energy and Surface Tension
4.7 Laplace Pressure and the Young–Laplace Equation
4.8 Contact Angles and Wetting: The Young–Dupré Equation
4.9 The Washburn Equation and Capillary Pressure
4.10 Measurement of Capillary Pressure
4.11 Pore Size Distribution
4.12 Mercury Porosimetry
4.13 Sorption in Porous Media
4.14 Pore Size Distribution from Small-Angle Scattering Data
4.15 Pore Size Distribution from Nuclear Magnetic Resonance
4.16 Determination of the Connectivity of Porous Media
4.17 Fractal Properties of Porous Media
Chapter 5: Characterization of Field-Scale Porous Media: Geostatistical Concepts and Self-Affine Distributions
Introduction
5.1 Estimators of a Population of Data
5.2 Heterogeneity of a Field-Scale Porous Medium
5.3 Correlation Functions
5.4 Models of Semivariogram
5.5 Infinite Correlation Length: Self-Affine Distributions
5.6 Interpolating the Data: Kriging
5.7 Conditional Simulation
Chapter 6: Characterization of Fractures, Fracture Networks, and Fractured Porous Media
Introduction
6.1 Surveys and Data Acquisition
6.2 Characterization of Surface Morphology of Fractures
6.3 Generation of a Rough Surface: Fractional Brownian Motion
6.4 The Correlation Function for a Rough Surface
6.5 Characterization of a Single Fracture
6.6 Characterization of Fracture Networks
6.7 Characterization of Fractured Porous Media
Chapter 7: Models of Porous Media
Introduction
7.1 Models of Porous Media
7.2 Continuum Models
7.3 Models Based on Diagenesis of Porous Media
7.4 Reconstruction of Porous Media
7.5 Models of Field-Scale Porous Media
Chapter 8: Models of Fractures and Fractured Porous Media
Introduction
8.1 Models of a Single Fracture
8.2 Models of Fracture Networks
8.3 Reconstruction Methods
8.4 Synthetic Fractal Models
8.5 Mechanical Models of Fracture Networks
8.6 Percolation Properties of Fractures
8.7 Models of Fractured Porous Media
Chapter 9: Single-Phase Flow and Transport in Porous Media: The Continuum Approach
Introduction
9.1 Derivation of Darcy’s Law: Ensemble Averaging
9.2 Measurement of Permeability
9.3 Exact Results
9.4 Effective-Medium and Mean-Field Approximations
9.5 Cluster Expansion
9.6 Rigorous Bounds
9.7 Empirical Correlations
9.8 Packings of Nonspherical Particles
9.9 Numerical Simulation
9.10 Relation between Permeability and Electrical Conductivity
9.11 Relation between Permeability and Nuclear Magnetic Resonance
9.12 Dynamic Permeability
9.13 Non-Darcy Flow
Chapter 10: Single-Phase Flow and Transport in Porous Media: The Pore Network Approach
Introduction
10.1 The Pore Network Models
10.2 Exact Formulation and Perturbation Expansion
10.3 Anomalous Diffusion and Effective-Medium Approximation
10.4 Archie’s Law and the Effective-Medium Approximation
10.5 Renormalization Group Methods
10.6 Renormalized Effective-Medium Approximation
10.7 The Bethe Lattice Model
10.8 Critical Path Analysis
10.9 Random Walk Method
10.10 Non-Darcy Flow
Chapter 11: Dispersion in Flow through Porous Media
Introduction
11.1 The Phenomenon of Dispersion
11.2 Mechanisms of Dispersion Processes
11.3 The Convective-Diffusion Equation
11.4 The Dispersivity Tensor
11.5 Measurement of the Dispersion Coefficients
11.6 Dispersion in Systems with Simple Geometry
11.7 Classification of Dispersion Regimes in Porous Media
11.8 Continuum Models of Dispersion in Porous Media
11.9 Fluid-Mechanical Models
11.10 Pore Network Models
11.11 Long-Time Tails: Dead-End Pores versus Disorder
11.12 Dispersion in Short Porous Media
11.13 Dispersion in Porous Media with Percolation Disorder
11.14 Dispersion in Field-Scale Porous Media
11.15 Numerical Simulation
11.16 Dispersion in Unconsolidated Porous Media
11.17 Dispersion in Stratified Porous Media
Chapter 12: Single-Phase Flow and Transport in Fractures and Fractured Porous Media
Introduction
12.1 Experimental Aspects of Flow in a Fracture
12.2 Flow in a Single Fracture
12.3 Conduction in a Fracture
12.4 Dispersion in a Fracture
12.5 Flow and Conduction in Fracture Networks
12.6 Dispersion in Fracture Networks
12.7 Flow and Transport in Fractured Porous Media
Chapter 13: Miscible Displacements
Introduction
13.1 Factors Affecting the Efficiency of Miscible Displacements
13.2 The Phenomenon of Fingering
13.3 Factors Affecting Fingering
13.4 Gravity Segregation
13.5 Models of Miscible Displacements in Hele-Shaw Cells
13.6 Averaged Continuum Models of Miscible Displacements
13.7 Numerical Simulation
13.8 Stability Analysis
13.9 Stochastic Models
13.10 Pore Network Models
13.11 Crossover from Fractal to Compact Displacement
13.12 Miscible Displacements in Large-Scale Porous Media
13.13 Miscible Displacements in Fractures
13.14 Main Considerations in Miscible Displacements
Chapter 14: Immiscible Displacements and Multiphase Flows: Experimental Aspects and Continuum Modeling
Introduction
14.1 Wettability and Contact Angles
14.2 Core Preparation and Wettability Considerations
14.3 Measurement of Contact Angle and Wettability
14.4 The Effect of Surface Roughness on Contact Angle
14.5 Dependence of Dynamic Contact Angle and Capillary Pressure on Capillary Number
14.6 Fluids on Rough Self Affine Surfaces: Hypodiffusion and Hyperdiffusion
14.7 Effect of Wettability on Capillary Pressure
14.8 Immiscible Displacement Processes
14.9 Mobilization of Blobs: Choke Off and Pinch Off
14.10 Relative Permeability
14.11 Measurement of Relative Permeabilities
14.12 Effect of Wettability on Relative Permeability
14.13 Models of Multiphase Flow and Displacement
14.14 Fractional Flows and the Buckley–Leverett Equation
14.15 The Hilfer Formulation: Questioning the Macroscopic Capillary Pressure
14.16 Two Phase Flow in Unconsolidated Porous Media
14.17 Continuum Models of Two Phase Flows in Unconsolidated Porous Media
14.18 Stability Analysis of Immiscible Displacements
14.19 Two Phase Flow in Large Scale Porous Media
14.20 Two Phase Flow in Fractured Porous Media
Chapter 15: Immiscible Displacements and Multiphase Flows: Network Models
Introduction
15.1 Pore Network Models of Capillary-Controlled Two-Phase Flow
15.2 Simulating the Flow of Thin Wetting Films
15.3 Displacements with Two Invaders and Two Defenders
15.4 Random Percolation with Trapping
15.5 Crossover from Fractal to Compact Displacement
15.6 Pinning of a Fluid Interface
15.7 Finite-Size Effects and Devil’s Staircase
15.8 Displacement under the Influence of Gravity: Gradient Percolation
15.9 Computation of Relative Permeabilities
15.10 Models of Immiscible Displacements with Finite Capillary Numbers
15.11 Phase Diagram for Displacement Processes
15.12 Dispersion in Two-Phase Flow in Porous Media
15.13 Models of Two-Phase Flow in Unconsolidated Porous Media
15.14 Three-Phase Flow
15.15 Two-Phase Flow in Fractures and Fractured Porous Media
References
Index
Muhammad Sahimi
Flow and Transport in Porous Media and Fractured Rock
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The Author
Prof. Muhammad SahimiUniversity of Southern CaliforniaDept. of Chemical [email protected]
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Dedicated to the memory of my parents
Habibollah Sahimi (1916–1997) and Fatemeh Fakour Rashid (1928–2006)
and to
Mahnoush, Ali and Niloofar
Preface to the Second Edition
Since 1995, when the first edition of this book was published, the science of porous media and the understanding of flow and transport phenomena that occur in them have been advanced greatly. Characterization of porous media – both fractured and unfractured – can now be done in great detail, that is, for samples on the laboratory scale. Reconstruction methods have made it possible to develop realistic models of porous media based on relatively limited experimental data. Many new approaches have been developed that have made it possible to compute various flow and transport properties of porous media with considerable precision. In particular, pore network models on the one hand, and advanced computational techniques, such as the lattice-Boltzmann method, on the other hand, have become invaluable tools for studying flow and transport in porous media and in fractures. New theoretical developments have made it possible to analyze field-scale data, such as, the various types of logs and seismic records, with precisions, hence yielding deeper insights.
The understanding of fractures – particularly the crucial effect of the roughness of their internal surface – and fracture networks – especially the effect of their connectivity on their effective (overall) properties – has deepened. Classical models of fractured porous media, for example, the double-porosity model, though still useful in some special limits, are no longer viewed as the only practicable models for simulating flow and transport in large-scale fractured porous media. Viable alternatives that are much more realistic have been emerging at a rapid pace.
On the experimental side, new instrumentations coupled with advanced theoretical developments have made it possible to measure the various morphological, flow, and transport properties of porous media. Use of such techniques as three-dimensional X-ray computed tomography and nuclear magnetic resonance for measuring the properties have almost become routine.
The new developments motivated the preparation of the second edition of the book. However, the new edition does not merely represent an updated version of the first edition. Almost all the chapters have been completely rewritten. The characterization of unfractured and fractured porous media has been separated, each described and developed in its own chapter. The classical continuum models of fluid flow and transport in porous media and the discrete approaches based on the network models have also been separated, with each subject having its own chapter. On the experimental side, many newer experimental techniques for measuring the important morphological, flow, and transport properties of porous media have been described. Additionally, instead of describing unconsolidated porous media separately, as in the first edition, they have been merged with consolidated porous media and given equal footing in order to described both types of porous media in a unified manner.
As the famous song by John Lennon and Paul McCartney goes, I get by with a little help from my friends, except that in my case, my students and colleagues have given me a lot of help. Many people have contributed to my understanding of the topics described in this book. First and foremost, I have been blessed by the many outstanding doctoral students that I have worked with throughout my academic career on the problems studied in this book. They include Drs. Sepehr Arbabi, Mitra Dadvar, Fatemeh Ebrahimi, Jaleh Ghassemzadeh, Hossein Hamzehpour, Mehrdad Hashemi, Abdossalam Imdakm, Mahyar Madadi, Ali Reza Mehrabi, Sumit Mukhopadhyay, Mohammad Reza Rasaei, and Habib Siddiqui. Over the years, I have also been fortunate enough to have fruitful collaborations with many friends and colleagues on research problems related to what is studied in this book, including Professors Joe Goddard, Manouchehr Haghighi, Barry Hughes, Mark Knackstedt, Charles Sammis, Dietrich Stauffer, and Theodore Tsotsis as well as Drs. Adel A. Heiba and Luigi Sartor.
In addition, the preparation of this edition of the book was greatly helped by several people. My former doctoral students, Drs. Faezeh Bagheri-Tar, Fatemeh Ebrahimi, and Mahyar Madadi provided much needed help for the figures. Dr. Mohammad Piri generously gave me the electronic file of his outstanding Ph.D. Thesis that I used as a great source for the discussions of multiphase flows as well as the electronic files of the figures, some of which I have used in Chapter 15.
Throughout my academic life, I have been blessed by great mentors. Dr. Hasan Dabiri, my first academic mentor when I was attending the University of Tehran in Iran, introduced me to the petroleum industry when he worked with me on the project for my B.S. degree, Evaporation Loss in the Petroleum Industry. Over 35 years after taking the first of many courses with him, I am still influenced by his outstanding qualities, both as an academic mentor and as a wonderful human being. My advisors for my Ph.D. degree at the University of Minnesota, the late Professors H. Ted Davis and L.E. (Skip) Scriven, introduced me to various porous media problems and percolation theory, and taught me the fundamental concepts.
Michael C. Poulson was the publisher of the first edition of this book (and my first book, Applications of Percolation Theory, when he was with Taylor and Francis) as well as a great friend. He passed away on December 31, 1996 at the age of 50. In preparing this edition of the book, I greatly missed his wise advice, humor, and cheerful personality. The staff of Wiley-VCH, particularly Ulrike Werner, were extremely patient with my long delay in delivering the 2nd edition.
I dedicate this book to the special people in my life. My mentors in life, my late parents, Habibollah Sahimi and Fatemeh Fakour Rashid, made me what I am. I will miss them until I see them again. My wife Mahnoush, son Ali, and daughter Niloofar put up with my long absence from family life, and my spending countless numbers of days, weeks, and months in front of the computer at home. This edition would not have been completed without their love and patience.
Los Angeles, August 2010
Muhammad Sahimi
Preface to the First Edition
Disordered porous media are encountered in many different branches of science and technology, ranging from agricultural, ceramic, chemical, civil and petroleum engineering, to food and soil sciences, and powder technology. For several decades, porous media have been studied both experimentally and theoretically. With the advent of precise instruments and new experimental techniques, it has become possible to measure a wide variety of physical properties of porous media and flow, and transport processes therein. New computational methods and technologies have also allowed us to model and simulate various phenomena in porous media, and thus a deep understanding of them has been gained.
Whether we like it or not, we have to accept the fact that many natural porous systems are fractured, the understanding of which requires new methodologies and ways of thinking. In the past two decades, the understanding of fractured rock has taken on new urgency since, in addition to oil reservoirs, many groundwater resources are also fractured. Thus, flow in fractured rock has attracted the attention of scientists, engineers and politicians as a result of growing concerns regarding pollution and water quality. Highly intense exploitation of groundwater, and the increase in solute concentrations in aquifers due to leaking repositories and use of fertilizers, have made flow in fractured rock a main topic of research.
I have been working on such problems for 15 years, and during this period, I have realized that there are two distinct approaches to modeling flow phenomena in porous media and fractured rock. Some of these approaches belong to a class of models that I call the continuum models. Largely based on the classical equations of flow and transport, the continuum models have been very popular with engineers. Although not as widely used as the continuum models, the second approach, which is based on the discrete models that represent a porous system by a discrete set of elements and use large-scale Monte Carlo simulations and various statistical methods to analyze flow phenomena in porous media and fractured rock, has also attracted wide attention. Many new ideas and concepts have been developed as the result of using this class of models, and new results have emerged that have helped us gain a much better understanding of porous systems. In addition, such ideas and concepts as percolation processes, universal scaling laws and fractals, the basic tools of the discrete models, have gradually found their rightful positions in the porous systems literature. Currently, such concepts are even taught in graduate courses on flow through porous media, and in courses on computer simulation of disordered media and statistical mechanical systems.
Realizing these facts, and given that there was no book that discusses and compares both approaches, I decided to write this book. Even a glance at the immense literature on these subjects reveals that it is impossible to discuss every issue and present an in-depth analysis of it in just one book. Percolation theory, fractals, Monte Carlo simulations, and similar topics have been, by themselves, the subjects of several books and monographs. Unless one explains the most important concepts and then provides references where the reader can find more material to read, such a book can easily contain over a thousand pages. Based on this realization and limitation, I selected the topics that are discussed in this book. Largely based on this limitation, I also had to ignore several important topics, for example, flow of non-Newtonian fluids in porous media, filtration, and dissolution of rock by an acid which creates large fractures in the rock. In spite of such limitations, this book represents, in my opinion, a comprehensive review and discussion of the most important experimental and theoretical approaches to flow phenomena in porous media and fractured rock. Therefore, it can be used both as a reference book and a text for graduate courses on the subjects that it discusses. Considering the fact that this book discusses experimental measurement of the most important morphological and transport properties of porous systems, and the fact that many topics, especially those of single-phase flow and transport, are discussed in great detail, I believe that roughly half of the book can also be used in a senior-level undergraduate course on porous media problems that is taught in many chemical, petroleum and civil engineering and geological science departments.
As the famous song by John Lennon and Paul McCartney goes, “I get by with a little help from my friends”, except that in my case, my friends and colleagues have given me a lot of help. Many people have contributed to my understanding of the topics discussed in this book, a complete list of whom would be too long to be given here. However, I would like to mention a few of them who have had greatly influenced my way of thinking. I would like to thank Professors H. Ted Davis and L. E. (Skip) Scriven of the University of Minnesota who introduced me to various porous media problems and percolation theory, and taught me the fundamental concepts when I was their doctoral student. For over a decade, Dietrich Stauffer has greatly influenced my way of thinking about percolation, disordered media, and critical phenomena. I am deeply grateful to him. I would like to thank all of my past collaborators with whom I have published many papers on flow in porous media, especially Adel A. Heiba and Barry D. Hughes. Three other persons helped me write and finish this book. Michael Poulson, my publisher at VCH Publishers, was very patient and helpful. Drs. Sherry Caine and Dalia Goldschmidt helped me to organize my thoughts, focus on writing this book, and have a more positive outlook on life.
My debts of gratitude to them, and to many more who taught and influenced me, thus making this book possible.
Los Angeles, August 1994
Muhammad Sahimi
Chapter 1
Continuum versus Discrete Models
Introduction
Flow and transport phenomena in porous media and fractured rock as well as industrial synthetic porous matrices arise in many diverse fields of science and technology, ranging from agricultural, biomedical, construction, ceramic, chemical, and petroleum engineering, to food and soil sciences, and powder technology. Fifty percent or more of the original oil-in-place is left behind in a typical oil reservoir after the primary and secondary recovery processes end. The unrecovered oil is the main target for the enhanced or tertiary oil recovery methods now being developed. However, oil recovery processes only constitute a small fraction of an enormous and still rapidly growing literature on porous media. In addition to oil recovery processes, the closely related areas of soil science and hydrology are perhaps the best-established topics related to porous media. The study of groundwater flow and the restoration of aquifers that have been contaminated by various pollutants are important current areas of research. The classical research areas of chemical engineers that deal with porous media include filtration, centrifuging, drying, multiphase flow in packed columns, flow and transport in microporous membranes, adsorption and separation, and diffusion and reaction in porous catalysts.
Lesser known, though equally important, phenomena involving porous media are also numerous. For example, for the construction industry, the transmission of water by building materials (bricks or concretes) is an important problem to consider when designing a new building. The same is true for road construction where penetration of water into asphatene damages roads. Various properties of wood, an interesting and unusual porous medium, have been studied for a long time. Some of the phenomena involving wood include drying and impregnation by preservatives. Civil engineers have long studied asphalts as water-resistant binders for aggregates, protection of various types of porous materials from frost heave, and the properties of road beds and dams with respect to water retention. Biological porous media with interesting pore space morphology and wetting behavior include skin, hair, feathers, teeth and lungs. Other types of porous media that are widely used are ceramics, pharmaceuticals, contact lenses, explosives, and printing papers.
In any phenomenon that involves a porous material, one must deal with the complex pore structure of the medium and how it affects the distribution, flow, displacement of one or more fluids, or dispersion (i.e., mixing) of one fluid in another. Each process is, by itself, complex. For example, displacement of one fluid by another can be carried out by many different mechanisms which may involve heat and mass transfer, thermodynamic phase change, and the interplay of various forces such as viscous, buoyancy, and capillary forces. If the solid matrix of a porous medium is deformable, its porous structure may change during flow or some other transport phenomenon. If the fluid is reactive or carries solid particles of various shapes, sizes, and electrical charges, the pore structure of the medium may change due to the reaction of the fluid with the pore surface, or the physicochemical interaction between the particles and the pore surface.
In this book, we describe and study various experimental, theoretical, and computer simulation approaches regarding diffusion, flow, dispersion, and displacement processes in porous media and fractured rock. Most of the discussions regarding porous media are equally applicable to a wide variety of systems, ranging from oil reservoirs to catalysts, woods, and composite materials. We study flow phenomena only in a static medium, that is, one with a morphology that does not change during a given process. Thus, deformable media as well as those that undergo morphological changes due to a chemical reaction, or due to physicochemical interactions between the pore surface and a fluid and its contents, are not studied here. The interested reader is referred to Sahimi et al. (1990) for a comprehensive discussion of transport and reaction in evolving porous media and the resulting changes in their morphology.
1.1 A Hierarchy of Heterogeneities and Length Scales
The outcome of any given phenomenon in a porous medium depends on several length scales over which the medium may or may not be homogeneous. By homogeneous, we mean a porous medium with effective properties that are independent of its linear size. When there are inhomogeneities in the medium that persist at distinct length scales, the overall behavior of the porous medium is dependent on the rate of the transport processes, for example, diffusion, conduction, and convection, the way the fluids distribute themselves in the medium, and the medium’s morphology. Often, the morphology of a porous medium plays a role that is more important than that of other influencing factors.
Consider, as an example, an oil reservoir, perhaps one of the the most important heterogeneous porous media. In principle, the reservoir is completely deterministic in that it has potentially measurable properties and features at various length scales. It could have been straightforward to obtain a rather complete description of the reservoir if only we could excavate each and every part of it. In practice, however, this is not possible and, therefore, a description of any reservoir, or any other natural porous medium for that matter, is a combination of the deterministic components – the information that can be measured – and indirect inferences that, by necessity, have stochastic or random elements in them. Over the past four decades, the statistical physics of disordered media has played a fundamental role in developing the stochastic component of description of porous media.
There are several reasons for the development. One is that the information and data regarding the structure and various properties of many porous media are still vastly incomplete. Another reason is that any property that we ascribe to a medium represents an average over some suitably selected volume of the medium. However, the relationship between the property values and the volume of the system over which the averages are taken remains unknown. The issue of a suitably selected volume reminds us that any proper description of a porous medium or fractured rock must have a length scale associated with it. In general, the heterogeneities of a natural porous medium are described at mainly four distinct length scales that are as follows (Haldorsen and Lake, 1984).
1. The microscopic heterogeneities are at the level of the pores or grains, and are discernible only through scanning electron microscopy or thin sections.
2. The macroscopic heterogeneities are at the level of core plugs, and are routinely collected in fields and analyzed. Such heterogeneities are found in every well with property values varying widely from core to core. In most theoretical studies, however, cores are assumed homogeneous and the average effective properties are assigned to them, notwithstanding their microscopic heterogeneities.
3. The megascopic or field-scale heterogeneities are at the level of the entire reservoir that may have large fractures and faults. They can be modeled as a collection of thousands, perhaps millions, of cores, oriented and organized in some fashion.
4. The gigascopic heterogeneities are encountered in landscapes that may contain many such reservoirs as described in the third example, along with mountains, rivers, and so on.
Not all of the aforementioned heterogeneities are important to all porous media. For example, porous catalysts usually only contain microscopic heterogeneities, and packed beds may be heterogeneous at both the microscopic and macroscopic levels. In this book, we consider the first three classes of heterogeneities and their associated length scales.
1.2 Long-Range Correlations and Connectivity
In the early years of studying flow phenomena in porous media and fractured rock, most researchers almost invariably assumed that the heterogeneities in one region or segment of the system were random and uncorrelated with those in other regions. Moreover, it was routinely assumed that such heterogeneities occur at length scales much smaller than the overall linear size of the system. Such assumptions were partly due to the fact that it was very difficult to model the system in a more realistic way due to the computational limitations and lack of precise experimental techniques for collecting the required information. At the same time, the simple conceptual models, such as random heterogeneities, did help us gain a better understanding of some of the issues. However, increasing evidence suggests that rock and soils do not conform to such simplistic assumptions. They exhibit correlations in their properties, and such correlations are often present at all the length scales. The existence of such correlations has necessitated the introduction of fractal distributions that tell us how property values of various regions of a porous medium depend on the length (or even time) scale of the observations, how they are correlated with one another, and how one can model such correlations realistically. Such concepts and modeling techniques are described in this book.
Once we accept that natural porous media and fractured rock are heterogeneous at many length scales, we also have to live with its consequences. As a simple, yet very important, example, consider the permeability of a porous medium, which is a measure of how easily a fluid can flow through it. In a natural porous medium and at large length scales (of the order of a few hundred meters or more), the permeabilities of various regions of the medium follow a broad distribution. That is, while parts of the medium may be highly permeable, other parts can be practically impermeable. If we consider a natural porous medium, then, the low permeability regions can be construed as the impermeable zones as they contribute little or nothing to the overall permeability, while the permeable zones provide the paths through which a fluid flows. Thus, the impermeable zones divide the porous medium into compartments according to their permeabilities. This implies that the permeable regions may or may not be connected to one another, and that there is disorder in the connectivity of various regions of the porous medium. Thus, if we are to develop a realistic description of a porous medium, the connectivity of its permeable regions must be taken into account.
The language and the tools for taking into account the effect of the connectivity of the permeable regions of a pore space are provided by percolation theory. Similar to fractal distributions, percolation has its roots in the mathematics and physics literature, although it was first used by chemists for describing polymerization and gelation phenomena. Percolation theory teaches us how the connectivity of the permeable regions of a porous medium affects its overall properties. Most importantly, percolation theory predicts that if the volume fraction of the permeable regions is below some critical value, the pore space is not permeable and its overall permeability is zero.
In the classical percolation that was studied over 50 years ago, it was assumed that the permeable and impermeable regions are distributed randomly and independently of each other throughout the pore space. Since then, more refined and realistic percolation models have been developed for taking into account the effect of correlations and many other influencing factors. Such ideas and concepts are developed and used throughout this book for describing various flow phenomena in porous media and fractured rock.
1.3 Continuum versus Discrete Models
Now that we know what kinds of heterogeneities one must deal with in studying porous media, it is also necessary to consider the types of models that have been developed over the past several decades for describing flow and transport phenomena in porous media and fractured rock. The analysis of flow, dispersion, and displacement processes in in porous and fractured media has a long history in connection with the production of oil from underground reservoirs. It was, however, only in the past 35 years that the analysis has been extended to include detailed structural properties of the media. Such studies are quite diverse in the physical phenomenon that they consider. In this book, we divide the models for flow, dispersion, and displacement processes in porous and fractured media into two groups: the continuum models and discrete or network models.
Continuum models represent the classical engineering approach to describing materials of complex and irregular geometry characterized by several distinct relevant length scales. The physical laws that govern flow and transport at the microscopic level are well understood. Thus, one can, in principle, write down differential equations for the conservation of momentum, energy, and mass and the associated initial and boundary conditions at the fluid-solid interface. However, as the interface in typical porous media is very irregular, practical and computationally and economically feasible techniques, while available, are often not feasible for solving such boundary-value problems – even when one knows the detailed morphology of the porous medium. Determination of the precise solid-fluid boundary remains a very difficult (if not impossible) task, particularly for large-scale porous media. The boundary within which one would have to solve the equations of change are so tortuous as to render the problem mathematically intractable. Moreover, even if the solution of the problem could be obtained in great detail, it would contain much more information than would be useful in any practical sense. Thus, it becomes essential to adopt a macroscopic description at a length scale much larger than the dimension of individual pores or fractures.
Effective properties of a porous medium are defined as averages of the corresponding microscopic values. The averages must be taken over a volume that is small enough compared with the volume of the system, but large enough for the equation of change to hold when applied to that volume. At every point in the medium, one uses the smallest such volume, thereby generating macroscopic field variables satisfying such equations as Darcy’s law of flow or Fick’s law of diffusion. The reasons for choosing the smallest suitable volume for averaging are to allow in the theory suprapore variations of the porous medium and to generate a theory capable of treating the usual macroscopic variations of the effective properties. In this book, we encounter several situations where the conditions for the validity of such an averaging are not satisfied. Even when the averaging is theoretically sound, the prediction of the effective properties is often difficult because of the complex structure of the pore space. In any case, with empirical, approximate, or exact formulae for the flow and transport coefficients and other effective properties, the consequences of a given phenomenon in a porous medium can be analyzed based on such a theory. As mentioned above, many of the past theoretical attempts to derive effective flow and transport coefficients of porous media from their microstructure entailed a simplified representation of the pore space, often as a bundle of capillary tubes. In this model, the capillaries were initially treated as parallel, and then later as randomly oriented. Such models are relatively simple, easy to use, and sufficiently accurate, provided that the relevant parameters are determined experimentally and the connectivity of the pore space does not play a major role. Having derived the effective governing equations and suitable flow and transport properties, one has the classical description of a porous medium as a continuum. We shall, therefore, refer to various models associated with the classical description as the continuum models.
The continuum models have been widely used due to their convenience and familiarity to the engineer. They do have some limitations, one of which was noted earlier in the discussion concerning scales and averaging. They are also not well-suited for describing those phenomena in which the connectivity of the pore space or the fracture network, or that of a fluid phase, plays a major role. Continuum models also break down if there are correlations in the system with an extent that is comparable with the linear size of the porous medium.
The second class of models, the discrete models, are free of the limitations of the continuum models. They have been advanced to describe phenomena at the microscopic level and have been extended in the last decade or so years to describe various phenomena at the macroscopic and even larger scales. Their main shortcoming, from a practical point of view, is the large computational effort required for a realistic discrete treatment of the pore space. They are particularly useful when the effect of the pore space or fracture network connectivity, or the long-range correlations, is strong. The discrete models that we consider in this book are mostly based on a network representation of porous media and fracture networks. The original idea for network representation of a pore space is rather old and goes back to the early 1950s, but it was only in the early 1980s that systematic and rigorous procedures were developed to map, in principle, any disordered porous medium onto an equivalent network. Once the mapping is complete, one can study a given phenomenon in porous media in great detail.
However, only in the past 35 years have ideas from the statistical physics of disordered media been applied to flow, dispersion, and displacement processes in porous and fractured media. The concepts include percolation theory, and fractal distributions and structures that are the main tools for describing the scale-dependence of the effective properties of disordered media and how long-range correlations affect them. What we intend to do in this book is to describe and review the relevant literature on the subject, define and discuss the ideas and techniques from the statistical physics of disordered media and their applications to the processes of interest in this book, and describe the progress that has been made as a result of such applications. In particular, we emphasize the important effect of the connectivity of the pores or fractures of a porous medium on the phenomena of interest. We also describe the characterization of fractured porous media and flow and transport in the fracture networks in great detail.
In summary, within this book we study models of porous media and fractured rock, explain various experimental techniques that are used for characterizing their morphology and flow and transport therein, describe the continuum models of flow and transport in such pore media, and compare them with the predictions of the discrete models. In all cases, we contrast the classical models and techniques with the modern approaches based on the discrete models. As such, we believe that the book is unique, as it treats the subjects of porous media and fractured rock on equal footing. We hope that this book can give the reader a clear view of where we stand in the middle of 2010.
