Percolation Theory for Flow in Porous Media

Book review: Percolation Theory for Flow in Porous Media (Lecture Notes in Physics), by A Hunt (Springer, Berlin, 2005).

Muhammad Sahimi

Percolation Theory for Flow in Porous Media (Lecture Notes in Physics)Percolation theory describes the effect of the connectivity of the small-scale, or microscopic, parts of a disordered system on its large-scale, or macroscopic, properties. The disordered system could be a polymer, a porous medium, a composite material, the society, etc. It was originally invented (though not under the name percolation theory) by Paul Flory in 1941 and Walter Stockmayer in 1943 to describe how chemical gelation of monomers (the small-scale or microscopic parts) leads to, and affects, the macroscopic properties of gel polymers (Flory received the chemistry Nobel Prize later on, partly for that work).

In its formal mathematical form and under its present name, percolation theory was formally introduced by S. R. Broadbent and John Hammersley in 1957 (Broadbent had actually talked about percolation in 1954), who were interested to understand how the connectivity of the pores of a porous material affects diffusion of gases in it.

Beginning in 1971 with a seminal paper by Scott Kirkpatrick (Kirkpatrick 1971), researchers began using percolation theory to describe transport in disordered materials and media, although earlier attempts had already been made in that direction. In 1977 a seminal paper of Ronald Larson, Skip Scriven, and Ted Davis of the University of Minnesota, USA (Larson et al 1977) demonstrated how percolation theory is used to describe two-phase flow of oil and water in porous media. Since then, hundreds, perhaps thousands, of papers have been published on the application of percolation theory to fluid flow and transport in porous media.

Allen Hunt’s book (Hunt 2005) describes some of the advances in the application of percolation theory to flow in porous media. The book is centered on Hunt’s own useful contributions to the subject. In particular, it describes how a key idea, originally developed by Vinay Ambegaokar, Bertrand Halperin, and James Langer in 1971 (Ambegaokar et al 1971), called the “critical path analysis” (CPA), may be used for describing flow in saturated and unsaturated porous media.

According to the CPA, in a porous medium in which the pore sizes, or the permeabilities, are broadly distributed, only a small fraction of the pores actually contributes significantly to the flow and the medium’s effective permeability. The small pores contribute negligibly, and the flow paths that are terminated in dead-end pores are irrelevant. Thus, the effective permeability of the porous sample should be proportional to the 4th power of the radius of the smallest pore in the sample-spanning flow path that has the largest small pore.

The application of this key idea to determining the permeability and electrical conductivity of a porous medium was actually first demonstrated by Katz and Thompson (1986, 1987) and others in the mid 1980s (e.g., Sahimi 1995). Its extension to flow of polymeric fluids in porous media was first proposed by the author (Sahimi 1993). Although after reading Hunt’s book one may get the impression that he was the first to apply the CPA to flow in porous media, his contributions represent, in fact, the extension of such original ideas. Perhaps, Hunt’s most important contribution has been popularizing the application of the CPA to hydrological phenomena, and bringing the power of the ideas to the attention of that community.

The book contains eight chapters. It begins with a chapter on the essentials of percolation theory. Various concepts are described, and important ideas such as finite-size scaling, cluster statistics, and the CPA, are explained.

Chapter 2 describes some basic properties of porous media such as the porosity, moisture content, soil morphology, hysteresis, and the representative elementary volume. The last property is heavily invoked in continuum models of porous media, when one attempts to average the microscopic equations over a portion of the medium that contains representative disorder and, therefore, its inclusion in chapter 2 is useful. The strange thing about the chapter is its title “Porous Media Primer for Physicists” as if the book has been written only for them.

Chapter 3 describes several examples of the application of the CPA. It starts with two problems totally unrelated to flow in porous media. One is the so-called r-percolation, i.e., one in which the separation between any two sites of a disordered medium is a randomly distributed value r, which was the model that was originally developed by Ambegaokar et al (1971) in their study of conduction in crystalline semiconductors.

This is followed by the description of a second model, the so-called rE-percolation, which is intended for the study of conductivity in amorphous semiconductors, in which the resistance between two points is a function of two variables, the distance r between them, and the energy E that should be overcome in order to hop from one site to the other. Both r and E are distributed randomly. These are followed by the application of the CPA to two problems in porous media, namely, estimation of the saturated and unsaturated hydraulic conductivities. Even though the description of the r- and rE-percolation models is quite useful and interesting to read, it was difficult to see the relevance of the models to flow in porous media.

Chapter 4 presents a description of the basic constitutive equations for unsaturated porous media. They include relations for the hydraulic and electrical conductivities, the permeability of air, and gas and solute diffusion in porous media. Such relations are typically in the forms of power laws, in which some exponents from percolation theory appear.

Some of such exponents are “universal”, in the sense that they only depend on the spatial dimension of the porous sample, and not on the details of its structure and morphology. Some of the exponents are not, however, universal. Hunt describes the conditions under which the exponents may be non-universal. He then presents the applications of the constitutive relations to several sets of actual data for the hydraulic conductivity, and demonstrates quite convincingly the usefulness and relevance of the CPA and percolation concepts.

Chapter 5 describes the application of percolation theory to modeling of the capillary pressure as a function of saturation in porous media, which is an important characteristic of porous media. Although such modeling represents one of the earliest applications of percolation in porous media and was undertaken in the 1970s, Hunt neglects the very extensive literature on the subject and, instead, describes his own work which is much more recent. This may be viewed as a significant shortcoming of the book.

Chapter 6 describes the application of the percolation correlation length to porous media problems. The correlation length defines the appropriate length scale for the macroscopic homogeneity of a porous medium. If the linear size of a porous medium is larger than the correlation length, then, the sample can be considered as homogeneous. Otherwise, the porous medium is not macroscopically homogeneous and is, in fact, self-similar. The self-similarity has major implications for describing flow and transport in such porous media. Clearly, then, the correlation length is closely linked to the representative elementary volume mentioned above. Chapter 6 highlights the application of this important concept to porous media problems.

Chapter 7 describes the application of cluster statistics. A cluster is a set of connected elements, e.g., pores, the most important of which is the sample-spanning cluster, through which macroscopic fluid flow occurs. Since the pores are distributed randomly, the statistics of the clusters that they form is important. Similar to chapter 3, the description begins with the problem of electrical conductivity in heterogeneous solids, unrelated to flow through porous media, which is then extended to porous media, and in particular the hydraulic conductivity. The book then ends with chapter 8 which presents a brief description of the ideas described in the previous chapters on porous media that contain a spectrum of heterogeneities—the so-called multi-scale porous media, which contain different types of heterogeneities at distinct length scales.

Overall, the book has two main features. One is that it presents the applications of percolation theory and the CPA to some problems in porous media, and compares the predictions with real data. This is the strength of the book. The second feature of the book is that it largely neglects a very significant part of the relevant literature on the applications of percolation theory to porous media which has, in fact, transformed percolation theory from an abstract mathematical theory to a useful and practical tool. As an example, no mention is made of the landmark paper of Larson et al. (1977).

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