Fractured porous media.

*(English)*Zbl 1266.74002
Oxford: Oxford University Press (ISBN 978-0-19-966651-5/hbk). viii, 175 p. (2013).

The study of fracture of a porous medium, as well as the study of flow and heat transfer through a porous medium is relevant in many practical fields such as petroleum technology and agriculture engineering. These processes have attracted the interest of many researchers because of their possible applications in many branches of science and technology. Some of the direct technological examples are drying processes, powder metallurgy, transpiration cooling, insulation of buildings and pipes, chemical catalytic reactors, geothermal energy, fiber and granular insulations, design of packed bed reactor, underground disposal of nuclear waste materials and many others.

This work presents a very careful and detailed introduction to the modern methodology of fractured porous media. Precise results in this area involve an exciting mixture of fluid mechanics, physics, chemistry, analytical, and related numerical methods. Important attention is paid to the isotropically oriented and uniformly distributed networks of fractures. The book contains 8 chapters, a preface, a notation, a list of references and an index. The description of these 8 chapters is, in short, as follows:

Chapter 1 “Introduction” describes the scope of the book with the aim to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities; description and terminology; objectives and organization of the book.

Chapter 2 “The geometry of a single fracture” presents the analysis of a fracture (three statistical characteristics of a fracture; two major autocorrelation functions and summary); generation of random fractures (generation of correlated random fields and generation of a random fracture); geometrical properties (contact zones and spatially periodic media); the concept of percolation (power laws; self-similarity and the percolation thresholds for fractures); extensions (generation of correlated fields by Fourier transforms and self-affine surfaces); exercises.

Chapter 3 “The geometry of fracture networks” contains an introduction; classical analysis of a fracture network (analysis of traces on plane surfaces and an example of three-dimensional field data); generation of a fracture network (complete analysis; deterministic models; random models; use of a library of data and some concluding remarks); statistical geometrical properties of fracture networks (a unifying concept: the excluded volume; the percolation threshold of identical convex fractures; solid blocks and concluding remarks); estimation of the dimensionless density from line data; estimation of the dimensionless density from surface date; stereological relations for convex fractures (the direct stereological relations; the inverse stereological relations and consistency relations); extensions (stereological relations for anisotropic networks of convex fractures and networks of heterogeneous fractures).

Chapters 4, 5 and 6 “Transport in a single fracture” (Chapter 4); “Transport in fracture networks” (Chapter 5) and “Transport in a fractured porous medium” (Chapter 6) describe the method of derivation of the local transmissivity of a fracture from the Stokes equations when its geometrical characteristics are known (Chapter 4); it is shown that on the Darcy scale the same equations govern phenomena such as diffusion which obey the Laplace equation on the pore scale (Chapter 5); an extension of the results to fractal porous media with power-law size distribution is presented in Chapter 6. Slightly compressible flows with application to pressure drawdown well tests are also summarized. These chapters end with a section called ‘Extensions’, which are devoted to more complex and more recent developments.

Chapters 7 “Two-phase flow through fractured porous media” is a short introduction to the field of two-phase flows through fractured porous media which are extremely important for industrial applications. The chapter contains the local equations on the Darcy scale (conservation equations and constitutive equations); numerical approach (spatial discretization and solution of the equations); regular fracture networks (multiple families of parallel plane fractures and sugar-box reservoir); isotropic and homogeneous fractured porous media (transients from various initial states; steady state macroscopic properties and influence of the parameters; comparison with a capillary dominated model).

Chapter 8 “Concluding remarks” presents an conclusion; numerical (two-dimensional meshes; three-dimensional meshes); single-phase flow; two-phase flow; and other phenomena.

The book ends with a section notation, a list of 88 references, and an index.

In the reviewer’s opinion, this book provides a solid fundamental and comprehensive presentation of the mathematical and physical theories of fractures, fracture networks, and fractured porous media. The book is excellently written and readable. Results of numerical solutions of the considered problems are given graphically and in tabular form. The book will be of great interest to a wide range of specialists working in the area of porous media, such as university students, graduate students, design engineers, physicists, chemical engineers, and also to researchers interested in the applied mathematical theory of porous media and connected topics. It can be also recommended as a text for seminars and courses as well as for an independent study. The topic of the book is exceptionally important, with fractured reservoirs containing at least half the word’s reserves of conventional oil, the possible use of fractured bedrock for nuclear waste storage as well as applications in contaminant transport and carbon dioxide storage.

However, the following basic books in fluid mechanics and porous media were not quoted in the present work:

1. [A. Nakayama, PC-Aided numerical heat transfer and convective flows. CRC Press, Tokyo (1995)].

2. [D. B. Ingham and the reviewer (eds.), Transport phenomena in porous media II. Amsterdam: Pergamon Press (2002; Zbl 1012.00023)].

3. [D. B. Ingham and the reviewer (eds.), Transport phenomena in porous media III. Elsevier, Oxford (2005)].

4. [the reviewer and D. B. Ingham, Convective heat transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon, Oxford (2001)].

5. [D. A. Nield and A. Bejan, Convection in porous media (3rd ed.). New York, NY: Springer (2006; Zbl 1256.76004)].

6. [K. Vafai (ed.), Handbook of porous media (2nd ed.). New York, NY: Marcel Dekker (2005; Zbl 0954.00016)].

7. [P. Vadasz, Emerging topics in heat and mass transfer in porous media. Springer, New York (2008)].

This work presents a very careful and detailed introduction to the modern methodology of fractured porous media. Precise results in this area involve an exciting mixture of fluid mechanics, physics, chemistry, analytical, and related numerical methods. Important attention is paid to the isotropically oriented and uniformly distributed networks of fractures. The book contains 8 chapters, a preface, a notation, a list of references and an index. The description of these 8 chapters is, in short, as follows:

Chapter 1 “Introduction” describes the scope of the book with the aim to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities; description and terminology; objectives and organization of the book.

Chapter 2 “The geometry of a single fracture” presents the analysis of a fracture (three statistical characteristics of a fracture; two major autocorrelation functions and summary); generation of random fractures (generation of correlated random fields and generation of a random fracture); geometrical properties (contact zones and spatially periodic media); the concept of percolation (power laws; self-similarity and the percolation thresholds for fractures); extensions (generation of correlated fields by Fourier transforms and self-affine surfaces); exercises.

Chapter 3 “The geometry of fracture networks” contains an introduction; classical analysis of a fracture network (analysis of traces on plane surfaces and an example of three-dimensional field data); generation of a fracture network (complete analysis; deterministic models; random models; use of a library of data and some concluding remarks); statistical geometrical properties of fracture networks (a unifying concept: the excluded volume; the percolation threshold of identical convex fractures; solid blocks and concluding remarks); estimation of the dimensionless density from line data; estimation of the dimensionless density from surface date; stereological relations for convex fractures (the direct stereological relations; the inverse stereological relations and consistency relations); extensions (stereological relations for anisotropic networks of convex fractures and networks of heterogeneous fractures).

Chapters 4, 5 and 6 “Transport in a single fracture” (Chapter 4); “Transport in fracture networks” (Chapter 5) and “Transport in a fractured porous medium” (Chapter 6) describe the method of derivation of the local transmissivity of a fracture from the Stokes equations when its geometrical characteristics are known (Chapter 4); it is shown that on the Darcy scale the same equations govern phenomena such as diffusion which obey the Laplace equation on the pore scale (Chapter 5); an extension of the results to fractal porous media with power-law size distribution is presented in Chapter 6. Slightly compressible flows with application to pressure drawdown well tests are also summarized. These chapters end with a section called ‘Extensions’, which are devoted to more complex and more recent developments.

Chapters 7 “Two-phase flow through fractured porous media” is a short introduction to the field of two-phase flows through fractured porous media which are extremely important for industrial applications. The chapter contains the local equations on the Darcy scale (conservation equations and constitutive equations); numerical approach (spatial discretization and solution of the equations); regular fracture networks (multiple families of parallel plane fractures and sugar-box reservoir); isotropic and homogeneous fractured porous media (transients from various initial states; steady state macroscopic properties and influence of the parameters; comparison with a capillary dominated model).

Chapter 8 “Concluding remarks” presents an conclusion; numerical (two-dimensional meshes; three-dimensional meshes); single-phase flow; two-phase flow; and other phenomena.

The book ends with a section notation, a list of 88 references, and an index.

In the reviewer’s opinion, this book provides a solid fundamental and comprehensive presentation of the mathematical and physical theories of fractures, fracture networks, and fractured porous media. The book is excellently written and readable. Results of numerical solutions of the considered problems are given graphically and in tabular form. The book will be of great interest to a wide range of specialists working in the area of porous media, such as university students, graduate students, design engineers, physicists, chemical engineers, and also to researchers interested in the applied mathematical theory of porous media and connected topics. It can be also recommended as a text for seminars and courses as well as for an independent study. The topic of the book is exceptionally important, with fractured reservoirs containing at least half the word’s reserves of conventional oil, the possible use of fractured bedrock for nuclear waste storage as well as applications in contaminant transport and carbon dioxide storage.

However, the following basic books in fluid mechanics and porous media were not quoted in the present work:

1. [A. Nakayama, PC-Aided numerical heat transfer and convective flows. CRC Press, Tokyo (1995)].

2. [D. B. Ingham and the reviewer (eds.), Transport phenomena in porous media II. Amsterdam: Pergamon Press (2002; Zbl 1012.00023)].

3. [D. B. Ingham and the reviewer (eds.), Transport phenomena in porous media III. Elsevier, Oxford (2005)].

4. [the reviewer and D. B. Ingham, Convective heat transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon, Oxford (2001)].

5. [D. A. Nield and A. Bejan, Convection in porous media (3rd ed.). New York, NY: Springer (2006; Zbl 1256.76004)].

6. [K. Vafai (ed.), Handbook of porous media (2nd ed.). New York, NY: Marcel Dekker (2005; Zbl 0954.00016)].

7. [P. Vadasz, Emerging topics in heat and mass transfer in porous media. Springer, New York (2008)].

Reviewer: Ioan Pop (Cluj-Napoca)

##### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

74Rxx | Fracture and damage |

76S05 | Flows in porous media; filtration; seepage |

80A20 | Heat and mass transfer, heat flow (MSC2010) |