nach oben

2024 | Buch

Kapitel lesen Erstes Kapitel lesen

Statistical Rock Physics

verfasst von: Gabor Korvin

Verlag: Springer Nature Switzerland

Buchreihe : Earth and Environmental Sciences Library

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

Einloggen, um Zugang zu erhalten

Über dieses Buch

The book is the first systematic and comprehensive treatise of stochastic models and computational tools that have emerged in rock-physics in the last 20 years. The field of statistical rock-physics is a part of rock-physics (Petrophysics). Its concepts, methods and techniques are borrowed from stochastic geometry and statistical physics. This discipline describes the interior geometry of rocks; derives their effective physical properties based on their random composition and the random arrangement of their constituents; and builds models to simulate the past geological processes that had formed the rock.

The aim of the book is to help the readers to understand the claims, techniques and published results of this new field and—most importantly—to teach them in order to creatively apply stochastic geometry and statistical physics in their own research tasks. For this purpose, the underlying mathematics will be discussed in all sections of the book; numerical solutions will be highlighted; a full set of references will be provided; and theory will go hand-in-hand with practical applications to hydraulic permeability, electric conduction, rock failure, NMR, mechanics of random grain packings, as well as the compaction of shale.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Random Functions and Random Fields, Autocorrelation Functions

Chapter Highlights

The brief survey of random functions and fields, and properties of the two-point and multi-point correlation functions (Sects. 1.1–1.3), introduces basic concepts used throughout the book. Section 1.4. describes how to estimate specific surface area and mean grain size from the 2-point correlation function.

Gabor Korvin

Chapter 2. Models of Tortuosity

Chapter Highlights

This short Chapter is devoted to the wonderful world of tortuous pathways inside the pore space of rocks, available for fluids, electric current, heat, sound waves, diffunding particles, and even to the “windings and turnings” of developing cracks around harder grains of the rock. The effect of tortuosity on permeability and other effective properties is treated in details.

Gabor Korvin

Chapter 3. The Internal Topology of Rocks

Chapter Highlights

We discuss the modern mathematical tools for studying the topology of pore space. In Sect. 3.1. Minkowski functionals are introduced through the paradigm of Mark Kac (Kac, Am Math Monthly 73:1–24, 1966) ‘Can one hear the shape of a drum?’. Euler characteristics, a powerful tool to study connectivity is treated in 3.2. The elegant and sophisticated research tool, Persistent Homology, is built up in a series of easy-to-follow steps in 3.3. The abstract constructions are illustrated with practical examples, and there are copious references in the text and in Tables (3.1, 3.4, 3.11) for their use in Petrophysics.

Gabor Korvin

Chapter 4. Random Network Models

Chapter Highlights

The common way to compute hydraulic and electric flow in porous media is to substitute the pore structure by a network consisting of nodes (for pores) and bonds (for the interconnecting throats). Different ways of constructing such models, and computation of their hydraulic and electric conductivity, are reviewed in Sect. 4.1. Section 4.2 introduces more than a dozen measures used to characterize the randomness and connectedness of such models. Section 4.3 treats three recently discovered phenomena arising in very large-scale networks: the emergence of connectivity and appearance of a giant component in evolving random networks; the scale-free distribution of the nodes’ coordination numbers; and the networks’ “small-world” property (Barabási 2018).

Gabor Korvin

Chapter 5. Menger Sponge Models

Chapter Highlights

Menger Sponge (in 3D) and Sierpiński Carpet (in 2D) are the most useful fractal models of the rocks’ pore space. Sections 5.1 and 5.2 treat these models, their combinatorial and topological properties. Random and multifractal generalizations are discussed in Sect. 5.3, where the permeability of these models is also studied. A mathematical Appendix 5.4 introduces a useful measure, lacunarity, a tool to distinguish between textures of the same fractal dimension but different geometry. Petrophysical applications are summarized in Table 5.1.

Gabor Korvin

Chapter 6. Coordination Number of Grains

Chapter Highlights

This chapter reviews the combinatorial geometry of granular packings in 2- and 3-D, and the average value of the coordination number, \(\langle Z\rangle \), for different cases. The mechanical stability of random packs requires the satisfaction of a set of linear equations, and the consistency of these equations determines the possible values of \(\langle Z\rangle \). The coordination number is also determined in the presence of friction and adhesive forces. Recent attempts to find an Equation of State (EOS) between packing fraction and mean coordination number are also discussed.

Gabor Korvin

Chapter 7. The Shape of Pebbles, Grains and Pores

Chapter Highlights

This short chapter offers a wealth of material, many of them recent developments. We discuss and tabulate (Table 7.2) the different classification schemes and numerical formulae used to describe the shape of grains, pebbles and rock fragments; acquaint the Reader with the efforts of a Hungarian research group (of Professor Gábor Domokos) to find the evolution equation that explains the characteristic ovoid shape of beach pebbles. Further topics include the number of stable and unstable points of equilibrium of a pebble or rock fragment, dependence of the settling velocity of sediments in water on particle shape, and the physical–mathematical fundamentals of the techniques measuring surface roughness and surface fractal dimension of irregular particles.

Gabor Korvin

Chapter 8. Entropy and Rock Physics

Abstract

The probabilistic concept of entropy as measure of information is introduced in Sect. 8.1 through a classic problem, “how to keep the forecaster honest?”.

Gabor Korvin

Chapter 9. Effective Properties of Rocks

Chapter Highlights

Effective Medium Theory (EMT) is a method to estimate the macroscopic properties of a composite material based upon the relative fractions and specific properties of its constituent parts. Its history, and the mathematical treatment of its different versions (Maxwell’s Approximation, the Self Consistent Method, and Differential Effective Method) are given in Sect. 9.1. Section 9.1.1 illustrates the use of EMT through (Kirkpatrick 1973) determination of the effective conductivity of a network of random conductors. Differential Effective Medium (DEM) approximation (Bruggeman 1935) is treated in Sect. 9.2, in the context of electric conductivity and elasticity. In Sect. 9.3 we discuss variational inequalities (Hashin and Shtrikman in Mech Phys Solids 1963, and in Phys Rev 1963), and a generalized mean value theorem (Korvin 1978, 1982), for effective rock properties, and present an example (Korvin 2012) for the anisotropic case. Section 9.4 reviews attempts to theoretically derive Archie’s law \({\sigma }_{rock}={\sigma }_{fluid}{\Phi }^{m}\) from basic principles, and to determine the cementation exponent m.

Gabor Korvin

Chapter 10. Markov Random Fields and Random Walks

Chapter Highlights

Markov Random Fields, which are spatial generalizations of Markov Chains in time, are mainly used in geophysical inverse problems and geologic image processing (Table 10.1). Discrete-time and continuous-time Random Walk models (Sect. 10.2, and Table 10.2) are among the most popular petrophysical simulation tools. Three important applications are discussed in details: anomalous diffusion, the NMR response of fluids in porous rocks, and an attempt (of Ioannidis et al. 1997) to derive the formation factor F in Archie’s equation. Mathematical details (such as: Derivation of the Gibbs Distribution; Proof of the Hammersley-Clifford Theorem; Polya’s Theorem and its proof; and Partial Differential Equations governing Continuous Time Random Walks) are discussed in appendices.

Gabor Korvin

Chapter 11. Thermodynamic Algorithms

Chapter Highlights

Four important computational tools are reviewed, each of them is based on thermodynamic principles. Simulated Annealing (Sect. 11.1) is presented through two applications: Reconstructing 3D pore space from 2D sections, and multiphase fluid distribution in porous rocks. Lattice Boltzman Methods (LBMs), recently widely used in Digital Rock Physics, are treated in Sect. 11.2, including their theoretical derivation, computational aspects, and generalizations. Their applications include simulation of multiphase flow, and Knudsen flow in tight shales. The powerful method of RNG (Renormalization Group, Sect. 11.3) has been borrowed from the Theory of Phase Transitions. After a theoretical introduction it is applied to determine the critical point of failure, and time of failure, of stressed rocks; to describe the erosion of tilted landscapes, and for permeability upscaling. Section 11.4 is devoted to discrete scale invariance, and its use for earthquake prediction and rock fragmentation.

Gabor Korvin

Backmatter

Titel: Statistical Rock Physics
verfasst von: Gabor Korvin
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-46700-4
Print ISBN: 978-3-031-46699-1
DOI: https://doi.org/10.1007/978-3-031-46700-4