In this paper, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed. Both macroscopically homogeneous and heterogeneous rocks are considered. The latter are characterized by large-scale spatial variations and correlations in their effective properties and include rocks that may be characterized by several distinct degrees of porosity, a well-known example of which is a fractured rock with two degrees of porosity---those of the pores and of the fractures. First, the diagenetic processes that give rise to the present reservoir rocks are discussed and a few geometrical models of such processes are described. Then, measurement and characterization of important properties, such as pore-size distribution, pore-space topology, and pore surface roughness, and morphological properties of fracture networks are discussed. It is shown that fractal and percolation concepts play important roles in the characterization of rocks, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Next, various structural models of homogeneous and heterogeneous rock are discussed, and theoretical and computer simulation approaches to flow, dispersion, and displacement in such systems are reviewed. Two different modeling approaches to these phenomena are compared. The first approach is based on the classical equations of transport supplemented with constitutive equations describing the transport and other important coefficients and parameters. These are called the continuum models. The second approach is based on network models of pore space and fractured rocks; it models the phenomena at the smallest scale, a pore or fracture, and then employs large-scale simulation and modern concepts of the statistical physics of disordered systems, such as scaling and universality, to obtain the macroscopic properties of the system. The fundamental roles of the interconnectivity of the rock and its wetting properties in dispersion and two-phase flows, and those of microscopic and macroscopic heterogeneities in miscible displacements are emphasized. Two important conceptual advances for modeling fractured rocks and studying flow phenomena in porous media are also discussed. The first, based on cellular automata, can in principle be used for computing macroscopic properties of flow phenomena in any porous medium, regardless of the complexity of its structure. The second, simulated annealing, borrowed from optimization processes and the statistical mechanics of spin glasses, is used for finding the optimum structure of a fractured reservoir that honors a limited amount of experimental data.

Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing

Porosity plays a clearly important role in geology. It controls fluid storage in aquifers, oil and gas fields and geothermal systems, and the extent and connectivity of the pore structure control fluid flow and transport through geological formations, as well as the relationship between the properties of individual minerals and the bulk properties of the rock. In order to quantify the relationships between porosity, storage, transport and rock properties, however, the pore structure must be measured and quantitatively described. The overall importance of porosity, at least with respect to the use of rocks as building stone was recognized by TS Hunt in his “Chemical and Geological Essays” (1875, reviewed by JD Dana 1875) who noted:

> “Other things being equal, it may properly be said that the value of a stone for building purposes is inversely as its porosity or absorbing power.”

In a Geological Survey report prepared for the U.S. Atomic Energy Commission, Manger (1963) summarized porosity and bulk density measurements for sedimentary rocks. He tabulated more than 900 items of porosity and bulk density data for sedimentary rocks with up to 2,109 porosity determinations per item. Amongst these he summarized several early studies, including those of Schwarz (1870–1871), Cook (1878), Wheeler (1896), Buckley (1898), Gary (1898), Moore (1904), Fuller (1906), Sorby (1908), Hirschwald (1912), Grubenmann et al. (1915), and Kessler (1919), many of which were concerned with rocks and clays of commercial utility. There have, of course, been many more such determinations since that time.

There are a large number of methods for quantifying porosity, and an increasingly complex idea of what it means to do so. Manger (1963) listed the techniques by which the porosity determinations he summarized were made. He separated these into seven methods for …

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Characterization and Analysis of Porosity and Pore Structures

The purpose of the present article is to review recent advances made in the determination and calculation of improved bounds on the effective properties of random heterogeneous media that depend upon the microstructure via n-point correlation functions. New breakthroughs made in the quantitative characterization of the microstructure of heterogeneous materials are also reviewed. The following four different effective properties shall be studied: (i) effective conductivity tensor (which includes, by mathematical analogy, the dielectric constant, magnetic permeability, and diffusion coefficient); (ii) effective stiffness tensor; (iii) diffusioncontrolled trapping constant; and (iv) fluid permeability tensor. It shall be demonstrated that improved upper and lower bounds can provide a relatively sharp estimate of the effective property even when the bounds diverge from one another. Although this article reviews stateof-the-art advances in the field, an attempt will be made to elucidate methods and principles for the nonexpert.

Random Heterogeneous Media: Microstructure and Improved Bounds on Effective Properties

The accelerating rate at which new materials are appearing, and transforming the engineering world, only serves to emphasize the vast potential for novel material structure, and related performance. Microstructure-sensitive design (MSD) aims at providing inverse design methodologies that facilitate design of material internal structure for performance optimization. Spectral methods are applied across the structure, property and processing design spaces in order to compress the computational requirements for linkages between the spaces and enable inverse design. Research has focused mainly on anisotropic, polycrystalline materials, where control of local crystal orientation can result in a broad range of property combinations. This review presents the MSD framework in the context of both the engineering advances that have led to its creation, and those that complement or provide alternative methods for design of materials (meaning ‘optimization of material structure’ in this context). A variety of definitions for the structure of materials are presented, with an emphasis on correlation functions; and spectral methods are introduced for compact descriptions and efficient computations. The microstructure hull is defined as the design space for structure in the spectral framework. Reconstruction methods provide invertible links between statistical descriptions of structure, and deterministic instantiations. Subsequently, structure–property relations are reviewed, and again subjected to representation via spectral methods. The concept of a property closure is introduced as the design space for performance optimization, and methods for moving between the closures and hulls are presented as the basis for the subsequent discussion on microstructure design. Finally, the spectral framework is applied to deformation processes, and methodologies that facilitate process design are reviewed.

Microstructure Sensitive Design for Performance Optimization

http://members.unine.ch/philippe.renard/articles/renard2013.pdf

Connectivity metrics for subsurface flow and transport

We introduce a two‐point cluster function C2(r1,r2) which reflects information about clustering in general continuum–percolation models. Specifically, for any two‐phase disordered medium, C2(r1,r2) gives the probability of finding both points r1 and r2 in the same cluster of one of the phases. For distributions of identical inclusions whose coordiantes are fully specified by center‐of‐mass positions (e.g., disks, spheres, oriented squares, cubes, ellipses, or ellipsoids, etc.), we obtain a series representation of C2 which enables one to compute the two‐point cluster function. Some general asymptotic properties of C2 for such models are discussed. The two‐point cluster function is then computed for the adhesive‐sphere model of Baxter. The two‐point cluster function for arbitrary media provides a better signature of the microstructure than does a commonly employed two‐point correlation function defined in the text.

Two‐point cluster function for continuum percolation

A bound on the fluid permeability k for viscous flow through a random array of N identical spherical particles (distributed with arbitrary degree of impenetrability) due to Weissberg and Prager [Phys. Fluids 13, 2958 (1970)] is rederived by averaging flow quantities with respect to the ensemble of particle configurations. The ensemble‐averaging technique enables one to obtain series representations of certain n‐point distribution functions that arise in terms of probability density functions which characterize a particular configuration of n(<N) spheres. The Weissberg–Prager bound on k is computed exactly through second order in the sphere volume fraction for arbitrary λ (where λ is the impenetrability parameter, 0≤λ≤1) for two different interpenetrable‐sphere models. It is found that, at the same sphere volume fraction, the permeability of an assembly of partially overlapping spheres is greater than that of one characterized by a higher degree of impenetrability. The results of this study indicate that the Weissberg–Prager bound will not only yield the best available bound on k for an assemblage of totally impenetrable spheres (λ=1) but will provide a useful estimate of k for this model for a wide range of sphere volume fractions. It is also demonstrated that bounds which incorporate a certain level of statistical information on the medium are not always necessarily sharper than bounds which involve less information.

Bounds on the permeability of a random array of partially penetrable spheres

The recently derived variational principle of Rubinstein and Torquato (submitted to J. Fluid Mech.) is applied to obtain new rigorous two‐ and three‐point upper bounds on the fluid permeability k for slow viscous flow around a random array of identical spheres which may penetrate one another in varying degrees. The n‐point bounds involve up to n‐point correlation function information. Both bounds are simplified and computed for the special case of mutually impenetrable spheres for a wide range of sphere volume fractions. The three‐point bound is sharp and provides significant improvement over the two‐point bound, especially at high sphere volume fractions (low porosities). It is the sharpest upper bound on k for a random array of impenetrable spheres developed to date and begins to approach the Kozeny–Carman empirical relation at low porosities.

J. D. Beasley

Papers

Two‐point cluster function for continuum percolation

Bounds on the permeability of a random array of partially penetrable spheres

New bounds on the permeability of a random array of spheres