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

In this paper, the concept of characteristic length introduced in the definition of the dynamic tortuosity by Johnson, Koplik, and Dashen [J. Fluid Mech. 176, 379 (1987)] is extended to express the frequency dependence of the bulk modulus of the saturating fluid at high frequencies. A general phenomenological frequency dependence for this dynamic bulk modulus is obtained using the expression for the dynamic tortuosity. The theoretical predictions for dynamic tortuosity and bulk modulus are compared with experimental results obtained from acoustic measurements on a rigid‐frame porous material saturated with air.

Dynamic tortuosity and bulk modulus in air‐saturated porous media

This review of process analytical chemistry is an update to the previous review on this subject published in 1995(A2). The time period covered for this review includes publications written or published from late 1994 until early 1999, with the addition of a few classic references pointing to background information critical to an understanding of a specific topic area. These older references have been critically included as established fundamental works. New topics covered in this review not previously treated as separate subjects in past reviews include sampling systems, imaging (via optical spectroscopy), and ultrasonic analysis.

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Process Analytical Chemistry

Permeability is a critical parameter for the petroleum geologist. By simulating the processes of compaction and cementation in a model porous medium, we have gained a new understanding of how permeability is controlled in reservoir sandstones. This understanding can be used predictively for simple sandstones. The geometry of the model pore system is completely defined, so permeability can be calculated directly using a network model for flow. The calculation is based on first principles and is physically rigorous. In contrast to many previous efforts to predict permeability, there are no adjustable parameters in the calculations and no additional measurements or correlations (e.g., capillary pressure data or pore system data from thin sections) are required. The model-derived porosity-permeability trend for a compacted or quartz cemented sandstone, or a sandstone having a combination of these processes, closely matches measurements on Fontainebleau sandstone samples whose permeabilities span nearly five orders of magnitude. The model also correctly predicts mercury injection measurements of pore throat size distribution for the Fontainebleau sandstone. Pore-scale geometric features of the model are found to be spatially correlated, and this departure from randomness significantly affects macroscopic properties such as permeability. The agreement between predictions and measurements suggests that spatial correlation is inherent in granular porous media and consequently, uncorrelated (or arbitrarily correlated) models of transport in such media are unlikely to be physically representative. We also discuss extending the model to predict properties of more complicated rocks.

Permeability Prediction from Geologic Models

The acoustic properties of first and second sound in He ii are used to probe the dynamic permeability k(ω), of porous media whose pores are large enough that the bulk properties of 4He are relevant. The measurements of phase velocity and attenuation, considered as functions of frequency and temperature, are in excellent agreement with the predictions based on a model k(ω) which involves four parameters: φ (porosity), α∞ (tortuosity), k0 (dc permeability), and Λ (dynamically connected pore size). The first three of these are measured by independent means; accurate values of Λ are extracted directly from the measurements in the high‐frequency region of the data where the observed temperature and frequency dependences of the modes are precisely those predicted on theoretical grounds. It is shown empirically that Λ is in fact closely related to the dc permeability k0 for all known data sets, thus supporting an earlier conjecture based on theoretical ideas.

Probing porous media with first and second sound. I. Dynamic permeability

We calculate from first principles the frequency-dependent permeability $\ensuremath{\kappa}(\ensuremath{\omega})$ in porous media for a variety of microstructures. It is found that when $\ensuremath{\kappa}(\ensuremath{\omega})$ is scaled by its static value ${\ensuremath{\kappa}}_{0}$ and the frequency by a characteristic ${\ensuremath{\omega}}_{c}$ particular to the sample, the resulting function exhibits excellent universal behavior independent of microstructure. We advance arguments that delineate the physical reason for the scaling behavior as well as the condition for its validity. Our predictions are supported by observed correlations in sedimentary rocks and experimental $\ensuremath{\kappa}(\ensuremath{\omega})$ measurements.

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Min-Yau Zhou

Papers

Dynamic permeability in porous media.