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

This paper discusses migration of radionuclides in the bedrock surrounding a repository. Currently available models use either a surface reaction or a bulk reaction concept to describe the retardation of migrating nuclides. The first model assumes that the nuclide reacts only with the surface of the fissures. This implies that the rock matrix is not utilized as a sink. The other model implies that the whole bulk of the rock is accessible to the nuclides. The paper analyzes the accessibility of the rock matrix to the radio-nuclides. The transport mechanisms are shown to be flow of water and nuclides in the fissures and transport of nuclides from the water in the fissures into water in the microfissures of the rock by pore diffusion. The diffusion of the nuclides into the rock matrix and their sorption onto the surfaces of the microfissures are the main mechanisms retarding migration from a repository. The diffusivity of the nuclide may be as important as its sorption equilibrium constant. Diffusivities in the pores and microfissures in such dense rocks as granite under confining pressure of hundreds of bars can be expected to be 6–20% of the diffusivity in water. These data are obtained from electrical resistivity measurements of saltwater-filled granites. Porosity of such granites varies from 0.4 to 0.9%. The apparent diffusivities in the granites will then vary between 0.25 · 10−12/Kdρp and 10 · 10−12/Kdρp m2/s, where Kdρp is the volume equilibrium constant. This varies from the porosity of the rock for nonsorbing species to up to and over 104. For a 100-year contact time a nonsorbing nuclide can be expected to penetrate tens of meters of the rock matrix and a strongly sorbing nuclide with Kdρp larger than 104 will penetrate a few millimeters. The diffusion into the rock matrix can enhance the retardation by many orders of magnitude as compared to retardation by surface reaction in fissures only. The retardation may, on the other hand, be many orders of magnitude smaller than the maximum value that could be obtained if all the rock matrix were accessible. This depends very much on the fissure widths and spacings.

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Diffusion in the rock matrix: An important factor in radionuclide retardation?

Effect of pore pressure and confining pressure on fracture permeability

Permeability of crystalline and argillaceous rocks

Permeability and diffusivity are critical parameters of tight reservoir rocks that determine their viability for commercial development. Current methods for measuring permeability and/or diffusivity may lead to erroneous results when applied to very tight rocks including gas shales, coal, and tight gas sands, as well as rocks considered as seals for nuclear waste repositories and strata for geological sequestration of CO2. The use of He as routinely applied to measure porosity, permeability, and diffusivity may result in non-systematic errors because of the molecular sieving effect of the fine pore structure to larger molecules such as reservoir gases. Utilizing gases with larger adsorption potentials than He, such as N2, and including all reservoir gases to measure porosity or permeability of rocks with high surface area is a viable alternative, but requires correcting for adsorption in the analyses. This study expands several approaches to measure permeability and diffusivity with considerations for gas adsorption, which has not been explicitly considered in previous studies. We present new models that explicitly correct for adsorption during pulse-decay measurements of core under reservoir conditions, as well as on crushed samples used to approximate permeability or diffusivity. We also present a method to determine permeability or diffusivity from on-site drill-core desorption test data as carried out to determine gas in place in coals or gas shales. Our new approach utilizes late-time data from experimental pressure-decay tests, which we show to be more reliable and theoretically (and practically) accurate than the early-time approach commonly used to estimate gas-transport properties.

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1468-8123.2009.00244.x

Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications

The permeability of Westerly granite was measured as a function of effective pressure to 4 kb. A transient method was used, in which the decay of a small incremental change of pressure was observed; decay characteristics, when combined with dimensions of the sample and compressibility and viscosity of the fluid (water or argon) yielded permeability, k. k of the granite ranged from 350 nd (nanodarcy = 10−17 cm2) at 100-bar pressure to 4 nd at 4000 bars. Based on linear decay characteristics, Darcy's law apparently held even at this lowest value. Both k and electrical resistivity, ρs, of Westerly granite vary markedly with pressure, and the two are closely related by k = Cρs−1.5±0.1, where C is a constant. With this relationship, an extrapolated value of k at 10-kb pressure would be about 0.5 nd. This value is roughly equivalent to flow rates involved in solute diffusion but is still a great deal more rapid than volume diffusion. Measured permeability and porosity enable hydraulic radius and, hence, the shape of pore spaces in the granite to be estimated. The shapes (flat slits at low pressure, equidimensional pores at high pressure) are consistent with those deduced from elastic characteristics of the rock. From the strong dependence of k on effective pressure, rocks subject to high pore pressure will probably be relatively permeable.

J. B. Walsh

Papers

Permeability of granite under high pressure