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

Enhanced oil recovery (EOR) techniques can significantly extend global oil reserves once oil prices are high enough to make these techniques economic. Given a broad consensus that we have entered a period of supply constraints, operators can at last plan on the assumption that the oil price is likely to remain relatively high. This, coupled with the realization that new giant fields are becoming increasingly difficult to find, is creating the conditions for extensive deployment of EOR. This paper provides a comprehensive overview of the nature, status and prospects for EOR technologies. It explains why the average oil recovery factor worldwide is only between 20% and 40%, describes the factors that contribute to these low recoveries and indicates which of those factors EOR techniques can affect. The paper then summarizes the breadth of EOR processes, the history of their application and their current status. It introduces two new EOR technologies that are beginning to be deployed and which look set to enter mainstream application. Examples of existing EOR projects in the mature oil province of the North Sea are discussed. It concludes by summarizing the future opportunities for the development and deployment of EOR.

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Recovery rates, enhanced oil recovery and technological limits

A detailed two-part computational investigation is conducted into the dynamical evolution of two-dimensional miscible porous media flows in the quarter five-spot arrangement of injection and production wells. High-accuracy direct numerical simulations are performed that reproduce all dynamically relevant length scales in solving the vorticity–streamfunction formulation of Darcy's law. The accuracy of the method is demonstrated by a comparison of simulation data with linear stability results for radial source flow.Within this part, Part 1 of the present investigation, a series of simulations is discussed that demonstrate how the mobility ratio and the dimensionless flow rate denoted by the Peclet number Pe affect both local and integral features of homogeneous displacement processes. Mobility ratios up to 150 and Pe-values up to 2000 are investigated. For sufficiently large Pe-values, the flow near the injection well gives rise to a vigorous viscous fingering instability. As the unstable concentration front approaches the central region of the domain, nonlinear interactions between the fingers similar to those known from unidirectional flows are observed, such as merging, partial merging, and shielding, along with secondary tip-splitting and side-branching instabilities. At large Pe-values, several of these fingers compete for long times, before one of them accelerates ahead of the others and leads to the breakthrough of the front.In contrast to unidirectional flows, the quarter five-spot geometry imposes both an external length scale and a time scale on the flow. The resulting spatial non-uniformity of the potential base flow is observed to lead to a clear separation in space and time of large and small scales in the flow. Small scales occur predominantly during the early stages near the injection well, and at late times near the production well. The central domain is dominated by larger scales.Taken together, the results of the simulations demonstrate that both the mobility ratio and Pe strongly affect the dynamics of the flow. While some integral measures, such as the recovery at breakthrough, may show only a weak dependence on Pe for large Pe-values, the local fingering dynamics continue to change with Pe.The increased susceptibility of the flow to perturbations during the early stages provides the motivation to formulate an optimization problem that attempts to maximize recovery, for a constant overall process time, by employing a time-dependent flow rate. Within the present framework, which accounts for molecular diffusion but not for velocity-dependent dispersion, simulation results indeed indicate the potential to increase recovery by reducing the flow rate at early times, and then increasing it during the later stages.

Miscible porous media displacements in the quarter five-spot configuration. Part 1. The homogeneous case

Interfacial or fingering instabilities have been studied recently in relation to contamination problems where a more dense plume is enclosed by and is moving along in a body of less dense fluid. Instabilities can play an important role in the mixing or dispersion process. Through the use of a variable density flow and transport code, we were able to study how the style of interfacial perturbation controls the pattern of instability development. Whether initial perturbations grow or decay depends mainly on the wavelength of the perturbing function. A critical perturbation wavelength must be exceeded for a perturbation to grow; otherwise the perturbation simply decays. Our work confirms earlier analyses that suggest that all stratified systems are inherently unstable, given some spectrum of the perturbing waves that exceed the critical wavelength. By implication, Rayleigh number stability criteria are inappropriate for evaluating the dense plume problem. Our study also demonstrates how numerical errors in a mass transport code can serve as a perturbing function and lead to the development of instabilities. However, these instabilities are not physically realistic and are essentially uncontrollable because their character depends on the extent to which numerical errors develop, as evidenced by the grid Peclet and Courant numbers.

On the generation of instabilities in variable density flow

Viscous fingering in miscible displacements in the presence of permeability heterogeneities is studied using two‐dimensional simulations. The heterogeneities are modeled as stationary random functions of space with finite correlation scale. Both the variance and scale of the heterogeneities are varied over modest ranges. It is found that the fingered zone grows linearly in time in a fashion analogous to that found in homogeneous media by Tan and Homsy [Phys. Fluids 31, 1330 (1988)], indicating a close coupling between viscous fingering on the one hand and flow through preferentially more permeable paths on the other. The growth rate of the mixing zone increases monotonically with the variance of the heterogeneity, as expected, but shows a maximum as the correlation scale is varied. The latter is explained as a ‘‘resonance’’ between the natural scale of fingers in homogeneous media and the correlation scale.

Viscous fingering with permeability heterogeneity

This paper describes applications of linear and nonlinear simulations to unstable miscible flooding. The first section describes a method of calculating linear growth of unstable modes by use of finite differences in the direction of flow and Fourier decomposition perpendicular to flow. This work extends the previous long- and short-wavelength analytic results to cover the whole wave-number range. Results obtained are used to help validate a two-dimensional (2D) code to study nonlinear evolution of viscous fingers and to identify likely fingering regimes in the computed solution by identifying the range of wave numbers that dominates the linear growth behavior. The second section describes the numerical scheme for calculations on a fine grid of the nonlinear development of an instability. Results are presented on calculations of nonlinear growth at several mobility ratios and levels of diffusion. Comparisons with Blackwell's experimental data are presented. Good agreement is obtained, suggesting that the physical processes governing fingering are being correctly modeled.

D.J. Bond

Papers

Detailed Simulation of Unstable Processes in Miscible Flooding