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

[1] We simulate two-fluid-phase flow at the pore scale using a lattice Boltzmann (LB) approach. Using a parallel processing version of the Shan-Chen model that we developed, we simulate a set of ideal two-fluid systems and a model two-fluid-phase porous medium system comprised of a synthetic packing with a relatively uniform distribution of spheres. We use the set of ideal two-phase systems to validate the approach and provide parameter information, which we then use to simulate a sphere-pack system. The sphere-pack system is designed to mimic laboratory experiments conducted to evaluate the hysteretic capillary pressure saturation relation for a system consisting of water, tetrachloroethylene, and a glass bead porous medium. Good agreement is achieved between the measured hysteretic capillary pressure saturation relations and the LB simulations when comparing entry pressure, displacement slopes, irreducible saturation, and residual entrapment. Our results further show that while qualitatively similar results are obtained when comparing systems consisting of 1200 spheres and 150 spheres, there is a significant difference between these two levels, suggesting a lower bound on the size of a representative elementary volume.

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Lattice‐Boltzmann simulation of two‐phase flow in porous media

In this literature survey, different aspects of dynamics of two-phase flow in porous media are discussed. This review is based on the results of developed dynamic pore-network models and their applications. Thus, those concepts of dynamics of two-phase systems are addressed that have been already discussed in previous studies. Since it is not always possible to study different aspects of laboratory experiments, dynamic pore-network models were developed to gain new insights into the process. This characteristic is the major advantage of pore-network models, which give a better understanding of the physics of a process at pore scale as well as at the scale of representative elementary volume. Dynamic pore-network models are reviewed under different classifications; structure, computational algorithm, and local rules and applications.

Analysis of Fundamentals of Two-Phase Flow in Porous Media Using Dynamic Pore-Network Models: A Review

The objective of this work is to assess the potential significance of deviations from local equilibrium for the exchange of mass between residual nonaqueous phase liquids and the aqueous phase in the saturated groundwater zone. A one-dimensional convection-dispersion mass balance equation incorporating a first-order interphase mass transfer rate relationship and temporal changes in blob configuration is used to model this system. Analytical and numerical methods are employed to examine the steady state and transient behavior of the system under a variety of hypothetical aquifer conditions and pumping remediation schemes. Sensitivity of the model to several parameters including mass transfer coefficient, blob size and shape, and Darcy velocity is explored. Results of the theoretical assessment indicate that nonequilibrium effects could play a significant role in some contamination scenarios, primarily for large blob sizes and relatively high velocities. Design of soil flushing techniques will be impacted by these conclusions. Uncertainty in several parameter values used in this analysis indicate the need for further experimental investigation of this process.

Theoretical Study of the Significance of Nonequilibrium Dissolution of Nonaqueous Phase Liquids in Subsurface Systems

Pore-Scale Viscous Fingering in Porous Media

To explore how the microscopic geometry of a pore space affects the macroscopic characteristics of fluid flow in porous media, the authors have used approximate solutions of the Navier-Stokes equations to calculate the flow of two fluids in random networks. The model pore space consists of an array of pores of variable radius connected by throats of variable length and radius to a random number of nearest neighbors. The various size and connectedness distributions may be arbitrarily assigned, as are the wetting characteristics of the two fluids in the pore space. The fluids are assumed to be incompressible, immiscible, Newtonian, and of equal viscosity. In the calculation, the authors use Stokes flow results for the motion of the individual fluids and incorporate microscopic capillary force via the Washburn approximation. At any time, the problem is mathematically identical to a random electrical network of resistors, batteries and diodes. From the numerical solution of the latter, the authors compute the fluid velocities and saturation rates of change, and use a discrete time-stepping procedure to follow the subsequent motion. The scale of the computation has so far restricted the authors to networks of modest size (100-400 pores) in two dimensions. Within these limitations,more » the authors discuss the dependence of residual oil saturations and interface shapes on network geometry and flow conditions.« less

J. Koplik

Papers

Two-phase flow in random network models of porous media