Capillary displacement and percolation in porous media
TL;DR: In this article, the authors consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate and find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic size distribution of the lattice elements.
Abstract: We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.
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TL;DR: In this paper, a new kind of percolation problem is described which differs from ordinary percolations in that it automatically finds the critical points of the system and is called invasion percolating.
Abstract: A new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system. The model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces, but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance. The name invasion percolation is proposed for this new process. Similarities to, and differences from, ordinary percolation theory are discussed.
1,151 citations
TL;DR: In this article, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed, and two different modeling approaches to these phenomena are compared.
Abstract: 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.
946 citations
TL;DR: In this paper, the mechanisms of displacement of one fluid by another were investigated in an etched network, where both fluids are simultaneously present in a duct, the wetting fluid remaining in the extreme corners of the cross-section.
Abstract: The mechanisms of displacement of one fluid by another are investigated in an etched network.Experiments show that both fluids are simultaneously present in a duct, the wetting fluid remaining in the extreme corners of the cross-section. Calculation of displacement pressures are in good agreement with experiments for drainage, imbibition and removal of blobs. The results may be related to some flow behaviour exhibited in porous media.
773 citations
Cites methods from "Capillary displacement and percolat..."
...…& Gulshan 1981 ; Koplik 1982) or using a statistical ‘ percolation-type ’ approach (Chatzis & Dullien 1977; de Gennes & Guyon 1978; Golden 1980; Lenormand & Bories 1980; Chandler et al. 1982; Larson, Scriven & Davis 1981 ; de Gennes 1982 private communication; Chatzis & Dullien 1982)....
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Book•
01 May 1985TL;DR: Six kinetic growth models that have witnessed an explosion of recent activity are defined and some of the approaches used to study these models are described, with emphasis on the re normalization group approach being developed by us.
Abstract: We define six kinetic growth models that have witnessed an explosion of recent activity:
1
Cancer Growth (Eden)
2
Colloid Growth (Langer-Muller-Krumbhaar; Witten-Sander)
3
Breakdown (Sawada et al.)
4
Crystallization (Rikvold)
5
Invasion Percolation (Schlumberger group)
6
Addition Polymerization (Manneville-de Seze; Herrmann et al.)
We also describe briefly some of the approaches used to study these models, with emphasis on the re normalization group approach being developed by us.
570 citations
TL;DR: A predictive calculation of two-phase relative permeabilities in granular porous media formed from a dense random packing of equal spheres, enabling the microstructure of the medium to be completely determined is presented.
Abstract: We present a predictive calculation of two-phase relative permeabilities in granular porous media formed from a dense random packing of equal spheres. The spatial coordinates of every sphere in the pack have been measured, enabling the microstructure of the medium to be completely determined. From these data we extract a network model that replicates the pore space. By compacting the packing or swelling individual spheres, we may generate model porous media of different porosities whose microstructure is also completely determined. We simulate both viscous- and capillary-dominated invasion of a nonwetting fluid into a wetting fluid contained in these media. During invasion we calculate the average hydraulic conductance of each phase to obtain the relative permeabilities as a function of fluid saturation. Because the microstructure is known, the calculations do not involve any adjustable parameters or supplementary measurements of pore structure. The computed relative permeabilities are successfully compared with experimental values previously measured on sand packs, bead packs, and a simple sandstone.
475 citations
References
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01 Jan 1973
4,243 citations
01 Jul 1957
TL;DR: In this paper, the authors study how the random properties of a medium influence the percolation of a fluid through it, in a general way, in which the treatment diifers from conventional diffusion theory.
Abstract: The paper studies, in a general way, how the random properties of a ‘medium’ influence the percolation of a ‘fluid’ through it. The treatment diifers from conventional diffusion theory, in which it is the random properties of the fluid that matter. Fluid and medium bear general interpretations: for example, solute diffusing through solvent, electrons migrating over an atomic lattice, molecules penetrating a porous solid, disease infecting a community, etc.
1,707 citations
TL;DR: Percolation theory, the theory of the properties of classical particles interacting with a random medium, is of wide applicability and provides a simple picture exhibiting critical behaviour, the features of which are well understood and amenable to detailed calculation.
Abstract: Percolation theory, the theory of the properties of classical particles interacting with a random medium, is of wide applicability and provides a simple picture exhibiting critical behaviour, the features of which are well understood and amenable to detailed calculation In this review the concepts of percolation theory and the general features associated with the critical region about the onset of percolation are developed in detail In particular, several dimensional invariants are examined which make it possible to unify much of the available information, and to extend the insights of percolation theory to processes which have not yet received numerical study The compilation of the results of percolation theory, both exact and numerical, is believed to be complete through 1970 A selective bibliography is given In a concluding chapter several recent applications of percolation theory to classical and to quantum mechanical problems are discussed
1,051 citations
01 Jul 1980
TL;DR: The problem of surface detection is translated into a problem of traversal of a directed graph, G, and it has been proven that connected subgraphs of G correspond to surfaces of connected components of Q, which allow the number of marked nodes to be kept to a small fraction of the total number of visited nodes.
Abstract: In many three-dimensional imaging applications the three-dimensional scene is represented by a three-dimensional array of volume elements, or voxels for short. A subset Q of the voxels is specified by some property. The objects in the scene are then defined as subsets of Q formed by voxels which are “connected” in some appropriate sense. It is often of interest to detect and display the surface of an object in the scene, specified say by one of the voxels in it.In this paper, the problem of surface detection is translated into a problem of traversal of a directed graph, G. The nodes of G correspond to faces separating voxels in Q from voxels not in Q. It has been proven that connected subgraphs of G correspond to surfaces of connected components of Q (i.e., of objects in the scene). Further properties of the directed graph have been proven, which allow us to keep the number of marked nodes (needed to avoid loops in the graph traversal) to a small fraction of the total number of visited nodes.This boundary detection algorithm has been implemented. We discuss the interaction between the underlying mathematical theory and the design of the working software. We illustrate the software on some clinical studies in which the input is computed tomographic (CT) data and the output is dynamically rotating three-dimensional displays of isolated organs. Even though the medical application leads to very large scale problems, our theory and design allows us to use our method routinely on the minicomputer of a CT scanner.
189 citations
TL;DR: In this paper, a new mathematical model for predicting permeabilities of porous media has been developed, which consists of a set of cubic networks of arbitrary orientations with respect to the macroscopic flow direction.
Abstract: In this study a new mathematical model for predicting permeabilities of porous media has been developed. The model consists of a set of cubic networks of arbitrary orientations with respect to the macroscopic flow direction. The networks consist of capillary tubes which are made up of segments of different diameters. It is shown that a cubic network of capillaries has isotropic permeability properties. The permeability model requires two pore size distributions and the porosity of the sample for the calculations. The calculated permeabilities agreed with experimental results for a wide range of permeabilities of highly compacted materials to within ± 23%, without using adjustable tortuosity factors.
120 citations