The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

Lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) are relatively new andpromising methods for the numerical solution of nonlinear partial differential equations. The bookprovides an introduction for graduate students and researchers. Working knowledge of calculus isrequired and experience in PDEs and fluid dynamics is recommended. Some peculiarities of cellularautomata are outlined in Chapter 2. The properties of various LGCA and special coding techniquesare discussed in Chapter 3. Concepts from statistical mechanics (Chapter 4) provide the necessarytheoretical background for LGCA and LBM. The properties of lattice Boltzmann models and amethod for their construction are presented in Chapter 5.

/pdf/lattice-gas-cellular-automata-and-lattice-boltzmann-models-2kkc77jjx8.pdf

Lattice-Gas Cellular Automata and Lattice Boltzmann Models

Mecanisme de digitation visqueuse. Ecoulements de Hele Shaw. Deplacements non miscibles en cellules de Hele Shaw

Viscous fingering in porous media

The concept of self-organized criticality was introduced to explain the behaviour of the sandpile model. In this model, particles are randomly dropped onto a square grid of boxes. When a box accumulates four particles they are redistributed to the four adjacent boxes or lost off the edge of the grid. Redistributions can lead to further instabilities with the possibility of more particles being lost from the grid, contributing to the size of each ‘avalanche’. These model ‘avalanches’ satisfied a power-law frequency‐area distribution with a slope near unity. Other cellular-automata models, including the slider-block and forest-fire models, are also said to exhibit self-organized critical behaviour. It has been argued that earthquakes, landslides, forest fires, and species extinctions are examples of self-organized criticality in nature. In addition, wars and stock market crashes have been associated with this behaviour. The forest-fire model is particularly interesting in terms of its relation to the critical-point behaviour of the sitepercolation model. In the basic forest-fire model, trees are randomly planted on a grid of points. Periodically in time, sparks are randomly dropped on the grid. If a spark drops on a tree, that tree and adjacent trees burn in a model fire. The fires are the ‘avalanches’ and they are found to satisfy power-law frequency‐area distributions with slopes near unity. This forest-fire model is closely related to the site-percolation model, that exhibits critical behaviour. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi steady-state cascade gives a power-law frequency‐area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees.

/pdf/self-organized-criticality-4v87dkoa15.pdf

Self-organized criticality

Immiscible displacements in porous media with both capillary and viscous effects can be characterized by two dimensionless numbers, the capillary number C, which is the ratio of viscous forces to capillary forces, and the ratio M of the two viscosities. For certain values of these numbers, either viscous or capillary forces dominate and displacement takes one of the basic forms: (a) viscous fingering, (b) capillary fingering or (c) stable displacement. We present a study in the simple case of injection of a non-wetting fluid into a two-dimensional porous medium made of interconnected capillaries. The first part of this paper presents the results of network simulators (100 × 100 and 25 × 25 pores) based on the physical rules of the displacement at the pore scale. The second part describes a series of experiments performed in transparent etched networks. Both the computer simulations and the experiments cover a range of several decades in C and M. They clearly show the existence of the three basic domains (capillary fingering, viscous fingering and stable displacement) within which the patterns remain unchanged. The domains of validity of the three different basic mechanisms are mapped onto the plane with axes C and M, and this mapping represents the ‘phase-diagram’ for drainage. In the final section we present three statistical models (percolation, diffusion-limited aggregation (DLA) and anti-DLA) which can be used for describing the three ‘basic’ domains of the phase-diagram.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112088000953

Numerical models and experiments on immiscible displacements in porous media

By sequentially adding line segments to a line or disks to a surface at random positions without overlaps, we obtain configurations of the one- and two-dimensional random sequential adsorption (RSA) problem. We have simulated the one- and two-dimensional problem with periodic boundary condition. The one-dimensional simulations are compared with the exact analytical solutions to give an estimate of the accuracy of the simulation. In two dimensions the geometrical properties of the RSA configuration are discussed and in addition known results of the RSA process are reproduced. Various statistical distributions of the Voronoi-Dirichlet (VD) network corresponding to the RSA disk configuration are analyzed. In order to characterize pores in the RSA configuration, we introduce circular holes. There is a direct correspondence between vertices of the VD network and these holes, and also between direct/indirect geometrical neighbors and these holes. The hole size distribution is found to be a parabola. We also find general relations that connect the asymptotic behavior of the surface coverage, the correlation function, and the hole size distribution.

Geometry of random sequential adsorption

THE idea of self-organized criticality1(SOC) is commonly illustrated conceptually with avalanches in a pile of sand grains. The grains are dropped onto a pile one by one, and the pile ultimately reaches a stationary 'critical' state in which its slope fluctuates about a constant angle of repose, with each new grain being capable of inducing an avalanche on any of the relevant size scales. Some numerical models of sand-pile dynamics do show SOC1–8, hut the behaviour of real sand piles remains ambiguous9–18. Here we describe experiments on a granular system—a pile of rice—in which the dynamics exhibit self-organized critical behaviour in one case (for grains with a large aspect ratio) but not in another (for less elongated grains). These results show that SOC is not as 'universal' and insensitive to the details of a system as was initially supposed1, but that instead its occurrence depends on the detailed mechanism of energy dissipation.

Avalanche dynamics in a pile of rice

Gas displacing a high-viscosity fluid in a two-dimensional porous disk intrudes in the form of ramified fingers similar to the structures obtained in diffusion-limited aggregation. We find that the resulting finger structures are described by a fractal dimension $D=1.62\ifmmode\pm\else\textpm\fi{}0.04$ consistent with $D$ for diffusion-limited aggregation clusters. This result confirms the analogy between diffusion-limited aggregation and two-fluid displacement in porous media introduced by Paterson.

Viscous fingering fractals in porous media.

A computer model for the dynamics of meandering rivers has been used to study the interplay between the migrating river and the changing sedimentary environment created by the meandering river itself. The model is based on the theory of Ikeda et al. [1981] and is closely related to that proposed by Howard [1983]. Coarser sands, which are often associated with high credibility, are deposited in the point bars formed when the river migrates away from its former bank. Fine-grained material eventually fills the oxbow lakes created by cutoff processes and forms erosion-resistant plugs. In the simulations, geometric forms of individual meanders observed in different natural sedimentary environments have been reproduced by changing the credibility of the corresponding sedimentary materials, such as point bar deposits, flood plain deposits, and oxbow lake deposits. The simulations indicate that the typical meander wavelength is determined mainly by hydraulic factors such as the flow in the channel and the inclination of the underlying flood plain and is independent of the difference in the credibilities of sedimentary deposits. The computational approach permits exploration of long-term changes in the floodplain geology, mediated by the meandering river. As an initial demonstration, the formation of meander belts is investigated using the model. The results suggest that a meander belt will be formed by a river's own cutoff loops only if the characteristic time of deposition and solidification of an oxbow lake is longer than the typical time that it takes the river to migrate downstream over the distance of a meander-loop wavelength.

Torstein Jøssang

Papers

Geometry of random sequential adsorption

Avalanche dynamics in a pile of rice

Viscous fingering fractals in porous media.

A Simulation Model for Meandering Rivers

The fractal nature of geochemical landscapes