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Percolation-type models

Percolation theory is a well-studied topic, with many excellent reviews [53,55,96,97,98]. The early studies of percolative structures and their effects on bulk physical properties were made on random lattice structures, thus making them relevant to structures seen in digital images, which are typically square or cubic lattice structures. A digital image approximates a continuum structure when the geometric features of interest each occupy many pixels. In this sense, a useful digital image is a lattice structure with spatial correlations among the pixels [99,100].

There has been less, but still substantial work, on generating 3-D continuum models using continuum objects placed at random or regular positions within the image frame. Many references can be found in the reviews cited above. One example is to build microstructures out of overlapping, randomly placed and oriented ellipsoids [101]. Figure 15 shows a 2-D section of such a 3-D model. The prolate ellipsoids used were of the same size with an aspect ratio of 10. The volume fraction of ellipsoids in Fig. 15 is approximately 7%. Even though in the image, the elliposids appear to be mostly isolated, over half of them are connected in 3-D and form a spanning cluster. Note that a similar model with ellipses in 2-D having the same aspect ratio would percolate at an area fraction of about 30% [102].

Figure 15: Slice through a 3-D model of overlapping prolate ellipsoids (white) having an aspect ratio of 10. The volume fraction of ellipsoids is 7%.

Building continuum models with other objects in 3-D has been reviewed by Balberg [103] and they are relevant to real processes. Cubes have been used to study percolation processes in the combustion of carbon [104], and regular lattice packings of spheres have been used to study capillary condensation hysteresis loops [105]. A regular lattice packing of spheres, that can consequently grow and overlap, called the grain consolidation model, has been used to gain insight into transport processes in sedimentary rocks [106]. Two-dimensional lattices of disks that can rub against each other have been used to simulate the elastic properties of sandstone [107]. Other discussions of percolative-type models can be found in Ref. [45].

A subset of this approach, mostly used for simulating fluid flow in porous materials, is the use of tube networks, both regular [108] and random [109]. In some sense, this is similar to using discrete conductor networks to solve the continuum Laplace's equation [100]. However, setting up a finite difference solution of the continuum Stokes' equation in a porous material does not result mathematically in equations that resemble those for a network of tubes. Nevertheless, a great deal can be found about flow in porous materials from this approach [110]. More details can be found in Chapter 2 of this book.