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## Structural models

The structural models studied in this paper are all digital image based [14]. A square array of pixels is defined such that each pixel represents either a solid or a fluid-filled pore phase. Assuming all pixels initially represent fluid, solid circular inclusions are introduced by centering a continuum circle on a pixel center, and changing all pixels to solids whose centers fall inside the boundary of the continuum circle [15]. For circles with diameter greater than 15 pixels, the area of the digital circle varies from the area of the generating continuum circle by less than 1%. Figure 1 shows a 41-pixel-diameter circle, centered on the middle of a pixel. Using circles of diameter 41 pixels or larger, we then construct random porous media by randomly depositing [16] many circles on a 1000x1000 unit cell such that no two circles overlap. Periodic boundary conditions are maintained by wrapping pixels that would extend outside the unit cell around to the opposite side. Using monosize circles a porosity of 50% can easily be achieved. The lowest porosity achievable for parking of monosize circles is 45%, called the "random jamming limit" [16,17]. Lower porosities are attainable using additional, smaller disc sizes. In our case, we tried adding additional disks with a diameter of 21 pixels. However, the spacing between disks at the lower porosities became too small, in terms of the number of pixels, to ensure adequate resolution and numerical accuracy in the electrical and fluid-flow computations. To be able to explore porosities less than or equal to 50%, we kept the unit cell at 1000x1000 pixels, but "rescaled" the pore space by using 83 pixel diameter circles as the large-size circle, and 43 pixel diameter circles as the small-size circle, so that there were fewer circles but more space between circles. Using fewer, larger circles of course meant that our sample statistics became somewhat worse.

Figure 1: 41-pixel diameter digital circle (gray), centered in a 51 x 51 pixel unit cell

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