Representation of Multiphase Media

The structure of a multiphase medium can be specified by the spatial distribution of the phases that form the medium. This spatial distribution can be generally represented by the phase function or, in specific situations, by the equivalent pore-network diagram, by the spatial distribution of particles or other constituents, and by the probability density function. Examples of the representation of the porous media are shown in Fig. 2. 

In the general case, the spatial distribution of the phases can be formally represented by the so-called phase function ft IR3 —s- 0 1 for each phase /. The phase function is defined as (Adler, 1992, 1994) 

By definition only one phase can be present at any point r e R3. It is further required that the set P, c IR3, P, = f ff) — 1 be a compact set, i.e., that the inter-phase boundaries are smooth in the mathematical sense. In a discrete form, the phase function f becomes the phase volume function which assigns 

In a practical implementation, the domain on which the phase volume functions are specified is typically a cubic grid of Nx x Ny x Nz voxels, which corresponds to real dimensions of Lx — hNx, Ly - hNy, and Lz — hNz, where h is the voxel size. We will further call this region of real space the computational unit cell. The relationship between the unit cell and the multiphase medium of interest depends on the absolute dimensions of the medium and on the spatial resolution at which the medium is represented (feature dimensions). The unit cell can either contain the entire medium and some void space surrounding it, as in the case of virtual granules described in Section IV.D below, or be a sample of a much larger (theoretically infinite) medium, as in the case of transport properties calculation, described in Section II.E below. 

In the latter case, the dimensions of the unit cell must be such that the unit cell is statistically representative of the entire medium. For spatially periodic regular media, the unit cell will coincide with one lattice unit. For random media, let Lm — max,( L,) be the maximum characteristic length-scale of all the phases forming the medium, where Lt is defined by Eq. (4) below. The unit cell dimensions (Lx, Ly, L ) should be much larger than the maximum characteristic length-scale Lm for transport properties calculated on the unit cell to become cell size-independent, and thus, representative of the entire medium (Adler, 1992). 

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