DMAP.m: A Mathematica® program for three-dimensional mapping of tortuosity and porosity of porous media.

We developed a Mathematica® program for three-dimensional mapping of the porosity and normalized apparent diffusion coefficient (tortuosity) of isotropic heterogeneous porous media. The program, DMAP.m, is a package-type program for Mathematica® version 4 or later. DMAP.m accepts three-dimensional (3D) digital image data for the porous media as an input. Such data may be obtained by, for example, X-ray computed tomography (CT) as a set of text files of two-dimensional contiguous CT slices (square matrices). DMAP.m reads the text files and divides the image set into sub-cubes, then executes a non-sorbing random walk (lattice walk) through the discrete pore space in each sub-cube. A specified number of voxels are chosen randomly as the start position of the random walk. If the chosen voxel falls within a pore, random walk simulation is carried out until the walker exits the sub-cube. If the chosen voxel falls within a solid, the random walk is not performed. The porosity of each sub-cube is calculated as the probability of a successful hit on a pore voxel in this random choice of the start position (Monte Carlo integral). The time required for the walkers to escape from each sub-cube is recorded in the random walk simulation, representing an “out-diffusion” or “out-leaching” numerical simulation. The tortuosity (apparent diffusion coefficient in the free space divided by that in porous media) is calculated by fitting the time-dependent cumulative number of walkers that have escaped from the sub-cube to a theoretical curve. DMAP.m was applied successfully here to the 3D X-ray CT image of a monosized sand pack. DMAP.m is available for free download on the author’s website (URL = http://staff.aist.go.jp/nakashima.yoshito/progeng.htm) to facilitate study on porous media by X-ray CT or nuclear magnetic resonance imaging.