Three-dimensional midpoint displacement algorithm for the generation of fractal porous media

We propose a novel method of generating fractal three-dimensional porous media geometry. The method is an extension of the two-dimensional midpoint displacement method, used to generate realistic looking terrain for graphics applications, to a third dimension. The extended algorithm generates a three-dimensional matrix of scalars and by selecting an appropriate cut-off value to produce the porosity of the resultant media. The specific surface area of the geometry can also be controlled by adjusting the decay of the random component of the midpoint displacement. The geometries generated are fully periodic, which will help to simplify boundary conditions for future simulations. Statistical properties such as the two-point probability function and lineal-path function are calculated for the generated geometries. These properties are shown to have similar features as those of rocks which have been digitized from 2D cross sections of a physical sample. The structures generated by the proposed method do not share the blocky appearance of previous fractal based methods. This gives them a more realistic appearance similar to those produced through an image reconstruction method. In future work the geometries generated using this method will be used to link pore scale properties such as specific surface area to macroscopic permeability and storage capacity.