Boundary Effects on the Tortuosity and Permeability of Idealized Porous Media

In this work, the tortuosity and the permeability of a porous medium whose geometry is idealized according to two Euclidean spatial dimensions is studied. Unit porous media solid matrices are taken as iterations toward the 2D Sierpinski carpet fractal geometry. Investigated porous media unit arrangements are either fully periodic (FP), or that of a periodic channel (PC), in which unit porous media geometries are infinitely stacked on both orthogonal directions or only in the flow direction inside a walled channel, respectively. A third case called adjusted fully periodic was also proposed as a counterpoint to the formation of the single preferential flow path of the FP case. The flow regime of interest is the Stokes (or creeping) one, and the numerical approach is done using the lattice Boltzmann method on a porous medium unit domain with a preset pressure drop. Moreover, the scale analysis technique is applied to obtain theoretical correlations for the permeability as a function of porous medium properties for the three cases. Good agreement is found between the correlations obtained herein with results available in the literature. A finding is that whenever the channel walls are removed, the PC case correlation recovers the one for FP. Finally, the results suggest a limit value for the iteration of the carpet beyond which the tortuosity and the permeability of the porous channel and the infinitely periodic (unwalled) porous media are equal.