The method of quasiperiodic fields for diffusion in periodic porous media

Abstract

In this paper, a new theoretical method for smoothing the diffusion equation in periodic porous media is presented. That strategy is named ‘the method of quasiperiodic fields’ because it is based on the postulate that the concentration is a quasiperiodic field throughout the porous medium. The method proceeds in four steps: (i) a first level averaging of the diffusion equation, (ii) the statement of a quasiperiodic problem, (iii) the statement of factorized quasiperiodic problems, and (iv) the development of a closed form for the averaged diffusion equation. A first application of these four steps on the initial diffusion equation in a periodic porous medium provides an averaged diffusion equation in which the effective diffusion coefficient varies periodically at the small scale. A second application of the four steps provides a smoothed diffusion equation in which the effective diffusion coefficient is constant. The final smoothed diffusion equation is stated in terms of the double average of the concentration field.