Application of p-Adic Wavelets to Model Reaction–Diffusion Dynamics in Random Porous Media

Abstract

Fourier and more generally wavelet analysis over the fields of p-adic numbers are widely used in physics, biology and cognitive science, and recently in geophysics. In this note we present a model of the reaction–diffusion dynamics in random porous media, e.g., flow of fluid (oil, water or emulsion) in a a complex network of pores with known topology. Anomalous diffusion in the model is represented by the system of two equations of reaction–diffusion type, for the part of fluid not bound to solid’s interface (e.g., free oil) and for the part bound to solid’s interface (e.g., solids–bound oil). Our model is based on the p-adic (treelike) representation of pore-networks. We present the system of two p-adic reaction–diffusion equations describing propagation of fluid in networks of pores in random media and find its stationary solutions by using theory of p-adic wavelets. The use of p-adic wavelets (generalizing classical wavelet theory) gives a possibility to find the stationary solution in the analytic form which is typically impossible for anomalous diffusion in the standard representation based on the real numbers.