DIFFUSION IN POROUS LAYERS WITH MEMORY

Abstract

The process of diffusion of fluid in porous media and biological membranes has usually been modelled with Darcy's constitutive equation, which states that the flux is proportional to the pressure gradient. However, when the permeability of the matrix changes during the process, solution of the equations governing the diffusion presents severe analytical difficulties because the variation of permeability is not known a priori.

A diverse formulation of the constitutive law of diffusion is therefore needed and many authors have studied this problem using various methods and solutions. In this paper Darcy's constitutive equation is modified with the introduction of a memory formalism. We have also modified the second constitutive equation of diffusion which relates the density variations in the fluid to the pressure, introducing rheology in the fluid represented by memory formalisms operating on pressure variations as well as on density variations. The memory formalisms are then specified as derivatives of fractional order, solving the problem in the case of a porous layer when constant pressures are applied to its sides.