Application of the fractional conservation of mass to Gas Flow diffusivity equation in heterogeneous porous media

TL;DR

This paper explores how fractional conservation of mass modifies the classical gas diffusivity equation in heterogeneous porous media, leading to linearized space-fractional models and exact solutions for pressure-dependent permeability.

Contribution

It introduces a fractional conservation of mass framework into the classical gas diffusion model, deriving linearized space-fractional equations and analytical solutions for specific permeability laws.

Findings

The fractional conservation of mass leads to a space-fractional diffusion equation.

Exact analytical solutions are obtained for power-law pressure-dependent permeability.

A Barenblatt-type solution for a space-fractional Boussinesq equation is derived.

Abstract

In this paper we reconsider the classical nonlinear diffusivity equation of real gas in an heterogenous porous medium in light of the recent studies about the generalized fractional equation of conservation of mass. We first recall the physical meaning of the fractional conservation of mass recently studied by Wheatcraft and Meerschaert (2008) and then consider the implications in the classical model of diffusion of a real gas in a porous medium. Then we show that the obtained equation can be simply linearized into a classical space-fractional diffusion equation, widely studied in the literature. We also consider the case of a power-law pressure-dependence of the permeability coefficient. In this case we provide some useful exact analytical results. In particular, we are able to find a Barenblatt-type solution for a space-fractional Boussinesq equation, arising in this context.