Lie group analysis of 2‐dimensional space‐fractional model for flow in porous media

Abstract

Using the Lie group analysis approach, we study the symmetry properties of 2‐dimensional space‐fractional filtration equation with the Riesz potential. This equation is derived by a fractional generalization of Darcy law and permits to describe the pressure evolution during 1‐phase fluid flow through naturally fractured porous medium. We construct the prolongation of the point transformation group on the Riesz potential and obtain a generalization of the Leibniz rule for the Riesz potential, which are necessary for investigating symmetry properties using the invariance principle. The Lie group of linearly autonomous point transformations is constructed for the considered equation. In a limiting case of zero‐order potential, the obtained symmetry group coincides with the group of point transformations admitted by the classical integer‐order 2‐dimensional linear heat equation. Also, the asymptotic behavior of a group invariant solution is investigated.