Analysis of a Mathematical Model for Fluid Transport in Poroelastic Materials in 2D Space

TL;DR

This paper develops and analyzes a mathematical model for fluid transport in 2D poroelastic materials, deriving analytical solutions for boundary-value problems with moving boundaries, and exploring symmetries of the governing equations.

Contribution

It introduces a new multidimensional poroelastic model with variable volume and provides analytical solutions for radially symmetric boundary-value problems.

Findings

The model admits infinite-dimensional Lie symmetries.

Analytical solutions are obtained for stationary boundary-value problems.

Correct boundary conditions for ring and annulus deformations are constructed.

Abstract

A mathematical model for the poroelastic materials (PEM) with the variable volume is developed in multidimensional case. Governing equations of the model are constructed using the continuity equations, which reflect the well-known physical laws. The deformation vector is specified using the Terzaghi effective stress tensor. In the two-dimensional space case, the model is studied by analytical methods. Using the classical Lie method, it is proved that the relevant nonlinear system of the (1+2)-dimensional governing equations admits highly nontrivial Lie symmetries leading to an infinite-dimensional Lie algebra. The radially-symmetric case is studied in details. It is shown how correct boundary conditions in the case of PEM in the form of a ring and an annulus are constructed. As a result, boundary-value problems with a moving boundary describing the ring (annulus) deformation are constructed. The relevant nonlinear boundary-value problems are analytically solved in the stationary case. In particular, the analytical formulae for unknown deformations and an unknown radius of the annulus are presented.