Dynamic properties of double porosity/permeability model

TL;DR

This paper investigates the transient behavior of double porosity/permeability models, establishing key properties like uniqueness, reciprocity, and a variational principle to enhance understanding and computational accuracy in fluid flow in porous media.

Contribution

It provides the first rigorous mathematical analysis of transient solutions in double porosity/permeability models, addressing a significant gap in existing steady-state-focused research.

Findings

Proves backward-in-time uniqueness of solutions

Establishes reciprocity property in transient regime

Derives a variational principle for dynamic flow analysis

Abstract

Understanding fluid movement in multi-pored materials is vital for energy security and physiology. For instance, shale (a geological material) and bone (a biological material) exhibit multiple pore networks. Double porosity/permeability models provide a mechanics-based approach to describe hydrodynamics in aforesaid porous materials. However, current theoretical results primarily address steady-state response, and their counterparts in the transient regime are still wanting. The primary aim of this paper is to fill this knowledge gap. We present three principal properties -- with rigorous mathematical arguments -- that the solutions under the double porosity/permeability model satisfy in the transient regime: backward-in-time uniqueness, reciprocity, and a variational principle. We employ the ``energy method'' -- by exploiting the physical total kinetic energy of the flowing fluid -- to establish the first property and Cauchy-Riemann convolutions to prove the next two. The results reported in this paper -- that qualitatively describe the dynamics of fluid flow in double-pored media -- have (a) theoretical significance, (b) practical applications, and (c) considerable pedagogical value. In particular, these results will benefit practitioners and computational scientists in checking the accuracy of numerical simulators. The backward-in-time uniqueness lays a firm theoretical foundation for pursuing inverse problems in which one predicts the prescribed initial conditions based on data available about the solution at a later instance.