Modeling flow in porous media with double porosity/permeability: Mathematical model, properties, and analytical solutions

TL;DR

This paper develops a rigorous mathematical model for fluid flow in porous media with double porosity and permeability, providing theoretical foundations, properties, and analytical solutions to better understand complex geo-material behavior.

Contribution

It introduces a thermodynamically grounded mathematical model for double porosity/permeability flow and proves key properties, filling a gap in the theoretical understanding of such systems.

Findings

Derived analytical solutions for canonical problems.

Identified differences from classical Darcy flow.

Provided properties to assess numerical solution accuracy.

Abstract

Geo-materials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores, which is commonly referred to as double porosity. To complicate things, the pore-network at each scale exhibits different permeability, and these networks are connected through fissure and conduits. Although some models are available in the literature, they lack a strong theoretical basis. This paper aims to fill this lacuna by providing the much needed theoretical foundations of the flow in porous media which exhibit double porosity/permeability. We first obtain a mathematical model for double porosity/permeability using the maximization of rate of dissipation hypothesis, and thereby providing a firm thermodynamic underpinning. We then present, along with mathematical proofs, several important mathematical properties that the solutions to the double porosity/permeability model satisfy. These properties are important in their own right as well as serve as good (mechanics-based) a posteriori measures to assess the accuracy of numerical solutions. We also present several canonical problems and obtain the corresponding analytical solutions, which are used to gain insights into the velocity and pressure profiles, and the mass transfer across the two pore-networks. In particular, we highlight how the solutions under the double porosity/permeability differ from the corresponding solutions under Darcy equations.