Modeling Flow in Porous Media With Double Porosity/Permeability: Mathematical Model, Properties, and Analytical Solutions

Abstract

Geomaterials 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 with different hydromechanical properties. Moreover, these pore-networks are connected through fissures and conduits. Although some models are available in the literature to describe flows in such media, they lack a strong theoretical basis. This paper aims to fill this gap in knowledge by providing the theoretical foundation for the flow of incompressible single-phase fluids in rigid porous media that exhibit double porosity/permeability. We first obtain a mathematical model by combining the theory of interacting continua and the maximization of rate of dissipation (MRD) hypothesis, and thereby provide a firm thermodynamic underpinning. The governing equations of the model are a system of elliptic partial differential equations (PDEs) under a steady-state response and a system of parabolic PDEs under a transient response. We then present, along with mathematical proofs, several important mathematical properties that the solutions to the model satisfy. We also present several canonical problems with 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 ones under Darcy equations.