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

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 with different hydro-mechanical properties. Moreover, these pore-networks are connected through fissures and conduits. Although some models are available in the literature to describe flow in such porous media, 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 that exhibit double porosity/permeability. We first obtain a mathematical model 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 model satisfy. 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 ones under Darcy equations.

Thermo-Chemo-Poromechanical Modeling of the Anode Mixture During the Baking Process: Constitutive Laws and Governing Equations

“…Let begin with the maximization of entropy production approach by following the similar procedures in works [47,48]. The mathematical statement of the maximization of the dissipation function can be written as follows:…”

“…These expressions correspond to those of expressions (50), (54), (64), (88), and (92) derived in this paper. Therefore, the approach used in this work to derive the constitutive relations is compatible with the maximization of entropy production approach suggested by Ziegler [46][47][48].…”

“…The approach followed in this work to develop the constitutive laws is based on the thermodynamic framework and the consideration of internal variables. However, there exist some other methodologies for the development of constitutive laws such as the principle of the maximization of entropy production [46][47][48]. In this context, one can show that using such a principle with an appropriate definition of the dissipation function, the present constitutive laws are recovered.…”

A global sensitivity analysis and reduced order models for hydraulically-fractured horizontal wells

“…Finally, we suggest three plausible future works. The first research effort can be towards combining fracture modeling with double porosity/permeability models (Nakshatrala et al, 2018;Joodat et al, 2018). The second effort is to incorporate inertial effects (e.g., Forchheimer-type models) and pressure-dependence viscosity (Chang et al, 2017;Mapakshi et al, 2018) on the hydraulic fracture propagation.…”

Hydro-mechanical coupling characteristics and damage constitutive model of low-permeability granite under triaxial compression

“…Ren et al (2017) carried out a study on the permeability similarity of a fracture network by proposing a density-reduced equivalent DFN (E-DFN) modeling method to simplify the density of the original DFN model. Barenblatt et al (1960) first proposed the concept of a dual-media model, but the anisotropy of fractured rock and pore water was not reflected accurately enough, and researchers (Nakshatrala et al, 2018;Panfilov et al, 2020;Warren and Root, 1963) subsequently continued to expand on this original basis, thereby making the model more consistent with actual rock conditions. The above model has a wide range of applications, however, it is difficult to describe the damage mechanism of the rock during hydro-mechanical coupling failure.…”