A Darcy‐Brinkman‐Biot Approach to Modeling the Hydrology and Mechanics of Porous Media Containing Macropores and Deformable Microporous Regions

Abstract

The coupled hydrology and mechanics of soft porous materials (such as clays, hydrogels, membranes, and biofilms) is an important research area in several fields, including water and energy technologies as well as biomedical engineering. Well‐established models based on poromechanics theory exist for describing these coupled properties, but these models are not adapted to describe systems with more than one characteristic length scale, that is, systems that contain both macropores and micropores. In this paper, we expand upon the well‐known Darcy‐Brinkman formulation of fluid flow in two‐scale porous media to develop a “Darcy‐Brinkman‐Biot” formulation: a general coupled system of equations that approximates the Navier‐Stokes equations in fluid‐filled macropores and resembles the equations for poroelasticity in microporous regions. We parameterized and validated our model for systems that contain either plastic (swelling clay) or elastic microporous regions. In particular, we used our model to predict the permeability of an idealized siliciclastic sedimentary rock as a function of pore water salinity and clay content. Predicted permeability values are well described by a single parametric relation between permeability and clay volume fraction that agrees with existing experimental data sets. Our novel formulation captures the coupled hydro‐chemo‐mechanical properties of sedimentary rocks and other deformable porous media in a manner that can be readily implemented within the framework of Digital Rock Physics.

Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy‐Brinkman‐Biot Approach

“…In multiphase porous systems with incompressible grains and no swelling pressure (i.e., ), Biot Theory states that , where and p c is the capillary pressure (Jha & Juanes, 2014; Kim et al., 2013). This expression is satisfied by the previous equation in the absence of capillary forces, where F c ,1 , F c ,2 , B cap , and p c equal zero (Carrillo & Bourg, 2019). In the presence of capillary forces, however, it imposes the following equality …”

“…Throughout this study and its of predecessors (Carrillo & Bourg, 2019;, we show that the Multiphase DBB model can be readily used to model a large variety of systems, from single-phase flow in static porous media, to elastic systems under compression, to viscosity-or capillarity-dominated fracturing systems, to multiscale wave propagation in poroelastic coastal barriers.…”

“…As described in Carrillo and Bourg (2019) and , these terms can be recast into the following expression through asymptotic matching to the standard multiphase Darcy equations:…”

“…Recently, a study by Carrillo and Bourg (2019) introduced a Darcy‐Brinkman‐Biot (DBB) approach capable of accurately representing single phase flow in multiscale deformable media including elastic porous membranes and plastic swelling clays. Simultaneously, studies by Soulaine et al.…”

Computational Microfluidics for Geosciences

“…where the overline notation denotes filtered variables, φ is porosity field, and k is the local permeability of the porous regions. Recent studies show that the micro-continuum framework is well-suited to solve problems involving two characteristic length-scales (e.g., fractured media, micro-porous rocks) (Arns et al, 2005;Scheibe et al, 2015;Soulaine et al, , 2021Carrillo et al, 2020;Menke et al, 2020;Poonoosamy et al, 2020) and also to describe the movement of fluid/solid interfaces caused by dissolution/erosion and precipitation/deposition (Soulaine and Tchelepi, 2016;Soulaine et al, 2017;Molins et al, 2020) or by solid deformation due to swelling or fracturing (Carrillo and Bourg, 2019). Microcontinuum models are also able to simulate two-phase flow processes (Soulaine et al, 2018(Soulaine et al, , 2019Carrillo et al, 2020).…”