The Biot-Darcy-Brinkman model of flow in deformable double porous media; homogenization and numerical modelling

Abstract In this paper we present the two-level homogenization of the flow in a deformable double-porous structure described at two characteristic scales. The higher level porosity associated with the mesoscopic structure is constituted by channels in a matrix made of a microporous material consisting of elastic skeleton and pores saturated by a viscous fluid. The macroscopic model is derived by the homogenization of the flow in the heterogeneous structure characterized by two small parameters involved in the two-level asymptotic analysis, whereby a scaling ansatz is adopted to respect the pore size differences. The first level upscaling of the fluid–structure interaction problem yields a Biot continuum describing the mesoscopic matrix coupled with the Stokes flow in the channels. The second step of the homogenization leads to a macroscopic model involving three equations for displacements, the mesoscopic flow velocity and the micropore pressure. Due to interactions between the two porosities, the macroscopic flow is governed by a Darcy–Brinkman model comprising two equations which are coupled with the overall equilibrium equation respecting the hierarchical structure of the two-phase medium. Expressions of the effective macroscopic parameters of the homogenized double-porosity continuum are derived, depending on the characteristic responses of the mesoscopic structure. Some symmetry and reciprocity relationships are shown and issues of boundary conditions are discussed. The model has been implemented in the finite element code SfePy which is well-suited for computational homogenization. A numerical example of solving a nonstationary problem using mixed finite element method is included.

[1]  H. Brinkman,et al.  On the permeability of media consisting of closely packed porous particles , 1949 .

[2]  Salah Naili,et al.  Multiscale modeling of a fluid saturated medium with double porosity: Relevance to the compact bone , 2012 .

[3]  Doina Cioranescu,et al.  The Periodic Unfolding Method in Homogenization , 2008, SIAM J. Math. Anal..

[4]  G. I. Barenblatt,et al.  Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata] , 1960 .

[5]  Claude Boutin,et al.  Deformable porous media with double porosity. Quasi-statics. I: Coupling effects , 1992 .

[6]  Gaffar Gailani,et al.  Ramp loading in Russian doll poroelasticity , 2011 .

[7]  James G. Berryman,et al.  Poroelastic Response of Orthotropic Fractured Porous Media , 2012, Transport in Porous Media.

[8]  Salah Naili,et al.  Double porosity in fluid-saturated elastic media: deriving effective parameters by hierarchical homogenization of static problem , 2016 .

[9]  D. Jamet,et al.  Jump Conditions and Surface-Excess Quantities at a Fluid/Porous Interface: A Multi-scale Approach , 2009 .

[10]  H. Brinkman.  A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles , 1949 .

[11]  U. Hornung.  Homogenization and porous media , 1996 .

[12]  Benoît Goyeau,et al.  On the Equivalence of the Discontinuous One- and Two-Domain Approaches for the Modeling of Transport Phenomena at a Fluid/Porous Interface , 2009 .

[13]  Jean-Louis Auriault,et al.  On the Domain of Validity of Brinkman’s Equation , 2009 .

[14]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[15]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[16]  Willi Jäger,et al.  Modeling Effective Interface Laws for Transport Phenomena Between an Unconfined Fluid and a Porous Medium Using Homogenization , 2009 .

[17]  Yalchin Efendiev,et al.  Homogenization of High-Contrast Brinkman Flows , 2015, Multiscale Model. Simul..

[18]  A. Marciniak-Czochra,et al.  Three-scale convergence for processes in heterogeneous media , 2012 .

[19]  Claude Boutin,et al.  Deformable porous media with double porosity. Quasi-statics. II: Memory effects , 1993 .

[20]  Ronaldo I. Borja,et al.  Stabilized mixed finite elements for deformable porous media with double porosity , 2015 .

[21]  Stephen Whitaker,et al.  Momentum transfer at the boundary between a porous medium and a homogeneous fluid—II. Comparison with experiment , 1995 .

[22]  P. Saffman.  On the Boundary Condition at the Surface of a Porous Medium , 1971 .

[23]  Todd Arbogast,et al.  Derivation of the double porosity model of single phase flow via homogenization theory , 1990 .

[24]  Vladimír Lukes,et al.  Modeling nonlinear phenomena in deforming fluid-saturated porous media using homogenization and sensitivity analysis concepts , 2015, Appl. Math. Comput..

[25]  Alfio Quarteroni,et al.  A multiscale Darcy–Brinkman model for fluid flow in fractured porous media , 2011, Numerische Mathematik.

[26]  Claude Boutin,et al.  Macroscopic modeling of double-porosity reservoirs , 1996 .

[27]  Robert Cimrman.  SfePy - Write Your Own FE Application , 2014, ArXiv.

[28]  Fredrik Larsson,et al.  Homogenization of coupled flow and deformation in a porous material , 2016 .

[29]  Jana Turjanicová,et al.  A Darcy-Brinkman model of flow in double porous media – Two-level homogenization and computational modelling , 2018, Computers & Structures.

[30]  Anna K. Marciniak-Czochra,et al.  Effective Pressure Interface Law for Transport Phenomena between an Unconfined Fluid and a Porous Medium Using Homogenization , 2011, Multiscale Model. Simul..

[31]  M. Biot.  THEORY OF ELASTICITY AND CONSOLIDATION FOR A POROUS ANISOTROPIC SOLID , 1955 .

[32]  James G. Berryman,et al.  Models for computing geomechanical constants of double‐porosity materials from the constituents' properties , 2002 .

[33]  Stephen Whitaker,et al.  Momentum transfer at the boundary between a porous medium and a homogeneous fluid-I. Theoretical development , 1995 .

[34]  G. Allaire.  Homogenization and two-scale convergence , 1992 .