A general approach for modeling interacting flow through porous media under finite deformations. (English) Zbl 1423.76447

Summary: In the last few decades modeling deformation and flow in porous media has been of great interest due to its possible application areas in various fields of engineering such as biomechanics, soil mechanics, geophysics, physical chemistry and material sciences. Due to the high complexity and in most cases also unknown geometry of porous media on the microscale, a fully resolved model is nearly impossible to obtain, but most of the times also not necessary to answer important questions. As a consequence, one switches to a macroscopic approach. Such a mathematical description of porous media on the macroscale leads to a volume-coupled multi-field problem, wherein the interface between the two phases is not resolved explicitly. In this work we propose a numerical approach for modeling incompressible flow through a nearly incompressible elastic matrix under finite deformations. After a short overview of physical and mathematical fundamentals, the system equations are formulated and different representations are introduced and analyzed. Based on thermodynamic principles, a general constitutive law is derived, which allows the integration of arbitrary strain energy functions for the skeleton. Discretization in space with three primary variables and discretization in time using the one-step-theta method lead to a complete discrete formulation, which includes both finite deformations as well as full coupling of structural and fluid phases. Therein, we include dynamic effects, especially a time and space dependent porosity. Due to the compressibility of the solid phase, the porosity and its time derivative is not depending on the determinant of the deformation gradient only, but also on the pore pressure, which is an effect that is neglected in many publications. Considering this and also a general version of Darcy’s law, we derive two finite element formulations in a straightforward way, which, along with the numerical illustrations, provide a new numerical scheme for solving large deformation porous media problems.


76S05 Flows in porous media; filtration; seepage
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Full Text: DOI


[1] Nield, D. A.; Bejan, A., Convection in porous media, (1998), Springer-Verlag New York · Zbl 1375.76004
[2] Sahimi, M., Flow and transport in porous media and fractured rock, (1995), VCH Weinheim · Zbl 0849.76001
[3] Gawin, D.; Baggio, P.; Schrefler, B., Coupled heat, water and gas flow in deformable porous media, Internat. J. Numer. Methods Fluids, 20, 969-978, (1995) · Zbl 0854.76052
[4] Callari, C.; Abati, A., Finite element methods for unsaturated porous solids and their application to dam engineering problems, Comput. Struct., 87, 485-501, (2009)
[5] You, L.; Hongtan, L., A two-phase flow and transport model for the cathode of PEM fuel cells, Int. J. Heat Mass Transfer, 45, 2277-2287, (2002) · Zbl 0993.76556
[6] Nemec, D.; Levec, J., Flow through packed bed reactors: 1.single-phase flow, Chem. Eng. Sci., 60, 6947-6957, (2005)
[7] Markert, B., Porous media viscoelasticity with application to polymeric foams, (2005), Institut für Mechanik Lehrstuhl II, Universität Stuttgart, (Ph.D. thesis)
[8] Almeida, E. S.; Spilker, R. L., Finite element formulations for hyperelastic transversely isotropic biphasic soft tissues, Comput. Methods Appl. Mech. Engrg., 151, 513-538, (1998) · Zbl 0920.73350
[9] Cowin, S. C., Bone poroelasticity, J. Biomech., 32, 217-238, (1999)
[10] Wall, W. A.; Wiechert, L.; Comerford, A.; Rausch, S., Towards a comprehensive computational model for the respiratory system, Int. J. Numer. Methods Biomed. Eng., 26, 807-827, (2010) · Zbl 1193.92068
[11] Bear, J.; Bachmat, Y., Introduction to modeling of transport phenomena in porous media, (1991), Kluwer Academic Publishers Dordrecht · Zbl 0780.76002
[12] Paul, M., Simulation of two-phase flow processes in heterogeneous porous media with adaptive methods, (2003), Institut für Wasserbau, Universität Stuttgart, (Ph.D. thesis)
[13] Süß, M., Analysis of the influence of structures and boundaries on flow and transport processes in fractured porous media, (2004), Institut für Wasserbau, Universität Stuttgart, (Ph.D. thesis)
[14] Chapelle, D.; Moireau, P., General coupling of porous flows and hyperelastic formulations - from thermodynamics principles to energy balance and compatible time schemes, Eur. J. Mech. B Fluids, 82-96, (2013) · Zbl 1297.76157
[15] Gajo, A.; Denzer, R., Finite element modelling of saturated porous media at finite strains under dynamic conditions with compressible constituents, Internat. J. Numer. Methods Engrg., 85, 1705-1736, (2011) · Zbl 1217.74120
[16] Coussy, O., Mechanics of porous continua, (1995), John Wiley and Sons West Sussex
[17] Coussy, O., Poromechanics, (2004), John Wiley and Sons West Sussex
[18] De Boer, R., Development of porous media theories - a brief historical review, Transp. Porous Media, 9, 155-164, (1992)
[19] Ehlers, W.; Bluhm, J., Porous media, (2002), Springer Verlag Berlin u.a
[20] Auriault, J.-L.; Geindreau, C.; Orgéas, L., Upscaling Forchheimer law, Transp. Porous Media, 70, 213-229, (2007)
[21] Hornung, U., Homogenization and porous media, (1997), Springer-Verlag New York · Zbl 0872.35002
[22] Sanchez-Palencia, E., Non-homogeneous media and vibration theory, (1980), Springer-Verlag Berlin · Zbl 0432.70002
[23] Gray, W.; Miller, C., Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 5. single-fluid-phase transport, Adv. Water Resour., 32, 681-711, (2009)
[24] Gray, W.; Schrefler, B., Thermodynamic approach to effective stress in partially saturated porous media, Eur. J. Mech. A Solids, 20, 521-538, (2001) · Zbl 1034.74019
[25] Schrefler, B.; Scotta, A., A fully coupled dynamic model for two-phase fluid flow in deformable porous media, Comput. Methods Appl. Mech. Engrg., 3223-3246, (2001) · Zbl 0977.74019
[26] Buhan, P. D.; Chateau, X.; Dormieux, L., The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach, Eur. J. Mech. A Solids, 17, 909-921, (1998) · Zbl 0936.74025
[27] Schreyer-Bennethum, L., Theory of flow and deformation of swelling porous materials at the macroscale, Comput. Geotech., 34, 267-278, (2007)
[28] Bear, J., Dynamics of fluids in porous media, (1988), Dover Publications New York · Zbl 1191.76002
[29] Diersch, H.-J. G.; Kolditz, O., Variable-density flow and transport in porous media: approaches and challenges, Adv. Water Resour., 25, 899-944, (2002)
[30] Pan, Y. A.N.; Horne, R. N., Generalized macroscopic models for fluid flow in deformable porous media I: theories, Transp. Porous Media, 45, 1-27, (2001)
[31] Weinstein, T.; Schreyer-Bennethum, L., On the derivation of the transport equation for swelling porous materials with finite deformation, Int. J. Solids Struct., 44, 1408-1422, (2006) · Zbl 1213.74122
[32] Whitaker, S., Flow in porous media I: A theoretical derivation of darcy’s law, Transp. Porous Media, 1, 3-25, (1986)
[33] Whitaker, S., The Forchheimer equation: A theoretical development, Transp. Porous Media, 25, 27-61, (1996)
[34] Mei, C., The effect of weak inertia on flow through a porous medium, J. Fluid Mech. 222, 647-663, (1991) · Zbl 0718.76099
[35] Whitaker, S., Flow in porous media III: deformable media, Transp. Porous Media, 1, 127-154, (1986)
[36] Chapelle, D.; Gerbeau, J.-F.; Sainte-Marie, J.; Vignon-Clementel, I., A poroelastic model valid in large strains with applications to perfusion in cardiac modeling, Comput. Mech., 91-101, (2010) · Zbl 1301.92016
[37] Eipper, G., Theorie und numerik finiter elastischer deformationen in fluidgesättigten porösen festkörpern, (1998), Institut für Mechanik Lehrstuhl II, Universität Stuttgart, (Ph.D. thesis) · Zbl 0925.73039
[38] Gajo, A., A general approach to isothermal hyperelastic modelling of saturatued porous media at finite strains with compressible solid constituents, Proc. R. Soc. A, 466, 3061-3087, (2010) · Zbl 1211.74094
[39] Badia, S.; Codina, R., Stabilized continuous and discontinuous Galerkin techniques for Darcy flow, Comput. Methods Appl. Mech. Engrg., 199, 1654-1667, (2010) · Zbl 1231.76134
[40] Badia, S.; Quaini, A.; Quarteroni, A., Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction, J. Comput. Phys., 228, 7986-8014, (2009) · Zbl 1391.74234
[41] Gee, M.; Küttler, U.; Wall, W., Truly monolithic algebraic multigrid for fluid-structure interaction, Internat. J. Numer. Methods Engrg., 85, 8, 987-1016, (2011) · Zbl 1217.74121
[42] Küttler, U.; Gee, M. W.; Förster, C.; Comerford, A.; Wall, W. A., Coupling strategies for biomedical fluid-structure interaction problems, Int. J. Numer. Methods Biomed. Eng., 26, 305-321, (2010) · Zbl 1183.92008
[43] Markert, B., A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua, Transp. Porous Media, 70, 427-450, (2007)
[44] Wong, J.; Kuhl, E., Generating fibre orientation maps in human heart models using Poisson interpolation, Comput. Methods Biomech. Biomed. Eng., 1-11, (2012)
[45] Sainte-Marie, J.; Chapelle, D.; Cimrman, R.; Sorine, M., Modeling and estimation of the cardiac electromechanical activity, Comput. Struct., 84, 1743-1759, (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.