A Model for Flow and Deformation in Unsaturated Swelling Porous Media

A thermomechanical theory for multiphase transport in unsaturated swelling porous media is developed on the basis of Hybrid Mixture Theory (saturated systems can also be modeled as a special case of this general theory). The aim is to comprehensively and non-empirically describe the effect of viscoelastic deformation on fluid transport (and vice versa) for swelling porous materials. Three phases are considered in the system: the swelling solid matrix s, liquid l, and air a. The Coleman–Noll procedure is used to obtain the restrictions on the form of the constitutive equations. The form of Darcy’s law for the fluid phase, which takes into account both Fickian and non-Fickian transport, is slightly different from the forms obtained by other researchers though all the terms have been included. When the fluid phases interact with the swelling solid porous matrix, deformation occurs. Viscoelastic large deformation of the solid matrix is investigated. A simple form of differential-integral equation is obtained for the fluid transport under isothermal conditions, which can be coupled with the deformation of the solid matrix to solve for transport in an unsaturated system. The modeling theory thus developed, which involves two-way coupling of the viscoelastic solid deformation and fluid transport, can be applied to study the processing of biopolymers, for example, soaking of foodstuffs and stress-crack predictions. Moreover, extension and modification of this modeling theory can be applied to study a vast variety of problems, such as drying of gels, consolidation of clays, drug delivery, and absorption of liquids in diapers.

[1]  A. Datta.  Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: Problem formulations , 2007 .

[2]  P. Flory,et al.  Thermodynamic relations for high elastic materials , 1961 .

[3]  John H. Cushman,et al.  Macroscale Thermodynamics and the Chemical Potential for Swelling Porous Media , 1997 .

[4]  Bernhard A. Schrefler,et al.  Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solutions* , 2002 .

[5]  John H. Cushman,et al.  Thermomechanical theories for swelling porous media with microstructure , 2000 .

[6]  John H. Cushman,et al.  Nonequilibrium Swelling- and Capillary-Pressure Relations for Colloidal Systems , 1994 .

[7]  J. H. Cushman,et al.  A Multiscale Theory of Swelling Porous Media: II. Dual Porosity Models for Consolidation of Clays Incorporating Physicochemical Effects , 1997 .

[8]  John H. Cushman,et al.  Three scale thermomechanical theory for swelling biopolymeric systems , 2003 .

[9]  J. H. Cushman,et al.  Clarifying Mixture Theory and the Macroscale Chemical Potential , 1996 .

[10]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[11]  Lynn Schreyer-Bennethum,et al.  Theory of flow and deformation of swelling porous materials at the macroscale , 2007 .

[12]  Martin R. Okos,et al.  On multicomponent, multiphase thermomechanics with interfaces , 1994 .

[13]  Moisture transport in shrinking gels during drying , 1995 .

[14]  John H. Cushman,et al.  Multiscale flow and deformation in hydrophilic swelling porous media , 1996 .

[15]  Patrick Perré,et al.  PROCESSES RELATED TO DRYING: PART II USE OF THE SAME MODEL TO SOLVE TRANSFERS BOTH IN SATURATED AND UNSATURATED POROUS MEDIA , 1991 .

[16]  K. Pruess,et al.  TOUGH2-A General-Purpose Numerical Simulator for Multiphase Fluid and Heat Flow , 1991 .

[17]  Effect of viscoelastic relaxation on fluid and species transport in biopolymeric materials , 2002 .

[18]  John H. Cushman,et al.  MODIFIED DARCY'S LAW, TERZAGHI'S EFFECTIVE STRESS PRINCIPLE AND FICK'S LAW FOR SWELLING CLAY SOILS , 1997 .

[19]  Ray W. Ogden,et al.  Nonlinear Elastic Deformations , 1985 .

[20]  John H. Cushman,et al.  Toward Rational Design of Drug Delivery Substrates: II. Mixture Theory for Three-Scale Biocompatible Polymers and a Computational Example , 2004, Multiscale Model. Simul..

[21]  J. H. Cushman,et al.  Two-scale, three-phase theory for swelling drug delivery systems. Part I: constitutive theory. , 2008, Journal of pharmaceutical sciences.

[22]  John H. Cushman,et al.  Multiscale, hybrid mixture theory for swelling systems—II: constitutive theory , 1996 .

[23]  William G. Gray,et al.  General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations , 1979 .

[24]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[25]  John H. Cushman,et al.  Multiscale fluid transport theory for swelling biopolymers , 2003 .

[26]  William G. Gray,et al.  Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries , 1990 .

[27]  B. Venkatesh,et al.  Simultaneous transport of water and solutes under transient unsaturated flow conditions — A case study , 2008 .

[28]  Lynn S. Bennethum,et al.  On the derivation of the transport equation for swelling porous materials with finite deformation , 2006 .

[29]  Ian Turner,et al.  A 3-D version of TransPore: a comprehensive heat and mass transfer computational model for simulating the drying of porous media , 1999 .

[30]  A. Cemal Eringen,et al.  A continuum theory of swelling porous elastic soils , 1994 .

[31]  I-Shih Liu,et al.  Method of Lagrange multipliers for exploitation of the entropy principle , 1972 .

[32]  John H. Cushman,et al.  Multiscale, hybrid mixture theory for swelling systems—I: balance laws , 1996 .

[33]  L. Treloar,et al.  The elasticity of a network of long-chain molecules.—III , 1943 .

[34]  John H. Cushman,et al.  A multi-scale theory of swelling porous media: I. Application to one-dimensional consolidation , 1995 .

[35]  Peter Lehmann,et al.  Unsaturated water flow across soil aggregate contacts , 2008 .

[36]  William G. Gray,et al.  General conservation equations for multi-phase systems: 1. Averaging procedure , 1979 .

[37]  L. Bennethum,et al.  Three Pressures in Porous Media , 2004 .