Márcio A. Murad

Electro-chemo-mechanical couplings in swelling clays derived from a micro/macro-homogenization procedure

Abstract A macroscopic model for highly compacted expansive clays composed of a charged solid phase saturated by a binary monovalent aqueous electrolyte solution is derived based on a rigorous scale-up of the microstructural behavior. The homogenization technique is applied to propagate information available in the pore-scale model to the macroscale. Macroscopic electrokinetic phenomena such as electro-osmotic flow driven by streaming potential gradients, electrophoretic motion of mobile charges and osmotically induced swelling are derived by homogenizing the microscopic electro-hydrodynamics coupled with the Nernst–Planck and Poisson–Boltzmann equations governing the flow of the electrolyte solution, ion movement and electric potential distribution. A notable consequence of the upscaling procedure proposed herein are the micromechanical representations for the electrokinetic coefficients and swelling pressure. The two-scale model is discretized by the finite element method and applied to numerically simulate contaminant migration and electrokinetic attenuation through a compacted clay liner underneath a sanitary landfill.

Improved accuracy in finite element analysis of Biot's consolidation problem

Abstract Numerical analysis and error estimates of finite element approximations of Biots consolidation problem are presented. Initially different orders of interpolation are employed, leading to lower orders of convergence for the pore pressure compared to the displacements of the porous medium. Lower accuracy also occurs in the approximation of the effective stress tensor, whether it is calculated directly from the constitutive equation, or through a primal mixed stress formulation. To improve the rates of convergence of the pore pressure and effective stresses, a sequential Galerkin Petrov-Galerkin post-processing technique is proposed.

Macroscale Thermodynamics and the Chemical Potential for Swelling Porous Media

The thermodynamical relations for a two-phase, N-constituent, swelling porous medium are derived using a hybridization of averaging and the mixture-theoretic approach of Bowen. Examples of such media include 2-1 lattice clays and lyophilic polymers. A novel, scalar definition for the macroscale chemical potential for porous media is introduced, and it is shown how the properties of this chemical potential can be derived by slightly expanding the usual Coleman and Noll approach for exploiting the entropy inequality to obtain near-equilibrium results. The relationship between this novel scalar chemical potential and the tensorial chemical potential of Bowen is discussed. The tensorial chemical potential may be discontinuous between the solid and fluid phases at equilibrium; a result in clear contrast to Gibbsian theories. It is shown that the macroscopic scalar chemical potential is completely analogous with the Gibbsian chemical potential. The relation between the two potentials is illustrated in three examples.

Macroscopic behavior of swelling porous media derived from micromechanical analysis

A new macroscopic model for swelling porous media is derived based on a rigorous upscaling of the microstructure. Considering that at the microscale the medium is composed of a charged solid phase (e.g. clay platelets, bio-macromolecules, colloidal or polymeric particles) saturated by a binary monovalent aqueous electrolyte solution composed of cations ‘+’ and anions ‘−’ of an entirely dissociated salt, the homogenization procedure is applied to scale up the pore-scale model. The microscopic system of governing equations consists of the local electro-hydrodynamics governing the movement of the electrolyte solution (Poisson–Boltzmann coupled with a modified Stokes problem including an additional body force of Coulombic interaction) together with modified convection–diffusion equations governing cations and anions transport. This system is coupled with the elasticity problem which describes the deformation of the solid phase. Novel forms of Terzaghis effective principle and Darcys law are derived including the effects of swelling pressure and osmotically induced flows, respectively. Micromechanical representations are provided for the macroscopic physico-chemical quantities.

Multiscale flow and deformation in hydrophilic swelling porous media

Abstract A three-scale theory of swelling porous media is developed. The colloidal or polymeric sized fraction and vicinal water (water next to the colloids) are considered on the macroscale. Hybrid mixture theory is used to upscale the colloids with the vicinal water to form mesoscale swelling particles. The mesoscale particles and bulk phase water (water next to the swelling particles) are then homogenized via an asymptotic expansion technique to form a swelling mixture on the macroscale. The solid phase on the macroscale can be viewed as a porous matrix consisting of swelling porous particles. Two Darcy type laws are developed on the macroscale, each corresponding to a different bulk water connectivity. In one, the bulk water is entrapped by the particles, forming a disconnected system, and in the other the bulk water is connected and flows between particles. In the latter case the homogenized equations give rise to a distributed model with microstructure in which the vicinal water is represented by sources/sinks at the macroscale. The theory is used to construct a three-dimensional model for consolidation of swelling clay soils and new constitutive relations for the stress tensor of the swelling particles are developed. Several heuristic modifications to the classical Terzaghi effective stress principle for granular (non-swelling) media which account for the hydration forces in swelling clay soils recently appeared in the literature. A notable consequence of the theory developed herein is that it provides a rational basis for these modified Terzaghi stresses.

A multi-scale theory of swelling porous media: I. Application to one-dimensional consolidation

A theory is developed which describes flow in multi-scale, saturated swelling media. To upscale information, both the hybrid theory of mixtures and the homogenization technique are employed. In particular, a model is formulated in which vicinal water (water adsorbed to the solid phase) is treated as a separate phase from bulk (non-vicinal) water. A new form of Darcys law governing the flow of both vicinal and bulk water is derived which involves an interaction potential to account for the swelling nature of the system. The theory is applied to the classical one-dimensional consolidation problem of Terzaghi and to verify Lows empirical, exponential, swelling result for clay at the macroscale.

MODIFIED DARCY'S LAW, TERZAGHI'S EFFECTIVE STRESS PRINCIPLE AND FICK'S LAW FOR SWELLING CLAY SOILS

Abstract Governing equations often used in soil mechanics and hydrology include the classical Darcys law, Terzaghis effective stress principle, and the classical Ficks first law. It is known that the classical forms of these relations apply only to non-swelling, granular materials. In this paper, we summarize recent generalizations of these results for swelling porous media obtained using hybrid mixture theory (HMT) by the authors. HMT is a methodical procedure for obtaining macroscopic constitutive restrictions which are thermodynamically admissible by exploiting the entropy inequality for spatially-averaged properties. HMT applied to the modeling of swelling clay particles, viewed as clusters of adsorbed water and clay minerals, produces additional terms necessary to account for the physico-chemical forces between the adsorbed water and clay minerals or, more generally, for swelling colloids. New directions for modeling consolidation of swelling clays are proposed based on our view of clay particles as a two-phase system.

Thermomechanical theories for swelling porous media with microstructure

Abstract Thermomechanical microstructural dual porosity models for swelling porous media incorporating coupled effects of hydration, heat transfer and mechanical deformation are proposed. These models are obtained by generalizing the three-scale system of Murad and Cushman [56] , [57] to accommodate heat transfer effects and their influence on swelling. The microscale consists of macromolecular structures (clay platelets, polymers, shales, biological tissues, gels) in a solvent (adsorbed water), both of which are considered as distinct nonoverlaying continua. These continua are homogenized to the meso (intermediate scale) in the spirit of hybrid mixture theory (HMT), so that at the mesoscale they may be thought of as two overlaying continua. Application of HMT leads to a two-scale model which incorporates coupled thermal and physico-chemical effects between the macromolecules and adsorbed solvent. Further, a three-scale model is obtained by homogenizing the particles (clusters consisting of macromolecules and adsorbed solvent) with the bulk solvent (solvent not within but next to the swelling particles). This yields a macroscopic microstructural model of dual porosity type. In the macroscopic swelling medium the mesoscale particles act as distributed sources/sinks of mass, momentum and energy to the macroscale bulk phase system. A modified Greens function method is used to reduce the dual porosity system to a single-porosity system with memory. The resultant theory provides a rigorous derivation of creep phenomena which are due to delayed intra-particle drainage (e.g. secondary consolidation of clay soils). In addition, the model reproduces a class of lumped-parameter models for fluid flow, heat conduction and momentum transfer where the distributed source/sink transfer function is a classical exchange term assumed proportional to the difference between the potentials in the bulk phase and swelling particles.

Micromechanical computational modeling of secondary consolidation and hereditary creep in soils

Abstract A computational modeling of a fluid-saturated deformable porous media characterized by two levels of hydrodynamics (flow in micro and macropores) is proposed based on a micromechanical analysis of dual porosity systems, i.e., media locally characterized by a porous matrix composed of permeable cells containing micropores and the surrounding system of macropores, void spaces or bulk flow paths (e.g., fissured rock or aggregated soil). The homogenization technique is applied to upscale the constitutive and geometric information available in the fine structure to the field scale leading to a microstructure model of dual porosity type, wherein the poroelastic cells act as distributed sources/sinks of mass and momentum to the global macroscopic medium. The theory provides a rigorous derivation of some secondary compression and hereditary creep effects in soils due to the delayed drainage of the fluid within the micropores under consolidation. Application of the Greens function method reduces the dual porosity system to a single-porosity viscoelastic integrodifferential system of Volterra type in which the constitutive law for the macroscopic stress tensor is given in terms of an hereditary integral with memory. A two-level finite element method is proposed to solve the coupled micro–macro governing equations of dual porosity type. Numerical experiments are performed showing the strong potential of the proposed formulation in solving consolidation problems with microstructure.

A Multiscale Theory of Swelling Porous Media: II. Dual Porosity Models for Consolidation of Clays Incorporating Physicochemical Effects

A three-scale theory of swelling clay soils is developed which incorporates physico-chemical effects and delayed adsorbed water flow during secondary consolidation. Following earlier work, at the microscale the clay platelets and adsorbed water (water between the platelets) are considered as distinct nonoverlaying continua. At the intermediate (meso) scale the clay platelets and the adsorbed water are homogenized in the spirit of hybrid mixture theory, so that, at the mesoscale they may be thought of as two overlaying continua, each having a well defined mass density. Within this framework the swelling pressure is defined thermodynamically and it is shown to govern the effect of physico-chemical forces in a modified Terzaghis effective stress principle. A homogenization procedure is used to upscale the mesoscale mixture of clay particles and bulk water (water next to the swelling mesoscale particles) to the macroscale. The resultant model is of dual porosity type where the clay particles act as sources/sinks of water to the macroscale bulk phase flow. The dual porosity model can be reduced to a single porosity model with long term memory by using Greens functions. The resultant theory provides a rational basis for some viscoelastic models of secondary consolidation.