Jolanta Lewandowska

Homogenization Modeling and Parametric Study of Moisture Transfer in an Unsaturated Heterogeneous Porous Medium

The classical mass balance equation is usually used to model the transfer of humidity in unsaturated macroscopically homogeneous porous media. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics. The formal homogenization method by asymptotic expansions is applied to derive the upscaled form of this equation in case of large-scale heterogeneities of periodic structure. The nature of such heterogeneities may be different, resulting in locally variable hydrodynamic parameters. The effective capillary capacity and the effective hydraulic conductivity are defined as functions of geometry and local characteristics of the porous medium. A study of a two-dimensional stone-mortar system is performed. The effect of the second medium (the mortar), on the global behavior of the system is investigated. Numerical results for the Brooks and Corey hydrodynamic model are provided. The sensitivity analysis of the parameters of the model in relation to the effective hydrodynamic parameters of the porous structure is presented.

Continuum Modelling of Contaminant Transport in Fractured Porous Media

This work is aimed towards deriving macroscopic models that describe pollutant migration through fractured porous media. A homogenisation method is used, that is, macroscopic models are deduced from the physical description over a representative elementary volume (REV), which consists of an open fracture surrounded by a porous matrix block. No specific geometry is at issue. The fractured porous medium is saturated by an incompressible fluid. At the REVs scale, the transport is assumed to be advective-diffusive in the porous matrix and due to convection and molecular diffusion in the fractures domain. It is also assumed that there is no diffusion in the solid. We demonstrate that the macroscopic behaviour is described by a single-continuum model. Fluid flow is described by Darcys law. Four macroscopic single-continuum models are obtained for the contaminant transport: a diffusive model, an advective-diffusive model and two advective-dispersive models. One of the two advective-dispersive models accounts for the advection process in the porous matrix. The domains of validity of these models are defined by means of the orders of magnitude of the local Péclet numbers in the porous matrix block and in the fractures domain.

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.

Two-scale modeling of unsaturated water flow in a double-porosity medium under axisymmetric conditions

In this paper the development and experimental validation of a numerical model of two-dimensional unsaturated flow in a double-porosity medium is presented. The model is based on the coupled formulation for flow in macro- and micropores obtained by homogenization. It was applied to simulate the axisymmetrical tension disk infiltration experiments that were carried out in a double-porosity medium. The physical model was a three-dimensional periodic structure, composed of porous spheres made of sintered clay and embedded in Hostun fine sand HN38. The hydraulic parameters of both porous materials were determined by inverse analysis of independent infiltration experiments performed on sand and sintered clay. The effective parameters of the double-porosity medium were calculated from the solution of the local boundary value problem, obtained from the homogenization procedure. The cumulative infiltration curve and the global dimensions of the humidified zone obtained from the numerical solution are in good agre...

Upscaling: Cell Symmetries and Scale Separation

To obtain the effective parameters of a heterogeneous medium from the microscale description, one has to solve a boundary value problem defined on a representative cell. For practical purposes, the medium is often considered as periodic and the period represents the representative cell. In some cases the cell posesses plane symmetries. The aim of this note is to investigate the simplifications that are introduced by such symmetries. The case of diffusion in a composite or porous medium, and the case of a Darcy flow in a porous medium are investigated. The consequences of plane symmetry on the condition of scale separation are addressed. It is shown that the scale separation condition does not always has to be fulfilled to determine effective parameters from experiments.

Humidity transfer in unsaturated heterogeneous porous media by homogenization

Abstract In the paper the one-equation model of humidity transfer in unsaturated macroscopically heterogeneous porous media is presented. The homogenization method by two-scale asymptotic expansions is used to derive the upscaled form of the Richard equation, which is commonly used when the medium is considered as macroscopically homogeneous. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics of the porous medium. The domain of validity of the model is explicitly given, namely: the length-scales separation, the characteristic time scale condition and the ratio of the hydrodynamic characteristics being of the same orders of magnitude. The effective capillary capacity and the effective hydraulic conductivity for an equivalent continuum are defined in terms of geometry and local hydrodynamic characteristics of the porous medium. A procedure of determination of the effective suction curve and the effective hydraulic conductivity curve as functions of the average water content for any type of the macroscopic heteregeneity for which the method can be applied, is provided. Since the problem is non-linear this procedure involves the resolution of a local boundary value problem formulated over a period for each value of suction. In two or three-dimensional cases, this problem can be solved using the numerical methods for any geometry of the medium. In a one-dimensional case it was shown that the analytical solution gives the well-known results of harmonic and arithmetic mean.

Experimental Evidence of the Double-Porosity Effects in Geomaterials

Double-porosity is an important characteristic of microstructure in a large range of geomaterials. It designs porous media with connected fissures/fractures or aggregated soils. The origin of double-porosity can be natural or/and it can result from mechanical, chemical or biological damage. The presence of double-porosity can significantly affect the behaviour of geomaterials. In this paper we provide an experimental evidence of the double-porosity effects by performing laboratory experiments. Series of tracer dispersion experiments (in saturated and unsaturated steady-state water flow conditions) in a physical model of double-porosity geomaterial were carried out. For the comparative purposes, experiments of the same type were also performed in a singleporosity model medium. The results clearly showed that the double-porosity microstructure leads to the non-Fickian behaviour of the tracer (early breakthrough and long tail) in both saturated and unsaturated cases.

Hydro-Mechanical Coupling in Damaged Porous Media Containing Isolated Cracks or/and Vugs: Model and Computations

In this paper we present the development of the macroscopic model describing the hydro-mechanical coupling of damaged porous media containing cracks or/and vugs, by using the asymptotic expansion method. The analysis starts at the mesoscopic scale at which we assume a generic microstructure and the validity of the Biot model in the micro-porous domain saturated by a fluid. In the crack/vug domain the Stokes equation is assumed. After estimation of orders of magnitude of different terms, the description is rendered non-dimensional and the homogenization process is carried out. It leads to an extended Biot model that possesses the same mathematical structure as the initial Biot model. However, the macroscopic poro-elasticity and the macroscopic Darcy conductivity are modified. In order to illustrate the performance of the model, numerical computations of a macroscopic boundary value problem were performed. The results show practical importance of modifications introduced in the Biot model.