Jean-Louis Auriault

Heterogeneous medium, is an equivalent macroscopic description possible ?

Abstract The aim of this paper is to answer the question: considering a finely heterogeneous medium submitted to some excitation, does an equivalent macroscopic description exist? An equivalent macroscopic description means here an intrinsic description, independent of the macroscopic boundary conditions. It is well known that the condition resides in a good separation of scales. This separation concerns both the structure of the medium and the excitation itself. The homogenization process using double scale asymptotic developments appears then to be the appropriate method giving the right answer to the question. This is emphasized in two simple examples.

Taylor dispersion in porous media: Analysis by multiple scale expansions

Convection-diffusion phenomena are analyzed by means of multiple scale expansion both in space and time when the macroscopic dimensions of the porous medium are much larger than the microscopic ones. Four different regimes are detailed and discussed depending upon the relative magnitude of the Peclet number. The results are compared with the prediction of the method of moments.

Dynamic behaviour of a porous medium saturated by a newtonian fluid

Abstract The macroscopic dynamical behaviour of a porous elastic medium the cavities of which are filled with a Navier-Stokes liquid can be described with equations obtained from certain assumptions and the homogenization method, based on the use of periodic solutions of space variables. The so-determined linear behaviour is a generalization of Biot and Levy works: particularly, strain rates of the solid are taken into account for the flow law of the liquid. Then, some special relationships are determined for coefficients which are involved in the macroscopic equations and determined by particular solutions of the boundary problem at the pore scale.

Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media

This work is concerned with deriving a macroscopic filtration law for describing transient linear viscoelastic fluid flow in porous media. This is performed using a homogenisation technique, i.e. by upscaling the heterogeneity scale description. The macroscopic filtration law is expressed in Fourier space as a generalised Darcys law with a dynamic permeability tensor. This model is valid at low Reynolds and Deborah numbers. Analytical results are determined in the particular case of the flow of an Oldroyd fluid in a bundle of capillary tubes and are compared to those obtained by a corresponding phenomenological model.

Effective diffusion coefficient : From homogenization to experiment

In this paper, an example of the application of the homogenization approach (asymptotic expansion technique) to predict the effective diffusion coefficient for an equivalent continuum, together with the experimental verification of the theoretical results is presented. The experimental setup was constructed for the measurements of diffusion in a model periodic porous medium made of Plexiglas. The computer program using the FEM was elaborated to solve the local boundary value problem for a period and to calculate the effective diffusion coefficient. The comparison between the theory and the experiment indicates good agreement between the numerical and experimental values of the effective diffusion coefficient. Interpretation of the test data from the point of view of the homogenization theory is also incorporated.

Microstructural effects on the flow law of power-law fluids through fibrous media

In this work, the flow of power-law fluids through anisotropic fibrous media is revisited, upscaling the fluid flow at the pore scale with the homogenization method of multiple scale expansions for periodic structures. This upscaling technique permits a quantitative study of the seepage law by performing numerical simulation with simple two-dimensional periodic arrays of circular solid inclusions. The significant role of the solid fraction, the fluid rheology and the porous media anisotropy on the resulting macroscopic flow law is underlined from the simulation.

Transport in Porous Media: Upscaling by Multiscale Asymptotic Expansions

Transport in porous media is investigated by upscaling the pore scale behaviour. We use the technique of multiscale asymptotic expansions which seems to be the most efficient method to obtaining macroscopic equivalent behaviours. Different transport phenomena are addressed: fluid flow through a saturated porous medium (Darcy’s law), diphasic flow (coupled Darcy’s laws), solute transport (diffusion, advection, dispersion) and fluid flow through deformable porous media (consolidation).

Coriolis Effects on Filtration Law in Rotating Porous Media

We investigate the filtration law of incompressible viscous Newtonian fluids in rigid non-inertial porous media, for example, rotating porous media. The filtration law is obtained by upscaling the flow at the pore scale. We use the method of multiple scale expansions which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. For finite Ekman numbers the filtration law is shown to resemble a Darcys law, but with a non-symmetric permeability tensor which depends on the angular velocity of the porous matrix. We obtain the filtration analog of the Hall effect. For large Ekman numbers the filtration law is a small correction to the classical Darcys law. The corrector is antisymmetric. In this case we recover a structure of law which is similar to phenomenological laws introduced in the literature, but with a dissimilar effective coefficient.

Transient quasi-static gas flow through a rigid porous medium with double porosity

The highly compressible fluid flow through a three-scales rigid porous medium (pore, fracture, macroscopic sample) is investigated using a homogenization method. The macroscopic description is strongly dependent on the separation of the different scales, and three cases are considered. The pores either play the role of a compressible fluid reservoir, introduce a memory effect, or are ignored, respectively. The homogenization result is compared to classical phenomenological models that are available in the case of slightly compressible fluids. Pseudo-steady state models are shown to give a rough description of the phenomenon.

Investigation of the viscoplastic behaviour of alloys in the semi-solid state by homogenization

The aim of this paper is to investigate the viscoplastic behaviour of alloys in the semi-solid state. Over a critical solid volume fraction, the metallic material is considered as a porous medium constituted by a viscoplastic skeleton and saturated by an incompressible Newtonian fluid. By using homogenization method of periodic structures, we determine the macroscopic equivalent medium. We show that the heterogeneous material displays three types of behaviour: biphasic, monophasic with fluid pressure effect and monophasic with fluid viscosity effect, according to the contrast of the mechanical properties of the two components. This contrast is measured by a small parameter e, ratio of the dimensions of pores or solid particles to the macroscopic characteristic length of the sample or of the excitation. The homogenization process applied to the viscoplastic saturated porous medium permits to validate the structure of constitutive laws already proposed in the phenomenological way to describe experimental data.