A Multiscale Model of Partial Melts 1: Effective Equations

In this paper a model for partial melts is constructed using two-scale homogenization theory. While this technique is well known to the mathematics and materials communities, it is relatively novel to problems in the solid Earth. This approach begins with a grain scale model of the medium, coarsening it into a macroscopic one. The emergent model is in good agreement with previous work, including D. McKenzie's, and serves as verification. This methodology also yields a series of Stokes problems whose solutions provide constitutive relations for permeability and viscosity. A numerical investigation of these relations appears in a companion paper.Comment: 55 pages. Submitted to JGR Solid Eart

Acoustique d'un fluide au voisinage du point d'ébullition

International audienceDans une étude précédente (Boutin et Auriault 1993), la propagation d'ondes acoustiques dans un fluide à bulles en concentration finie était analysée au moyen de la méthode des échelles multiples. Trois comportements différents étaient mis en évidence suivant la taille des bulles, grande, moyenne ou petite. Dans la présente note, nous étendons cette étude par la prise en compte d'un possible changement de phase. Nous montrons que les effets de celui-ci sont négligeables dans le cas de grandes bulles, et modifient fortement le comportement des bulles moyennes en diminuant de plusieurs orders de grandeur la rigidité effective. Pour les petites bulles la capillarité devient le phénomène prépondérent

Waves in bubbly liquids with phase change

International audienceIn a previous paper (C. Boutin, J.-L. Auriault, Acoustics of a bubbly fluid at large bubble concentration, Eur. J. Mech. B/Fluids, 12(3) (1993) 367-399), the homogenization technique was used to investigate how acoustic waves propagate in a bubbly fluid at finite concentration. Three different equivalent macroscopic behaviours were shown to exist, for "large"-, "medium"- and "small"-size bubble systems, respectively. In the present paper, we extend the analysis by taking into consideration possible phase change effects. We show that phase change effects are negligible in the case of large-size bubbles, whereas they strongly modify the medium-size bubble system behaviour. For small-size bubbles capillarity dominates the process

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

Long wavelenght inner-resonance cut-off frequencies in elastic composite materials

International audienceWe revisit an ancient paper (Auriault and Bonnet, 1985) which points out the existence of cut-off frequencies for long acoustic wavelength in high-contrast elastic composite materials, i.e. when the wavelength is large with respect to the characteristic heterogeneity length. The separation of scales enables the use of the method of multiple scale expansions for periodic structures, a powerful upscaling technique from the heterogeneity scale to the wavelength scale. However, the results remain valid for non-periodic composite materials which show a Representative Elementary Volume (REV). The paper extends the previous investigations to three-component composite materials made of hard inclusions, coated with a soft material, both of arbitrary geometry, and embedded in a connected stiff material. The equivalent macroscopic models are rigorously established as well as their domains of validity. Provided that the stiffness contrast within the soft and the connected stiff materials is of the order of the squared separation of scales parameter, it is demonstrated (i) that the propagation of long wave may coincide with the resonance frequencies of the hard inclusions/soft material system and (ii) that the macroscopic model presents a series of cut-off frequencies given by an eigenvalue problem for the resonating domain in the cell. These results are illustrated in the case of stratified composites and the possible microstructures of heterogeneous media in which the inner dynamics phenomena may occur are discussed

Flow of low pressure gas through dual-porosity media

Using the theory of homogenization we derive macroscopic models for describing flow of gas at low pressure in dual-porosity media. The case of a fractured porous medium is under consideration for the study, and the existence of a representative elementary volume that consists of open connected fractures surrounded by porous matrix blocks is assumed. The local flow is governed by either Klinkenberg's law or Knudsen's diffusion law in the matrix while either a non-slip flow or a slip flow occurs in the fractures. Six new models are derived by homogenization, which are compared to the three models which were obtained for Darcy's regime in an earlier work. Each of these nine models is characterized by its macroscopic flow regime and by the type of macroscopic behavior it describes. Besides Darcy's and Klinkenberg's macroscopic flow regimes, a transition regime between Klinkenberg's and Knudsen's regimes is identified. The types of macroscopic behaviors include a dual and a single porosity description and an intermediate behavior that describes a single-porosity behavior, but in which the porosity of the entire fractured porous medium is accounted for