Plan du manuscrit

Le manuscrit suit un ordre logique mais ne décrit pas les travaux dans l’ordre chronologique dans lequel ils ont été effectués. Il est composé de trois parties qui regroupent des études liées à une même thématique :

- Partie I, la modélisation macroscopique des phénomènes de transport. - Partie II, l’étude des propriétés effectives de transport.

- Partie III, le suivi des métamorphoses au cours du temps. Partie I

Dans cette partie, on utilise la méthode d’homogénéisation pour exprimer le modèle macroscopique équivalent à partir de la description microscopique du transport de chaleur et de vapeur, couplé au changement de phase vapeur-glace. La description microscopique utilisée est différente entre les chapitres 2 et 3.

• Chapitre 2

Dans cette étude on considère localement les phénomènes de conduction et de diffusion, ainsi que le changement de phase. Ce chapitre est constitué de l’article Macroscopic

modeling for heat and water vapor transfer in dry snow by homogenization, actellement

• Chapitre 3

On considère ici un écoulement d’air dans les pores à micro-échelle. Le transport de vapeur et de chaleur se fait donc par diffusion/conduction mais aussi par convection. Partie II

La Partie II est consacrée à l’estimation des propriétés effectives de transport de chaleur et de vapeur d’eau impliquées dans les métamorphoses de la neige sèche à partir d’images 3D couvrant toute la gamme de masses volumiques et de types de neige.

• Chapitre 4

Ce chapitre se focalise sur la conductivité thermique effective et est constitué de l’article

Numerical and experimental investigations of the effective thermal conductivity of snow,

publié dans Geophysical Research Letters en 2011. • Chapitre 5

Ce chapitre se consacre à l’étude de la perméabilité intrinsèque. Il est constitué de l’article 3-D image-based numerical computations of snow permeability: links to specific

surface area, density, and microstructural anisotropy, publié dans The Cryosphere en

2012. Partie III

La Partie III est constituée de deux études dédiées à l’évolution de la neige au cours du temps, i.e. lors des métamorphoses.

• Chapitre 6

On étudie ici l’évolution des propriétés microstructurales et effectives d’une couche de neige au cours d’une métamorphose de gradient de température à l’aide d’images 3D. Ce chapitre est constitué de l’article Study of a temperature gradient metamorphism of

snow from 3-D images: time evolution of microstructures, physical properties and their associated anisotropy, soumis à The Cryosphere.

• Chapitre 7

Dans ce chapitre, on présente une cellule cryogénique opérant à température ambiante qui permet le suivi grain à grain des métamorphoses d’un échantillon de neige par tomographie au cours du temps. Il est constitué de l’article CellIVM: a room temperature

operating cryogenic cell for the in vivo monitoring of snow metamorphism by X-ray microtomography (soumis).

Conclusion générale et perspectives

On rappelle les principaux résultats obtenus dans le cadre du travail de thèse et quelques perspectives sont proposées.

Part I

Equivalent macroscopic modelings

of heat and vapor transport by

As mentioned in Chapter 1, the physical description of the heat and vapor transfers of the current snowpack models have been proposed in a phenomenological way. The different physical phenomena (thermal conduction, water vapor diffusion, convection, source terms due to phase change, etc) involved in these models vary from one model to another, as well as the definition of the effective properties (enhanced effective properties due to the phase change, etc).

In this part, we derive rigorously the macroscopic modeling for the heat and water vapor transport through a snow layer from the physics involved at the grain scale using an upscaling method, namely the homogenization of multiple scale expansions [Bensoussan

et al., 1978; Sanchez-Palencia, 1980; Auriault, 1991; Geindreau and Auriault, 2001; Auriault et al., 2009]. Under the condition of separation of scales, the macroscopic

equivalent modeling is obtained without any prerequisite at the macroscopic scale and is intrinsic to the geometry of the medium and the phenomenon. The formulation of such macroscopic modeling is driven by an analysis of the relative importance of the physical phenomena considered at the grain scale. It includes the exact expression of the effective properties, source terms, etc. To our knowledge, this upscaling approach has never been applied to snow physics.

Part I includes mathematical developments to derive the macroscopic modeling from the description of the physics at the micro scale, as well as numerical simulations using the finite element code Comsolmultiphysics. Those simulations were performed on a 2D geometry to illustrate and evaluate the macroscopic modelings with respect to the “real” physics operating at the micro-scale. This part is divided in two chapters. In Chapter 2, we suppose that the heat and water vapor transfer at the grain scale are driven by conduction, diffusion and phase changes at the ice grain interface. In Chapter 3, the microscopic description is enriched by adding the heat and vapor convection by an air flow through the pore spaces.

2

Macroscopic modeling for heat and water vapor

transfer in dry snow by homogenization

Contents

2.1 Introduction . . . 32 2.2 Derivation of the macroscopic modeling . . . 33

2.2.1 Upscaling method . . . 33

2.2.2 Description of the physics at the microscopic scale . . . 34

2.2.3 Dimensionless pore scale description: normalization . . . 36

2.2.4 Estimation of the dimensionless numbers . . . 36

2.2.5 Asymptotic analysis . . . 37

2.2.6 Macroscopic equivalent description . . . 38

2.3 2D numerical example . . . 40

2.3.1 Problem definition . . . 40

2.3.2 Results . . . 42

2.4 Effective diffusion tensors from 3D images of snow . . . 45 2.5 Discussion . . . 48 2.6 Conclusion . . . 51

This chapter corresponds to the paper entitled Macroscopic modeling for heat and

water vapor transfer in dry snow by homogenization by Calonne N., Geindreau C. and Flin

F., published in the “as soon as publishable“ version of the Journal of Physical Chemistry

Abstract

Dry snow metamorphism, involved in several topics related to cryospheric sciences, is mainly linked to heat and water vapor transfers through snow including sublimation and deposition at the ice-pore interface. In this paper, the macroscopic equivalent modeling of heat and water vapor transfers through a snow layer was derived from the physics at the pore scale using the homogenization of multiple scale expansions. The microscopic phenomena under consideration are heat conduction, vapor diffusion, sublimation, and deposition. The obtained macroscopic equivalent model is described by two coupled transient diffusion equations including a source term arising from phase change at the pore scale. By dimensional analysis, it was shown that the influence

of such source terms on the overall transfers can generally not be neglected, except

typically under small temperature gradients. The precision and the robustness of the proposed macroscopic modeling were illustrated through 2D numerical simulations. Finally, the effective vapor diffusion tensor arising in the macroscopic modeling was

computed on 3D images of snow. The self-consistent formula offers a good estimate of

the effective diffusion coefficient with respect to the snow density, within an average relative error of 10%. Our results confirm recent work that the effective vapor diffusion is not enhanced in snow.

2.1 Introduction

The macroscopic modeling of heat and water vapor transfer through a snowpack is important to describe dry snow metamorphism and has important applications in snowpack stability [Schweizer et al., 2003], transport of chemical species[Grannas et al., 2007], snow energy balance [Sokratov and Barry, 2002]... Currently, the numerical resolution at the pore scale of the heat and water vapor transfer including phase change processes at the ice-pore interface is possible over small volumes only [Christon et al., 1994; Flin et al., 2003;

Kaempfer et al., 2005; Brzoska et al., 2008; Flin and Brzoska, 2008; Kaempfer and Plapp,

2009; Vetter et al., 2010]. At the scale of a snow layer, a macroscopic equivalent model is more relevant.

In the literature, different macroscopic models have been proposed to describe such phenomena. Using a phenomenological approach, Albert and McGilvary [1992] proposed to describe the heat and water vapor transfer through the snowpack by two coupled advection-diffusion equations including a source term due to the sublimation and deposition. Thanks to these equations, they investigated thermal effects due to air flow and vapor transport for different snow types and conditions. Following this work, Neumann et al. [2009] performed several experiments in a cold room to determine the mass transfer coefficient that arises in the source term and plays an important role on the coupling between the temperature and the water vapor fields. In this way, they proposed an empirical expression of the mass transfer coefficient.

In parallel, several models[Jordan, 1991; Brun et al., 1992; Lehning et al., 1999] have been also developed in order to simulate the physical phenomena of the snowpack. As an example, in the Crocus model [Vionnet et al., 2012] the effective vapor diffusion is not described and the temperature is determined in dry snowpacks from a classical heat

transfer equation without taking into account the latent heat due to phase change. The effective thermal conductivity is assumed constant within a snow layer and estimated with respect to density using the parametrization of Yen [1981]. In the SNOWPACK model [Lehning et al., 2002b], the effective vapor flux is simulated using an enhanced diffusion coefficient of snow (=8.5×10−5 m2s−1, http://models.slf.ch).

As already mentioned, the above models have been established in a phenomenological way. Consequently the link between these macroscopic modelings (e.g. effective properties or geometrical parameters involved) and the physical phenomena occurring at the pore scale is not always clearly established. As fully explained in Pinzer et al. [2012], the macroscopic description of the water vapor transfer in such models remains controversial. Studies by Yosida et al. [1955]; Colbeck [1993]; Sturm et al. [1997], and Satyawali [2000] indicate that the effective vapor diffusion in snow is greater than in a free air space, whereas

Sokratov and Maeno [2000] and Pinzer et al. [2012] suggest a nonenhanced coefficient. In particular, from measurements and numerical simulations Pinzer et al. [2012] indicate that the macroscopic vapor transfer through snow is not significantly influenced by the ice structure but mainly depends on the temperature field.

In this paper, we propose to derive the macroscopic equivalent modeling of the heat and vapor transfer through snow from its description at the pore scale (at the representative elementary volume (REV) scale) using the homogenization of multiple scale expansions [Bensoussan et al., 1978; Sanchez-Palencia, 1980; Auriault, 1991; Geindreau and Auriault, 2001; Auriault et al., 2009]. Under the condition of separation of scales, the macroscopic equivalent modeling is obtained without any prerequisite at the macroscopic scale and is intrinsic to the geometry of the medium and the phenomenon. The method also provides the definitions of the effective parameters arising at macro scale and the domains of validity of the macroscopic modeling, given by the order of magnitude of the dimensionless numbers that characterize the intensity of the physical phenomena at the pore scale.

The paper is organized as follows: a step by step description of the homogenization method applied to the heat and water vapor transfer through snow is presented in the first section. The obtained macroscopic equivalent modeling is discussed. In a second section, the precision and robustness of the proposed macroscopic equivalent modeling is evaluated. For that purpose, a comparison of the numerical results for the heat and vapor transfer through a 2D snow layer obtained in the case of a fine scale modeling (i.e., by taking into account all the heterogeneities) and in the case of the corresponding macroscopic equivalent modeling is performed. Finally, the influence of the snow microstructure on the effective vapor diffusion tensor arising in the macroscopic modeling is investigated by performing numerical simulations on 3D images of different snow types.