Professor Olof Nilsson

Dr Peter Mcdonald

Professor Karen Scrivener

Professor Mette Geiker Peter Mc- Donald

Mrs Marie-

Alix Dalang

Pauline Huart

Biyun Antoine Xiaomeng

Valérie Guillaume

Aurélie Nicolas Assia

Othman, Rim, Jennifer Thao

Patrick Héla

Keywords: Cementitious materials, Moisture transport, Hysteresis, Water vapour sorption isotherms, Drying and wetting cycles, Numerical simulation, Experimental investigations, Gas pressure variations, Transport modes, Knudsen eect, Apparent diusivity, Mass boundary region Résumé microstructure, modélisation

The durability of reinforced concrete structures and their service life are closely related to the simultaneous occurrence of many physical and chemical phenomena.

These phenomena are diverse in nature, but in common they are dependent on the moisture properties of the material.

Therefore, the prediction of the potential degradation of cementitious materials requires the study of the movement of liquid-water and gas-phase transport in the material which is considered as a porous medium. In natural environment, structures are always aected by periodic variations of external relative humidity (RH). However, most moisture transport models in the literature only focus on the drying process. There are few studies considering both drying and wetting, although these conditions represent natural RH variations. Much less studies take into account hysteresis in moisture transport. Therefore, this work is devoted to a better understanding of how the moisture behaviour within cementiitous materials responds to the ambient RH changes through both experimental investigations and numerical modelling. In particular, the consideration of hysteresis will be included in numerical modelling.

In the research, a multiphase continuum model was recalled rstly. According to a theoretical analysis and experimental verications, a simplied model was obtained for the case that the intrinsic permeability to liquid-water is much smaller than the intrinsic permeability to gas-phase.

The review of commonly-used hysteresis models enabled to conclude a set of best models for the prediction of water vapour sorption isotherms and their hysteresis. The simplied model was coupled with the selected hysteresis models to simulate moisture transport under drying and wetting cycles. Compared with experimental data, numerical simulations revealed that modelling with hysteretic eects provided much better results than non-hysteresis modelling. Among different hysteresis models, the use of the conceptual hysteresis model, which presents closed form of scanning loops, showed more accuracy predictions. Further simulations for dierent scenarios were also performed. All comparisons and investigations enhanced the necessity of considering i Abstract hysteresis to model moisture transport for varying RH at the boundary. The investigation of moisture penetration depth could provide a better understanding of how deep moisture as well as ions can move into the material. Furthermore, this research investigated dierent methods to determine the liquid-water intrinsic permeability, including the inverse analysis with dierent boundary conditions and tting measured apparent diusivity values for dierent equations to calculate relative permeabilities.

expérimentale et de modélisations numériques. Un soin sera apporté an de tenir compte dans les modèles numériques des eets d'hystérésis. Dans ce travail, nous détaillerons, tout d'abord, un modèle multi-phasiques complet. Un modèle simplié est obtenu, sur la base de considérations théoriques et de vérications expérimentales dans le cas où la perméabilité intrinsèque à l'eau liquide reste très inférieure à la perméabilité intrinsèque au gaz. Une étude comparative des modèles d'hystérésis couramment utilisés permet d'obtenir un jeu de modèles proposant les meilleures prédictions d'isothermes de sorption d'eau et de leurs hystérésis. Par la suite, le modèle de transport simplié est couplé avec les modèles d'hystérésis sélectionnés an de simuler les transferts hydriques dans des bétons soumis à des cycles d'humidication-séchage. La comparaison avec des données expérimentales révèle que la prise en compte de l'hystérésis de l'isotherme de sorption d'eau ne peut pas être négligé. De plus, il est montré que les prédictions obtenues iii Résumé avec des modèles d'hystérésis théoriques, sont les plus cohérentes avec les résultats expérimentaux, en particulier, pour des chemins secondaires d'hystérésis. Plusieurs scénarios (conditions environnementales, bétons diérents) sont également simulés. Les résultats obtenus pointent à nouveau la nécessité de tenir compte de l'hystérésis lors de la modélisation des transferts hydriques à travers des matériaux cimentaires soumis à des variations d'humidité relative. La

2.4

Comparison of k rl and k rg calculated by Method-1 and Method-2 for Paste3. . .

3.3

Comparison of R 2 adj and NME for tting of the main desorption isotherms.

. . .

3.4

Comparison of R 2 adj and NME for the main adsorption isotherms. . . . . . . . . .

3.5

Fitting results for Paste2 material to compare results tted by selected models. .

5.8

Evolutions of k rl and f (φ, S) at dierent depths simulated by VGMa for Paste3.

5.9

Comparisons of simulation results with measured data for PCEMI6. . . . . . . . Fitted D a curves by four models for Paste5 in the desorption process. . . . . . .

6.3

Fitted D a curves for Paste2 during both desorption and adsorption processes. .

6.4

Fitted D a curves for Paste3 during both desorption and adsorption processes. .

6.5

Denition of pore radii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6

The t a curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.7

The pore size distribution calculated by BJH method. . . . . . . . . . . . . . . .

6.8

The calculated Knudsen eect for dierent materials. . . . . . . . . . . . . . . . .

6.9

Schematic representation of the system in case of our experimental conditions. . .

2.2

Values of a V and m V for the selected materials tted by Eq. (2.4). . . . . . . . .

2.3

The intrinsic permeabilities tted by inverse analysis for dierent materials. . . .

2.4

The intrinsic permeabilities to liquid-water validated by experimental data for dierent materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

The intrinsic permeabilities to liquid-water validated by experimental data for two versions of simplied models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

Proposed multi-level approach to model hysteresis. . . . . . . . . . . . . . . . . .

3.2

Collected datasets of WVSIs measured by the saturated salt solution method . .

3.3

Proposed models for the multi-level approach to model hysteresis. . . . . . . . . .

Validated liquid intrinsic permeabilities (×10

-21 m 2 ) for dierent materials. . . .

6.1

Fitted K l (×10 -21 m 2 ) for the desorption and adsorption processes. . . . . . . . .

6.2

The tted x D for the desorption process. . . . . . . . . . . . . . . . . . . . . . . .

6.3

The tted x D for the adsorption process. . . . . . . . . . . . . . . . . . . . . . . .

6.4

The calculated RH threshold values for using VGMa model with Knudsen eect.

J • K -1 • mol -1 R 2 Coecient of determination. - R 2 adj Adjusted R 2 .
-

RH

Background

For a variety of reasons, concrete has become the mostly used build materials for centuries. In fact, for every person one ton of concrete is used annually. It is a strong and durable material that is versatile and economical. Designers of structures have been mostly interested in how to make structures more durable. One reason is that the cement industrials constitute 5-7% of man-made CO 2 emissions over the world [1, 2, 3]. More durable structures mean that less cement will be used in construction, which could be an indirect way to reduce global CO 2 emissions.

Another reason is that it has been estimated that in developed countries about 40% of the total resources of the construction industry are being applied to repair and maintenance of existing structures [START_REF] Mehta | Concrete -Microstrucure, Properties and Materials[END_REF]. This cost forces engineers to pay serious attention to durability issues.

Concrete structures are sub jected to either biological, physical or chemical degradation processes. But in common, water, as the primary agent of both creation and destruction of the material, attributes to be central to most durability problems in concretes [START_REF] Nilsson | Water and the hygro-thermal characteristics of hardened concrete[END_REF]. Indeed, liquid-water can transport chemicals such as chloride ions, alter mechanical properties, and cause freeze-thaw damage. To predict the degradation of concrete structures, it is important to better understand the moisture behaviour of the material and its responses to external ambient environment.

Hence, the European Community (Framework Programme 7) funded a Marie Curie Initial

Training Network, called TRANSCEND, to carry out research aiming at understanding water transport in cementitious materials. This thesis is one part of the TRANSCEND project, which stands for understanding TRANSport for C oncrete which is Eco-friendly iN novative and Durable.

Natural conditions and importance of moisture transport

In the natural environment, the moisture behaviour is very complicated (e.g., tidal zone, uctuating groundwater, weather changes), and the moisture exchanges occur continually at the interface between materials and environment. The material can lose moisture caused by low environment RH or gain water due to high external RH. In this process, moisture is able to move into or out the porous media, and this process keeps going along with the varying external conditions.

In porous solids, the cyclic changes of water content are known to be the cause of many types of chemical and physical degradation processes. The moisture movement is generally due to capillary suction, condensation, gradient of density of water vapour, etc. But one thing can be sure is that rarely pure water is involved. Water is noted for its ability to dissolve more substances than any other known liquid. When liquid-water enters the material, aggressive ions and salts will penetrate into along with water, and when moisture leaves the material, some chemical components will move out with liquid-water. During this repetitive process, salts and ions will accumulate in the zone where drying and wetting cycles frequently occur. Therefore, the repeated wetting and drying regimes can increase the risk of deteriorations of cement-based materials.

For concrete structures which are exposed to those conditions, materials are rarely in a state of full saturation. In that case, strongly capillary pressure will apply on pore walls and further induce shrinkage (during drying) or swelling (during wetting) which is able to damage the microstructure of cementitious materials. As the essential factor, moisture transport can induce many other durability issues, such as corrosions and carbonation, whose kinetics is correlated to the moisture states. Hence, it is of great importance to study moisture transport in cementitious materials under drying and wetting cycles conditions.

Modelling of drying and wetting cycles

Modelling of moisture transport under drying-wetting cycles conditions for cementitious materials has been studied more than two decades [START_REF] Cunningham | The moisture performance of framed structures -a mathematical model[END_REF]. In the early studies, most moisture transport models used a simple way, which employed the same sorption curve for both drying and wetting processes; thus, sorption hysteresis between these two processes was neglected [START_REF] Cunningham | The moisture performance of framed structures -a mathematical model[END_REF][START_REF] Arfvidsson | A new algorithm to calculate the isothermal moisture penetration for periodically varying relative humidity at the boundary[END_REF][START_REF] Arfvidsson | Isothermal moisture ow in building materials: modelling, measurements and calculations based on Kirchho 's potential[END_REF]. However, sorption hysteresis has been recognised as one main characteristic of moisture behaviour for cementitious materials [START_REF] Anderberg | Moisture properties of self-levelling ooring compounds[END_REF][START_REF] Espinosa | Inkbottle pore-method: Prediction of hygroscopic water content in hardened cement paste at variable climatic conditions[END_REF][START_REF] Espinosa | Inuence of the age and drying process on pore structure and sorption isotherms of hardened cement paste[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF][START_REF] Åhs | Sorption scanning curves for hardened cementitious materials[END_REF]. Recent studies performed by Maekawa et al. [START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF], Johannesson and Nyman [START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF][START_REF] Johannesson | A numerical approach for non-linear moisture ow in porous materials with account to sorption hysteresis[END_REF] and Derluyn et al. [START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF] have proved that this simple approach cannot provide high accurate predictions and suggested to take into account sorption hysteresis. This suggestion leads to a more complicated modelling method, which needs to couple a sorption hysteresis model with a moisture transport model.

In the literature, researchers used dierent ways to study modelling of sorption hysteresis.

One way tried to understand reasons and mechanisms of sorption hysteresis.

Espinosa and

Franke [START_REF] Espinosa | Inkbottle pore-method: Prediction of hygroscopic water content in hardened cement paste at variable climatic conditions[END_REF] proposed an approach called ink-bottle pore method, assuming ink-bottle pores as the reason of hysteresis, in order to predict the main desorption, main adsorption and scanning curves. However, there are more than eight undetermined parameters which have to be analysed rstly by tting experimental data such as molecular adsorption data. Another model presented by Ranaivomanana et al. [START_REF] Ranaivomanana | Toward a better comprehension and modeling of hysteresis cycles in the water sorptiondesorption process for cement based materials[END_REF] was developed based on the description of the pore size distribution by considering oblong-shaped pores. Pinson et al. [START_REF] Pinson | Hysteresis and percolation in continuum modelling of water transport in cementitious materials[END_REF][START_REF] Pinson | Inferring pore size and network structure from sorption hysteresis[END_REF] further assumed dierent levels of pore network connectivity to explain sorption hysteresis. These theoretical studies can present good explanations of hysteresis but they rarely provide an easy-to-use calculation method that can be combined with moisture transport model to simulate drying and wetting cycles.

Other researchers proposed most straightforward methods which are generally taken from modelling of hysteresis in soil science. These models t the measured main desorption and main adsorption curves by using empirical models and further predict scanning loops, such as studies in [START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF][START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF][START_REF] Johannesson | A numerical approach for non-linear moisture ow in porous materials with account to sorption hysteresis[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF]. Models used in this method may lack physical interpretation, but they are very easy to couple with moisture transport models and can provide good tting and prediction results.

Since the current research has a focus on modelling of drying and wetting cycles, it will only review and apply of hysteresis models which were developed in the second case mentioned above.

Objective, methods and chapter outline

The ob jective of this study is set to better understand how the moisture behaviour inside cementitious materials responds to varying relative humidity at the material boundary. In the current research, both a simple method (without sorption hysteresis) and a more complicated one (with sorption hysteresis) for modelling of drying and wetting cycles will be compared and discussed.

The arrangement for each chapter is presented as follows.

For modelling of drying and wetting cycles, the rst step focuses on the review of models which were initially developed to simulate moisture transport for varying relative humidity at the boundary. Three methods can be summarized: using dierent transport coecients for drying and wetting, modelling without hysteretic eects and modelling with hysteresis.

The second step focuses on understanding and simplifying a multiphase continuum model.

The simplied model will be further used for modelling of drying and wetting cycles. This continuum model considers that the mass transport includes advection of liquid-water and gasphase (dry air and vapour), and diusion of gas-phase.

The liquid intrinsic permeability is determined by tting drying kinetics, while the gas intrinsic permeability can be either set equal to the liquid intrinsic permeability or obtained from measurements. In the latter case, the gas intrinsic permeability is normally several orders of magnitude higher than the liquid intrinsic permeability. The discussion and analysis of both cases can lead to simplied moisture transport models which will be used for the implementation of hysteresis models into moisture transport.

The third step is to nd an appropriate hysteresis model. A detailed review of hysteresis models available in the literature is proposed in this chapter. Hysteresis models which have been initially developed for water transport in soils and sands are selected, including conceptual and empirical models. A comprehensive statistic comparison with experimental data is performed for cement pastes and concretes. A multi-level approach to simulate hysteresis is proposed as well. Evaluations and recommendations for use of this multi-level approach are provided for an application purpose of the compared models.

The fourth step is to prepare the input data for modelling, experimental data for calibration of the liquid intrinsic permeability and data for verication of modelling results of drying and wetting cycles. Meanwhile, experimental investigations of eects of cyclic RH at the boundary on moisture distribution within cementitious materials are also reported in this part. As the main input data, water vapour sorption isotherms for various cement pastes (made by CEM I, CEM III and CEM I + silica fume) are measured by using a dynamic vapour sorption instrument (DVS) and pressure plate. Measured mass loss curves during drying are used to determine the liquid intrinsic permeability. Mass loss curves measured in following processes or wetting and re-drying and saturation proles measured at dierent times are used to verify the modelling results.

The most important part of this work is to perform modelling of moisture transport under drying and wetting cycles conditions. Modellings with and without hysteretic eects are implemented and compared based on experimental data. Simulations in various situations, such as dierent initial conditions, cycle durations, RH amplitudes, etc., are conducted to check the applicability of the proposed modelling methods. In the cyclic drying and wetting regime, the moisture penetration depth, which is very useful for studies on durability issues, such as ions penetration, can be determined by modelling.

The last step introduces two research perspectives which can provide further information to understand moisture transport properties in cementitious materials, especially for the assessment of the liquid-water intrinsic permeability K l . The rst perspective is to determine K l by tting measured apparent diusivity values taken from the literature [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF]. The resistance factor to vapour diusion can also be determined likewise. To improve the tting results, Knudsen eect is taken into account for the vapour diusion. The second perspective is to check if using dierent boundary conditions has an eect on moisture transport in the material under our experimental conditions. The liquid intrinsic permeabilities can be determined by inverse analysis for dierent boundary conditions. These K l values are further compared with these determined by other methods (Katz-Thompson [START_REF] Thompson | The microgeometry and transport properties of sedimentary rock[END_REF] and tting apparent diusivity values). The work in this part aims to illustrate the robustness and the reliability of using inverse analysis to determine K l .

Declaration of modelling codes and experimental data

Numerical modellings in Chapter 2 of this thesis were conducted by using BIL, a modelling platform based on nite element/volume methods dedicated to coupled problems involved in environmental engineering, geomechanical engineering, material sciences, etc. BIL is released by Patrick Dangla (see http://perso.lcpc.fr/dangla.patrick/bil/) and distributed under the terms of the GNU General Public License (GPL). Other calculations (in Chapters 3, 5 and 6) were carried out by the authors themselves by using MATLAB, Microsoft Excel and other free softwares.

The experimental data used to feed and verify the models were either measured in the previous work by Baroghel-Bouny [START_REF] Baroghel-Bouny | Characterization of cement pastes and concretes-methods, analysis, interpretations[END_REF][START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF] and Nguyen [START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF] performed at IFSTTAR (former LCPC)

or conducted by the authors themselves at IFSTTAR and Lund University.

Chapter 1

Review of modelling of drying and wetting cycles

Summary

Cementitious materials are composite materials, with cement, sands and gravels, whose microstructure is random over wide range of length scales, which makes the major dierence to other porous media such as soils and rocks. Of course, the moisture (liquid-water + vapour) states, transport properties, measurement methods, etc., are also dierent.

This chapter will focus on discussing moisture in cementitious materials, which is associated with several questions: Where is moisture from? How does moisture stay in the material? How does external moisture become internal moisture and vice versa? How does moisture move within the material? What are the moisture transport mechanisms? After discussions about these questions, a multiphase moisture transport model for porous media will be introduced.

The most important part is the review of models initially developed for varying external RH at the boundary to predict moisture behaviour under drying and wetting cycles conditions.

Then, the discussion about moisture interactions between materials and their surrounding will be presented.

Introduction

For materials like concretes, the moisture transport properties are not easy to determine because of the unpredictable sources of water and dierent states of water in the material.

Sources of water

Water is a necessary ingredient for the cement hydration reactions. As an agent, it facilitates the mixing of the components of concrete. The amount of added water is more than the amount water needed for the chemical reactions, since only a water-to-cement ratio (w/c) = 0.25 is enough for all chemical reactions in case of Portland cements [START_REF] Delmi | Modelling the coupled evolution of hydration and porosity of cement-based materials[END_REF][START_REF] Scrivener | Quantitative study of Portland cement hydration by X-ray diraction/Rietveld analysis and independent methods[END_REF]. Besides, a cementitious material normally cannot reach a fully hydrated state, so more water beyond estimation will be left in the material and become physically-bound and free water. This is the initial source of water in hardened cement-based materials.

Gradually, depending on the ambient conditions and the size of the material, most of the evaporable water in concrete is lost, leaving empty or unsaturated pores. When the ambient relative humidity increases, the material can capture external moisture. This is the second source of moisture in the material.

Water states in cementitious materials

Water states and water transport are related to the microstructure of the material. It needs to know the pore size distribution to understand water states and their mobility. Owing to the complex structure and connectivity of the network, pores are in dierent shapes and randomly distribute which makes the quantication of the exact pores size distribution (PSD) extremely dicult. By assuming pores as cylinders, techniques can be used to measure PSD. Based on measurements using such as Scanning Electron Microscopy, Optical Microscopy, Impedance Spectroscopy and Mercury Intrusion Porosimetry (MIP), movable water in the pores in cementitious materials can be classied into dierent levels (see Table 1.1). • C).

The water states in the material are not only related to the pore size distribution but also related to other microstructure properties such as geometry shapes of pores and pores layout. Researchers create dierent simplistic models for hardened cement paste to represent the structure.

For example, Powers' model [START_REF] Powers | Structure and physical properties of hardened portland cement paste[END_REF] considered that a representative part of the material consists of either two or three layers of C-S-H, which could roll into bres. Feldman and Sereda [START_REF] Feldman | A new model of hydrated portland cement paste as deduced from sorption -length change and mechanical properties[END_REF][START_REF] Feldman | A new model for hydrated portland cement and its practical implications[END_REF] regarded gel as a three-dimensional assemblage of C-S-H layers, which tends to form a subparallel group with a few layers of thickness. In Munich model [START_REF] Wittmann | Surface tension shrinkage and strength of hardened cement paste[END_REF], the paste was described as a dispersed gel, together with the crystalline constituents. Jennings [START_REF] Jennings | A model for the microstructure of calcium silicate hydrate in cement paste[END_REF] proposed a model for the structure of C-S-H, considering roughly spherical C-S-H units, which occulate to form larger units.

Figure 1.1: Water states in cementitious material (adapted from Feldman and Sereda [START_REF] Feldman | A new model of hydrated portland cement paste as deduced from sorption -length change and mechanical properties[END_REF][START_REF] Feldman | A new model for hydrated portland cement and its practical implications[END_REF]).

The layered irregular structure of the Feldman and Sereda model [START_REF] Feldman | A new model of hydrated portland cement paste as deduced from sorption -length change and mechanical properties[END_REF][START_REF] Feldman | A new model for hydrated portland cement and its practical implications[END_REF], which was developed based on studies of sorption properties, porosity and relations between water content and mechanical properties, clearly shows three types of water in the material: interlayer, adsorbed and capillary water (see Fig. 1.1). Interlayer water is highly combined to C-S-H sheets, so it needs very strong driving force to move. In atmospheric condition, this water is rarely aected by varying RH at the boundary. Adsorbed water and gel pore water can be released under low RH.

Capillary water is more movable and evaporable, and is the main source of transported water.

However, capillary water is generally not continuous because of the complex geometry structure.

It evaporates into vapour at the menisci between liquid and vapour. Partially continuous liquidwater only exits in the high water content range. It is distributed in dierent locations and is locally separated by the gas-phase in the material.

1.2 Moisture sorption and hysteresis

Moisture sorption processes

When the pore is not fully lled by liquid-water, the material gains or releases moisture through the sorption process. Sorption, including adsorption and desorption, is the accumulation of molecules onto pore walls from the surrounding moisture [START_REF] Gregg | Adsorption, surface area and porosity[END_REF]. The adsorption is brought about by forces acting between the solid and water molecules. These forces are of two kinds -physical and chemical -and give rise to physical sorption and chemisorption, respectively. The main mechanisms of adsorption are given as follows.

1) Exchange sorption: electrostatic interaction of ions due to the dierent charges between the pore walls and water molecules.

2) Physical sorption: van der Waals attraction between adsorbate (moisture) and adsorbent (pore walls). The attraction is not xed to a specic site and the adsorbate is relatively free to move on the pore walls. This sorption is relatively weak and reversible, and can lead to a multilayer of adsorbed water.

3) Chemical sorption: some degree of chemical bonding between adsorbate and adsorbent characterized by strong attractiveness. Adsorbed molecules are not free to move on the pore walls. There is a high degree of specicity, and typically a monolayer is formed.

The adsorption and desorption of a gas on the solid surface is always accompanied by the evolution of heat.

It is dependent on the adsorbent adsorbate and adsorbate -adsorbate interactions [START_REF] Gregg | Adsorption, surface area and porosity[END_REF]. The adsorption releases heat, while the desorption requires heat. This makes these two processes quite dierent.

When menisci are formed in pores, capillarity is involved. The interface between liquidwater and gas-phase is maintained due to the surface tension γ between these two phases. At equilibrium, a balance of force at this interface can be described by the Young-Laplace equation:

P c = P g -P l = 2γ r * (1.1)
where P c , P g and P l represent capillary pressure, gas pressure and liquid pressure, respectively.

r * denotes the mean radius of curvature.

Moisture sorption and hysteresis

When the concentration of vapour in gas-phase changes, vapour can condense or release from solid surfaces, and menisci appear in big or small pores. There should be a strong relationship between vapour concentration in gas-phase and water saturation in the material. At the macroscopic scale, vapour concentration in gas-phase is closely related to the external relative humidity. In physical and chemical science, the relationship between external relative humidity RH and water saturation S in the material at equilibrium is known as sorption isotherms or water vapour sorption isotherms (WVSIs) [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. For more porous materials, such as soils, a great amount of menisci can be formed in the material, so the relationship between macroscopic capillary pressure and water saturation is also called retention curves [START_REF] Bear | Modeling Groundwater Flow and Contaminant Transport[END_REF].

P c = P c (S) (1.2)
At equilibrium, the macroscopic capillary pressure in the material can be calculated based on the external relative humidity. Hence, for cementitious materials, sorption isotherms can be measured by exposing the material to water vapour with a denite RH and temperature. When RH changes from a high value to a low value, a desorption curve can be measured. In an inverse process, an adsorption isotherm is obtained.

In the low RH range, the sorption process plays the main role for sorption isotherms. Sorption models, such as Linear [START_REF] Erbil | Surface Chemistry of Solid and Liquid Interfaces[END_REF], Freundlich [START_REF] Freundlich | ber die adsorption in lösungen[END_REF], Langmuir [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids. Part I. solids[END_REF], BET [START_REF] Brunauer | Adsorption of gases in multimolecular layers[END_REF], Dent [START_REF] Dent | A multilayer theory for gas sorption. Part I: Sorption of a single gas[END_REF], etc., can provide good tting of gas sorption isotherms. Langmuir's equation is used for the adsorption of a single layer of vapour molecules (monomolecular). BET and Dent's equations are capable of accounting for multilayer molecules adsorption (polymolecular). However, comparisons performed by Kumar et al. [START_REF] Kumar | Water vapor sorption in cementitious materials measurement, modeling and interpretation[END_REF] show that these models are only valid for isotherms in the low RH range, i.e., smaller than 45% [START_REF] Wadsö | Transcend course 3 at Lund University: Sorption and Transport[END_REF]. In the high RH range, these models produce big errors. Nevertheless, most concrete structures suer durability issues under moderate and high RH levels, generally between 50% and 100%. In this range, water vapour sorption isotherms not only include sorption but also is involved by capillarity. Hence, for the purpose of modelling, researchers are apt to t measured sorption isotherms by using empirical equations, which can be widely found in soil science to t retention curves [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF][START_REF] Baroghel-Bouny | Modelling of isothermal coupled moistureion transport in cementitious materials[END_REF][START_REF] Baroghel-Bouny | Prediction of chloride binding isotherms of cementitious materials by analytical model or numerical inverse analysis[END_REF]. The review of these equations, which were initially developed for soils and sands, will be presented in Chapter 3.

Sorption hysteresis

In measurements, adsorption curves show dierent paths to desorption curves. This phenomenon, dierent sorption isotherms during drying and wetting, is called sorption hysteresis. Experimental data from the literature [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] reveal that hysteresis between desorption and adsorption is found in the whole RH range. It is more signicant in the middle RH range, in which the capillary condensation and evaporation is assumed to be the mechanism which governs the thermodynamic equilibrium between liquid-water and water vapour.

Actually, in dierent pore scales, various reasons contribute to hysteresis. For capillary pores, the main reason, known as ink-bottle eect, results from the dierent shapes of connected pores.

One example is shown in Figure 1.2. If considering the pore is initially lled with water, during drying, capillary pressure increases gradually from P c (r 5 ) to P c (r 4 ). At the beginning, because the meniscus at the narrow throat r 0 blocks moisture inside, at capillary pressure P c (r 5 ), P c (r 3 )

and P c (r 1 ), no moisture can be transported outside the pore. Until capillary pressure is larger than P c (r 0 ), the meniscus cannot support the same capillary pressure and moisture in r 0 and r 1 suddenly loses. The same process will happen when capillary pressure increases to P c (r 2 ) and P c (r 4 ). This kind of sudden change is called Haines jump [START_REF] Haines | Studies in the physical properties of soil. V: The hysteresis eect in capillary properties, and the modes of moisture distribution associated therewiht[END_REF]. During wetting, the throat will be lled by water in the sequence of r 4 , r 2 , r 0 , r 1 , r 3 and r 5 . Therefore, hysteresis is observed between the drying curve and the wetting curve (see Fig. 1.2b). It is clear that the drying process depends on the narrow throats, while the wetting process is more related to the pore bodies [START_REF] Bear | Modeling Groundwater Flow and Contaminant Transport[END_REF]. The second most known reason of sorption hysteresis is the dierent contact angle [START_REF] Aligizaki | Pore Structure of Cement-Based Materials Testing Interpretation and Requirements -Modern Concrete Technology[END_REF][START_REF] Bear | Modeling Groundwater Flow and Contaminant Transport[END_REF] which is able to aect hysteresis in various ways. Indeed, dierent chemical components in cementitious materials have dierent contact angles (see Fig. Theoretically, moisture in C-S-H pores induces very small hysteresis. Hence, there should be linear relationship between RH and water content in the low RH range, and no hysteresis between desorption and adsorption should be observed. However, experimental data show hysteresis when water is the adsorbate, while there is no hysteresis for nitrogen or other gas [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. The main reason is that in room temperature moisture is in liquid form which has stronger interactions with pore walls. During drying, because the removal of interlayer water which is strongly bound to the C-S-H [START_REF] Feldman | A new model of hydrated portland cement paste as deduced from sorption -length change and mechanical properties[END_REF], interlayer pores collapse. During adsorption, the collapsed pores will expand again, but they cannot reach their usual size [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. The reason of shrinkage and swelling has been explained in more detail in the literature [START_REF] Guyer | Hysteresis in modeling of poroelastic systems: quasistatic equilibrium[END_REF]. The existing hysteretic behaviour leads to the need of hysteresis models when performing modelling of moisture transport. Some hysteresis models, which were developed based on the conceptual theories, are known as conceptual models [START_REF] Everett | A general approach to hysteresis -Part 3: a formal treatment of the independent domain model of hysteresis[END_REF][START_REF] Everett | A general approach to hysteresis -Part 4: an alternative formulation of the domain model[END_REF][START_REF] Preisach | Uber die magnetische nachwirkung[END_REF][START_REF] Mayergoyz | Hysteresis models from the mathematical and control theory points of view[END_REF][START_REF] Mualem | Modied approach to capillary hysteresis based on a similarity hypothesis[END_REF][START_REF] Mualem | A conceptual model of hysteresis[END_REF][START_REF] Mualem | Dependent domain model of capillary hysteresis[END_REF][START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF]. Others, which were initially used to t the measured sorption curves, are called empirical models [START_REF] Hanks | A numerical method for estimating inltration, redistribution, drainage and evaporation of water from soil[END_REF][START_REF] Jaynes | Comparison of soil-water hysteresis models[END_REF][START_REF] Scott | Hysteresis eects on net inltration[END_REF][START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF][START_REF] Nyman | A numerical method for the evaluation of non-linear transient moisture ow in cellulosic materials[END_REF][START_REF] Wei | Formulation of capillary hysteresis with internal state variables[END_REF]. Both types of hysteresis models will be compared and discussed in Chapter 3.

r 0 r 1 r 2 r 3 r 4 r 5 r < r < r < r < r < r 4 2 0 1 3 5 P (r )> r )> r )> r )> r )> r )
Solid A Solid B Θ A Θ B Impurity Θ A Θ A Θ A Impurity ( a ) ( b ) ( c )

Mechanisms of moisture transport

Moisture transport in cementitious materials is related to many complex mechanisms which occur more or less jointly. In porous materials, several phenomena such as permeation, diusion, adsorption-desorption-condensation and evaporation are very important. They depend on the internal relative humidity prevailing in the material.

Diusion

The diusion processes are those of ordinary diusion and Knudsen diusion as put forward by

Mason and Malinauskas [START_REF] Mason | Gas Transport in Porous Media. The Dusty-Gas Model[END_REF] and surface diusion as reported by Higashi et al. [START_REF] Higashi | Surface diusion phenomena in gaseous diusion, (I) surface diusion of pure gas[END_REF]. The dierent diusion processes are illustrated in Fig. 1.4 [START_REF] Gudmundsson | An approach to determining the water vapour transport properties of building materials[END_REF].

Ordinary diffusion

Knudsen diffusion

Surface diffusion

Adsorbed layers

Film flow

.4: Representation of dierent diusion and lm ow processes (adapted from [START_REF] Gudmundsson | An approach to determining the water vapour transport properties of building materials[END_REF]).

Ordinary diusion

Ordinary diusion is commonly found in both low RH and high RH ranges. It is caused by the gradient of concentration, so water molecules tend to diuse from the high concentration region to the low concentration zone. There are two main eects which can slow down ordinary diusion. One is collisions between water molecules, which are able to oset some molecular momentum. The other one is the molecules exchange with adsorbed layer on pore walls. Water molecules adsorbed by pore walls decrease the ordinary diusion process. When the relative humidity exceeds a threshold value, a meniscus is formed at each end of the path. In this condition, moisture transport becomes a process of condensation and evaporation at menisci. This mechanism is dominant in pores with size between 50 nm and 10 µm [START_REF] Xi | Moisture diusion in cementitious materials moisture capacity and diusivity[END_REF] which is typically the size of capillary pores in cementitious materials (see Table 1.1).

Knudsen diusion

According to the denition, Knudsen diusion occurs when the pore size is the same as or smaller than the mean free path of the water molecules. This kind of diusion is governed by collisions against the pore walls, which are the main source of resistance to Knudsen diusion. Knudsen diusion can be quantied by the Knudsen number K n = λ/2r p ≥ 1 where λ is the mean free path. For Knudsen diusion, the value of K n is equal to or greater than 1. The gives free path in air at atmospheric pressure and room temperature (for instance 23

Surface diusion

This mechanism primarily takes place when water molecules are mainly adsorbed by pore walls (which is the case if RH is low). On the one hand, when water molecules are forced to the solid walls, they have the trend to be adsorbed by walls; on the other hand, due to the strong dierence of molecules concentration at dierent distances from the pore walls, water molecules have another trend to leave the solid walls and to diuse to lower concentration region. The result is that water molecules move near pore walls. It is governed by leaps of water molecules between dierent adsorption sites.

In fact, during desorption, when there are less and less water molecules in pores, collisions between molecules become weaker and weaker, and Knudsen diusion and surface diusion are apt to be ordinary diusion in capillary pores. All those phenomena will progressively speed up ordinary diusion. Besides, ordinary diusion is faster compared to other diusion processes, so it represents the main part of mass diusion. That is why, most researchers generally focus on ordinary diusion in the literature. More discussions about Knudsen diusion will be presented in Chapter 6.

Capillary transport

Capillary transport is the movement of liquid-water under the gradient of capillary pressure in porous media when there is no external hydraulic head. This is the result of interactions between liquid-water and pore walls. Capillary transport generally occurs when the liquid body is continuous. For high water content, the pores are almost lled with liquid-water (see Fig. 1.5a).

When a driving force is available, capillary ow is established through the liquid body in the porous network. At the beginning of drying, the continuous liquid-water is driven by strong capillary forces with high liquid permeability. Thus, moisture transport is dominated by capillary transport. Along with drying time, liquid-water will progressively become discontinuous (see Fig. 1.5b). In this period, the capillary transport only remains preponderant in regions of the material where liquid-water is continuous, such as in segments AB and CD, whilst in the part BC the main transport is diusion.

Capillary pressure is so powerful and can create severe damages more than any other transport mechanisms. When the liquid phase is extracted by capillary pressure from the spaces between solids, the collapse will appear in the small pores due to no liquid phase supporting solid walls in cementitious materials (see Fig. 1.5b). By contrast, when the collapsed microstructure gains liquid-water, the volume of the system will expand and try to return to its normal state (see Fig. 1.5c).

Figure 1.5: Illustration of capillary transport and evaporation-condensation is a single pore (adapted from [START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF]).

Evaporation-Condensation

Evaporation-Condensation occurs at the interface between liquid-water and gas. For instance, in Fig. 1.5, moisture evaporates from surface B and becomes vapour, so it can be transported by diusion through the pore and condenses on surface C. This process is due to the non-equilibrium between vapour pressure (in segment BC) and capillary pressure (segments AB and CD) in the vicinity of the interfaces.

When there is no continuous liquid-water in the pore, a thin lm of water may form on the pore walls above the adsorbed layers (see the middle pore in Figs. 1.5b and 1.5c). The thickness of this lm is related to the van der Waals forces between water molecules and pore walls [START_REF] Iwamatsu | Capillary condensation and adhesion of two wetter surfaces[END_REF][START_REF] Tuller | Hydraulic conductivity of variably saturated porous media: Film and corner ow in angular pore space[END_REF].

This thin water lm is linked with menisci, so it can move due to capillary pressure. This kind of transport is called lm ow which is more obvious for rocks and porous media containing hydrophilic particles [START_REF] Dullien | Hydraulic continuity of residual wetting phase in porous media[END_REF][START_REF] Dullien | The eects of surface roughness on the capillary pressure curves and the heights of capillary rise in glass bead packs[END_REF].

In addition, in more porous materials, such as soils, rocks, cracked concretes, etc., hydraulic ow may be signicant when the material is directly in contact with liquid-water. This transport is driven by capillary pressure or external applied liquid pressure. But this process only exits 1.4. Continuum approach to model unsaturated moisture transport in porous media in the largest pores and for specic boundary conditions; thus, it will not be considered in this study.

1.4 Continuum approach to model unsaturated moisture transport in porous media

Calculation of water transport in unsaturated systems was initially developed based on the saturated ow in porous media. In 1907, Buckingham [START_REF] Buckingham | Studies on the movement of soil moisture[END_REF] extended Darcy's law to the case of unsaturated ow. In 1931, Richards further improved Buckingham's idea and combined the unsaturated ux law with mass balance principles to derive a governing equation for unsaturated moisture transport which is known as Richards' equation [START_REF] Richards | Capillary conduction of liquids in porous mediums[END_REF]. Since then, a number of models were developed for modelling of moisture transport in porous media. The development of theoretical basis models started in the late 1950s in the research provided by Philip and de Vries [START_REF] Philip | Moisture movement in porous materials under temperature gradients[END_REF].

Later on, in a series work performed by Bear [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF][START_REF] Bear | Modelling of Groundwater Flow and Pollution[END_REF][START_REF] Bear | Introduction to Modeling of Transport Phenomena in Porous Media[END_REF][START_REF] Bear | Modeling Groundwater Flow and Contaminant Transport[END_REF], the multiphase transport model for unsaturated porous media started to be used. Meanwhile, the principle of multiphase, including liquid-water, vapour and drying air, was also applied to studies on mechanics and physics of porous solids [START_REF] Coussy | Mechanics of porous continua[END_REF][START_REF] Coussy | Mechanics and physics of porous solids[END_REF]. In the eld of civil engineering materials, several models, which described moisture taking into account vapour and liquid-water mobility, have been developed for the prediction of the service-life of concrete structures [START_REF] Bazant | Nonlinear water diusion in non-saturated concrete[END_REF][START_REF] Daïan | Condensation and isothermal water transfer in cement mortar. Part I: pore size distribution, equilibrium water condensation and imbibition[END_REF][START_REF] Nilsson | Long-term moisture transport in high performance concrete[END_REF][START_REF] Janssen | Conservative modelling of the moisture and heat transfer in building components under atmospheric excitation[END_REF]. In 2001, the multiphase model was further used to simulate drying of cementitious materials [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF].

Nevertheless, the permeability of most materials of civil engineering range from very large values (10 -10 m 2 for sands) to very low ones (10 -21 m 2 for concretes). This feature explains that there exist as many ways of modelling as kinds of materials, since the models are generally adapted to specic microstructure and even to particular initial and boundary conditions (initial water content within the material, relative humidity and temperature prevailing in the surrounding environment, etc.). When engineers are trying to numerically predict moisture transport within building materials, the fundamental question which generally raises is which mechanisms govern the transport process: advection of liquid-water by capillarity, evaporation of liquid-water inside the material and/or water vapour diusion, variations of total gas pressure, etc. To answer this question, models need to simulate all above mechanisms simultaneously.

In this section, a multiphase model which has been used in the work of Coussy [START_REF] Coussy | Mechanics of porous continua[END_REF][START_REF] Coussy | Mechanics and physics of porous solids[END_REF], and complemented by other researchers [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Baroghel-Bouny | Modelling of isothermal coupled moistureion transport in cementitious materials[END_REF][START_REF] Baroghel-Bouny | Prediction of chloride binding isotherms of cementitious materials by analytical model or numerical inverse analysis[END_REF] will be recalled.

Mass balance equations

At the macroscopic level, most porous media can be viewed as homogeneous isotropic materials.

The deformation of the porous solid is neglected. There are no ongoing chemical reactions. Because moisture transport in porous materials is occurring with a very low velocity, the momentum of each phase can be assumed in a quasi-equilibrium state. Hence, the transport process is only governed by mass conservation laws. By letting ml→v represent the rate of liquid transforming into vapour, the continuity equations related respectively to phases of liquid (l), vapour (v) and dry air (a) are expressed in a 1D form as follows:

∂[ρ v φ(1 -S)] ∂t = - ∂ ∂x (ρ v v v ) + ml→v (1.3) ∂[ρ l φS] ∂t = - ∂ ∂x (ρ l v l ) -ml→v (1.4) ∂[ρ a φ(1 -S)] ∂t = - ∂ ∂x (ρ a v a ) (1.5)
where ρ v (ρ a resp.) is the density of water vapour (dry air resp.). φ is the porosity through which the ows of liquid-water and water-vapour occur, and S is the liquid-water saturation which can include both capillary water (interstitial water) and bound water (physically-adsorbed water).

ρ l is the density of this mixture of these two types of water (capillary and adsorbed) which are encompassed in liquid-water as a generic designation. v v (v a resp.) represents the velocity of vapour (dry air resp.) within the porous material. v l is the macroscopic velocity of liquid-water.

It should be remarked that in Eqs. (1.3)-(1.5), ρ v φ(1 -S), ρ a φ(1 -S), and ρ l φS correspond to apparent volumetric mass of vapour, dry air and liquid-water.

Transport laws

Vapour and dry air are transported both by advection of the gaseous mixture (according to Darcy's law) and by molecular diusion (according to Fick's law). The velocity of the vapour-air mixture v g is dened as a molar-averaged velocity v g = c a v a + c v v v where c v or c a represents the molar fractions related to vapour or dry air (c v + c a = 1). If the gaseous mixture is considered as ideal, the molar fractions express themselves according to the ratios of pressure (c v = P v /P g and c a = P a /P g where P v and P a are the partial pressure of vapour and dry air, respectively). In agreement with the second law of thermodynamics about entropy, the relationships governing the 1.4. Continuum approach to model unsaturated moisture transport in porous media transport of water vapour and dry air must satisfy the positiveness of the dissipation associated with the transport of both phases [START_REF] Coussy | Mechanics of porous continua[END_REF]:

-v g ∂P g ∂x -(v v -v a ) ∂c v ∂x ≥ 0 (1.6)
The rst term accounts for the dissipation related to the advective transport of the mixture, while the second term represents the dissipation associated with diusion. Using the kinetic theory of gases and the previous condition related to the second law of thermodynamics, the following relations governing the transport of vapour and dry air can be proposed in the case of unsaturated porous materials:

v v = v g -φ(1 -S)α(S)τ D 0 P atm P v ∂ ∂x P v P g (1.7) v a = v g -φ(1 -S)α(S)τ D 0 P atm P a ∂ ∂x P a P g (1.8)
In Eqs. (1.7) and (1.8), D 0 is the diusion coecient of water vapour or dry air in the wet mixture. τ is the tortuosity coecient which is aected by a relative tortuosity α(S) taking into account the fact that the material is not saturated (the quantity α(S)τ represents the tortuosity of the sole gaseous phase in the porous medium). It should be noted that α(S) has to satisfy α(S = 0) = 1 and α(S = 1) = 0. The term φ(1 -S) represents the available space for gas transport.

In this description of the movement of the gas mixture, the motors of diusion are the gradients of the molar fractions of water vapour (c v ) and dry air (c a ) in the gaseous mixture. In addition, assuming that wet air is an ideal gas mixture (i.e., P g

= P v + P a ⇔ 1 = c a + c v ),
Eq. (1.8) can be simplied as:

v a = v g + φ(1 -S)α(S)τ D 0 P atm P a ∂ ∂x P v P g (1.9)
The velocities v g and v l of the gaseous mixture and of the liquid-water phase are governed by the generalized Darcy law which is written for an unsaturated porous medium and which meet the dissipation related to the advective transport of gas and liquid-water [START_REF] Coussy | Poromechanics[END_REF]:

v g = - K 0 η g k rg (S) ∂P g ∂x (1.10) v l = - K 0 η l k rl (S) ∂P l ∂x (1.11)
where K 0 represents the intrinsic permeability. k rg (S) and k rl (S) are the relative permeabilities of the liquid and gaseous phases, respectively. η l and η g are the molecular viscosities of liquid and gas phases, respectively.

In this model, the gas intrinsic permeability is assumed equal to the liquid intrinsic permeability K g = K l = K 0 . Actually, this assumption is reasonable because the intrinsic property is independent to the type of uid and only related to the structure of pore network. However, many studies have already shown the necessity to make a dierence between gas and liquid intrinsic permeabilities in order to satisfy experimental observations, especially pointing out that K l is much smaller than K g [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF][START_REF] Hamami | Inuence of mix proportions on microstructure and gas permeability of cement pastes and mortars[END_REF][START_REF] Loosveldt | Experimental study of gas and liquid permeability of a mortar[END_REF]. Coussy [START_REF] Coussy | Mechanics and physics of porous solids[END_REF] considers that researchers are obliged to use a lower value of K l than K g in the framework of continuum modelling like the one presented here.

Both cases K l = K g and K l K g will be used to simulate moisture transport in cementitious materials in Chapter 2.

State equations

The local thermodynamic equilibrium between the liquid-water and vapour is assumed. This equilibrium is governed by Kelvin's law written in the following form:

-P c + P g -P atm = ρ l RT M v ln P v P vs (1.12)
in which P c (S) appears as the macroscopic capillary pressure dened as the dierence between macroscopic pressure of the gaseous phase and that of the liquid one (P c = P g -P l ). The relationship of P c as a function of the saturation degree of liquid-water S is called the capillary pressure relationship. The relative humidity is dened as the fraction of vapour pressure to the saturated vapour pressure, RH = P v /P vs , where P vs is the saturated vapour pressure. For most porous materials, the entry pressure P ent exits at very high saturation S ≈ 0 because in this range water in large pore can not be maintained by capillary pressure. In the case of weakly permeable materials such as cement-based materials, the entry pressure is very small. Indeed, many studies have highlighted that for cementitious materials (concrete and mortars) the moisture content proles, measured by gamma ray attenuation, X-ray projection [START_REF] Leech | Validation of mualem's conductivity model and prediction of saturated permeability from sorptivity[END_REF] or Nuclear Magnetic Resonance imaging (NMR imaging) [START_REF] Carmeliet | Determination of the liquid water diusivity from transient moisture transfer experiment[END_REF] show a negligible height of the fully saturated area near the surface in contact with liquid-water.

For cement-based materials, the capillary pressure curve is indirectly measured by means of sorption experiments performed for isothermal conditions (the so-called sorption isotherms [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF],

see Section 1.2). Capillary pressure is written in a dimensionless form:

P c = Mπ c (S) (1.13)
where M stands for a capillary modulus or for a reference pressure. π c (S) is a dimensionless function satisfying π c (0) = +∞ and π c (1) = 0.

The partial pressure of water vapour P v is much smaller compared to the total gaseous pressure P g since P v is bounded by the saturated vapour pressure P vs at a given temperature.

Thus, the assumption that the gaseous mixture behaves as an ideal gas is appropriate, yielding the following relationships between pressure of each phase and its density:

ρ v = P v M v RT (1.14)
ρ a = P a M a RT

(1.15)

P g = P v + P a (1.16)
In Eqs. (1.14) and (1.15), M v and M a are the molar mass of vapour and dry air. R and T stands for the ideal gas constant and absolute temperature.

Concerning the liquid-water phase, the assumption of incompressibility is made because in the proposed modelling temperature is considered as constant.

Applications of the multiphase model for cementitious materials

In the literature, the above multiphase model has been applied to cementitious materials more than a decade ago in the research by Mainguy et al. [START_REF] Mainguy | Modèles de diusion non linéaire en milieu poreux. Application à la dissolution et au séchage des matériaux cimentaires[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF]. In that research, the liquid relative permeability k rl was calculated by van Genuchten-Mualem model (VGMa, see Eq. 2.7 in Chapter

2) [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF]. The equation for k rg was taken from the proposal of Parker et al. [START_REF] Parker | A model for hysteretic constitutive relations governing multiphase ow: 1. Saturation-pressure relations[END_REF] and Kuckner et al. [START_REF] Luckner | A consistent set of parametric models for the two-phase-ow of immiscible uids in the subsurface[END_REF] (see Eq. 2.9). The tortuous eects on vapour diusion was estimated by the equation proposed by Millington [START_REF] Millington | Gas diusion in porous media[END_REF] (see Eq. 2.37). The studied materials were imposed to constant external RH.

Simulation results showed a gas overpressure could be observed in the part near the material surface. Ignoring the transport of the gas-phase (including eects of gas pressure, vapour diusion and advection of gas-phase, and evaporation between liquid and vapour), Mainguy et al. [START_REF] Mainguy | Modèles de diusion non linéaire en milieu poreux. Application à la dissolution et au séchage des matériaux cimentaires[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF] reported that the multiphase model turns to a single diusion-like equation only with moisture transport in its liquid form. This simplication concerns the weakly permeable materials, such as cement-based materials:

∂S ∂t = ∂ ∂x D l (S) ∂S ∂x (1.17)
where D l (S) represents the diusivity of the liquid-phase.

Mainguy et al. concluded that Eq. (1.17) can provide a high accuracy prediction when it is used to determine K 0 by tting drying kinetics (mass loss vs. time). They also recalled that most standard diusion-like drying models [START_REF] Bazant | Nonlinear water diusion in non-saturated concrete[END_REF][START_REF] Daïan | Condensation and isothermal water transfer in cement mortar. Part I: pore size distribution, equilibrium water condensation and imbibition[END_REF][START_REF] Xi | Moisture diusion in cementitious materials moisture capacity and diusivity[END_REF] in the literature are developed based on the assumption of constant gas pressure. Actually, the multiphase model can be simplied as a standard diusion equation according to two main assumptions:

• Gas pressure in the material is constant and always equal to the atmospheric pressure.

• The vapour transport is a pure diusion process which means that advection of vapour is negligible.

These two assumptions lead the multiphase model to appear as:

∂S ∂t = ∂ ∂x (D l (S) + D v (S)) ∂S ∂x (1.18)
where D v (S) represents the diusivity of vapour. In fact, Eq. (1.17) is a special case of Eq. (1.18) when vapour diusion is negligible. Most models mentioned above [START_REF] Bazant | Nonlinear water diusion in non-saturated concrete[END_REF][START_REF] Daïan | Condensation and isothermal water transfer in cement mortar. Part I: pore size distribution, equilibrium water condensation and imbibition[END_REF][START_REF] Xi | Moisture diusion in cementitious materials moisture capacity and diusivity[END_REF] and in the next section [START_REF] Li | Numerical analysis of moisture inuential depth in concrete during drying-wetting cycles[END_REF][START_REF] Li | Inuential depth of moisture transport in concrete sub ject to drying-wetting cycles[END_REF][START_REF] Cunningham | Eective penetration depth and eective resistance in moisture transfer[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF][START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF] can be considered as a simplied form of the multiphase transport model.

However, Mainguy et al. [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF] did not really state the relationship between K 0 and gas pressure variations (i.e., the denition of a weakly permeable porous medium), and the reasons they made two assumptions about gas pressure to obtain Eq. (1.18). In Chapter 2, a systemic study will present the whole scope of modelling moisture transport for cementitious materials according to the possibility to make a dierence between liquid intrinsic permeability and gas intrinsic 1.5. Modelling of moisture transport under varying ambient relative humidity permeability through a theoretical analysis and experimental verications.

Modelling of moisture transport under varying ambient relative humidity

To study the internal moisture behaviour responding to external humidity changes, there are two processes that should be taken into account: how is moisture transport inside the materials? and how are moisture interactions between the material surface and the environment? The review of these two aspects will be presented in the following subsections.

In the literature, most of models were initially developed for drying, when the material is exposed to the surrounding with lower RH compared with RH in the material. However, one main characteristic of the natural environment is the arbitrary ambient humidity changes, which should include both drying and wetting. Wetting means a higher RH at the boundary than the one prevailing in the material. Drying and wetting appear alternatively in natural conditions. This is considered as the most unfavourable environmental situation for concrete structures exposed to high ions content surrounding, because drying and wetting cycles can accelerate the penetration of ions [START_REF] Hong | Eects of cyclic chloride exposure on penetration of concrete cover[END_REF]. Therefore, to understand durability issues, it needs to model moisture behaviour under both drying and wetting processes. The terms drying and wetting used here are specied to moisture transport occurring in the hygroscopic range; namely, the present research only focuses on the research of the material exposed to the humid air, but not directly in contact with liquid-water.

In this section, the purpose is to review the model which can be used for periodic boundary conditions. To model drying and wetting cycles by using a continuum model, there are generally two ways which can be used: (1) A sorption isotherm is one of essential input data for the continuum model, so using the same sorption isotherm for dierent processes is a simplied approach to simulate drying and wetting cycles; (2) The more complicated method is to use distinct sorption isotherms for dierent processes, so that a hysteresis model has to be used to predict hysteresis loops. In the latter case, a moisture transport model needs to be coupled with a hysteresis model. Since there are many moisture transport models and dierent hysteresis models in the literature, this research cannot enumerate all combinations. Thus, the current section focuses on reviewing the model: (a) which has already been used in the eld of cementitious materials, and (b) for hysteresis modelling, which has provided the combination of a moisture Chapter 1. Review of modelling of drying and wetting cycles transport model and a hysteresis model.

Cunningham's method -analytical solution

Cunningham [START_REF] Cunningham | Eective penetration depth and eective resistance in moisture transfer[END_REF] investigated diusion which is driven by both the gradient of moisture concentration and temperature in a building material when it is subjected to periodic moisture and temperature boundary conditions. In other words, two-variable driving potentials with nonlinear diusion coecients were used in this method. The diusion equation for Cunningham's method is:

∂S ∂t = ∂ ∂x D S ∂S ∂x + D T ∂T ∂x (1.19)
where D S and D T are moisture diusivities under saturation gradient and under temperature gradient respectively. The part under saturation gradient is the same as Eq. (1.18), so D S should contain both contributions of liquid-water and vapour.

To get an analytical solution to Eq. (1.19), three assumptions were made [START_REF] Cunningham | Eective penetration depth and eective resistance in moisture transfer[END_REF]:

(1) In practical, the time of response for temperature variations is very much shorter than for the moisture concentration, so that the Fourier number can assumed far bigger than 1.

F o = κ ρ v l 2 C T 1 (1.20)
where C T is temperature energy capacity and l is the length of the material.

(2) Because of the rapid temperature diusion, the second assumption is that the thermal conductivity κ is constant, meaning the gradient of temperature is not considered as a driving force.

(3) The terms ∂D S ∂S and ∂D T ∂S are assumed to be constant.

The initial condition corresponds to uniform moisture content and temperature. Boundary conditions are dened as sinusoidal moisture content and temperature uctuations. Thus, analytical solutions can be obtained, inspired by the work from Carslaw and Jaeger [START_REF] Carslow | Conduction of heat in solids[END_REF].

Clearly, this model is only valid for specied initial and boundary conditions. Like most analytical solutions, the applicability is restricted.

EMPD model

The EMPD (Eective Moisture Penetration Depth) model was developed by Kerestecioglu et al. [START_REF] Kerestecioglu | Theoretical and computational investigation of simultaneous heat an d moisture transfer in buildings: Eective penetration depth theory[END_REF] and Cunningham [START_REF] Cunningham | The moisture performance of framed structures -a mathematical model[END_REF], and later has been used in studies [START_REF] Steeman | Modelling indoor air and hygrothermal wall interaction in building simulation: Comparison between CFD and a well-mixed zonal model[END_REF][START_REF] Steeman | On coupling 1d non-isothermal heat and mass transfer in porous materials with a multizone building energy simulation model[END_REF][START_REF] Janssen | Qualitative and quantitative assessment of interior moisture buering by enclosures[END_REF][START_REF] Woloszyn | The eect of combining a relative-humidity -sensitive ventilation system with the moisture-buering capacity of materials on indoor climate and energy eciency of buildings[END_REF][START_REF] Woods | Evaluation of the eective moisture penetration depth model for estimating moisture buering in buildings[END_REF]. It is derived from considering cyclic variations of the external relative humidity. It also has been veried having good practical applications [105,[START_REF] Trnsys | Transient system simulation program[END_REF] and has shown good agreement between predicted relative humidity and measured ones [START_REF] Woloszyn | The eect of combining a relative-humidity -sensitive ventilation system with the moisture-buering capacity of materials on indoor climate and energy eciency of buildings[END_REF].

The EMPD model was developed based on the assumption that moisture transport takes place between the environmental air and a thin ctitious layer (also called the moisture buering layer) of material with uniform moisture content. This moisture content is a function of RH and is characterised by the moisture penetration depth x p (the thickness of the moisture buering layer) which can be calculated based on the material properties. The governing equation in the EMPD model is expressed as [START_REF] Woods | Evaluation of the eective moisture penetration depth model for estimating moisture buering in buildings[END_REF]:

ρ m x p dθ dt = k m (RH p -RH e ) (1.21)
where ρ m is the dry density of the material, θ is water content, k m is the moisture transport coecient in the airside (kg • m -2 • s -1 The easiest way to solve this equation is to assume a series of x p to calculate RH p since x p only exits on the left-hand side of Eq. (1.21). Besides, there are some variations using a little more complex equations [START_REF] Steeman | Modelling indoor air and hygrothermal wall interaction in building simulation: Comparison between CFD and a well-mixed zonal model[END_REF][START_REF] Steeman | On coupling 1d non-isothermal heat and mass transfer in porous materials with a multizone building energy simulation model[END_REF] which further separate the moisture buering layer into a surface layer and a deep layer.

ρ m x p ∂θ ∂RH p dRH p dt = (P v,p -P e v ) 1 k m + x p 2K v (RH p ) + P v,r -P v,p x p 2K v (RH p ) + x r -x p 2K v (RH r ) (1.22) ρ m x r ∂θ ∂RH r dRH r dt = P v,r -P v,p x p 2K v (RH p ) + x r -x p 2K v (RH r ) (1.23)
where K v is the vapour permeability, P v,p and P e v are the vapour pressures in the material (at depth x p ) and in the environment, respectively. The surface layer (subscript p) and the deep layer (subscript r) correspond to the rst and the second term on the right side of Eq. (1.22).

The surface layer is based on the short term RH uctuations such as a day, and the deep layer is based on the longer term RH uctuations which can be weeks to months. The prior parameters Chapter 1. Review of modelling of drying and wetting cycles in this model are x p and x r .

For a perfectly periodic boundary condition, based on the assumptions of constant K v and knowing the sorption isotherm, the penetration depth is given as [START_REF] Cunningham | Eective penetration depth and eective resistance in moisture transfer[END_REF][START_REF] Steeman | Modelling indoor air and hygrothermal wall interaction in building simulation: Comparison between CFD and a well-mixed zonal model[END_REF]: Li et al. [START_REF] Li | Numerical analysis of moisture inuential depth in concrete during drying-wetting cycles[END_REF][START_REF] Li | Inuential depth of moisture transport in concrete sub ject to drying-wetting cycles[END_REF] have performed a modelling of drying-imbibition cycles and their inuence on the moisture inuential depth. Li et al. [START_REF] Li | Inuential depth of moisture transport in concrete sub ject to drying-wetting cycles[END_REF] pointed out that the transport mechanisms are dierent during drying and imbibition: during drying, moisture transport to the surface is driven by a combined evaporation-diusion-convection process, while an imbibition process is dominated by capillary absorption of liquid-water. If using water content as potential, considering 1D problem and without gas pressure variations, the mass balance equation according to Eq. (1.18) is given as:

EM P D =    K v P vs t p ρ m π dθ dRH    1/
∂θ ∂t = ∂ ∂x D(θ) ∂θ ∂x (1.27)
The diusivity D(θ) is dierent during the drying and imbibition processes. The diusivity during drying is taken from [START_REF] Bazant | Nonlinear water diusion in non-saturated concrete[END_REF].

D d (θ) = D d0          α 0 + 1 -α 0 1 + 1 -θ 1 -θ c N          (1.28)
where D d0 is the diusivity at saturated condition, and α 0 , θ c and N are parameters to be determined by experiments. D d (θ) should include both contributions of liquid-water and vapour.

The diusivity during imbibition is taken from Hall [START_REF] Hall | Water sorptivity of mortars and concretes: A review[END_REF].

D w (θ) = D w0 exp(nθ) (1.29)
where D w0 is the diusivity at fully-dried condition and n is a regression coecient. D w (θ) only includes the contribution of liquid-water.

Besides, in their later studies [START_REF] Li | Hydro-ionic transport in concrete near surface under dying-wetting cycles[END_REF], the eective diusivity, reported by Mainguy [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF] on the basis of the works of Philip and de Vries [START_REF] Philip | Moisture movement in porous materials under temperature gradients[END_REF], was also used for modelling of drying and wetting cycles. But, hysteretic eects were not considered. The main reason, according to [START_REF] Li | Hydro-ionic transport in concrete near surface under dying-wetting cycles[END_REF], is that D(θ) for main desorption and main adsorption does not show signicant hysteresis.

For cementitious materials, Baroghel-Bouny's experimental results have shown that no obvious hysteresis can be observed in apparent diusivity D a (S) [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF] for cementitious materials. In contrast, hysteresis in D a (RH) is much clearer. Hence, no hysteresis in D a (θ) does not mean no hysteresis in D a (RH). In the literature, hysteresis has been proved important for modelling moisture transport in drying-wetting conditions [START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF][START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF]. Therefore, it seems not relevant to neglect hysteretic eects.

Hysteresis modelling methods

As discussed above, hysteresis is a common behaviour for moisture transport in porous media.

Some models neglect hysteresis and use the main desorption isotherm to simulate moisture transport in both drying and wetting processes (e.g., [START_REF] Arfvidsson | Isothermal moisture ow in building materials: modelling, measurements and calculations based on Kirchho 's potential[END_REF]). More recently, modelling considering hysteresis has become a topic of interest.

For hysteresis modelling, both moisture transport and hysteresis models are needed. This subsection does not review these two kinds of models separately. Instead, it only focuses on models which have originally coupled moisture transport and hysteresis together. Chapter 3 will present a more detailed review of hysteresis models.

Method from Johannesson et al.

Johannesson et al. [START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF][START_REF] Johannesson | A numerical approach for non-linear moisture ow in porous materials with account to sorption hysteresis[END_REF] employed a two state-variable model to investigate moisture transport in building materials. The state-variable for liquid phase is the saturation S and for vapour phase it is the relative humidity RH. The mass balance equations are coupled to each phase through the mass exchange term which is related to either evaporation or to condensation.

Modelling of moisture transport under varying ambient relative humidity

The equation for liquid phase is written as:

ρ l ∂θ ∂t = - ∂ ∂x D l ∂θ ∂x -E m (θ eq -θ) (1.30)
where D l = D l (θ) is the diusivity of liquid-water, which is assumed to be an increasing function of S. The rate of mass exchange E m is constant. The superscript eq represents the variable at equilibrium.

The governing equation for the vapour phase is:

(φ -θ) ∂ρ v ∂t -ρ v ∂θ ∂t = - ∂ ∂x D v ∂ρ v ∂x + E m (θ eq -θ) (1.31)
where

D v = D v (ρ v )
is the diusivity of water vapour, which is assumed to be an decreasing function of RH.

In this model, the use of term E m (θ eq -θ) means that this model takes into account nonequilibrium or dynamic eects. Unlike the multiphase model, the assumption of local equilibrium is not applicable in this model.

A third degree polynomial function is used to express the both main desorption and adsorption curves, as well as scanning curves. This method was originally developed for wood materials.

θ = b 1 RH + b 2 RH 2 + b 3 RH 3 (1.32)
where b 1 , b 2 and b 3 are parameters, needing to be determined by experimental data for the main sorption curves. For a scanning curve, it starts at the beginning point (RH 0 , θ eq 0 ) and is tangent to the main curve at the junction point (RH junc , θ eq junc ). For a desorption scanning curve, there are several rules to calculate parameters.

At the beginning point:

dθ eq dRH RH=RH 0 = k d dθ eq 0 dRH (RH = RH 0 ) 0 < k d ≤ 1 (1.33)
θ eq (RH 0 ) = θ eq 0 (RH 0 )

(1.34)

At the junction point:

dθ eq dRH RH=RH junc = dθ eq d dRH (RH = RH junc ) (1.35) θ eq (RH junc ) = θ eq d (RH junc ) (1.36)
where the subscript d represents the main desorption curve and k d is a material constant that is used to determine the initial slope of the desorption scanning curve. It should be calibrated by experimental data. RH 0 and RH junc are relative humidity at the starting point of the scanning curve and the junction point, respectively.

Equation (1.33) means that the slope of a scanning curve at the beginning point is the fraction of the slope of the previous scanning curve at the end point. Equation (1.35) shows that the slopes of the scanning curve and the main desorption curve are the same at the junction point.

Equations (1.34) and (1.36) are used to make sure a continuous scanning curve. Totally, for each scanning curve, there are four unknowns (three in Eq. 1.32 and RH junc (or θ eq junc ) for the junction point) in four equations, so these equations can be mathematically solved. After the junction point, the scanning curve should be same with the main desorption curve. In the same manner, the rules for an adsorption scanning curve are given as:

At the beginning point:

dθ eq dRH RH=RH 0 = k a dθ eq 0 dRH (RH = RH 0 ) 0 < k a ≤ 1 (1.37) θ eq (RH 0 ) = θ eq 0 (RH 0 ) (1.38)
At the junction point:

dθ eq dRH RH=RH junc = dθ eq w dRH (RH = RH junc ) (1.39)
θ eq (RH junc ) = θ eq w (RH junc )

(1.40)
where the subscript w represents the main adsorption curve and k a is a material constant that is used to determine the initial slope of the adsorption scanning curve. proposed in [START_REF] Nyman | A numerical method for the evaluation of non-linear transient moisture ow in cellulosic materials[END_REF]. The scanning curve starts at point A and is tangent to the adsorption curve at the junction point B. The material constant k a = 0.2.

Modelling of moisture transport under varying ambient relative humidity

An example for the rst wetting scanning curve is shown in Fig. 1.6. This model is more suitable for the main sorption curves with low curvature and small hysteresis. If the main sorption curves have a high curvature, there may not have solutions to equations for the scanning curves.

For example in Fig. 1.6, the calculated scanning curve may not be tangent to the adsorption curve and point B cannot be found. It indicates that this hysteresis model is generally suitable for more porous materials because these materials have small hysteresis and two main sorption curves are close to each other. But cementitious materials show bigger hysteresis, which can cause problems for the use of this model.

Method from Derluyn et al.

Derluyn et al. [START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF] assumed that moisture transport in concrete is induced by various RH and temperature conditions, so they coupled moisture and heat transport for porous building materials by adopting capillary pressure P c and temperature T as potentials. Moisture transport is described as vapour transport using a vapour permeability K v by considering constant gas pressure in the material.

C ∂P c ∂t = - ∂ ∂x K v (P v ) ∂P v ∂x (1.41)
where the moisture capacity C = ∂θ ∂P c and vapour pressure P v can be calculated by using Kelvin's equation based on RH. The vapour permeability is formulated as a function of RH and it is also considered as the fraction of the vapour permeability in the air K v0 .

K v = K v0 [α 1 + α 2 exp(α 3 RH)] (1.42)
where α 1 , α 2 and α 3 are three parameters. Derluyn et al. [START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF] argued that K v includes transport in both liquid and gas phases. Equation (1.41) is one form of Eq. (1.18) by using the moisture capacity C on both sides of the equation.

To t the main sorption isotherms, equations were introduced by Derluyn et al. [START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF] by summing a concave curve and a convex curve. Expressions to the main desorption and main adsorption isotherms are given as: Ads. The scanning curves are predicted by an independent domain model, called PM model [START_REF] Preisach | Uber die magnetische nachwirkung[END_REF][START_REF] Mayergoyz | Hysteresis models from the mathematical and control theory points of view[END_REF], which was originally developed for studies on magnetism eld and has been used for the research of moisture transport in wood [START_REF] Derome | Model for hysteretic moisture behaviour of wood[END_REF]. The PM model assumes that a domain (pores in a porous material) consists of a number of sorption sites, which control the variation of water content according to the decrease or increase of RH. A triangular diagram was introduced to represent the PM domain (see Fig. 1.7 for the main sorption curves). The density distribution of these sites is represented by the function Φ(RH w , RH d ) and the corresponding water content distribution is denoted by Θ(RH w , RH d ). Therefore, water content at a given RH can be calculated by integrating two distribution functions along x-axis (wetting) and y-axis (drying) in a domain Ω.

θ d = θ max ω l (1 + a d )RH d 1 + a d RH d + (1 -ω l ) 1 -exp(b d RH d ) 1 -exp(b d ) (1.43) θ w = θ max ω l (1 + a w )RH w 1 + a w RH w + (1 -ω l ) 1 -exp(b w RH w ) 1 -exp(b w ) (1.
θ(RH w , RH d ) = Ω Φ(RH w , RH d ) Θ(RH w , RH d ) dRH w dRH d (1.45)
The integral equation is not easy to use, so a function H(RH w , RH d ) is introduced instead.

For a desorption scanning curve, a complex equation is given as:

1.5. Modelling of moisture transport under varying ambient relative humidity

θ(RH w , RH d ) = H 1 (RH w , RH d ) + H 2 (RH w , RH d ) (1.46) with H 1 (RH w , RH d ) = θ max ω l 1 - (1 + a d )RH d 1 + a d RH d (1 + a w + c h RH d ) RHw-RH d 1-RH d 1 + (a w + c h RH d ) RHw-RH d 1-RH d (1.47) H 2 (RH w , RH d ) = θ max (1 -ω l ) 1 - 1 -exp(b d RH d ) 1 -exp(b d ) 1 -exp (b w + d h RH d ) RHw-RH d 1-RH d 1 -exp(b w + d h RH d ) (1

DuCOM model

The DuCOM (Du rability COncrete M odel) computer codes, rst released in 1999, is a thermohygro-physics coupled model which aims at integrating macroscopic material properties and durability performances. Furthermore, it also combines carbonation, chloride ion penetration, steel corrosion and calcium leaching [START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF].

Moisture transport model is developed on 10 -10 -10 -6 m scale in DuCOM. The moisture mass balance includes both vapour and liquid-water. The conservation equation is expressed with moisture capacity, moisture transport conductivity and sink term on the referential volume.

ρ l ∂φ t S t ∂P c ∂P c ∂t = ∇ (J l + J v ) -ρ l S t ∂φ t ∂t + W p ∂β ∂t (1.49)
where the total porosity φ t is the sum of gel porosity φ gl , interlayer porosity φ il and capillary porosity φ cap . The total saturation S t is related to the adsorbed water saturation S ads , the condensed water saturation S con and the interlayer water saturation S il . S t is calculated by Eq. (1.64). J l and J v are the uxes of liquid-water and vapour. The second term on the righthand side is related to the pore structure development. The last term W p ∂β ∂t is the sink term accounting for the moisture loss consumed by hydration.

Knudsen diusion theory was applied for vapour transport. Under isothermal conditions, the vapour ux is described as [START_REF] Maekawa | Modelling of Concrete Performance: Hydration, Microstructure Formation, and Mass Transport[END_REF]:

J v = - φD 0 τ ∞ rc dV 1 + N k ∇ρ v (1.50)
where V is the pore volume, τ is a parameter representing the tortuous eect of the pore network (=(π/2)

2 ). r c is the threshold pore radius, and pores with radii below r c are considered to be fully lled by liquid-water. N k is the Knudsen number of the water molecule.

If assuming cylindrical pores, the ux of liquid-water is derived as [START_REF] Maekawa | Modelling of Concrete Performance: Hydration, Microstructure Formation, and Mass Transport[END_REF]:

J l = - ρ l φ 2 50η l rc 0 rdV 2 ∇P c (1.51)
where η l is the viscosity of liquid.

The moisture capacity (see the rst term in Eq. 1.49) enumerates moisture content with regard to capillary pressure. Quasi-equilibrium is assumed at any time in micro-pores. So, capillary pressure can be obtained by Kelvin's equation.

To solve Eq. (1.49), the relation between S t and RH (sorption isotherms) must be known.

In DuCOM, the liquid-water is viewed in three states: adsorbed, condensed and interlayer. To calculate the water content in these dierent states, it needs to know the pore size distribution, which is simply represented by the Raleigh-Ritz distribution function.

V = 1 -exp(-rB), dV = rB exp(-rB)d lnr

(1.52)
where B is a distribution parameter, which in fact represents the peak of pore size distribution on a logarithmic scale.

In natural conditions, drying and wetting cycles have small inuence on the adsorbed water, so DuCOM assumes that adsorption is the perfectly reversed drying. If regarding pores are cylinders, the contribution of adsorbed water to the total liquid-water is obtained as:

S ads = ∞ rc 1 - r -t a r 2 dV (1.53)
where t a is the thickness of the adsorbed layer which is calculated by the BET theory [START_REF] Brunauer | Adsorption of gases in multimolecular layers[END_REF] modied by Hillerborg [START_REF] Hillergorg | A modied absorption theory[END_REF].

DuCOM considers that hysteresis in sorption isotherms for cementitious materials is mainly due to the ink-bottle eect. The amount of trapped water in ink-bottles and free water can be described by considering the pore size distribution (see Fig. 1.8). Trapped water means water in the pore whose size is below the ink-bottle neck (threshold radius r c ).

Figure 1.8: Water in pores trapped by ink-bottle necks (taken from [START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF]).

Under equilibrium conditions, the pores with radii smaller than r c would be completely saturated. The main wetting curve starts from the totally dried state, so r c also begins from the minimum pore size (see Fig. 1.9). The condensed water saturation is the integration of the pore size distribution function from the minimum pore size to r c . where S c is the contribution of trapped water.

Figure 1.9: Main desorption and adsorption curves calculated based on the pore size distribution [START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF]. Hysteresis between two main sorption curves is also shown in this gure.

The main drying curve starts from the totally saturated state, so r c also begins from the maximum pore size. The saturation S ink due to the additional water present in trapped pores can be calculated by summing water in pores having a radius greater than r c :

S ink = ∞ rc V rc V r dV = -S c lnS c (1.55)
where V rc is the volume of the pores with radii less than r c . V r is the volume of pores with radii less than r in the pore size distribution. Therefore, saturation for the main drying curve can be obtained according to Fig. 1.9.

S con = S c + S ink = S c [1 -lnS c ] (1.56)
Obviously, the two main sorption curves are not enough to describe varying environment conditions. The model has to be able to simulate scanning curves from wetting to drying and from drying to wetting. For a drying scanning curve, the ink-bottle eect according to Fig. 1.10a can be obtained as:

S ink = rmax rc V rc V r dV = S c (ln S rmax -ln S c ) (1.57)
where r max is the radius of the largest pore which has been reached during the previous wetting process. Thus, the saturation for the drying scanning curve is written as: In the same way, a wetting scanning curve can be obtained on the basis of Fig. 1.10b.

S con = S c + S ink = S c (ln S rmax -ln S c ) (1.
S con = S c + S ink = S c + ∞ rc V rc V r dV = S c -S r min lnS c (1.59)
where r min is the radius of the smallest pore which has been reached during the previous drying process.

In DuCOM, subsequent hysteresis in the scanning curves is disregarded in order to getting more practical applicability; namely, no hysteresis after the rst scanning curves will be consid-1.5. Modelling of moisture transport under varying ambient relative humidity ered. The inner loops are assumed to be reversible to make sure that closed form loops can be obtained.

However, hysteresis in condensed water can only explain hysteresis behaviour in the high RH range. For the low RH range, hysteresis is mainly aected by interlayer water. But the amount of interlayer water is very small and the behaviour has not been fully understood. DuCOM denes four empirical equations to correlate the interlayer saturation to relative humidity for the above mentioned four drying-wetting scenarios.

For main wetting curve

S il = RH (1.60)
For main drying curve

S il = RH 0.05 (1.61)
For the rst drying scanning curve

S il = 1 + (RH -1) S r min RH r min (1.62)
For the rst wetting scanning curve

S il = S rmax RH 0.05 (1.63)
The total saturation can be calculated based on the contribution of each state of water.

S t = (φ gl + φ cp )(S ads + S con ) + φ il S il φ t (1.64)
where φ gl , φ il , φ cp and φ t are porosities of gel pores, interlayer, capillary and total pores, respectively.

In DuCOM, a complete set of equations have been proposed to calculate hysteresis, and further to simulate drying and wetting cycles. However, the pore size distribution function used in Eq. (1.52) is too simplistic to calculate the main sorption isotherms, and has not been veried by experimental data.

Interactions between materials and their surrounding

To study moisture transport, it needs to understand moisture interactions between the material and its surrounding. Normally, the boundary condition is used to represent those interactions.

In the literature, there are two common boundary conditions. The Dirichlet type boundary condition (also called the rst type boundary condition) xes the value of RH (or P v ) on the exposed surface. For the case of RH chosen as the driving force for moisture transport, this boundary condition is formulated as:

RH 0 = RH e (1.65)
where RH 0 and RH e represent the relative humidity at the material surface and in the environment, respectively.

The rst type boundary condition can be found in earlier moisture transport research, such as in [START_REF] Arfvidsson | A new algorithm to calculate the isothermal moisture penetration for periodically varying relative humidity at the boundary[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF]. However, RH in the natural environment varies a lot and uctuates rapidly, which make the surface of the material is hardly in equilibrium with its surrounding. Therefore, the rst type boundary condition, imposing RH, seems less applicable.

The other type of boundary condition, which has become more common in recent years [START_REF] Pont | Modeling concrete under severe conditions as a multiphase material[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF], is to impose a moisture ux at the boundary [START_REF] Pont | Modeling concrete under severe conditions as a multiphase material[END_REF]. The total moisture ux J total is given as:

J total x=0 = E(P 0 v -P e v ) = EP vs (RH 0 -RH e ) (1.66)
The surface emissivity E (kg On the contrary to Azenha's experimental results [START_REF] Azenha | Drying induced moisture losses from mortar to the environment. Part II: numerical implementation[END_REF], some researchers reported that E seems to be independent to the environmental RH [START_REF] Ayano | Drying, moisture distribution, and shrinkage of cementbased materials[END_REF]. A modication to Eq. (1.66) has been used in previous studies [START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF][START_REF] Zhang | Analysis of moisture transport in cementitious materials and modelling of drying-wetting cycles[END_REF].

• m -2 • s -1 • Pa -1 )
J total x=0 = φSEP vs (RH 0 -RH e ) (1.68)
Equation (1.68) includes both eects of material property (porosity φ) and the current saturation S. φS represents the accessibility of moisture in the porous material to the ambient environment. This boundary condition has been criticized in the case of the natural condition [START_REF] Defraeye | Stomatal transpiration and droplet evaporation on leaf surfaces by a microscale modelling approach[END_REF],

in which the inuence of wind and temperature makes the relationship between moisture ex-1.6. Critical review and conclusion change rate and saturation quite non-linear. Since in the current study no wind or temperature eects will been considered, Eq. (1.68) is still applicable.

From a numerical simulation point of view, the advantage of using the ux boundary condition compared to using the rst type boundary condition is that it reduces the eect of a sharp relative humidity gradient at the beginning of drying, thus obtaining a faster convergence and a considerable reduction of oscillations in the solution.

Even though dierent boundary conditions have been considered in the modelling, they generally represent a simplication of complex interactions between the material and its surrounding.

Further investigations, which include both mass transports in the material and in its surrounding, will be carried out in Chapter 6.

Critical review and conclusion

Compared to other models, the continuum model is the most rigorous one. It is expected to be more physical for modelling moisture transport and moisture storage within building materials.

Thus, to model drying and wetting cycles, the continuum models should be the rst choice. Only concerning continuum models, the above review summarises three dierent methods to simulate moisture transport under drying-wetting conditions:

(1) Using dierent transport coecients for the drying and wetting processes [START_REF] Li | Numerical analysis of moisture inuential depth in concrete during drying-wetting cycles[END_REF][START_REF] Li | Inuential depth of moisture transport in concrete sub ject to drying-wetting cycles[END_REF].

So, this kind model does not need sorption isotherm to address the equilibrium between liquid-water and vapour. All transport properties are only related to the water content.

(2) Non-hysteresis modelling which uses the same sorption isotherm for both drying and wetting. The main desorption curve is generally used in the literature [START_REF] Cunningham | Eective penetration depth and eective resistance in moisture transfer[END_REF][START_REF] Arfvidsson | A new algorithm to calculate the isothermal moisture penetration for periodically varying relative humidity at the boundary[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF]. There is a one-to-one relation between S and P c .

(3) Hysteresis modelling, in which a hysteresis model is involved to calculate inner scanning curves (e.g., [START_REF] Maekawa | Multi-Scale Modeling of Structural Concrete[END_REF][START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF][START_REF] Johannesson | A numerical approach for non-linear moisture ow in porous materials with account to sorption hysteresis[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF]). This leads to a non-unique relation between S and P c . This relation is highly dependent upon the sorption history.

The diusion coecients in the rst method are highly dependent on the calibration of experimental data. Actually, for dierent cycles, transport coecients are also dierent. Hence, for this method, it is denitely dicult to calibrate dierent coecients by experimental data for uctuating and unstable drying and wetting cycles. Those limitations restrict the application of this method.

Non-hysteresis modelling is an easy way to achieve our purpose for modelling of drying and wetting cycle.

Avoiding the calculation of scanning curves for each time step can speed up the numerical calculations. However, this method has shown low agreement with experimental data [START_REF] Zhang | Analysis of moisture transport in cementitious materials and modelling of drying-wetting cycles[END_REF], and has been questioned by researchers when it is compared with hysteresis modelling. Johannesson et al. [START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF] reported that the error (between non-hysteresis and hysteresis modelling) in determining the water content may be as high as 30-35% . Derluyn et al. [START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF] concluded durability risks may be underestimated when omitting moisture hysteresis. Both studies emphasised the necessity of considering hysteresis for the prediction of the moisture behaviour of cementitious materials.

All of these conclusions indicate that if the model is able to take into account hysteresis, more accurate results will be obtained. But, the conclusion has been only veried by limited supporting experimental data. It inspires us to carefully analyse hysteretic eects on moisture transport in cementitious materials through both experimental investigations and numerical modelling.

Chapter 2

Modelling of isothermal drying of unsaturated cementitious materials

Summary

In this chapter, the multiphase transport model described in Chapter 1 will be used to simulate isothermal drying of cementitious materials (mainly cement pastes). First of all, this model will be veried by means of experimental data selected from the literature. Asymptotic analysis concludes that cementitious materials can be viewed as weakly permeable materials in the case K g = K l which is in agreement with the theoretical construction of the modelling. Simulation results show that the gas overpressure and underpressure can be observed in the material. Gas pressure variations are highly dependent on the initial saturation within the material. However, it is recognised that K l K g when comparing K l provided by inverse analysis of drying kinetics and K g directly measured by experiments. This provides the second case for numerical modelling which shows that gas pressure in the material is constant. By neglecting advection of the gasphase, a simplied model, only including advection of liquid-water and vapour diusion, can be obtained. In the last part of this chapter, the dierent moisture transport modes will be discussed according to the above two cases illustrating that quite dierent transport regimes can be achieved.

Introduction

The study on drying is of great importance for moisture transport within building materials.

Porous building materials such as concretes and mortars are inevitably exposed to natural conditions. These conditions denitely have inuence on their performances related to their mechanical properties, such as strength, shrinkage, creep, cracking, etc., as well as their durability since the chemical ingresses occurring in cementitious materials are strongly dependent on the moisture content. The importance of moisture transport in the eld of civil engineering is thus particularly marked when the prediction of the service-life of concrete structures is required [START_REF] Bazant | Nonlinear water diusion in non-saturated concrete[END_REF][START_REF] Daïan | Condensation and isothermal water transfer in cement mortar. Part I: pore size distribution, equilibrium water condensation and imbibition[END_REF][START_REF] Nilsson | Long-term moisture transport in high performance concrete[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF].

Drying of cementitious materials has been widely studied by numerical modelling [START_REF] Bazant | Nonlinear water diusion in non-saturated concrete[END_REF][START_REF] Daïan | Condensation and isothermal water transfer in cement mortar. Part I: pore size distribution, equilibrium water condensation and imbibition[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Nilsson | Long-term moisture transport in high performance concrete[END_REF]. By using the multiphase model described in Chapter 1 (see Section 1.4), Mainguy et al. [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF] analysed two assumptions that gas pressure in the material is constant or non-constant and reported a simplied moisture transport model in the case of non-constant gas pressure (see Eq. 1.17). The current chapter will adopt this multiphase model to further discuss the impact of two assumptions about the concept of intrinsic permeability (K l = K g or K l K g ) on gas pressure variations, and the regimes of moisture transport. Firstly, the case K l = K g is use to simulate moisture transport during drying, and then the case K l K g will be discussed.

Modelling of drying porous media in the case of cementitious materials

The governing equations of the multiphase model have been given in Section 1.4 in Chapter 1 (see Eqs. 1.3-1.11). To perform numerical simulations, the initial and boundary conditions and equations for transport coecients have to be dened.

Initial and boundary conditions and geometry of the problem

The ux boundary condition [START_REF] Gustafson | The third boundary condition -was it Robin's?[END_REF][START_REF] Pont | Modeling concrete under severe conditions as a multiphase material[END_REF] which has been introduced in Chapter 1 is used to account for an imperfect moisture transport between the environment and the surface of the material.

If only focusing on the 1D transport problem for a given size material x ∈ [0, l], the boundary conditions are expressed as:

for t > 0, at x = 0 : J total = φSEP vs (RH 0 -RH e ); P g = P atm

(2.1)
for t > 0, at x = l :

J total = 0 (2.2)
State variables are considered to be uniformly distributed in the material at the initial.

for x > 0, at t = 0 : RH = RH 0 ; P g = P atm

(2.3)
where RH 0 = P v,0 /P vs is the initial RH.

An idealized 1D drying experiment is considered. A sample of porous material is lying between x = 0 and x = l (see Fig. 2.1). Initially, relative humidity RH 0 in the material is uniform. The initial liquid-water saturation S 0 is calculated from RH 0 according to the desorption isotherm. At time t = 0, the face x = 0 comes into contact with an ambient environment whose relative humidity RH e is lower than RH 0 . the surface that is exposed to the ambient environment.

Materials properties

The drying experimental data were collected from the literature [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF][START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF]. The properties of the selected materials are given in Table 2.1, including three cement pastes and one concrete, which were made from the same cement (CEM I 52.5, according to EN 197-1 European standard), but mixed with dierent water-to-cement ratios (w/c) or water-to-binder ratio (w/b). Paste3 and Concrete3 were cast in cylindrical moulds with a 16 cm-diameter [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF]. Smaller moulds were used for Paste4 and Paste5 with a 7 cm-diameter [START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF].

Before drying experiments, materials had been sealed more than 200 days for curing. After Paste3 was also submitted to drying at RH e = 63.2% (ammonium nitrate, NH 4 NO 3 ). The total porosity was calculated based on the mass dierences between the weight of saturated samples and the weight of fully dried samples (at 105 • C) [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF][START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF].

One important property of the material is its capillary pressure curve. As mentioned in Section 1.2 in Chapter 1, capillary pressure curve addresses the equilibrium between liquid-water and vapour. For drying, the desorption curve is required for numerical simulations. At a given temperature, the desorption curve is experimentally determined by the saturated salt solution method [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. However, measurements can only provide limited points for a capillary pressure curve, such as 11 points in Fig. 2.2. From a practical point of view, in the purpose of numerical modelling, an analytical equation is necessary to represent the capillary curve. The equation can be obtained by tting the measured curve. Dierent tting equations can be found in the literature. Some of them will be compared and analysed in Chapter 3 (see Section 3.4). Following the work carried out by Mainguy et al. [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF], the current study tentatively chooses a simple and commonly-used equation, proposed by van Genuchten [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF].

P c (S) = a V S -m V -1 1/n V (2.4)
where a V , m V and n V are parameters, which will be evaluated by experimental data. In practice,

m V = 1 -1/n V is

Transport coecients

Due to the lack of experimental data, tortuous and unsaturated eects on gas diusion introduced in Eqs. (1.7) and (1.8) were calculated by the equation proposed by Millington [START_REF] Millington | Gas diusion in porous media[END_REF].

α(S)τ = φ x D -1 (1 -S) x D +1 (2.5)
The parameter x D was xed at 4/3 in Millington's study for soils. However, soils are more porous than cementitious materials and therefore the resistant eects should be more signicant in the case of cementitious materials. Thiéry et al. [START_REF] Thiéry | Modélisation du séchage des bétons -Analyse des diérents modes de transfert hydrique[END_REF] suggested x D = 2.74 based on the tting of experimental data for O 2 and CO 2 diusion measurements conducted by Papadakis et al. [START_REF] Papadakis | Physical and chemical characteristics aecting the durability of concrete[END_REF].

The comparison of α(S)τ calculated by these two proposals for the studied three materials are shown in Fig. 2

.3. Thiéry's suggestion provides smaller resistance factor values as it has been

shown in studies [START_REF] Baroghel-Bouny | Modelling of isothermal coupled moistureion transport in cementitious materials[END_REF][START_REF] Baroghel-Bouny | Prediction of chloride binding isotherms of cementitious materials by analytical model or numerical inverse analysis[END_REF][START_REF] Wang | Modélisation du transport multi-espèces dans les matériaux cimentaires saturés ou non saturés et éventuellement carbonatés[END_REF][START_REF] Morandeau | Carbonatation atmosphérique des systémes cimentaires à faible teneur en portlandite[END_REF], which may be closer to the real condition of cementitious materials compared to the original x D value. For cementitious materials, measuring the permeability to liquid-water for dierent RH is very dicult due to the fact that advective liquid transport and vapour diusion always occur together [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF]. Hence, analytical equations are normally used to calculate k rl . One most common method is to consider k rl as a function of S (k rl = k rl (S)), which was rst reported for two-phase ow by Geen et al. [START_REF] Geen | Experimental investigation of factors aecting laboratory relative permeability measurements[END_REF] and Osaba et al. [START_REF] Osoba | Laboratory measurements of relative permeability[END_REF]. The equation for the gas relative permeability k rg can be deduced based on the equation for k rl .

Relative permeabilities -Method-1

Mualem proposed an equation for porous materials based on the known capillary curve [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF].

k rl (S) = S     S 0 dS P c 1 0 dS P c     γ (2.6)
In Eq. (2.6), the term 1 0 dS/P c is related to the total pore volume which is available for liquid-water transport, and the term S 0 dS/P c stands for the actual pore volume which has been occupied by liquid-water. The term S is a correction factor, which accounts for the inuence of tortuous eects of pores on the liquid transport. Dierent suggestions of parameter have been proposed by researchers [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Parker | A model for hysteretic constitutive relations governing multiphase ow: 1. Saturation-pressure relations[END_REF]. In Mualem's studies, varies between -1 and 3, and nally the value 0.5 was considered as the best choice. The parameter γ is generally taken as γ = 1/ .

These values of and γ have been adopted for cementitious materials [START_REF] Savage | Soil physics principles validated for use in predicting unsaturated moisture movement in portland cement concrete[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF].

If putting the capillary curve equation (Eq. 2.4) into Eq. (2.6), a simple and easy-to-use analytical equation is provided [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF]:

k rl (S) = S 0.5 1 -1 -S 1/m V m V 2 (2.7)
where the parameter m V is the same as one used in Eq. (2.4).

Equation (2.6) indicates that the pore volume which has been lled by liquid-water is used for liquid transport, so that it is easy to understand that the pore volume which has not been occupied by liquid-water is available for gas transport. Hence, the expression of the gas relative permeability can be deduced following the form of Eq. (2.6) [START_REF] Lenhard | A model for hysteretic constitutive relations governing multiphase ow: 2. Permeability-saturation relations[END_REF]:

k rg (S) = (1 -S)     1 S dS P c 1 0 dS P c     γ (2.8)
In a similar manner, if plugging Eq. (2.4) into Eq. (2.8) and letting = 0.5, an analytical equation [START_REF] Parker | A model for hysteretic constitutive relations governing multiphase ow: 1. Saturation-pressure relations[END_REF][START_REF] Luckner | A consistent set of parametric models for the two-phase-ow of immiscible uids in the subsurface[END_REF] appears as:

k rg (S) = (1 -S) 0.5 1 -S 1/m V 2m V (2.9)
One can notice that Eqs. (2.7) and (2.9) are denitely easier to use in numerical modelling due to analytical forms. This version of Mualem's model [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF] indicates that there is a relation between k rl and k rg if merging Eqs. (2.6) and (2.8).

k rl S 1/γ + k rg (1 -S) 1/γ = 1 (2.10)
Therefore, if either k rl or k rg is known, the other one can be determined by using Eq. (2.10).

Relative permeabilities -Method-2

Based on the simplied independent domain theory [START_REF] Mualem | A conceptual model of hysteresis[END_REF], Mualem [START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF] developed a comprehensive model to calculate the relative permeability. The main feature of this model is that it has taken into account hysteresis in the relation k rl (S). Thus, the equations to calculate k rl corresponding to the main sorption (desorption and adsorption) and scanning curves are dierent.

The principle of the independent domain theory will be described in Chapter 3 (see Section 3.2).

The complete equations for this model will be given in Chapter 5 (see Section 5.6.5). Since there are two versions of Mualem's models for k rl , let's denote Eq. (2.6) as Mualem's model-a [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF] and the model mentioned in this paragraph as Mualem's model-b [START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF]. These two models are combined with van Genuchten's equation to calculate k rl . Thus, the combination of van Genuchten's equation + Mualem model-a is further denoted by VGMa (see Eq. where L and H are two cumulative pore water distribution functions. Their expressions are:

k rl (S) = S 0.5 [KL -(1 -KL) KH]
L(P c ) = S w (P c ) H(P c ) = S d (P c ) -S w (P c ) 1 -S w (P c ) (2.13)
where S w (P c ) and S d (P c ) are the saturation of the main adsorption and desorption curves.

Since Eq. (2.11) cannot be solved analytically, it is dicult to apply the same theory as the one used in Mualem model-a (see Eq. (2.10) to directly infer k rg . Here, an alternative method is chosen. Thiéry et al. [START_REF] Thiéry | Modelling of isothermal drying process in cementitious materials -Analysis of the moisture transfer and proposal of simplied approaches[END_REF] adopted the formula based on the theory of Mualem model-a [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF] for k rg but proposed dierent values for and γ. By the validation of experimental data taken from Villain et al. [START_REF] Villain | Mesure de la perméabilité aux gaz en fonction du taux de saturation des bétons[END_REF], Monlouis-Bonnaire et al. [START_REF] Monlouis-Bonnaire | Prediction of the relative permeability to gas ow of cement-based materials[END_REF] and Thiéry [START_REF] Thiéry | Modélisation de la carbonatation atmosphérique des matériaux cimentaires avec prise en compte des eets cinétiques et des modications microstructurales et hydriques[END_REF], Thiéry et al. [START_REF] Thiéry | Modelling of isothermal drying process in cementitious materials -Analysis of the moisture transfer and proposal of simplied approaches[END_REF] concluded that = 7.15 and γ = -8.02 should be the best choice for cementitious materials. experimental data [START_REF] Thiéry | Modélisation du séchage des bétons -Analyse des diérents modes de transfert hydrique[END_REF], while there are no measured data available in the literature to verify k rl . Therefore, the next subsection will discuss eects of dierent choices to k rl and k rg on moisture transport in unsaturated porous materials.

For cementitious materials, directly measuring the intrinsic permeability to liquid-water K 0 is extremely dicult because the measurement requires very high pressure gradient, which may destroy the material. Since K 0 is the only undetermined parameter in the multiphase moisture transport model in the case K l = K g = K 0 , a method called inverse analysis can be used to determine K 0 through tting the measured mass loss curve obtained in a drying experiment.

The inverse determination of K 0 has been used in [START_REF] Coussy | Assessment of the water permeability of concretes from their mass loss during drying (in french)[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Baroghel-Bouny | Assessment of transport properties of cementitious materials. A major challenge as regards durability?[END_REF]. The modelling results in the case of distinguishing K l and K g will be shown in Section 2.4.

Simulation results veried by experimental data of drying

Experimental data include mass loss curves and saturation proles [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF]. During experiments, samples were weighed to get mass loss curves and saturation proles were measured by gamma-ray attenuation.

During the sealed-curing period, the cement hydration processes consumes water molecules and self-desiccation occurs in the material. This means that the material undergoes a decrease of its internal RH. After moulds had been removed, the internal RH, which should be the initial condition for numerical modelling, was measured for all the selected materials (see Table 2.1) [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF].

Since the material has been assumed as a homogeneous porous medium, the internal RH is also considered to be distributed uniformly in the material. (taken from the reference [START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF]) are also shown in this table. Obviously, K 0 determined based on Method-1 are larger than that calibrated by Method-2. The dierences can be up to 500 times.

The order of magnitude of the intrinsic permeabilities determined by Method-1 is around 10 -21 m 2 , which is close to results provided in [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF], but smaller than results calculated by the Katz-Thompson model. Experiments, such as the beam-bending method [START_REF] Vichit-Vadakan | Measuring permeability and stress relaxation of young cement paste by beam-bending[END_REF][START_REF] Scherer | New methods to measure liquid permeability in porous materials[END_REF] and the dynamic pressurization method [START_REF] Grasley | Dynamic pressurization method for measuring permeability and modulus: II. cementitious materials[END_REF], show that the intrinsic permeability is around 10 -20 m 2 for cement pastes with equivalent w/c. Thus, it is not easy to say which method to calculate k rl and k rg is better than the other one. Nevertheless, Method-1 seems to be easier to use since all equations are analytical.

Gas pressure variations during drying of unsaturated cementitious materials

When applying the multiphase model to simulate moisture transport in cementitious materials, it reveals the existing of gas overpressure (above the atmospheric pressure) in the material.

This phenomena has been reported by Mainguy et al. [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF], but these authors used Millington's proposal of x D for the resistance factor (see Eq. 2.5) and chose Method-1 to calculate k rl and k rg . This section will utilize dierent equations to calculate the resistance factor, k rl and k rg and to analyse gas pressure variations during drying. and merging them, it is possible to derive the following mass balance equation for the gas mixture:

φ ∂ ∂t [P g (1 -S)] = - ∂ ∂x (P g v g ) + RT M v ml→v (2.14)
The left hand side of the above equation is divided into two parts leading to the following equation:

φ(1 -S) ∂P g ∂t = φP g ∂S ∂t - ∂ ∂x (P g v g ) + RT M v ml→v (2.15)
Equation (2.15) shows that there are three components which contribute to gas pressure variations: (a) The term involving the evaporation rate ml→v causes an increase of gas pressure.

The evaporated vapour is forced to diuse toward the surrounding air because the maximum vapour pressure is far below vapour pressure in saturated condition (P vs ). According to Fick's law, a symmetric diusion of dry air occurs simultaneously to diusion of vapour, and may engender an increase of pressure by provoking a net arrival of fresh air. This phenomenon was studied in detail in [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF]. (b) The term involving the derivative of S with respect to time is negative during drying, and thus it entails a decrease of pressure due to the release of liquid-water in the pores. (c) The divergence term stands for the advective Darcean ow of the gas-phase which can dissipate any variations of pressure. The existence of an overpressure or underpressure of gas during drying depends on the time scales at which the three above-mentioned mechanisms can occur.

Substitution of Darcy's law (see Eq. 1.11) into the mass balance equation (see Eq. 1.4), the following expression of the evaporation rate ml→v can be derived:

ml→v = -ρ l φ ∂S ∂t + ρ l η l K 0 ∂ ∂x k rl (S) ∂P g ∂x -M dπ c dS ∂S ∂x (2.16)
Putting the previous expression of ml→v into the mass balance equation of gas (see Eq. 2.14), the equation governing the advective transport of the vapour-dry air mixture is given as:

∂ ∂t [P g (1 -S)] + ρ l RT M v ∂S ∂t = 1 2η g K 0 φ ∂ ∂x k rg (S) ∂P 2 g ∂x (2.17) + ρ l RT M v η l K 0 φ ∂ ∂x k rl (S) ∂P g ∂x -M dπ c dS

∂S ∂x

By putting the expression of ml→v in Eq. (2.16) into Eq. ( 1.3) and by accounting for Eq. (1.7), it comes the equation governing the transport of moisture in both liquid and vapour form:

∂ ∂t [(1 -S)P v ] + ρ l RT M v ∂S ∂t = - 1 η g K 0 φ ∂ ∂x P v k rg (S) ∂P g ∂x (2.18) +D 0 P atm τ ∂ ∂x (1 -S)α(S) ∂ ∂x P v P g + ρ l RT M v η l K 0 φ ∂ ∂x k rl (S) ∂P g ∂x -M dπ c dS

∂S ∂x

Among the three thermodynamic variables S, P v and P g involved in Eqs. (2.17) and (2.18), only two are actually independent since the equilibrium between liquid-water and vapour must be ensured. This equilibrium state is governed by the Kelvin's law (see Eq. 1.12).

Asymptotic analysis

According to studies performed by Coussy and Thiery [START_REF] Coussy | Mechanics and physics of porous solids[END_REF][START_REF] Coussy | Asymptotic[END_REF], it is possible to identify three characteristic times on the basis of analysis of Eqs. (2.17) and (2.18):

t F = l 2 D 0 τ t g D = φη g l 2 P atm K 0 t l D = φη l l 2 MK 0 (2.19)
The characteristic time t F scales the rate at which diusion of vapour and dry air occur, and t g D and t l D scale the rates at which the advective ows of gas and liquid-water take place, respectively. Then, the following dimensionless quantities are introduced:

x = x l P g = P g P atm P v = P v P atm (2.20)
To simplify the expressions, another parameter Λ is introduced:

Λ = 1 P atm ρ l RT M v (2.21)
This parameter represents the ratio of pressure that the water vapour would have if its density was the same value as the density of the liquid-water ρ l to the atmospheric pressure P atm . Adopting an absolute temperature of 293 K, the value of the parameter Λ is around 1350.

The time scale at which the coupled transports (liquid-water, vapour and dry air) successively occur depends on the relative range of the characteristic times t l D , t g D and t F . The drying mechanism will be explored according to the intrinsic permeability K 0 of the material.

The value M stands for a reference capillary pressure related to the whole pore size distribution corresponding to the entry radii of the pores. By considering Laplace's equation with a zero wetting angle, a relationship between the capillary modulus M and a radius of pore representing the pore size distribution appears as r ∼ σ/M where σ is the air-water interfacial energy (7.3 × 10 -2 J • m -2 ). Moreover, it is well recognized that the intrinsic permeability is proportional to the square of a characteristic length related to the porous volume where the ow takes place. By adopting the percolation theory [START_REF] Thompson | The microgeometry and transport properties of sedimentary rocks[END_REF], K 0 is demonstrated to be proportional to r 2 φτ . Thus, it follows:

M ∼ σ φτ K 0 (2.22) Adopting T = 293 K, η g = 1.8 × 10 -5 Pa • s, η l = 1.0 × 10 -3 Pa • s, P atm = 10 5 Pa and D 0 = 2.5 × 10 -5 m 2 • s -1
, the three following ratios can be assessed in the form:

t F t g D ∼ 2.2 × 10 14 K 0 φτ (2.23) t l D t g D ∼ 7.7 × 10 7 K 0 φτ (2.24) t F t l D ∼ 2.9 × 10 6 K 0 φτ (2.25)
A quite permeable porous material can be dened when the ratio t F /t g D 1 (⇔ K 0 /(φτ ) 4.5×10 -15 m 2 ). Notice that this condition also implies the fact that the ratio t l D /t g D (see Eq. 2. 24) is far larger than 1 (⇔ K 0 /(φτ ) 1.7 × 10 -16 m 2 ). The situation of a quite permeable porous material corresponds to the fact that the advective transport of the gaseous phase occurs much faster than transports of both liquid-water by capillarity and water vapour by diusion.

In contrast, because of a suciently low value of intrinsic permeability, i.e., K 0 /(φτ ) 4.5 × 10 -15 m 2 , vapour diusion occurs much faster than the advective transport of gas (t F /t g D

1)

and of liquid-water (t

F /t l D 1 ⇔ K 0 /(φτ ) 1.2 × 10 -13 m 2
). However, the condition K 0 /(φτ ) 4.5 × 10 -15 m 2 does not imply that the ratio t l D /t g D is much lower than 1. Because of the much lower value of the liquid-water viscosity compared to that of the gaseous mixture, it could be expected that advective of liquid-water occurs slower than advection of gas-phase in the material. Indeed, as shown by Eq. (2.24), for the weakly permeable porous media, advection of liquid-water takes place either at the same rate as advection of the gaseous mixture or much faster.

Actually, the huge dierence of viscosity between the liquid-water and the gas is compensated by the huge value of capillary pressure driving the liquid ow. High values of capillary pressures are associated with a large value of capillary modulus M and low values of permeability.

On the whole, two cases are distinguished in this asymptotic analysis of drying:

-Quite permeable porous materials:

Simulated gas pressure variations for cementitious materials

According to the denition of quite or weakly permeable material in the previous section (see Eq. 2.23), all the selected materials in this chapter are considered as weakly permeable materials by using K 0 values in Table 2.3 and considering τ in the order of magnitude of 0.01. Therefore, advection of the gas-phase is slower than diusion of the gas-phase (vapour + dry air), t g D t F , which causes that gas pressure in the material cannot dissipate in a short time. Meanwhile, since advection of liquid-water is slower than the gas diusion, t l D t F , the space for gas-phase does not increase too much. Hence, gas pressure variations are expected during drying of cementitious materials.

To demonstrate the above explanation, simulations of gas pressure during drying need to be carried out.

Here, let's take Paste4 as an example. Two dierent initial conditions are considered: S 0 = 0.9 and 0.99. The external RH e = 53.5% is used as the same as experiments (see Section 2.2.2). Simulation results of gas pressure proles are shown in Fig. 2.11. It is clear that a signicant gas overpressure appears at the border layer of the specimen.

The predicted gas pressure can reach up to 1.5 times higher than the initial value of gas pressure (atmospheric pressure). This is rather unusual because the gas conductivity of porous materials is generally so high that any change of gas pressure vanishes quasi-instantaneously. Nevertheless, numerical simulations [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF] have already shown that the advective Darcean transport of gas (the second term on the right-hand side of Eq. (2.15)) is low for cementitious materials. Thus, the overpressure phenomena related to liquid-water evaporation are not easily dissipated so that the role of the third term on the right-hand side of Eq. (2.15) is predominant.

It is worth noting that when the initial condition S 0 = 0.99, an obvious gas underpressure (below the atmospheric pressure) behaviour is observed within the material, but not in the vicinity of the surface (see Fig. 2.11b). It gradually moves toward the inner part of the sample, while for a lower initial condition S 0 = 0.90, there is no observed gas underpressure. It shows that the previous asymptotic analysis remains questionable since it does not include the eects of S on the transport parameters and is not valid near the surface under the inuence of the boundary condition. Thus, one may expect a role of the initial moisture state of the material, as well as the boundary condition. On one hand, drying of a material which is initially close to saturation shows signicant movements of liquid-water due to the strong gradient of capillary pressure. On the other hand, the diusion coecient of water vapour is lower because of the high water content according to Eq. (2.5) and therefore diusion of water vapour is restricted and cannot move to inner of the sample concurrently. In this process, the desaturation of the porosity causing by liquid-water ow contributes to the decrease of gas pressure in the material (the rst term on the right-hand side of Eq. (2.15)). shown that, for a given material (such as Paste4), the major factor determining the maximum overpressure during drying is the external RH prevailing in the atmosphere. For example, almost the same curves for maximum overpressure varying with external RH are given in Fig. 2.12 for the two dierent studied initial conditions. On the contrary, the curves for the minimum gas pressure in the same gure are also illustrating that they mainly depends on the initial saturation. If the initial saturation is very high (such as 0.99), underpressure behaviour appears easily according to results shown in Fig. 2.12.

Modelling of drying cementitious materials in the case K l K g

The multiphase model, described in Section 1.4 in Chapter 1, forced K l = K g = K 0 . However, the measured data by CEMBUREAU method (e.g., [START_REF] Kollek | The determination of the permeability of concrete to oxygen by the cembureau methoda recommendation[END_REF][START_REF] Kameche | Measurement of the relative gas permeability of ordinary concrete: inuence of saturation degree and the sample size[END_REF]) and other techniques (e.g., [START_REF] Hamami | Inuence of mix proportions on microstructure and gas permeability of cement pastes and mortars[END_REF][START_REF] Loosveldt | Experimental study of gas and liquid permeability of a mortar[END_REF])

show that intrinsic permeabilities to gas-phase corresponding to values in Actually, it is reasonable to observe K l K g , because: 1) Liquid-water has higher viscous eects from pore walls and therefore its transport should be slower than gas-phase. 2) Ions in liquid-water are able to decrease the chemical activity of liquid-water and to slow down advection of liquid-water. 3) Generally, there are some impurities in pore network, such as unhydrated cement particles, which aect the velocity of liquid transport, but have small inuence on gas transport. 4) Liquid phase has stronger electricity charges exchanges with pore walls than gas phase (see explanation in Section 1.2), so that perfect Darcy transport is not valid in micropores.

Besides, the faster gas advection can also result from the slipping contact between gas and pore walls (also known as Klinkenberg's concept [START_REF] Klinkenberg | The permeability of porous media to liquids and gases, in drilling and production practice[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF]), while there is no slipping between liquidwater and pore walls.

Since the value of K g has been xed at around 1 × 10 -17 m 2 , K l becomes the only unknown in the multiphase model. By using the same inverse analysis as Section 2.2.3, K l can be obtained for the case K l K g for the studied materials (see Table 2

Gas pressure variations

In the case K l K g , three characteristic times in Eq. (2.19) can be rewritten as equations which should be involved by the eect of K g .

t F = l 2 D 0 τ t g D = φη g l 2 P atm K g t l D = φη l l 2 MK l (2.26)
Like in Eq. 2.22, the capillary modulus can be considered as a function of K l .

M ∼ σ φτ K l (2.27)
Following Eqs. (2.23), (2.24) and (2.25), the relationships between the three characteristic times can be rewritten as:

t F t g D ∼ 2.2 × 10 14 K g φτ (2.28) t l D t g D ∼ 7.7 × 10 7 K g K l K l φτ (2.29) t F t l D ∼ 2.9 × 10 6 K l φτ (2.30)
Thus, the criterion to distinguish quite or weakly permeable porous materials is related to the ratio of K g to φτ : Above simulations and comparisons conclude that whether gas pressure variations in the material can be observed depends on the relationship between two intrinsic permeabilities: the case K l = K g means non-constant gas pressure and the case K l K g corresponds to constant gas pressure.

Simplication of moisture transport models

A dimensionless time is introduced t = t/t l D . Thus, the time is scaled by the characteristic time of advection of liquid-water. The equation governing the transport of the gaseous mixture (see Eq. 2.17) can be rewritten in a dimensionless form:

∂ ∂t [P g (1 -S)] + Λ ∂S ∂t = t l D 2t g D ∂ ∂x k rg (S) ∂P 2 g ∂x (2.31) +Λ ∂ ∂x k rl (S) P atm M ∂P g ∂x - dπ c dS

∂S ∂x

In the case K l K g , diusion of vapour occurs much slower than advection of gas-phase.

Keeping terms which have the same order of magnitude with regard to t l D /t g D , Eq. (2.31) for t ∼ 1 reduces to:

∂ ∂x k rg (S) ∂P 2 g ∂x = 0 (2.32)
Owing to the symmetry condition in Eq. (2.1) and the boundary condition, the integration of Eq. (2.32) yields the simple equality P g ∼ P atm . Taking benet of this result, the rst term on the right-hand side of Eq. (2.18) can be neglected.

Noting that Λ 1 results in

P v P atm ρ l RT M v
, meaning that the moisture mass is mainly represented by liquid phase.

Therefore, the mass balance equation of moisture (liquid and vapour, see Eq. (2.18) is written in the following simplied form:

∂S ∂t = ∂ ∂x (D l (S) + D v (S)) ∂S ∂x (2.33)
where

D l (S) = - K l φη l k rl (S)M dπ c dS (2.34) D v (S) = M v ρ l RT D 0 τ α(S)(1 -S) dP v dS (2.35)
Actually, Eq. (2.33) is the same as the one that has been reported by Daïan [START_REF] Daïan | Condensation and isothermal water transfer in cement mortar. Part I: pore size distribution, equilibrium water condensation and imbibition[END_REF] (see Eq. 1.18).

The sum of D v and D l is the so-called apparent diusivity D a [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF]. The ratio of D v and D l is

D v D l = M v ρ l RT 2 η l D 0 P vs RH(S) f (φ, S) K l k rl (S) (2.36)
where f (φ, S) is the resistance factor of vapour diusion.

f (φ, S) = φτ α(S)(1 -S) (2.
D v D l = 3 × 10 -21 K l f (φ, S)RH(S) k rl (S) (2.

38)

The ratio of D v and D l depends on the relationship between 3×10 -21

K l and f (φ,S)RH(S) k rl (S)
. The value 3×10 -21 is in the same order of magnitude with K l validated by using VGMa model [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF] to calculate the liquid relative permeability (see Table 2.3). If assuming

3×10 -21 K l ≈ 1 such as for
Paste4, the ratio of D v and D l is largely dependent on the relationship between f (φ, S)RH(S)

and k rl (S). Calculated curves for the relation between these two parameters are shown in For most natural conditions (such as in the range of RH > 56%), the liquid-water movement is the main transport mode in unsaturated porous media like cementitious materials. Therefore, if ignoring the gas-phase transport, the moisture transport model is further simplied as a single diusion equation only with liquid-water as it has been reported by Mainguy et al. [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF] (see Eq. 1.17). Using the same K l validated by the multiphase model (see Table 2.3), mass loss curves simulated by three versions of models for Paste4 are shown in Fig. 2.16. Actually, the dierences between these three curves are not signicant, which implies that if one only focuses on the mass transport, these three versions of models are applicable. But notice that the rst simplied model is only suitable for the high RH range (e.g., RH> 56% for Paste4). For the low RH range, this version cannot provide a high enough apparent diusivity as shown in experimental data [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF] because the liquid permeability K l k rl (S) is a monotonically increasing function with RH or S (see Eq. 2.34). Therefore, the second simplied model will be used in Chapter 5 for modelling of drying and wetting cycles.

Transport modes analysis

According to the above simulations and comparisons, drying of a porous material is mainly controlled by two transport modes: (1) Liquid-water is transported to the surface layer and then evaporates to the surrounding environment (see Fig. 2.17a); (2) Along with advection, liquidwater also evaporates within the material to water vapour, which diuses through the porous network to the surrounding environment (see Fig. 2.17b). To identify the dierent moisture transport modes, one not only needs to quantify the contribution of each phase to the total mass transport within the material but also to know how much liquid-water or vapour can be transported to the surface layer. This section will employ the multiphase model to investigate the dierent transport modes.

Solid

Gas

Liquid

Liquid advection Evaporation

Since the case using Method-1 to calculate k rl and k rg has been already studied in [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF], only Method-2 is used here for the simulations (see Section 2.2.3).

Contribution of mass transport in the material

The contribution of liquid-water transport to the total mass transport in the material can be easily achieved by calculating the ratio of liquid ux J l to the total ux J l + J v + J a . By taking Paste4 as an example, the contribution of liquid-water transport is shown in Fig. 2.18 in cases

K l = K g and K l K g .
In Fig. 2.18, the ratio remains near 1 for the inner part of the material, implying that the mass transport is mainly in its liquid form. But along with drying, the ratio near the boundary decreases; namely, contributions of vapour diusion and advection of gas-phase increase with the decrease of S (see saturation proles in Fig. 2.7). For the case K l K g , the contribution of liquid-water transport drops more quickly compared to the case K l = K g (see bold black curves at 371 days in Fig. 2.18). At the end of simulations, the contribution of liquid-water transport at the surface is around 0.5 in Fig. 2.18a, but it is only 0.2 in Fig. 2.18b. This may mean that the exchange rate from liquid to vapour ( ml→v in Eqs. (1.3) and (1.4)) is enhanced for the case The simulation results show that the transport mode in the material is greatly related to the drying time and the current saturation. In the part with high saturation, advection of liquidwater is denitely predominant. However, at the boundary the contribution of liquid advection decreases with drying time due to strong interactions between liquid-water and gas phase. Hence, it should be more interesting to investigate the contribution of liquid advection by an integration at the boundary to identify the dierent transport modes.

K l K g .

Contribution of mass transport by an integration at the boundary

At the boundary, moisture is transported out the material through evaporation. In a same way with the calculation of mass loss curve, one can calculate the mass loss due to the movement of liquid-water (time integration of liquid ux J l at the boundary) to the total mass loss (time integration of uxes of liquid, vapour and dry air J l + J v + J a ) at the boundary during drying.

Q l Q t = t 0 J l dt t 0 (J l + J v + J a ) dt (2.39)
Simulation results for drying of Paste4 are shown in Fig. 2.19, which include both cases: c) The third aspect from Fig. 2.20 is the dierences between the two cases K l = K g and K l K g . The case K l = K g provides a larger contribution of liquid-water transport compared to the case K l K g in the whole range of external RH. This can be also seen in Fig. 2.19. These dierences are much clearer in the moderate range of external RH between 40% and 80%. One possible reason is that the gas overpressure (in the case K l = K g ) in the border layer of the material can reduce the evaporation of liquid-water, which forces the mass transport mainly in its liquid form. By contrast, vapour transport should be a little more signicant in the case K l K g .

K l = K g and K l K g .
Through the above analysis, it can be concluded that transport modes ( 1) and ( 2) occur simultaneously in most situations. The fact that which transport mode is predominant is dependent on the boundary condition (external RH), material properties (e.g., porosity and PSD), the initial condition and the duration of drying.

Conclusion

In this chapter, a multiphase moisture transport model for isothermal conditions was recalled, which considers that the mass transport consists of advection of liquid-water, advection of gasphase (vapour + dry air) and diusion of gas-phase. Since the intrinsic permeabilities to liquidwater K l and gas-phase K g are only two undetermined parameters in this model, two relationships were introduced: K l = K g = K 0 and K l K g . The rst case naturally comes from the theoretical analysis. The second case is given based on the fact that measured value of K g is much larger than the value of indirectly-determined K l by inverse analysis of drying kinetics.

For both cases, the simulation results for four cementitious materials (three cement pastes and one concrete) showed that they provided good agreements with measured mass loss curves. Quite permeable and weakly permeable porous materials were dened according to an asymptotic analysis. For cementitious materials, the case K l = K g implies that materials are considered as weakly permeable, while materials are considered as quite permeable when K l K g . Gas pressure variations, including overpressure and underpressure, were observed for weakly permeable materials. But for quite permeable materials, gas pressure in the material is always constant (equal to the atmospheric pressure), which is because gas pressure variations can dissipate very quick due to the signicant gas-phase advection.

For the case K l K g , a simplied model was obtained, including advection of liquid-water and vapour diusion. Further simplication was achieved if ignoring vapour diusion, so moisture transport is only in its liquid form. Even though the latter version was veried by experimental data, it is not recommended because advection of liquid-water is only predominant in the high water content range. In the low RH range, liquid-water is disconnected and vapour diusion should be more signicant.

Dierent transport modes were identied for moisture transport both at the boundary and in the material. At the beginning of drying, due to the high initial saturation, advection of liquid-water is more signicant. Thus, during drying, liquid-water near the surface layer loses quickly and the vapour transport gradually becomes the main transport mode. In a word, the transport mode highly depends on saturation. In the high water content range, advection of liquid-water is predominant, while in the low water content range, the vapour transport plays a main role. But for most cases, two transport modes always occur simultaneously.

Chapter 3

Review of hysteresis models

Summary

Hysteresis of water vapour soption isotherms (WVSIs) has been widely recognized as one of main factors which can signicantly aect moisture transport within building material in the natural environmental conditions. In experiments, only a limited number of WVSI loops can be measured. Nevertheless, in natural conditions, concrete structures undergo arbitrary drying and wetting cycles. Thus, using relevant models to predict these curves for cementitious materials is necessary for modelling of moisture transport.

A detailed review of hysteresis models, including for the tting of main sorption isotherms and the prediction of scanning loops, has been carried out in the study [START_REF] Zhang | A review and statistical study of existing hysteresis models for cementitious materials[END_REF], which has been published in Cement and Concrete Research journal. The main content of the current chapter has been included in this paper. Comparison results conclude a best set of equations to calculate hysteresis for cementitious materials. By the verication of experimental data, it shows that for the prediction of the rst scanning isotherms, hysteresis models (either physical or empirical) do not have signicant dierent. But a non-physical behaviour, called pumping eect produced by empirical models is problematic and needs more studies to fully understand its eect on moisture transport.

Introduction

For the modelling of moisture transport in cementitious materials, such as the continuum model proposed in [START_REF] Coussy | Mechanics of porous continua[END_REF], WVSIs, representing the relationship between relative humidity RH (or capillary pressure P c ) and water content θ (or degree of saturation S), are used to describe the equilibrium between liquid-water and vapour. A group of typical WVSIs are illustrated in Fig. 3.1, which contains two main isotherms (adsorption AB and desorption BA) and several scanning curves (the rst and secondary wetting scanning curves CB and EB, and the rst scanning curve in drying DE). For cementitious materials, the characteristics of the hysteretic behaviour might be dierent to soils or sands because cementitious materials contain a wider range of magnitude of pore sizes (from gel pores nm to capillary pores µm) [START_REF] Maekawa | Modelling of Concrete Performance: Hydration, Microstructure Formation, and Mass Transport[END_REF]. Experiments showed that in the low RH range (≤ 33%, according to [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]), hysteresis between the desorption and adsorption isotherms is much smaller than in the high RH range. This results from the nanostructure of C-S-H [START_REF] Feldman | A new model of hydrated portland cement paste as deduced from sorption -length change and mechanical properties[END_REF] which is not found in soils and sands.

Although dierent types of materials have dierent pore systems, it is of interest to test whether hysteresis models from soil science are also applicable to cementitious materials, as there are few studies on the hysteresis behaviour of such materials. Carlier et al. [START_REF] Carlier | Performance evaluation of models describing sorption isotherm in cementitious materials between saturation and oven dryness[END_REF] evaluated several dierent closed-forms of analytical expressions used in soil science for the main desorption curve. The authors concluded that all models were able to correctly represent the desorption isotherms for mortars and concretes. Recently, Johannesson et al. [START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF] and Derluyn et al. [START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF] have used hysteresis models to investigate hysteretic eects on moisture behaviour in cementitious materials. Hence, more work is needed to investigate methods which are able to predict sorption isotherms and their hysteresis for cementitious materials.

History of development of hysteresis models

The rst documented description of the relation P c -θ in porous media was provided by

Haines [START_REF] Haines | Studies in the physical properties of soil. V: The hysteresis eect in capillary properties, and the modes of moisture distribution associated therewiht[END_REF]. Since then, the hysteretic behaviour of this relation was widely recognized and several models had been developed in the purpose of prediction of this behaviour. Those models can roughly be classied into two groups: conceptual and empirical models [START_REF] Pham | A study of hysteresis models for soil-water characteristic curves[END_REF][START_REF] Maqsoud | Hysteresis eects on the water retention curve: A comparison between laboratory results and predictive models[END_REF].

The conceptual models are mainly developed based on the independent or dependent domain theories, which usually employ distribution diagrams to demonstrate the theory. They assume that a domain is made up of groups of pores in a porous medium. Poulovassilis was one of the rst to adopt the independent domain model from magnetism science to the research of water transport in porous media [START_REF] Poulovassilis | Hysteresis of pore water, an application of concept of independent domains[END_REF]. This theory, including Preisach space and Néel's diagram, was developed and improved by Preisach [START_REF] Preisach | Uber die magnetische nachwirkung[END_REF], Néel [START_REF] Néel | Theorie des lois d'aimantation de lord rayleigh[END_REF], Everett [START_REF] Everett | A general approach to hysteresis -Part 3: a formal treatment of the independent domain model of hysteresis[END_REF][START_REF] Everett | A general approach to hysteresis -Part 4: an alternative formulation of the domain model[END_REF], etc. However, it was reported that those models showed discrepancies with experimental data [START_REF] Topp | Hysteretic moisture characteristics and hydraulic conductivities for glass-bead media[END_REF][START_REF] Topp | Soil-water hysteresis: The domain model theory extended to pore interaction conditions[END_REF][START_REF] Poulovassilis | The hysteresis of pore water: the non-independence of domains[END_REF]. Hence, researchers tried to modify and develop new models for porous materials. Based on the similarity hypothesis proposed by Philip [START_REF] Philip | Similarity hypothesis for capillary hysteresis in porous materials[END_REF], Mualem [START_REF] Mualem | Modied approach to capillary hysteresis based on a similarity hypothesis[END_REF][START_REF] Mualem | A conceptual model of hysteresis[END_REF] introduced the simplied independent domain models. The basic idea of Mualem's independent models is that no interactions between pores in the domain are considered. In other words, each pore is independent of its neighbours.

So, the drying or wetting of the pore system is only determined by the pore geometry itself, in which the pore necks and pore bodies are used to characterise the pore geometry. Two normalized variables, r and ρ varying from 0 to 1, are used to represent the normalized radii of pore necks and pore bodies, respectively. When r (or ρ) = 1, it corresponds to the minimum capillary pressure P c min and r (or ρ) is 0 when P c = P c max . At a given capillary pressure, θ can be obtained by integrating the pore water distribution function f (r, ρ). In independent domain models, f (r, ρ)

is formulated by the contribution of two independent pore water distribution functions h(r) and l(ρ), which describe the radii of pore necks and pore bodies, respectively. θ = f (r, ρ)drdρ = l(ρ)dρ h(r)dr and the vertical axis represents the function l(ρ). At equilibrium, capillary pressure throughout the pore network is constant [START_REF] Bertotti | The Science of Hysteresis[END_REF], either for pore necks or in pore bodies. So, h(r) and l(ρ) always change at the same capillary pressure, which is illustrated as the diagonal line in Mualem's diagram (see Fig. 3.2). Hence, the diagram is able to describe the total water distribution (f (r, ρ)) through the product of h(r) and l(ρ). The water content at P c during the wetting process is obtained by integrating the water distribution function from the smallest pore size (at P c max ) to current pore size (at P c ):

θ w (P c ) = ρ(Pc) 0 l(ρ)dρ 1 0 h(r)dr (3.2)
To formulate water content clearly and to avoid using integral forms, two cumulative pore water distribution functions were introduced instead of the integral forms for each axis [START_REF] Mualem | Modied approach to capillary hysteresis based on a similarity hypothesis[END_REF].

L (P c ) = ρ(Pc) 0 l(ρ)dρ H (P c ) = r(Pc) 0 h(r)dr (3.3)
According to Mualem's diagram, L(P c max ) = 0, L(P c min ) = θ max (maximum water content), and H(P c max ) = 0, H(P c min ) = θ max . Hence, Eq. (3.2) can be rewritten as:

θ w (P c ) = θ max L (P c ) (3.4)
The expression of water content during the drying process is obtained in a similar manner by an integration of Mualem's diagram.

θ d (P c ) = ρ(Pc) 0 l(ρ)dρ 1 0 h(r)dr + 1 ρ(Pc) l(ρ)dρ r(Pc) 0 h(r)dr = θ w (P c ) + [θ max -θ w (P c )] H (P c ) (3.5)
From Eq. (3.5), the expression of H is given as follows:

H(P c ) = θ d (P c ) -θ w (P c ) θ max -θ w (P c ) (3.6)
If both main isotherms are known (θ d (P c ) and θ w (P c )), functions H and L can be determined easily by using Eqs. (3.4) and (3.6). Then, they are used to calculate scanning loops.

The assumption of independent domain means that the pore systems are fully-dried or

Θ min Θ max Θ min Θ max Θ min Θ max (b)
The main adsorption process where p d represents the volumetric ratio of the actually dried pores to pores supposed to be dried in the independent domain theory. Similarly, p w is the ratio for the wetting process. Introducing these two weighting functions implies that interactions between pores and their neighbours are considered. This is the basic idea of the dependent domain models.

Through a comparison with experimental data, Mualem and Dagan [START_REF] Mualem | Dependent domain model of capillary hysteresis[END_REF] found that p d played the major role during both drying and wetting processes, while the eect of p w could be neglected (p w = 1). Obviously, p d is the key factor for Mualem's dependent models. Various methods attempting to determine p d have been reported in the literature [START_REF] Mualem | A hysteresis model based on an explicit domain-dependence function[END_REF][START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF][START_REF] Lehmanna | Eect of hysteresis on water ow in a sand column with a uctuating capillary fringe[END_REF].

Empirical models, which have been applied to the modelling of moisture transport in building materials (e.g., [START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF]), are developed based on tting the shapes of experimental WVSIs. Normally, an empirical was proposed without physical meanings. In the literature, there are a number of empirical models, such as linear models (e.g., [START_REF] Hanks | A numerical method for estimating inltration, redistribution, drainage and evaporation of water from soil[END_REF]), slope models (e.g., [START_REF] Jaynes | Comparison of soil-water hysteresis models[END_REF]) and scalingdown models (e.g., [START_REF] Scott | Hysteresis eects on net inltration[END_REF]). Initially, they were developed for one type of porous medium (such as soils and sands) and not based on the physical representation of hysteresis. This thesis has chosen recently developed and easy-to-apply empirical models which can provide reasonable tting results for experimental curves of cementitious materials. Such models usually have one or two parameters which need to be determined from experimental data. For example, Li [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF] proposed an incremental relation to calculate hysteresis, with one parameter controlling the shape of the curve. In the methods introduced by Wei et al. [START_REF] Wei | Formulation of capillary hysteresis with internal state variables[END_REF] and Nyman et al. [START_REF] Nyman | A numerical method for the evaluation of non-linear transient moisture ow in cellulosic materials[END_REF], parameters are used to initialise the slope of the scanning curves.

Adopted comparison strategy

A strategy of comparison of models for WVSIs and hysteresis to experimental data has to be introduced. Statistical analysis tools will be presented in the following sections.

Procedure of comparisons

According to the availability of experimental data, the normal procedure for the calculation of hysteresis is like as follows:

• If experimental data for both main adsorption and desorption branches are measured, a model is needed to t them, and then another model can be used to predict scanning curves;

• If only one main branch is measured, the other one has to be determined by means of a model, and then scanning curves can be predicted by another model.

Based on above procedure, a multi-level approach to model hysteresis is proposed in Table 3.1.

It depends on the amount of available experimental data. This approach may lead to dierent levels of prediction accuracy, which will be revealed through comparisons in next sections. and then predicting scanning loops Table 3.1 indicates that a complete set of hysteresis models consists of three parts: 1 models to t two main sorption isotherms; 2 models to determine one main sorption isotherm on the basis of the other measured main branch; 3 models to predict scanning loops. In next sections, hysteresis models will be compared following these three stages.

Experimental data collection

In order to test the performance of hysteresis models, measured WVSIs were collected from the literature. The requirements for data collection are:

• The collected data contain at least both main desorption and main adsorption isotherms. It is better if some scanning curves have been measured.

• The number of measurement data for each isotherm must be larger than the number of undetermined parameters in models.

WVSIs can be measured by dierent methods. The saturated salt solution method is a common gravimetric one in which RH is controlled by the saturated salt solution in a small volume, such as in a desiccator. The mass of a specimen decreases (during desorption) or increases (during adsorption) until equilibrium (mass stabilization) is reached. Other methods, such as dynamic vapour sorption (DVS) [START_REF] Johannesson | Microstructural changes caused by carbonation of cement mortar[END_REF][START_REF] Åhs | Sorption scanning curves for hardened cementitious materials[END_REF], use a small specimen, usually less than 1 g, and a microbalance to monitor mass changes. The advantage of this kind of instrument is that the entire desorption and adsorption loop can be measured in one month or less, more rapidly than with the saturated salt solution method, for which a method can take many months and even years. However, a potential problem with small specimens is that they cannot contain aggregates.

Measurements probably only quantify the sorption behaviour for the paste in concrete or mortar, rather than the global sorption behaviour of the material. For instance, the inuence of pasteaggregate interfacial transition zone (ITZ) on the sorption is not investigated in this method.

This may lead to dierent measurement results. The crushed materials can also increase the surface area, which let more water molecules can adsorb on the material surface. Another risk is that small specimens can be carbonated quickly during the preparation.

Table 3.2: Collected datasets of WVSIs measured by the saturated salt solution method [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. Besides, the equilibrium state for each RH step was carefully dened as that the mass change is less than 0.001 g after one month. At least three specimens were measured individually to determine one point on the sorption curve for each material.

Comparison criteria

For the tting of the main sorption isotherms, all parameters in the models are optimized to t the experimental curves using a nonlinear least squared optimization procedure, which aims to minimize the sum of the squares of the errors.

Two kinds of statistical criteria have been used to evaluate the accuracy of the modelling results in the literature, the coecient of determination (R 2 ) and a dierential residual-based error metrics (see [START_REF] Pham | A study of hysteresis models for soil-water characteristic curves[END_REF][START_REF] Carlier | Performance evaluation of models describing sorption isotherm in cementitious materials between saturation and oven dryness[END_REF]). R 2 is dened as the ratio between the dispersion predicted by the model and the total dispersion of the measured data [START_REF] Pham | A study of hysteresis models for soil-water characteristic curves[END_REF].

The accuracy of tting (or prediction) is related to the number of parameters used in a model. More parameters usually yields a higher R 2 value, but low robustness and low sensitivity to parameters. To take into account the eect of the number of parameters, this study employs

an adjusted R 2 [145]: R 2 adj = (n -1)R 2 -(m -1) n -m (3.8)
where m is the number of parameters, n is the number of measured data.

Equation (3.8) implies that R 2 adj < R 2 when m > 1, R 2 adj = R 2 when m = 1 and R 2 adj > R 2 when m = 0.
The latter case shows the advantage of using a model without undetermined parameter.

The other statistical criterion is known as the normalized mean error (NME):

NME = 1 n n i=1 |S pr i -S ms i | S ms i (3.9)
where S ms i and S pr i are measured and tted (or predicted) values, respectively. Notice that the absolute value is used in Eq. (3.9). It can avoid the cancellation of positive and negative residuals. If the value of NME is close to 0, it indicates a better result.

Comparisons of main isotherms tting models

Numerous models have been proposed to t the main branches of WVSIs. Each of these equations is applicable to one or several groups of materials (e.g., sands and soils). Models were selected in this research primarily based on whether they can t both desorption and adsorption isotherms for cementitious materials.

On the whole, two types of methods to t WVSIs can achieve our purpose. They are unimodal and multi-modal models. As the name suggests, uni-modal models assume that the material consists of one pore system. Thus, a simple single equation can satisfactorily t experimental data. In the research of Carlier et al [START_REF] Carlier | Performance evaluation of models describing sorption isotherm in cementitious materials between saturation and oven dryness[END_REF], the Kosugi model (K) [START_REF] Kosugi | Lognormal distribution model for unsaturated soil hydraulic properties[END_REF] seems to be the better one for the tting of the main desorption isotherm. Besides, the van Genuchten's model [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF], including two-parameter (VG2) and three-parameter (VG3) versions, will be compared. The former version has been used in studies on concretes [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Poyet | Assessment of the unsaturated water transport properties of an old concrete: Determination of the pore-interaction factor[END_REF]. Two more models, (Feng and Fredlund's model (FF) [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] and Fredlund and Xing's model (FX) [START_REF] Fredlund | Equations for the soil-water characteristic curve[END_REF]) performing well in soil science, will be validated for cementitious materials. Multi-modal models consider that there are two or even more pore systems in the material [START_REF] Durner | Hydraulic conductivity estimation for soils with heterogeneous pore structure[END_REF]. This consideration can improve the tting accuracy. A brief description of each selected model will be presented below.

Feng and Fredlund's model (FF)

The FF model is an empirical relationship which was used to t desorption and adsorption curves of a ceramic material [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF]. A simple equation is used in this model.

S = a F a F + P c b F (3.10)
Two parameters (a F (Pa) and b F ) in Eq. (3.10) need to be determined. The research performed in [START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF] showed that this curve-tting equation is most applicable for low swelling materials (e.g., clay loam).

Van Genuchten's model

The equation proposed by van Genuchten [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF] has been given in Chapter 2 (see Eq. 2.4). Both three-parameter (VG3) and two-parameter (VG2) versions are compared in this chapter.

Fredlund and Xing's model (FX)

When Fredlund and Xing [START_REF] Fredlund | Equations for the soil-water characteristic curve[END_REF] studied the pore size distribution on the basis of the VG2 model, they found that this model was not suitable in the high capillary pressure region. The authors introduced a modied pore size distribution function. Integration of this function yielded a new equation:

S = ln e + P c a X n X -m X (3.11)
where e is the Euler number (a mathematical constant).

Kosugi's model

Using a lognormal pore size distribution, Kosugi [START_REF] Kosugi | Lognormal distribution model for unsaturated soil hydraulic properties[END_REF] deduced the following expression:

S = Q σ p -1 ln P c P cm (3.12)
where P cm (Pa) is capillary pressure related to the medium pore radius. σ p is the standard deviation of log-transformed pore radii which is related to the width of the pore size distribution.

If the pore size distribution is unknown, P cm and σ have to be obtained by tting experimental sorption isotherms. In Eq. (3.12), Q denotes the complementary normal distribution function, which is dened as:

Q(x) = (2π) -1/2 ∞ x exp - x 2 2 dx (3.13)

Multi-modal model

The generalized form of multi-modal models is written as:

S = N i=1 w i S i (P c ) (3.14)
where N is the number of pore systems and w i ∈ [0, 1] ( N i w i = 1) is a weighting factor characterizing the pore system i and w i represents the water content contribution of the pore system i to the total water content in the material. If N = 2, Eq. (3.14) becomes a bi-modal model. One well-known bi-modal model was proposed by Durner [START_REF] Durner | Hydraulic conductivity estimation for soils with heterogeneous pore structure[END_REF], adopting VG2 equation for each pore system.

Comparison results and discussion

All materials in the collected database have been tted by above models. Two statistical criteria R 2 adj and NME for the tting of the two main sorption isotherms are provided in Figs. The bi-modal model [START_REF] Durner | Hydraulic conductivity estimation for soils with heterogeneous pore structure[END_REF] shows better tting results than uni-modal models. For all models, the tting of the adsorption isotherm is better than the tting of the desorption isotherm.

Concrete2 Paste2

Concrete2 Paste2 shows the same shape as the ones described with other models (see Fig. 3.6a). For Paste2, this model performs much better in higher RH range, while the predicted desorption isotherm does not reach S = 0 at RH = 0. This is clearly an non-physical tting result (see Fig. 3.5a). The result is likely caused by the very ne microstructure of the high performance cement pastes (e.g., Paste2) and concretes (e.g., Concrete2). Because those materials can hold much water in the low RH range, the measured points in the low RH range are higher compared to the ordinary materials, such as the point at RH = 12% in Fig. 3.5. This leads the non-physical tting. Thus, it is worth tting WVSIs by using higher modal models.

A comparison of uni-, bi-and tri-modal models is made in Fig. However, the number of parameters in multi-modal models may be several times higher than in uni-modal models. This results in a more dicult tting procedure and can even lead to instability. However, uni-modal models show acceptable tting results for the selected database, except for high performance materials. The choice of WVSI model is thus highly dependent on the complexity of the measured isotherm.

Above, some commonly-used WVSIs models have been compared to measured isotherms.

In many cases, these models do not have a clear physical meaning. The models proposed by

Fredlund and Xing [START_REF] Fredlund | Equations for the soil-water characteristic curve[END_REF], van Genuchten [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF] and Brutsaert [START_REF] Brutsaert | Probability laws for pore-size distribution[END_REF] originally come from Brooks and Corey's power type function [START_REF] Brooks | Hydraulic properties of porous media[END_REF], in which two parameters are involved: the air-entry pressure and a dimensionless parameter referring to the pore-size distribution [START_REF] Brooks | Hydraulic properties of porous media[END_REF]. When van Genuchten developed the closed form equation, the introduced parameter a V was disconnected to the air-entry pressure [START_REF] Ippisch | Validity limits for the van genuchten-mualem model and implications for parameter estimation and numerical simulation[END_REF]; the same is true for a X in the FX model. So, a V and a X should be only seen as air-entry pressure related parameters. In the same way, n V , m V , n X and m X are parameters related to the pore size distribution. Kosugi's model [START_REF] Kosugi | Lognormal distribution model for unsaturated soil hydraulic properties[END_REF] was developed for the soil and two parameters (P cm and σ p ) have to be determined based on the measured pore size distribution. Due to the lack of experimental data of the pore size distribution for cementitious materials, the tted P cm and σ p in this model cannot be veried whether they still retain the same physical meaning as when they were rst introduced. The FF model is a pure empirical equation for the purpose of tting experimental data, without any physical meaning being mentioned.

To conclude the above comparisons, this section recommends that uni-modal models, FX and VG3, can be used as a rst tentative tting. If one does not get satisfying tting results, multi-modal models can be used. In next sections of current chapter, the FX model will be used to t the main sorption isotherms because it shows a slightly better performance to t desorption isotherms than the VG3 model (see Fig. 3.3).

Prediction of one main isotherm from the other main branch

Most hysteresis models require at least both main curves to predict scanning loops. Nonetheless, measuring both desorption and adsorption isotherms for cementitious materials is time consuming and quite costly. In practice, it is still under debate whether the initial fully dried state can change the microstructure during measurements of adsorption isotherms. Instead, the desorption isotherm is easier to determine from an initially saturated state, which is also closer to the condition after the concrete structure formworks are removed. From this point of view, only the models to predict the adsorption isotherm from the desorption isotherm are evaluated herein.

In the literature, some comparisons have been done for sands and soils. Pham et al. [START_REF] Pham | A study of hysteresis models for soil-water characteristic curves[END_REF] compared and ranked ves selected models.

Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] (improved by Pham et al. [START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF]) appeared to be the most accurate one, followed by Mualem Model IV [START_REF] Mualem | Prediction of the soil boundary wetting curve[END_REF].

Meanwhile, in Maqsoud's comparison [START_REF] Maqsoud | Hysteresis eects on the water retention curve: A comparison between laboratory results and predictive models[END_REF], Parlange's model [START_REF] Parlange | Capillary hysteresis and the relationship between drying and wetting curves[END_REF] (modied by Braddock et al. [START_REF] Braddock | Application of a soil water hysteresis model to simple water retention curve[END_REF]) showed the best prediction for silty sand and ne sand.

According to the requirements of experimental data, comparisons will here be carried out in two groups: models only based on the measured main desorption branch and models needing additional data besides the main desorption branch.

Models only based on the measured main desorption branch

Mualem Model II-1 [START_REF] Mualem | Extension of the similarity hypothesis used for modelling the soil water characteristics[END_REF] The Mualem Model II-1 [START_REF] Mualem | Extension of the similarity hypothesis used for modelling the soil water characteristics[END_REF] is the extension of Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF]. In a totally homogeneous porous medium, two pore water distribution functions h and l can be considered identical for pore necks and pore bodies. This leads to h(ρ) = l(ρ) (or h(r) = l(r), see Eq. 3.1). Hence, the same distribution function is applied to both pore necks and pore bodies. The water distribution function (see Eq. 3.1) can be rewritten as:

f (r, ρ) = h(r)h(ρ) (3.15)
The water content during the wetting process (see Eq. 3.4) is rewritten as:

θ w (P c ) = θ max H(P c ) (3.16)
This simplication does not aect the expression for water content during a drying process (see Eq. is derived as:

θ w (P c ) = θ max -[(θ max ) 2 -θ max θ d (P c )] 1/2 (3.17)
The equivalent equation as a function of S appears as:

S w (P c ) = 1 -[1 -S d (P c )] 1/2 (3.18)
Parlange's model [START_REF] Parlange | Capillary hysteresis and the relationship between drying and wetting curves[END_REF], modied by Braddock et al. [START_REF] Braddock | Application of a soil water hysteresis model to simple water retention curve[END_REF] Based on Parlange's hysteresis model [START_REF] Parlange | Capillary hysteresis and the relationship between drying and wetting curves[END_REF], an expression of the adsorption isotherm can be obtained as:

dS w dP c = S w -S d P c (3.19)
Taking VG2 equation to describe the desorption isotherm and integrating Eq. (3.19), an analytical form of the adsorption isotherm was provided by Braddock et al. [START_REF] Braddock | Application of a soil water hysteresis model to simple water retention curve[END_REF].

S w = - P c a dV + 1 + P c a dV n dV 1/n dV (3.20)
where a dV and n dV are tting parameters for the desorption isotherm.

Models needing additional data besides the main desorption branch

Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF], improved by Pham et al. [START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF] In Section 3.4, it showed that there are two coecients in FF model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] (see Eq. 3.10). This indicates that two points on the adsorption isotherm are enough to determine these two parameters and to calculate the entire adsorption isotherm. Based on this idea, Pham et al. [START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF] introduced a method to nd the positions of two such points (denoted by A and B). The position of point A corresponds to a capillary pressure:

P c,wA = a d 10 1/b d (3.21)
where a d and b d are parameters used in FF equation [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] for the desorption isotherm.

Point B on the adsorption isotherm is dened as the point having a capillary pressure which is symmetrical to capillary pressure of point A with respect to a horizontal line passing through point E on the adsorption isotherm (see Fig. S = 0.5:

P c,dC = (a d ) 1/b d (3.23)
The other point D is on the desorption curve and has the same saturation value as point A:

P c,dD = a d S wA - 1
Mualem Model IV [START_REF] Mualem | Prediction of the soil boundary wetting curve[END_REF] The Mualem Model IV [START_REF] Mualem | Prediction of the soil boundary wetting curve[END_REF] is an improved version of Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF][START_REF] Mualem | Extension of the similarity hypothesis used for modelling the soil water characteristics[END_REF]. In addition to the main desorption branch, a wetting scanning cure is needed to predict the main adsorption branch. The prediction is divided into two parts: from the highest capillary pressure to the starting point of the wetting scanning curve, and for capillary pressure lower than the starting point of the wetting scanning curve. The rst part is predicted using the same method as Mualem

Model II-1 [START_REF] Mualem | Extension of the similarity hypothesis used for modelling the soil water characteristics[END_REF]. The second part is calculated by using both the main desorption branch and the wetting scanning curve. Thus, for the main adsorption branch, the saturation can be expressed as follows:

S w (P c ) =      1 -[1 -S d (P c )] 1/2 for P c,1 P c P max c 1 - 1 -S 1,w (P c ) [1 -S d (P c,1 )] 1/2 for P min c P c < P c,1 (3.29)
where P c,1 is capillary pressure at the starting point of the additional wetting scanning curve.

Comparison results and discussion

The prediction results for Paste2 are shown in Figs. 3.8. For the models only using the main desorption curve (Mualem Model II-1 [START_REF] Mualem | Extension of the similarity hypothesis used for modelling the soil water characteristics[END_REF] and Parlange's model [START_REF] Parlange | Capillary hysteresis and the relationship between drying and wetting curves[END_REF] modied by Braddock et al. [START_REF] Braddock | Application of a soil water hysteresis model to simple water retention curve[END_REF]), the standard deviation of the statistical criteria R 2 adj and NME are very large (see Fig. 3.9). Furthermore, R 2 adj values are low and NME values are high. Figure 3.8 also displays that the prediction of these two models do not show good agreements with measured curves. As a result, one can conclude that only using one main curve is not sucient to predict the other main curve in the case of cementitious materials. (b) Parlange's model [START_REF] Parlange | Capillary hysteresis and the relationship between drying and wetting curves[END_REF] (modied by Braddock et al. [START_REF] Braddock | Application of a soil water hysteresis model to simple water retention curve[END_REF]).

Mualem Model

Mualem Model IV

(c) Mualem Model IV [START_REF] Mualem | Prediction of the soil boundary wetting curve[END_REF], RH1 = 12%.

Improved Feng and Fredlund model (d) Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] (improved by Pham et al. [START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF]). With regard to models needing additional data, Fig. 3.9 clearly shows that the improved Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] yields the best prediction with the highest R 2 adj and the lowest NME. This is due to the fact that adding two additional measured points signicantly improves the prediction accuracy. In spite of needing one wetting scanning curve, by contrast, Mualem

Model IV [START_REF] Mualem | Prediction of the soil boundary wetting curve[END_REF] does not provide good performance regardless of the level of RH.

The requirement of additional experimental data limits the application of the improved Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF]. In fact, the positions of points A and B depend on the shape of the IV [START_REF] Mualem | Prediction of the soil boundary wetting curve[END_REF] of the wetting scanning curve starting at RH 1 = 12% (MIV1), RH 1 = 33% (MIV2) and RH 1 = 53% (MIV3). desorption isotherm; thus, they vary according to the studied materials. If these two points are xed independently of the material, it will be more interesting from a practical point of view to determine the adsorption isotherm. The mean RH values for these two points are calculated according to the prediction results, which are around 67% and 86%. They almost correspond to the RH values (63.2% and 90.4%) used in the measurements of the sorption isotherms in [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF].

Thus, these two points can be xed at 63.2% and 90.4%. The main adsorption isotherms can be determined by using Eq. (3.10). The predicted results are compared with other methods in Fig. 3.9. It is clear that the xed points method has an equivalent accuracy as Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] (improved by Pham et al. [START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF]). This implies that if one wants to know the entire adsorption isotherm, it is enough to determine two adsorption points at RH = 63.2% and 90.4%.

The above comparisons reveal that it may be not appropriate to predict one main isotherm by using the other main isotherm only (e.g., Mualem Model II-1 [START_REF] Mualem | Extension of the similarity hypothesis used for modelling the soil water characteristics[END_REF] and Parlange's model [START_REF] Parlange | Capillary hysteresis and the relationship between drying and wetting curves[END_REF] modied by Braddock et al. [START_REF] Braddock | Application of a soil water hysteresis model to simple water retention curve[END_REF]). Two additional measured points on the adsorption isotherm can provide a better prediction (improved Feng and Fredlund model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF]) than using one wetting scanning curve (Mualem Model IV [START_REF] Mualem | Prediction of the soil boundary wetting curve[END_REF]).

Prediction of scanning isotherms

Comparisons of dierent hysteresis models can be found in the literature [START_REF] Jaynes | Comparison of soil-water hysteresis models[END_REF][START_REF] Viaene | A statistical analysis of six hysteresis models for the moisture retention characteristic[END_REF][START_REF] Si | Unied solution for inltration and drainage with hysteresis: Theory and eld test[END_REF][START_REF] Pham | A study of hysteresis models for soil-water characteristic curves[END_REF][START_REF] Izady | Hysteresis: Phenomenon and modeling in soilwater relationship[END_REF] for a large variety of porous media, not including cementitious materials. By statistical analysis, Viaene et al. [START_REF] Viaene | A statistical analysis of six hysteresis models for the moisture retention characteristic[END_REF] concluded that models based on domain theories give the best predictions. A similar conclusion has been drawn in the comparisons performed by Pham et al. [START_REF] Pham | A study of hysteresis models for soil-water characteristic curves[END_REF]. However, besides the domain models, several recently developed empirical models were also selected for comparisons due to their simplicity of application.

Mualem Model II

In Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF], a series of equations to calculate scanning curves have been proposed based on Mualem's diagram (see Fig. 3.2). They use two basic functions H (see Eq. 3.3) and L (see Eq. 3.6), which are determined by two main sorption isotherms (desorption and adsorption).

In practice, expressions to H and L are also written as the function of S (see Eq. 2.13). The rst scanning curve in drying (scanning curve of order N = 1), starting on the adsorption isotherm at the point (S w (P c,1 ), P c,1 ), is formulated according to Mualem's diagram: The rst scanning curve in wetting (order N = 1), starting on the desorption isotherm at the point (S d (P c,1 ), P c,1 ), is calculated by: S 1,w (P c ) = S d (P c,1 ) + [L(P c ) -L(P c,1 )] H(P c,1 )

(3.31)
It can be noted that the expression for the rst drying scanning is dierent to the rst wetting scanning curve. The expression for the wetting scanning curve of order N (odd number and > 1) isotherms (order N = 0). For example in Fig. 3.11, if the third scanning curve in wetting CB continues to decrease after point B, it must do so on curve BD, which is the extension line of the rst scanning curve in wetting AB. This ensures that all scanning loops are closed and enclosed in the main loop.

Θ min Θ max Θ min Θ max Θ min Θ max Θ min Θ max Θ min Θ max Θ min Θ max Θ min Θ max Θ min Θ max
In the same manner, the expression for the drying scanning curve of order N (even number) is written as:

(0)

Mualem dependent model

Based on the assumption of dependent of the neighbouring pores in Mualem Model III [START_REF] Mualem | Dependent domain model of capillary hysteresis[END_REF],

Mualem and Miller [START_REF] Mualem | A hysteresis model based on an explicit domain-dependence function[END_REF] and Mualem [START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF] have provided dierent weighting functions p d to improve the prediction of the scanning curves (see Mualem's diagram for dependent model in Fig. 3.12). The improved version in [START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF] seems to provide a better agreement with experimental data for soils [START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF][START_REF] Lehmanna | Eect of hysteresis on water ow in a sand column with a uctuating capillary fringe[END_REF][START_REF] Izady | Hysteresis: Phenomenon and modeling in soilwater relationship[END_REF]. This version (hereafter called Mualem dependent model), as the same as Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF], only needs both main sorption isotherms. The formula of p d is given by [START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF]: The rst scanning curve in wetting (order N = 1), starting on the main desorption isotherm, can be calculated by:

p d (S) = 1 -S [1 -S w (P + c )] 2

Incremental models

Elastoplastic models have been used in porous media eld to describe the relation between P c and S [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF][START_REF] Wei | Formulation of capillary hysteresis with internal state variables[END_REF]. Usually, a simple incremental relation is used. It is generally assumed that the dierential of P c on the scanning curve is related to the dierential of P c or S on the main isotherm. Two typical models are described below.

(1) Li's model [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF] In this method, the scanning curve is obtained by integrating the incremental relation dP c -dS. A proposition for the incremental form of P c (S) is given in [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF] as follows:

One can notice that in this method the scanning curve always starts with an innite slope (i.e., |dP c /dS| → ∞). (2) Wei's model [START_REF] Wei | Formulation of capillary hysteresis with internal state variables[END_REF] Based on reference [START_REF] Dafalias | Bounding surface plasticity. I: Mathematical foundation and hypoplasticity[END_REF], Wei et al. [START_REF] Wei | Formulation of capillary hysteresis with internal state variables[END_REF] suggested a new and simpler incremental relation, expressed as:

dP c = -K p dS (3.42)
where -K p is the slope of the scanning curve.

According to Wei's theory, when P c changes, the water content in the material is assumed to be additively decomposed into two parts, a reversible part and an irreversible part. The irreversible part of water content is dependent on capillary pressure, while the reversible part is independent. Accordingly, the slope of each scanning curve can be divided into two parts, one is an irreversible part aected by the main isotherm (e.g., the rst term on the right side of Eq. (3.43)), and the other is based on the eect of both reversible and irreversible water content (e.g., the second term on the right side of Eq. 3.43). At current P c , the slope of the drying scanning curve can be calculated by:

K p = - dP c,d (S) dS + c d |P c -P c,d (S)| P c,d (S) -P c,w (S) -|P c -P c,d (S)| (3.43)
For a wetting scanning curve, it is: One can notice that K p depends on the current position of the scanning curve. An iterative method is needed to compute the current capillary pressure.

K p = -

Improved Rubin's empirical hysteresis model

The basic idea for the calculation of the scanning curve is to determine the distance between the scanning curve and the main isotherm which the scanning curve approaches. Here, considering the saturation as an independent variable, this thesis takes the prediction of the drying scanning curve as an example of how to develop an empirical hysteresis model.

As above models show, the distance between the drying scanning curve and the main desorption curve should be a function of the current saturation (S d ) and the starting point of the scanning curve, which is simply formulated as:

P c,d -P c = F (S d , P c,1 , S 1 ) (3.47)
Various F functions can be found in the literature. For instance, Rubin [START_REF] Rubin | Numerical method for analyzing hysteresis-aected, post-inltration redistribution of soil moisture[END_REF], as well as Feng and Fredlund [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF], proposed expressions for the rst drying scanning curve, considering F as proportional to the distance between the two main isotherms. Nevertheless, these two models did not take into account the position of the starting point in their expressions. Thus, these expressions were only valid for the rst scanning curve in drying and not able to simulate scanning loops. Here, a new function F is introduced by adapting the original form of [START_REF] Rubin | Numerical method for analyzing hysteresis-aected, post-inltration redistribution of soil moisture[END_REF].

The exponential relation is used to control how fast the scanning curve approaches the main isotherm. Finally, the following expression is given for the drying scanning curve:

P c (S) = P c,d (S) -[P c,d (S) -P c (S 1 )] exp [γ d (S -S 1 )] (3.48)
The negative value of γ d (S -S 1 ) limits the exponential expression from 1 (at the starting point) to near 0 (almost on the main isotherm). Similarly, the expression for the wetting scanning curve is derived as:

P c (S) = P c,w (S) + [P c (S 1 ) -P c,w (S)] exp [γ w (S 1 -S)]

(3.49)

The constant γ d and γ w are used to determine the shape of the scanning curve and needs to be assessed from experimental data.

Comparison results and analysis

Overall, ve materials, (Concrete1, Concrete2, Concrete3, Paste2 and Paste3, see Table 3. Empirical models reveal that the predicted wetting scanning curves increase slowly at the beginning where the curves are almost parallel to the horizontal axis (see Fig. 3.15), so that these curves can approach the main adsorption branch in a shorter range of RH. This has been observed

by Åhs [START_REF] Åhs | Sorption scanning curves for hardened cementitious materials[END_REF] in experimental results on hardened cementitious materials. As a consequence, the rst wetting scanning curves probably reach the main curve before RH = 100%, as illustrated by modied Wei's model and improved Rubin's model (see Fig. 3.15). The wetting scanning curves predicted by Mualem's models [START_REF] Mualem | A conceptual model of hysteresis[END_REF][START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF] increase more smoothly and reach the main wetting curve at RH close to 100%. It results in a poorer agreement with measured data than empirical models.

Experimental data for the rst drying scanning curves are only available for Paste2 and Paste3 materials in our database [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. Predictions from Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF] show that curves decrease with a lower slope and turn towards the desorption isotherm more gently than curves predicted by empirical models. Therefore, empirical models yield predictions with a lower accuracy compared The two criteria R 2 adj and NME for the prediction of the rst wetting scanning curves are shown in Fig. 3.16. They illustrate that empirical models can give higher R 2 adj and lower NME values than the Mualem Model II. This is denitely due to that the two additional parameters in empirical models can be adjusted to t experimental data, while Mualem's models essentially depend on the shape of the two main sorption isotherms. In contrast, in the prediction of the rst drying scanning curves for Paste3, Mualem Model II reveals a better prediction than the empirical models.

Figure 3.16 also shows that all empirical models have almost equal accuracy for the estimation of the rst scanning curves, either wetting or drying. None of the selected hysteresis models is considerably better than the others. This conclusion is similar to that reported by Jaynes [START_REF] Jaynes | Comparison of soil-water hysteresis models[END_REF] when he compared hysteresis models for soils. Hence, it needs the comparison of higher order scanning curves.

M-II

Predictions of scanning loops

Scanning loops have been measured for Paste2 and Paste3 [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. Here three measured loops, one for Paste2 and two for Paste3, have been chosen to verify the applicabilities of the selected hysteresis models. One of the prediction result is shown in Fig. 3.17 (other comparison results are included in the document supplementary materials).

Because of the poor prediction of the rst wetting scanning curves, Mualem dependent model [START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF] do not show better prediction results than Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF]. Among empirical models, Li's model [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF] appears to be the best one to predict the shape of the measured scanning loops. Besides, the improved Rubin's model [START_REF] Rubin | Numerical method for analyzing hysteresis-aected, post-inltration redistribution of soil moisture[END_REF] shows a better prediction of secondary drying scanning curves than the modied Wei's model [START_REF] Wei | Formulation of capillary hysteresis with internal state variables[END_REF] (see Fig. 3.17).

Analysis of the pumping eect

One hysteretic behaviour, called the pumping eect, should be noticed for empirical models. It refers to the non-closure of scanning loops when the secondary and higher order scanning curves are calculated. Researchers consider that pumping eect is an artifact of the algorithm and does not correspond to an actual material property [START_REF] Huang | A novel hysteresis model in unsaturated soil[END_REF][START_REF] Werner | Articial pumping errors in the kool-parker scaling model of soil moisture hysteresis[END_REF]. Thus, this behaviour is also called the pumping errors (PEs), which is dened and illustrated in Fig. 3.18. Point A is the starting (a) Mualem's models [START_REF] Mualem | A conceptual model of hysteresis[END_REF][START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF]. (symbols) from [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. Simulated RH cycle: 97% → 53% → 99%. point of the scanning loop. Point B is on the secondary scanning curve in drying at the same RH as point A. The dierence of S between A and B is dened as the drying pumping error (DPE). Similarly, a wetting pumping error (WPE), originating from wetting scanning curves (dierence between point C and point D in Fig. 3.18), is also observed for empirical models. The evolution of PEs is not only dependent upon the initial state but also upon the magnitude of RH oscillations. Thus, to assess the evolution of DPE and WPE, two initial states of RH cycles are tested: an initial drying state and an initial wetting state. Paste3 material is taken as an example because many measured scanning curves are available for this material, including the rst and secondary drying curves and the rst wetting scanning curves. Two RH oscillations have been chosen for each initial state. The rst one has been provided by experiments, with RH changing between 63% and 97% (see Figs 3.19a and 3.19c). The second one reduces the RH dierence to 20%, such as for an RH oscillating between 63% and 83% (see Figs 3.19b and 3.19d). constant value after about 2 cycles for large RH oscillations and about 5 cycles for small RH oscillations. Li's [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF] and improved Rubin's models [START_REF] Rubin | Numerical method for analyzing hysteresis-aected, post-inltration redistribution of soil moisture[END_REF] provide similar PEs (less than 0.1) for large RH oscillations, while improved Rubin's model yields larger PEs (around 0.3) than Li's model (around 0.2) for small RH oscillations. It is also clear that WPEs are larger than DPEs in both empirical models for the same kind of RH oscillations.

DPE

For large RH oscillations, PEs of empirical models are acceptable. Nevertheless, for small RH oscillations, if the model fails to eliminate PEs, the cumulative errors associated with oscillations of hysteresis loops are probably signicant and lead to unrealistically simulated results. Some works have been done to avoid PEs when a hysteresis model was developed. By assuming that scanning curves have the same shape as the main branches, Parker and Lenhard [START_REF] Parker | A model for hysteretic constitutive relations governing multiphase ow: 1. Saturation-pressure relations[END_REF] and Huang et al. [START_REF] Huang | A novel hysteresis model in unsaturated soil[END_REF] enforced closure of scanning loops to eliminate pumping errors. The problem of these methods is that they need to rescale and reformulate each scanning curve if the direction changes, and this may be inecient for modelling of frequent drying and wetting cycles.

Discussion

Compared with empirical models, Mualem's models (Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF] and dependent model [START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF])

have the advantage to avoid PEs, even though their predictions are not as good as empirical

Prediction of scanning isotherms

Starting point

Starting point

(a) Li's model [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF] for the 1st RH oscillation (a) Li's model [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF]. models. They may be more applicable for smaller RH oscillations comparing with empirical models. Another advantage of Mualem's models is that they need only the two main sorption isotherms to predict scanning loops, unlike empirical models that require at least two scanning curves to determine parameters.

Notably, from a practical point of view, equations of Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF], Li's model [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF] and improved Rubin's model are expressed in explicit forms which are easier to implement in a numerical modelling of moisture transport. In particular, the improved Rubin's model uses very simple equations and provides an acceptable prediction. The modied Wei's model [START_REF] Wei | Formulation of capillary hysteresis with internal state variables[END_REF] and Mualem dependent model [START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF] correspond to implicit formulas which signicantly increase the diculty of application. A drawback for Li's model [START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF] is that the scanning curve always starts in a horizontal line, which leads to the moisture capacity C = ∂S ∂Pc = 0 for the beginning point.

It will denitely numerical problem when this model is implemented into a moisture transport model.

To conclude the above analysis about hysteresis models, this thesis recommends Mualem

Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF] and improved Rubin's models to be used for moisture transport modelling within cementitious materials exposed to cyclic boundary conditions. The empirical one (improved Rubin's models) is more suitable for large RH oscillations, in which pumping eect can be reduced.

3.7. Models for the proposed multi-level approach to simulate hysteresis 3.7 Models for the proposed multi-level approach to simulate hysteresis Based on experimental verications, models for the tting of main isotherms, the prediction of one main curve from the other one and the simulation of scanning loops have been evaluated.

However, from a practical point of view, it needs to organise those models in the framework of a multi-level approach capable of providing dierent possibilities to predict hysteresis according to the available data and the level of sophistication which is required. Dierent combinations of those models for this multi-level approach are provided in Table 3.3. 3.8 A method to t both main curves by using the fewest measured data

Since measuring sorption isotherms is quite time-consuming when using the saturated salt solution method. The DVS measurement can signicantly save time, but no every researcher is able to aord the expensive apparatus. Thus, it could be interesting to nd a way requiring the fewest measured data to predict both main desorption and main adsorption curves. The similar idea can be found in the literature, such as in [START_REF] Carmeliet | Determination of the moisture capacity of porous building materials[END_REF], while only the main desorption was considered.

Another main advantage of the proposed method in current study is that it is developed based on data measured by the saturated salt solution method, and will be applied for DVS data (see Chapter 4). Section 3.5 showed that improved Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF][START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF] needs two measured points on the main adsorption curve and the entire main desorption curve to predict the main adsorption curve. These two points are not random and have been xed at RH = 63.2% and 90.4%

in the current study. However, comparisons in Section 3.4 revealed that the FF model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF], which is used in improved Feng and Fredlund's model, is not the best model to t the main sorption curves. This may result in improved Feng and Fredlund's model less accurate. As Section 3.4

suggested, FX or VG3 models [START_REF] Fredlund | Equations for the soil-water characteristic curve[END_REF][START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF] can provide better tting than FF. Hence, it would be necessary to use FX or VG3 instead of FF to improve the prediction accuracy.

The above discussion inspires us to think about a more interesting question: can the number of experimental data for the tting of the main sorption curves be reduced? In particular from a practical point of view, it is worth nding a method which needs the fewest experimental data to predict both main desorption and adsorption isotherms. In the literature, there is a lack of research trying to estimate how many experimental data are required for the tting of the main sorption isotherms.

This attempt indicates that two steps need to be done: determine the number of points needed to predict each main curve and nd the most relevant positions of those points.

A proposed new statistical criterion

Statistical criteria R 2 , R 2 adj and NME were introduced in Section 3.3. In this section, only three- parameter models are chosen so there is no need of R 2 adj . R 2 will be used for the comparison purpose. In fact, NME yields the same results with another criterion, so-called the mean multiplicative error (MME). The only dierence between these two criteria is that the value of NME is in the range of [0, +∞], but MME is in the range of [1, +∞]. If plotting MME and R 2 in a same gure, it will allow a synthetic representation of the relevancy of the model in a unique gure. Thus, comparison will be easier and clearer. The equation of MME is expressed as [START_REF] Moog | Analysis of reaeration equations using mean muliplictive error[END_REF]:

MME = exp n i=1 | ln(S pr i /S ms i )| n (3.50)
Notice that the above equation uses the absolute value which is able to avoid the cancellation of positive and negative residuals. If the value of MME is close to 1, it indicates a good prediction.

The above description implies that for a high accuracy tting, both values of R 2 and MME should be close to 1 (R 2 from a low value and MME from a high value). In a gure including both MME and R 2 , one only needs to compare the distance between R 2 and MME to assess models. If these two values are close, it implies a good tting result.

Determination of the number of points for the tting

Adopting a three-parameter model to t both main sorption curves means totally six undetermined parameters (three for each main curve). In fact, an alternative approach can reduce the number of undetermined parameters. The parameter a X in Eq. (3.11) (or a V in Eq. 2.4) is an air-entry pressure-related parameter. For cementitious materials, the air-entry pressure is very low and is negligible. Thus, the rst simplication for the tting is proposed: a X,w ( or a V,w ) for wetting is assumed to be equal to a X,d ( or a V,d ) for drying.

a X,w = a X,d ; a V,w = a V,d (3.51)
To use the rst simplication, the main desorption curve should be tted rstly by using all measured data, and then a X,d ( or a V,d ) is used to determine two other undetermined parameters for the main adsorption curve by tting measured data. Thus, to check the tting accuracy, one only needs to compare the main adsorption curves tted by using the rst simplication and by using the original method (with three undetermined parameters). Comparisons of tted curves for Paste2, Paste3, Concrete2 and Concrete3 are presented in Fig. 3.22 by using the FX equation. Good agreement between the rst simplication and the original method is observed in this gure.

According to the rst simplication, the number of undetermined parameters for the tting of both main sorption curves is reduced to ve: three for the main desorption and two for the main adsorption isotherms. This may indicate that much less data are able to t two main sorption curves. Therefore, the second simplication is proposed here: ve points are enough to predict both main sorption isotherms (three points on the main desorption curve and two points on the main adsorption curve). Mathematically, this simplication is no problem, but it needs to carefully examine the accuracy of the tting. The priority for the second simplication is to determine the positions for these ve points.

Determination of the relevant positions of those ve points

A simple way is used to determine the positions of those points. Firstly, a model (FX or VG3) is used to t all the collected materials based on the rst simplication, and then the mean residual (MR) at each measured point can be calculated.

MR = 1 M M i=1 |S pr i -S ms i | (3.52)
where M is the number of materials. In this study, eight materials from the literature [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] are used so M = 8 (see Table 3.2).

The MR results for FX and VG3 models at each measured point are presented in Fig.

Relevant three points for the main desorption curve

MR results from FX model show that three most relevant points for the desorption isotherm are at RH = 22%, 44% and 71%, which correspond to the low, medium and high RH range. However, three points for VG3 are at RH = 22%, 71% and 80%, which gives two points in the high RH range. Actually, the MR value at RH = 44% is no signicant dierent to RH = 80%.

Hence, three points on the desorption isotherm are determined at RH = 22%, 44% and 71% for the purpose of getting the same positions for two models.

Relevant two points for the main adsorption curve

Results for the most relevant two points are at RH = 44 and 80% for the adsorption isotherm.

FX and VG3 models reveal a high consistency for these two points.

Fitting results and comparisons

In the previous study on Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] (see Section 3.5.2), two points on the main adsorption curve were xed at RH = 63% and 90%. In this section, this method is also compared with the newly proposed method. Firstly, let's focus on the results calculated by the rst simplication. Bearing in mind that this simplication only aects the tting of the main adsorption curve. Figure 3.22 shows that adsorption curves tted by using the rst simplication are nearly identical to curves tted by the original method. Then, let's check the applicability of the second simplication.

For most materials, this simplication works quite well for both main desorption and adsorption curves. However, for Paste2, it shows clear disagreement with the original model for the tting of the main desorption curve. It is mainly due to the shape of measured desorption isotherm. For Paste2, the desorption curve remains at high water content in the range of RH > 63%. But when RH < 63%, the curve drops more rapidly than other materials.

Values of the new criterion in Figs. 3.23 and 3.24 clearly show that the second simplication has the equivalent accuracy as the original method for both FX and VG3 equations when they are used to t the main desorption curve. Only Paste2 shows the poor performance for the second simplication (see the highlighted diamonds in Figs. 3.23 and 3.24). For the tting of the main adsorption curves, the distance between mean MME and mean R 2 increases from the original method to the second simplication. Nonetheless, the mean value of MME for the second simplication remains around the same order as that in the prediction of desorption isotherm. Thus, the slight dierence between the original method and the second simplication is acceptable.

Lastly, let's focus on the prediction accuracy of improved Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF][START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF] whose MME and R 2 are presented in Figs. [START_REF] Mehta | Concrete -Microstrucure, Properties and Materials[END_REF].23b and 3.24b. Obviously, this model provides a high mean value of MME, meaning that it performs poorer than the two simplications which were proposed in this section. It is concluded that two points which were previously suggested for Feng and Fredlund's model [START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] cannot be used for FX or VG3.

Discussion on the proposed approach

A method to t the main desorption and adsorption curves by using ve points has been proposed.

There are some details which need to be discussed.

Comments on the necessary number of points

Results from the second assumption reveal that this assumption is valid for most materials, except Paste2. This may imply that more points are needed for this kind of material. Paste2 has more C-S-H pores which mainly control the shape of sorption isotherm in the low RH range.

Thus, one more measured data is added in this range for the main desorption curve. Here, the point with lowest RH in the measured data is chosen (at RH = 12%). Therefore, two points (RH = 12% and 22%) in the low RH range are used for the tting. Results tted by FX and VG3 models are shown in Fig. 3.25.

Notice that after adding one point for the main desorption curve, it will aect the value of a X,d ( or a V,d ). This may have an indirect inuence on the tting of the main adsorption curve.

But it still needs two points for the main adsorption curve. Obviously, Fig. 3.25 show that one more point can signicantly improve the prediction accuracy. The calculated curves (either for main desorption or main adsorption) show extremely high agreements with the original method.

Hence, it requires four points to predict the main desorption curve for Paste2.

Comments on the relevant positions of points

Three points on the main desorption curve can be considered as three parts: low RH range (represented by RH = 22%), medium RH range (represented by RH = 44%) and high RH range (represented by RH = 71%). In the low RH range, moisture transport is dominated by C-S-H pores, while in the high RH range it is controlled by capillary pores. The medium range shows a mixed inuence from both C-S-H and capillary pores. Those positions may vary according to dierent methods to measure WVSIs (the saturated salt solution, DVS, etc.). However, these three parts of a desorption isotherm should be always remained.

The point at RH = 44%, which is a special point that has been emphasised in the literature [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF], is used for the tting of both main desorption and main adsorption curves. It was reported that almost all the studied materials have the same shape of adsorption isotherms when RH<44% [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]. It can roughly be considered as the threshold value which is able to separate C-S-H 3.9. Conclusion and recommendations and capillary porosities.

Conclusion and recommendations

Detailed comparisons of hysteresis models for cementitious materials were performed in the current chapter. Two quantitative criteria, R 2 adj and MME, were used to quantitatively evaluate comparisons. A new way to t the main desorption and main adsorption curves by using the fewest experimental data was proposed. A number of conclusions about hysteresis models are drawn from this study.

1) Although the cementitious materials have quite dierent pore structure compared to soils, sands and rocks, most models from these elds can be used to t the main curves and to predict hysteresis loops for cementitious materials.

2) Among the selected ve main isotherm tting equations, the three-parameter tting models provided better results than the two-parameter models. The equation proposed by Fredlund and Xing [START_REF] Fredlund | Equations for the soil-water characteristic curve[END_REF] provided the best tting results. It is able to t the shape of WVSIs very well and to capture the physical relationship between the desorption and adsorption isotherms.

3) There is no relevant model which is able to infer one main isotherm based on the other main isotherm.

By using more experimental data, one wetting scanning isotherm for Mualem [START_REF] Mualem | Prediction of the soil boundary wetting curve[END_REF] and two uncertain points on the adsorption isotherm for improved Fend and Fredlund's model by Pham et al. [START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF], they show a high accuracy prediction. But these requirements restrict the use of these two models.

4) Concerning hysteresis models, empirical ones have a slight better prediction for the rst scanning isotherms than the domain theory models. For multi-loops hysteresis simulations, empirical models also showed the simplicity of use. However, the non-physical behaviour, pumping eect, is a controversial issue for empirical hysteresis models.

5) The domain theory models is able to avoid the pumping eect autogenously due to the assumption used in the diagram such as Neel diagram [START_REF] Néel | Theorie des lois d'aimantation de lord rayleigh[END_REF] and Mualem diagram [53]. This is one of main advantages to use conceptual models. In this chapter, the recommended models are Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF] and improved Rubin's model [START_REF] Rubin | Numerical method for analyzing hysteresis-aected, post-inltration redistribution of soil moisture[END_REF][START_REF] Zhang | A review and statistical study of existing hysteresis models for cementitious materials[END_REF] for further use.

6)

To predict both desorption and adsorption isotherms, for ordinary materials, suggested ve points (three for the main desorption isotherm and two for the main adsorption isotherm) can provide the same tting accuracy as using the original model. For some high performance materials (e.g., Paste2), the use of one more point at low RH is able to greatly improve the tting accuracy.

7) The positions of those points were veried by collected experimental data. Finally, those points are given at RH = 22%, 44% and 71% for the main desorption curve and at RH = 44% and 80% for the main adsorption curve. One more point for the main desorption curve for Paste2 is chosen at RH = 12%.

In this chapter, although a complete set of hysteresis models was concluded, it still needs more experimental data to rene these conclusions. Because only experimental data measured by the saturation salt solution were chosen in the current chapter, one main task for the next chapter is to carry out measurements of sorption curves by using DVS method.

Materials preparation

In previous studies led by [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF] and Nguyen [START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF], many experiments for various mixtures of cementitious materials have been done. In this study, cements and mixture designs are dierent from experiments mentioned above. Three water-to-cement (or binder) ratios (w/c or w/b) were used for all binders. Materials and their porosities are presented in The procedure of preparing paste materials followed European standard EN 196-1. The pastes were cast in cylindrical plastic bottles with 7 cm-diameter and 14 cm-height. After the casting, these bottles were kept at room temperature for curing (23 • C).

Drying-wetting experiments

During drying-wetting experiments, materials were subjected to cyclic RH at the boundary. In order to quantitatively measure mass loss and saturation proles, the geometry of samples has to be properly dened. After more than four months of sealed curing, the cylindrical bottles were demoulded and cut into samples of 10 cm-length (7 cm-diameter).

a Measured by weighing.

Experimental procedure

The drying-wetting experiments started with the saturated state because it is the easiest way to be sure that the materials have uniform water content. The saturation method corresponds to the vacuum saturation. The procedure to conduct the vacuum saturation has already been used in the literature [START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF][START_REF] Wang | Modélisation du transport multi-espèces dans les matériaux cimentaires saturés ou non saturés et éventuellement carbonatés[END_REF][START_REF] Saillio | Interactions physiques et chim iques ions -matrice dans les bétons sains et carbonatés. Inuence sur le transport ionique[END_REF][START_REF] Morandeau | Carbonatation atmosphérique des systémes cimentaires à faible teneur en portlandite[END_REF]. After the vacuum saturation, all sides of the sample were tightly sealed by at least double layers of adhesive aluminium sheet, except one face which is exposed to the ambient environment (see Fig. 2

Materials

RH cycles (%)

All Pastes

Saturation→33.0 →80.0→53.5

The weight of samples was regularly measured. Meanwhile, the gamma-ray attenuation measurements were conducted to provide data for calculation of porosity and saturation proles.

It was carried out monthly or when RH was switched to next step. The description of gamma-ray attenuation technique can be found in references [START_REF] Villain | Mesure de la perméabilité aux gaz en fonction du taux de saturation des bétons[END_REF][START_REF] Villain | Gammadensimetry: A method to determine drying and carbonation proles in concrete[END_REF]. This technique has been widely-used in studies on cementitious materials, especially at IFSTTAR (e.g., [START_REF] Thiéry | Modélisation de la carbonatation atmosphérique des matériaux cimentaires avec prise en compte des eets cinétiques et des modications microstructurales et hydriques[END_REF][START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF][START_REF] Wang | Modélisation du transport multi-espèces dans les matériaux cimentaires saturés ou non saturés et éventuellement carbonatés[END_REF][START_REF] Saillio | Interactions physiques et chim iques ions -matrice dans les bétons sains et carbonatés. Inuence sur le transport ionique[END_REF][START_REF] Morandeau | Carbonatation atmosphérique des systémes cimentaires à faible teneur en portlandite[END_REF]).

The last step of the experiment was to dry the sample at 105

• C to get the dried mass in order to determine porosity and liquid-water saturation.

Porosity proles

Due to the absorption of gamma-ray photons by the material, the intensity of gamma rays suers a loss of intensity while passing through the material. The amount of lost radiation depends on the type of the radiation source, the density of the material and the traversed thickness. The relationship between the intensity of incident and transmitted photons is given by Beer-Lamber's law [START_REF]Guidebook on non-destructive testing of concrete structures[END_REF],

I = I 0 exp(-µρ m d) (4.1)
where I 0 is the incident photon intensity (the number of photons measured in the air). I is the transmitted photon intensity (the amount of photons that passes through the material with thickness d (m)). ρ m (kg • m -3 ) is the material density and µ (m 2 • s -1 ) is the mass absorption coecient of the material (also called attenuation coecient). If parameters µ, d, I and I 0 are known, the material density ρ m at the measured point can be calculated.

φ = ρ sat -ρ 105 ρ l = 1 ρ l µ l d ln I sat 0 I 105 I sat I 105 0 (4.2)
where superscripts sat and 105 represent measured data at the saturated state and the dried state, respectively. Another thing for all curves is that error bars at the surface are much larger than them inside materials. This is clearer for CEM I pastes (all three w/c). The possible reason is that the latent hydration process was still going on inside materials, but hydration for the part near the surface has been already stopped due to less water in this part when the material was subjected to low RH for drying.

The global porosity for the sample is approximately calculated through the known porosity at each depth by gamma-ray attenuation.

φ γ = 1 l φ i ∆x i (4.3)
where l is the length of the sample and ∆x i is the length associated with the ith measured depth.

Besides, when the sample was vacuum saturated, the global porosity is also determined by the weight of the sample in the air, in the water and after dried at 105 • C. 3a and4.3b show that some measured data are slightly higher than 1. This is due to the problem that materials were not fully saturated when the experiments started. For PSF05, this mistake is much clearer than PSF06.

Data consistency analysis

Actually, the total mass loss at the dierent times can be calculated based on the saturation proles measured by the gamma-ray attenuation method.

m loss = m loss,i = (S 0,i -S i )∆x i φ i Aρ l (4.5)
where S 0,i and S i represent the initial saturation and measured saturation at each depth, respectively. φ i is the porosity of the ith depth. A is the area of the cross section. As mentioned before, the mass loss curve has been measured by weighing and therefore data consistency between the global weighing and the integration of gamma-ray attenuation data can be obtained. In other words, two mass loss curves measured by two methods should match. Good agreements between these curves for PSF05 and PSF06 are shown in Fig. 4.4. It proves the applicability of using these two methods to measure mass changes of cementitious materials.

Results of drying and wetting experiments

Two sets of experimental results, including mass loss curves and saturation proles, have been shown in Figs. 4.4 and 4.3 for PSF05 and PSF06.

Even if RH uctuates at the boundary from drying to wetting, and then to re-drying, saturation proles inside the material constantly decrease. The main reason is that RH at the boundary is smaller compared to RH in the inner part; therefore, moisture is always moving to the surface part. However, because the boundary RH (80%) during the second step is higher than that during the rst (33%) step, moisture is able to transport from the surrounding to the surface region of the material. Owing to eects of these two opposite transports, mass loss curves are quite at during the period at RH e = 80%.

Another reason is that it needs time to show eects of RH changes at the boundary on internal moisture transport due to the very slow moisture transport in the material. Longer experiments would be able to show more obvious dierences between dierent measured saturation proles.

Sorption isotherms measurements

As mentioned in Chapter 3 (see Section 3.3.2), both saturated salt solution method and DVS are able to measure WVSIs. Compared to the saturated salt solution method, DVS uses tiny specimens, usually less than 1 g, so that it takes much shorter time to obtain a sorption curve.

Besides, this research only uses cement pastes and therefore no eect of aggregates needs to be considered. Hence, the choice of a DVS instrument to measure WVSIs should be acceptable.

However, DVS is only valid in the hygroscopic range, up to around RH = 95%. For higher RH, such as above 97%, water content was determined by another method called pressure plate.

DVS method

DVS is a relatively new technique for water vapour sorption analysis in the eld of cementitious materials. Three main aims that a DVS apparatus needs to achieve are: constant temperature, stable RH and accurate mass weighing [START_REF] System | DVS Advantage operation manual[END_REF][START_REF] Janz | Methods of measuring the moisture diusivity at high moisture levels[END_REF][START_REF] Johannesson | Verication of the bet-theory by experimental investigations on the heat of adsorption[END_REF][START_REF] Janz | Measurement of the moisture storage capacity using sorption balance and pressure extractors[END_REF][START_REF] Anderberg | Moisture properties of self-levelling ooring compounds[END_REF][START_REF] Arlabosse | Comparison between static and dynamic methods for sorption isotherm measurements[END_REF][START_REF] Wadsö | An experimentally simple method for measuring sorption isotherms[END_REF][START_REF] Johansson | Water absorption in two-layer masonry systems -Properties, proles and predictions[END_REF]. The equipment is placed in an incubator, permitting the temperature to be kept constant. The required RH is created by mixing two gas ows with known RH: one is a completely dry nitrogen gas and the other with 100% RH. Each ow is controlled by thermal conductivity mass ow controllers. The relative humidity of the mixed gas ow could be varied between 0 and 98%. Nitrogen gas can also prevent the further carbonation during measurements.

The other main component of a DVS system is the micobalance, which continuously weighs the mass of the sample as a function of time. The microbalance is protected against high relative humidity by a continuous ow of dried nitrogen gas.

Procedure of DVS measurements

Before measurements, the large sample was vacuum saturated, and then crushed into small pieces. Usually, 30-70 mg materials (pieces mixed with powders) were randomly chosen and put in the sample pan. All measurements were performed at 23 ± 0.1 • C.

A full cycle of measurements starts with desorption at RH = 95% and continues stepwise to a low RH = 30%. After equilibrium, the inverted process was performed, until RH = 95%, and then the direction would be changed again until the completely dried gas ow (RH = 0%, see Fig. 4.5). As mentioned above, a maximum RH approximately 98% can be used. However, RH = 95% was chosen since the time needed to reach equilibrium at a higher RH is estimated that the balance was insuciently stable to allow use of the relative mass dierence in time as an equilibrium criterion [START_REF] Johansson | Water absorption in two-layer masonry systems -Properties, proles and predictions[END_REF]. The swing of the sample because of the gas ow made the measurement dicult to reach an equilibrium state. Therefore, a relative mass dierence criteria (dm/dt = 0.0001 %/min) coupled with a xed maximum duration (2000 minutes) was used for each RH step.

Extrapolation of the mass loss curve

Sometimes, the mass loss curve for an RH step may not reach equilibrium. This could make the measured total relative mass change less than the realistic one. The curve extrapolation is needed to improve the measured results. Drying of the porous media, an exponential decay function is usually used to describe the mass loss as a function of time [START_REF] Åhs | Sorption scanning curves for hardened cementitious materials[END_REF]:

m(t) -m 0 m f -m 0 = 1 -exp(-pt) (4.6)
where m(t) is the measured mass of the sample, m 0 and m f are the initial mass and the nal mass, respectively. p is a parameter. m f and p are needed to be tted by the measured curve.

The nal mass m f used to improve the measured results. Weibull [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF] has proposed a cumulative distribution function which has been used to predict the mass loss of drying food [START_REF] Díaz | Modelling of dehydrationrehydration of orange slices in combined microwave/air drying[END_REF][START_REF] Dadali | Eect of drying conditions on rehydration kinetics of microwave dried spinach[END_REF][START_REF] Doymaz | Evaluation of some thin-layer drying models of persimmon slices (diospyros kaki l.)[END_REF]. The advantage of this method is that it is able to provide very good tting results for drying of irregular shapes of materials like crushed cement pastes. This distribution function is written as:

m(t) -m 0 m f -m 0 = 1 -exp -(t/a W ) b W (4.7)
where parameters a W and b W , and the nal mass m f are adjusted to obtain good tting with the measured curve. One result in Fig. 4.6 shows that Eq. (4.7) can provide very good tting with measured mass loss curve. equilibrium is not easy to be reached due to unstable moisture transport. This may indicate that in this RH range more strict equilibrium criteria should be used during a DVS measurement.

Sorption isotherms in high RH -pressure plate measurements

The sorption isotherm above the hygroscopic range is not only related to the sorption of water vapour, but also the suction of liquid-water. This part of curve shows the relationship between moisture content and pore water pressure. The simplest way to determine the curve in this part is to expose the material to a negative pressure -P , which is equal to the macroscopic capillary pressure P c within the material at equilibrium. Therefore, liquid-water in the material can move under external pressure. This method has been widely used for more porous materials such as soils. By using Kelvin's equation, the relationship between this pressure and relative humidity can be established.

RH = exp - M v RT ρ l P (4.8)
Pores in cementitious materials are smaller than pores in soils, so the external pressure for cementitious materials is higher. The traditional measurement methods, which are only for smaller pressure, cannot be used for cementitious materials. This makes researchers to develop a new apparatus, called pressure plate (see Fig. 4.8) [START_REF] Johansson | Water absorption in two-layer masonry systems -Properties, proles and predictions[END_REF].

The detailed description of the theory and the measurement procedure of pressure plate have been reported in [START_REF] Johansson | Water absorption in two-layer masonry systems -Properties, proles and predictions[END_REF]. Here, only several important points are mentioned. Specimens for cement pastes are prepared as very thin slices, between 3-4 mm, and the surface size is about 3 mm × 3 mm. Before the tests, all specimens were vacuum saturated.

(a) Schematic sketch of the pressure plate extractors for pressure 1 MPa [START_REF] Janz | Methods of measuring the moisture diusivity at high moisture levels[END_REF].

(b) Deployed specimens in the pressure plate extractor for pressure 3 MPa. For the tests, two dierent pressures were used: 1 MPa and 3 MPa, corresponding to RH = 99.26 and 97.81%, respectively. Tests for two pressures were run in two separate pressure plates.

The schematic design of the pressure plate is depicted in Fig. 4.8a. For the low pressure test (1 MPa), to ensure hydraulic contact between the ceramic plate and the specimen, a thin layer of kaolin clay was applied to the ceramic plate. The thickness of kaolin clay is around 2 mm. Above the kaolin, a nylon cloth was put to prevent the specimen from being contaminated by direct contact with clay. For the high pressure tests (3 MPa), a cellulose membrane with considerably smaller pores was used instead of the ceramic plate to avoid that the high pressure may break the ceramic plate. A thin disc-shaped metal grid was located under and kaolin clay the cellulose membrane to allow water ow through the cellulose membrane. On the bottom, a Plexiglas plate was placed to provide a smoother and at bed for specimens.

After the metal lid was sealed, the gas was let in to provide a certain pressure. Nitrogen was used instead of air to prevent further carbonation. At the beginning, the gas had to be provided very gently to avoid the displacement of specimens. Figure 4.8b shows that the suddenly increased gas pressure can induce high speed gas ow to blow the specimens away. After the pressure was applied, the water for the system owed out through the outow tube. The extracted water was continuously measured by a pipette. The equilibrium is dened when the water level in the pipette was constant for 48 hours. At this point, it is considered that the pressure in the plate is equal to the macroscopic capillary pressure within the specimens.

The measurement for each pressure lasted around 8 weeks. The mass of specimens was measured after the pressure plate opened. Those specimens were then dried at 105

• C in an oven to get dried mass. Thus, water content of specimens for measurements at each pressure was determined.

Sorption isotherm measurements results

An example of the measured results for CEM I materials with three dierent w/c is illustrated in Fig. 4.9. From this gure, several behaviours can be observed:

(1) When RH decreases from 95% to 50%, desorption curves for the three studied materials are nearly parallel. This phenomenon is also found for desorption curves between RH = 30% and 5%, and for adsorption curves from RH = 30% to 95% which indicates that the drying behaviour (the process of releasing water) and the wetting behaviour (the process of adsorbing water) of these three materials are quite similar. This can only be explained by the fact that these three CEM I materials have some common features in microstructure properties, such as pore network connectivity which lets these materials release or adsorb the same amount of amount of water molecules in the same RH interval.

(2) For all these three materials, there is a large drop when RH changes from 40% to 30% (from 50% for w/c = 0.5 and 0.6 due to no measured data at 40%). Actually, the same trend is also found in experimental data measured by saturated salt solution method [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] between RH = 44% and 33%. This may be due to the fact that above this RH range, the capillary water presents in the material, but below this RH range there is no capillary water (even discontinuous capillary water) and water is mainly located in gel and interlayer pores.

(3) Points measured by pressure plate at RH = 99.26% are greatly higher than points measured by DVS at RH = 95%. It is acceptable because RH = 99.26% is above the hygroscopic range (capillary water range) and most large pores are lled up by liquid-water. However, there are some obvious problems for points at RH = 97.81%. For CEM I with w/c = 0.5, this point is very high, but for CEM I with w/c = 0.4 this point is slightly low compared to the point at RH = 95%. This may be due to some unpredicted measurement mistakes.

Hence, the point measured at RH = 97.81% will not be used for modelling.

Determination of the main adsorption curve

The above measurements show that no entire main sorption curve has been measured, either for desorption or adsorption. For the desorption process, two separated curves were measured in two steps, from RH = 99.26% to 30% and from RH = 30% to 0%. In this study, these two parts are presumed to merge into one main desorption curve because scanning loops are always closed for cementitious materials. However, there are no measured data corresponding to the main adsorption curve. As a matter of fact, the adsorption curve measured between RH = 30% and 95% can be considered as the rst wetting scanning curve which starts at RH = 30%. Therefore, the main adsorption curve can be approximately estimated by using the method proposed in Chapter 3 (see Sections 3.5 and 3.8).

Indeed, when discussing the use of the fewest data to predict the main adsorption isotherm (see Section 3.8), the conclusion was given that two points for the main adsorption curve in the high RH range and the middle RH (or low) range are enough to predict the entire curve. In the high RH range, the last point of the rst wetting scanning curve measured at RH = 95% is supposed to be on the main adsorption curve. For the other point, the extended Mualem Model II (named Mualem Model II-1 in Chapter 3, see Section 3.5) is employed to estimate this point based on the point on the main desorption curve for the same RH. Even though this method has been criticised when it was used to predict the entire main adsorption curve, results in Chapter 3 presented that for points at RH = 44%, 33% and 22% this method showed smaller errors. Thus, Overall, the predicted sorption isotherms have the similar trend to ones measured by the satu-rated salt solution method in the literature (e.g., [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF]). In the low RH range, hysteresis between two main sorption curves is smaller than hysteresis in the high RH range. This gure also clearly shows that CEM I materials have smaller hysteresis than CEM III and CEM I+SF materials. This is because the pore connectivity in CEM I materials is higher than that in CEM III and CEM I+SF materials.

If comparing sorption isotherms measured in this research with results obtained by the saturated salt solution method in [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] (see Fig. 3.6a for Paste3 in Chapter 3), one is able to nd an obvious dierence between these two measurement methods. DVS results show much smaller hysteresis between the two main sorption curves than results in [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF].

This dierence is certainly related to the dierent principle and procedure of these two measurement methods. DVS method is known as a dynamic approach which adopts owing vapour to dry or wet the specimen. Flowing vapour can accelerate the drying or wetting rates. The real equilibrium between external RH and internal RH is not reached to some extent due to the disturbance of velocity of owing vapour. Another important reason is the size of specimens.

Tiny specimens were used for DVS measurements. In particular, these tiny pieces increase the total surface area of the material, so more water molecules can be adsorbed on the surface during wetting and released during drying. Besides, crushed materials contain less capillary pores so that ink-bottle eect is not so signicant when DVS method is used.

Conclusion

In this chapter, the description of two experiments (drying-wetting experiments and WVSIs measurements) was presented. Based on the analysis of some results, the following conclusions are given.

1) Porosity proles measured by gamma-ray attenuation method showed that most of the cement paste materials can be regarded as homogeneous porous media.

2) Comparisons of mass loss curves calculated based on data from gamma-ray attenuation and measured by weighing showed that gamma-ray attenuation technique provided very reliable results. Furthermore, the total porosities measured by gamma-ray attenuation also showed good agreements with these measured by weighing.

3) Weibull's equation is a good way to extrapolate the mass loss curve measured by DVS when this curve has not reached equilibrium for an RH step.

4) The proposed method to estimate the main adsorption curve when it is not measured by DVS has provided satisfactory results. It should be further veried by more experimental data.

5) Compared to sorption isotherms measured by the saturated salt solution method, hysteresis between the two main sorption curves measured by DVS is smaller.

There are two problems for drying and wetting measurements. One problem is that samples were not fully saturated during vacuum saturation. This leads to experimental data for PCEMIII04, PCEMIII05, PSF04 and PSF05 less accurate. The other problem is that the duration for drying, wetting and re-drying was short. Hence, the measured saturation proles are not distinctly dierent from one to another.

All experimental data obtained in this chapter will be used in the next chapter for validation of the proposed method to simulate moisture transport under varying boundary conditions. They will enrich the data previously obtained at IFSTTAR [START_REF] Baroghel-Bouny | Characterization of cement pastes and concretes-methods, analysis, interpretations[END_REF][START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF][START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF] and already used in Chapters 2 and 3. The use of the sorption isotherms measured by DVS instead of the saturated salt solution method and verication with more elaborated drying and wetting cycles will be clearly able to test the applicability of the proposed method.

Introduction

In Chapter 1, a multiphase moisture transport model has been presented (see Section 1.4). In Chapter 3, commonly-used hysteresis models have been compared and veried by experimental data (see Section 3.6). It has concluded that Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF] and improved Rubin's model [START_REF] Rubin | Numerical method for analyzing hysteresis-aected, post-inltration redistribution of soil moisture[END_REF][START_REF] Zhang | A review and statistical study of existing hysteresis models for cementitious materials[END_REF] could be further integrated with moisture transport model. The current chapter is devoted to nding an appropriate modelling method to simulate moisture transport under varying ambient humidity loads. Hysteresis and non-hysteresis modelling methods will be implemented. Comparison of dierent modelling methods will be veried by experimental data, including mass loss curves and saturation proles, from both literature and Chapter 4.

One important factor to evaluate the durability of concrete structures under varying RH at the boundary is called moisture penetration depth x p which has ma jor importance for engineers.

This factor is used to quantify how deep moisture is able to penetrate into the material under cyclic RH at the boundary. It indicates as well the depth that ions can reach into the material with moisture variations. This is particularly important for the description of ions penetration under drying and wetting cycles. Earlier studies concluded that x p is dependent on the material properties (porosity, diusivity, etc.), the cycle duration and the external RH amplitude, except the initial condition [START_REF] Arfvidsson | A new algorithm to calculate the isothermal moisture penetration for periodically varying relative humidity at the boundary[END_REF]. Nevertheless, this conclusion was based on the non-hysteresis modelling.

Results in the case of the hysteresis modelling have not been investigated yet. As a result, the current chapter will study eects of hysteresis on the moisture penetration depth.

Moisture transport models

In Chapter 2, a simplied moisture transport model has been introduced (see Eq. 2.33 in Section 2.4.2) based on the assumptions of constant gas pressure (P g = P atm ) and negligible gas advection. Here, the mass balance equation is written as:

∂S ∂t = ∂ ∂x D a (S) ∂S ∂x (5.1)
in which, the apparent diusivity D a (S) has contributions from liquid-water and water vapour (D a (S) = D l (S) + D v (S), see Eq. 2.33).

As described in Chapter 1, Eq. ( 5.1) is one form of Richards' equation, which is known as the S-based form. If selecting other state variables, Richards' equation can be expressed in dierent 5.2. Moisture transport models forms. A pressure-based form is obtained when the time derivative of S (∂ S/∂t) on the left side of Eq. (5.1) is rewritten as follows: C∂P c /∂t, where the moisture capacity C is dened as the slope of the sorption isotherm, that is dS/dP c .

C ∂P c ∂t = ∂ ∂x D a (S)C ∂P c ∂x = ∂ ∂x K a (P c ) ∂P c ∂x (5.2)
where K a (P c ) = D a (S)C is the apparent permeability. In Eq. (5.2), P c is used as the pressure potential. Actually, there is no dierence to use P c or P l as the potential since P g = P atm .

A mixed form rises when both S and the pressure variables appear simultaneously in the governing equation.

∂S ∂t

= ∂ ∂x K a (P c ) ∂P c ∂x (5.3)
These three forms of Richards' equations (Eqs. 5.1, 5.2 and 5.3) can be transformed from one to another based on the moisture capacity C. However, there are some dierences in the numerical approaches to solve them. Milly [START_REF] Milly | A mass-conservative procedure for time stepping in models of unsaturated ow[END_REF] presented two mass-conservative schemes for computing nodal values of the moisture capacity in the pressure-based form to force global mass balance. In the study by Hills et al. [START_REF] Hills | Modeling one-dimensional inltration into very dry soils, 1. Model development and evaluation[END_REF], the S-based form of Richards' equation can show fast and accurate convergence for moisture inltration. Kirkland et al. [START_REF] Kirkland | Algorithms for solving richards' equation for variably saturated soils[END_REF] expanded the work of Hills et al. by combining the S-based and pressure-based models to yield a transformation method applicable to unsaturated and saturated systems. Celia et al. [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated ow equation[END_REF] suggested that numerical solutions based on the standard pressure-based form of Richards' equation often present poor results, characterized by relatively large mass balance errors and incorrect predictions of the pressure distributions in the saturation prole. They solved the mixed form of Richards' equation that can reach mass-conserving properties for both nite dierence and nite element spatial approximations. Additional transformations of Richards' equation, with the common goal of decreasing its non-linearity and increasing the eciency of the numerical solution, were reviewed by Williams et al. [START_REF] Williams | Transformation approaches for simulating ow in variably saturated porous media[END_REF]. Testing Dirichlet and Neumann boundary conditions for these three forms or equations, de Vasconcellos and Amorim [START_REF] De Vasconcellos | Numerical simulation of unsaturated ow in porous media using a mass conservative model[END_REF] conrmed the conclusion by Celia et al. [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated ow equation[END_REF].

Hence, from the point of view of the numerical solution, S-based and mixed forms are better choices. The S-form allows for very ecient numerical convergence, even for inltration into initially dry porous materials such as soils. However, moisture in porous materials does not necessarily ow under the gradient of S because the driving force for moisture movement is not only the gradient of S, but also the pressure gradient. Another interesting thing between using S-based form and mixed form is quite special for modelling of moisture transport with hysteretic eects because these two forms can present dierent evolutions of scanning loops. This phenomenon will be discussed later in the current chapter (see Section 5.6.4).

In this research, the mixed form is adopted to simulate moisture transport. The potential in this form is capillary pressure (see Eq. 5.3).

Modelling methods

The main dierence between non-hysteresis and hysteresis modelling is the use of sorption isotherms. Thus, methods to implement these two modelling methods are also dierent.

Modelling without hysteretic eects

Modelling of moisture transport without hysteretic eects is easily achieved by using the same sorption isotherm for both drying and wetting. Generally, the main desorption curve is used [START_REF] Arfvidsson | A new algorithm to calculate the isothermal moisture penetration for periodically varying relative humidity at the boundary[END_REF][START_REF] Arfvidsson | Isothermal moisture ow in building materials: modelling, measurements and calculations based on Kirchho 's potential[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF]. This method has been proved that simulation results do not agree with experimental data for the wetting process [START_REF] Zhang | Analysis of moisture transport in cementitious materials and modelling of drying-wetting cycles[END_REF]. A new expression for sorption curves is proposed here on the basis of the measured main desorption and main adsorption curves involving a weighting factor ω = [0, 1].

S = ωS d + (1 -ω)S w (5.4)
In Eq. (5.4), if ω = 1, it corresponds to the main desorption curve; if ω = 0, the main adsorption curve is used, and if ω = 0.5 the sorption curve is the average curve. The parameter ω increases the exibility of the use of this non-hysteresis modelling method.

Modelling with hysteretic eects

In order to incorporate a hysteresis model to simulate drying and wetting cycles, two issues need to be solved: how to check whether the node is going to change the state, and how to keep the node on the current state? Inspired by the work of Gillham et al. [START_REF] Gillham | Measurement and numerical simulation of hysteretic ow in a heterogeneous porous medium[END_REF], two hysteresis indexes U and V are introduced to deal with each above issue. These two indexes have to be assigned to each node during the numerical calculations.

Modelling methods

The index U is used to mark the current state. The initial values of U for all nodes are zero. If the initial step of a node is drying (the case of initial drying), U will keep on 0 until the state changes. If the initial step is wetting (initial wetting), U must change to 1. For the rst node (the node on the surface of the material), the initial step can be determined by the sign of the RH dierence between the boundary condition and the initial condition. The negative sign means initial drying, and the positive sign means initial wetting. For inner points, the initial step will be given during the numerical calculations. In the calculation, U will be added by 1 after the state changes once. So, the even numbers of U represent the drying process and the odd numbers stand for the wetting process.

The hysteresis index V is set as 0 at the beginning of the calculation. This index is used to notify the programme if one node has to change its state in next time step through checking whether the current state is consistent with the sign of the saturation dierence between the current step and the previous step ∆S(i). If ∆S(i) is smaller than a criterion (such as ξ = 10 -10 ), but the current state is wetting (U is an odd number), this means that the node will turn to drying, and V is set as -1. If ∆S(i) is larger than a criterion (ξ = 10 -10 ), but the current state is drying (U is an even number), this implies that the node is going to change to wetting, and

V is assigned to 1. If there is no state changes in the next time step, V has to become 0.

The combinations of these two indexes V and U are able to represent all states of one node during arbitrary drying and wetting cycles. After a number of cycles, index U can tell how many drying or wetting processes to which one given node has been sub jected, and the N th drying or wetting scanning curves on which the current node is. This is very useful for hysteresis modelling due to the strong sorption history-dependence.

Figure 5.1 demonstrates the use of these two indexes for one node. In natural conditions, the reinforced concrete covers are nearly saturated and exposed to a low RH environment which induces drying after the formwork removal. In this process, the P c -S relation is considered on the main desorption curve (see point A in Fig. 5.1) and it will stay on this curve until the next state changes. In this process, both V and U are zero and the programme checks if the state of this node is going to change by comparing the consistency between ∆S and U in each time step.

Point B is where RH increases. At this point, ∆S is positive, but U is an even number, so the node will turn to wetting and the programme must assign 1 to V . After the state changes, V

has to become 0. The scanning curve BC is used instead of the main desorption curve for this node. At the same time, U increases to 1 to note that the node is on the rst wetting scanning curve. In a similar way, when the state change occurs at point C, V becomes -1 to show that this node will be on drying in the next time step, and U increases to 2 to record the second drying scanning curve. In this way, the program can continue the calculation.

In this implementation method, reverse states (starting points of scanning curves) for each node have to be recorded, either for conceptual or empirical models. The programme can distinguish which state the node is on and continue the calculation through the combinations of these two indexes. For most hysteresis models, it is obviously easy to obtain reverse points for all nodes, so this method has a high applicability. Overall, three modelling methods, including the hysteresis modelling (Mualem Model II and improved Rubin's model, see Section 3.6) and the non-hysteresis modelling, are able to simulate drying and wetting cycles. They will be compared in the following sections.

Numerical method

To solve the non-linear equations, a numerical algorithm has to be used. The spatial discretization is performed by the nite dierence method, since only 1D problem is considered in this thesis.

An implicit approximation (backward Euler method) in time step is employed to ensure the stability of the scheme. In each time step, the Picard iterative scheme is used to reach the numerical convergence criteria [START_REF] Celia | A general mass-conservative numerical solution for the unsaturated ow equation[END_REF]. The time step is adaptive according to the convergence rate to solve the partial dierential equation eciently.

Verication with experimental data 5.4.1 Experimental data collection

Two main sources of drying and wetting experimental data will be used to verify the above proposed modelling methods.

The rst source of experimental data were collected from the literature [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF][START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF]. These materials are named Paste3, Paste4 and Paste5 (see Table 2.1). The drying experiments have been described in Section 2.2.2. After drying, all samples were exposed to RH e = 97% (potassium sulfate, K 2 SO 4 ) for wetting (see Fig. 5.2).

The water vapour sorption isotherms, including two main sorption curves and drying/wetting scanning curves, were measured by the saturated salt solution method as described in Chapter 3 (see Section 3.3.2). Saturation proles were measured by the gamma-ray attenuation method [START_REF] Villain | Gammadensimetry: A method to determine drying and carbonation proles in concrete[END_REF][START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF].

Determination of transport coecients

The resistance factor involved in D v (see Eq. 2.35) is calculated by the formula proposed by Millington [START_REF] Millington | Gas diusion in porous media[END_REF] and x D suggested by Thiéry et al. [START_REF] Thiéry | Modélisation du séchage des bétons -Analyse des diérents modes de transfert hydrique[END_REF] (see Eq. 2.37 in Chapter 2).

Another important transport coecient is the relative permeability k rl . The critical question about k rl is whether hysteresis exists in relation k rl (S). The independent domain theory described by Poulovassilis [START_REF] Poulovassilis | The eect of hysteresis of pore-water on the hydraulic conductivity[END_REF] implies that k rl in wetting should be larger than k rl in drying for the same water content. This theory was supported by Mualem [START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF], in which a sophisticated hysteretic model for the prediction of k rl was proposed. However, experimental data for glass-beam (a very porous material) have shown that there is no hysteresis in k rl (S) [START_REF] Topp | Hysteretic moisture characteristics and hydraulic conductivities for glass-bead media[END_REF]. This conclusion was further proved in the cases of sands and soils [START_REF] Topp | Soil-water hysteresis: The domain model theory extended to pore interaction conditions[END_REF][START_REF] Poulovassilis | The hysteresis of pore water: the non-independence of domains[END_REF]. A multiscale network model for liquid-water and vapour transport in porous media also showed that obvious hysteresis in k rl (RH) but no hysteresis in k rl (S) [START_REF] Carmeliet | A multiscale network model for simulating moisture transfer properties of porous media[END_REF]. This chapter will consider both cases, non-hysteresis and hysteresis in k rl (S). Because it is much simpler to implement, the rst case will be applied rstly, and the second case will be discussed later (see Section 5.6.5). For the rst case, it assumes that k rl is a unique function of S. Following the choice in Chapter 2, the well-known VGMa model is used to calculate k rl (S) (see Eq. 2.7 in Section 2.2.3). Notice that the parameter m V in Eq. (2.7) should be dependent on the sorption isotherm, while because there is no such parameter for a scanning curve, m V from the main desorption curve is used for all scanning cases.

The intrinsic permeability K l in Eq. (2.34) was inversely determined by tting drying kinetics measured during a drying experiment [START_REF] Coussy | Assessment of the water permeability of concretes from their mass loss during drying (in french)[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Baroghel-Bouny | Assessment of transport properties of cementitious materials. A major challenge as regards durability?[END_REF]. Results of K l for the three studied materials from the literature have been reported in Table 2.4.

The same values of K l are also used for modelling the wetting process. In other words, besides the boundary conditions, the only dierence between modelling of drying and wetting is the use of dierent sorption isotherms.

The ux boundary condition (see Eq. 1.68) is used for all simulations of drying and wetting cycles. The constant value of emissivity (E = 2.188

× 10 -8 kg • m -2 • s -1 • Pa -1
) is used for all drying and wetting conditions since E seems to be independent to the environmental RH [START_REF] Ayano | Drying, moisture distribution, and shrinkage of cementbased materials[END_REF]. Simulations performed by the non-hysteresis modelling are presented in two cases, ω = 0.5 and 1, which correspond to the average sorption curve and the main desorption curve, respectively.

Mass loss curves measured during the drying process are used to determine K l . Thus, to evaluate the dierent modelling methods, it is necessary to compare simulation results during wetting.

Mualem Model II provides the best agreement with measured mass loss curves for the wetting process, either with a boundary conditions of RH e = 53.5% or 63.2% for drying. Meanwhile, improved Rubin's model overestimates the total moisture mass obtained during wetting. Both cases of non-hysteresis modellings (ω = 0.5 or 1) unexpectedly provide very similar results which clearly underestimate the mass variation and cause that curves drop sharply during wetting. Table 2.4 shows that two tted K l values by using data from drying experiments at RH e = 53.5% and 63.2% are very close. These two K l values are used to simulate moisture transport for the experimental condition at RH e = 63.2 Two calculated mass loss curves are shown in Fig. 5.5.

The dierence between these two curves is not signicant (see Fig. Because both cases of non-hysteresis modelling show similar results, the case of using the main desorption curve (ω = 1) will be adopted to represent the non-hysteresis modelling in the following sections.

In terms of saturation proles, Fig. All gures clearly show that the hysteresis modelling certainly provides better simulation results than the non-hysteresis modelling. In Figs. 5.10a and 5.11a, during wetting, mass loss curves predicted by the non-hysteresis modelling drop sharply (reasons will be explained in Section 5.5), while in Fig. 5.9a this curve has a little bit increase. In fact, this behaviour is only observed in short term. For a long term simulation, the mass loss curve computed by the non-hysteresis modelling will increase slowly and goes underneath the curve calculated by the hysteresis modelling. In general, measured saturation proles at the ends of drying, wetting and re-drying are very close because the duration for each process is not long enough. Three saturation proles predicted by the hysteresis modelling show the same trends as measured curves. The drying process continuously occurs inside materials during experiments, because saturation inside is higher than that in the part near the surface. However, at the end of wetting, saturation proles calculated by the non-hysteresis modelling are far above the measured curves. The same conclusion as experimental verications by using data from the literature can be drawn here: a hysteresis modelling can provide better simulation results than the non-hysteresis modelling.

However, it should also be noted that the dierences between the non-hysteresis and hysteresis modellings at the end of wetting (both in mass loss curves and saturation proles) are larger than the dierences at the end of drying. Liquid intrinsic permeabilities in Table 5.1 are inversely determined by using mass loss curves during the rst drying process. CEM I materials show much larger K l than the materials blended with slag and silica fume (CEM III and CEM I+SF) which are known as able to improve the microstructure of cementitious materials (lower connectivity, lower tortuousity values, ner PSD, etc.). If comparing K l for PCEMI06 with K l for Paste5 in Table 2.4 (both materials have the same w/c), values in Table 5.1 are around twice larger. This may be due to the dierent types of cements. The cement used in TRANSCEND pro ject is a low grade cement (CEM I-32.5 R, see Table 4.1), while the reference [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] used CEM I-52.5 (see Table 3.2).

Investigation of reasons of dierences between three modelling methods

The above verications show the dierences between hysteresis and non-hysteresis modellings, and the dierences between two hysteresis models (Mualem Model II and improved Rubin's model). In the following subsections, reasons of these dierences will be analysed.

Dierences between hysteresis and non-hysteresis modellings

Comparisons show that the non-hysteresis modelling (ω = 1) yields higher S at the surface of the material during wetting than the hysteresis modelling. A gure has been drawn to explain the reason of this dierence (see Fig. 5.12). Because the non-hysteresis modelling uses the same sorption curve, when drying is switched to wetting, S at the boundary will increase to the point corresponding to RH e on the main desorption isotherm, as shown in Fig. 5.12. However, for the hysteresis modelling, the wetting boundary condition is much lower than the point on the main desorption curve for the same RH. This boundary condition is close to the main adsorption curve (see Fig. Besides, Fig. 5.12 also provides evolutions of sorption isotherms for two inner nodes (points 3 and 6 in Fig. 5.12). In the hysteresis modelling, these two nodes are always at lower S than the non-hysteresis modelling for the same RH. This directly causes a lower k rl for these two points (see curves 3 and 6 in Fig. 5.8a). The mass transport in a porous medium like cement-based material is mainly governed by liquid-water movement [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Zhang | Analysis of moisture transport in cementitious materials and modelling of drying-wetting cycles[END_REF]. Thus, the mass transport with the hysteresis modelling is slower than the mass transport in the non-hysteresis modelling. As a consequence, the mass variations predicted by the hysteresis modelling during wetting is smaller than that computed by the non-hysteresis modelling (see Figs. 5.4 and 5.6). This also results in a slower increase of saturation proles during the wetting process if hysteretic eects are considered (see Fig. 5.7).

Dierences between two hysteresis models

Comparisons also show signicant dierences between the case of using the conceptual model (Mualem Model II) and using the empirical model (improved Rubin's model). That is mainly due to the pumping eect, which can cause pumping errors (denoted PEs, see Section 3.6). It is recognised as the main dierence between conceptual and empirical hysteresis models [START_REF] Zhang | A review and statistical study of existing hysteresis models for cementitious materials[END_REF].

In this section, the PEs are further investigated through their inuence on moisture transport.

Simulations are performed on Paste3 by using the initial RH 0 = 75%, drying at RH e = 53% and wetting at RH e = 97%. Firstly, the material is submitted to drying, and then changed to wetting. Each process has the same duration, 15 days (30 days for a cycle). Three cycles are simulated. [START_REF] Mualem | A conceptual model of hysteresis[END_REF] and improved Rubin's model [START_REF] Rubin | Numerical method for analyzing hysteresis-aected, post-inltration redistribution of soil moisture[END_REF][START_REF] Zhang | A review and statistical study of existing hysteresis models for cementitious materials[END_REF] to investigate the PEs. The subgures (a) and (b) show sorption isotherms (scanning loops) simulated by both hysteresis models.

Mualem Model

The subgure (c) presents mass loss curves.

In Fig.

Discussions about eects of dierent modelling conditions

Compared to the hysteresis modelling, the non-hysteresis modelling is much easier to implement and is able to save computing time. From an engineer's point of view, the non-hysteresis modelling is a handy method. That is why it is worth carefully analysing the applicability of these two dierent methods under various situations. Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF] will be used as a benchmark for comparisons because it has been conrmed to be the more relevant one in comparisons with experimental data in the laboratory (see Section 5.4).

Meanwhile, eects of other factors on modelling of moisture transport, such as the dierent forms of Richards' equations, dierent RH loads curves at the boundary and hysteresis in k rl will be discussed. The last part has a focus on the analysis of the moisture penetration depth.

Eect of the drying and wetting cycle duration

This subsection focuses on discussion about the inuence of the drying and wetting cycle duration.

In real conditions, RH varies daily, monthly and seasonally, so for modelling purposes, these variations will be tested as dierent cycle durations: 1 day, 30 days and 90 days. The initial and drying/wetting boundary conditions used here are the same as the ones used for the analysis of the PEs (see Section 5.5.2). The reason using these initial and boundary conditions is that Obviously, the time to reach mass equilibrium is dierent between both modellings and depends on many factors (durations of the drying/wetting cycles, boundary conditions, initial RH, etc.).

For the tested case in the current section, the mass loss curve provided by the hysteresis modelling reaches equilibrium quickly, while for the case of non-hysteresis modelling, this process takes longer time. After 3000 days of simulation, only approximate equilibrium is observed for 30-and 90-day cycle durations.

For the 1-day cycle duration, the dierences of mass loss curves calculated by these two methods are very small, as well as saturation proles, which always uctuate around the initial

Eect of the boundary RH amplitude

The RH amplitude is another factor which signicantly aects moisture transport under drying and wetting cycles. In the previous discussion about the cycle duration, an RH amplitude of 44% (53% -97%) was used. Another RH amplitude is introduced here between 64% and 86%, which is the half of the RH amplitude used in the previous discussion. The initial RH is still 75% and a moderate cycle duration of 30 days is chosen.

Eect of the initial step

To perform a numerical modelling for materials after self-desiccation, the initial position of the relationship RH-S needs to be determined. In sealed-cured conditions, the process of selfdesiccation consumes free water in the material. Even though the hydration process is involved in self-desiccation, the initial position is considered to be located on the main desorption curve.

This position should correspond to the stabilized state from the point of view of hydration.

Experimental data shows that the internal RH varies a lot depending on the duration of the sealed-curing period. For example, in measurements carried out by Baroghel-Bouny et al. [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF] for a large variety of cementitious materials, the internal RH is between 69% and 94% after a 1-year sealed curing period. Thus, based on these data and concerning the necessity to x the same initial RH for all the studied materials for the comparison purpose, the choice of a middle RH (e.g., RH 0 = 75%) on the main desorption curve as the initial condition for modelling seems a most relevant choice. In the central subgure, the red dash-dotted lines represent the saturation proles at the end of drying and at the end of wetting of the rst cycle. The explanations of the legend are the same as in Fig. 5.14.

The next question has to be answered before performing the simulation is whether the rst step is drying or wetting. In experiments (see Section 5.4), materials were rstly exposed to a low RH, while in natural conditions, the external RH can be either higher or lower than the initial RH. So, both initial drying and wetting should be considered to represent the real conditions. Simulations in the case of initial drying have been done in the above sections (see Sections 5.5.2, 5.6.1 and 5.6.2). This subsection proposes to carry out the simulation in the case of initial wetting. For the purpose of comparison, the initial conditions is RH 0 = 75%, a 30-day cycle duration and RH amplitude = 44% (53%-97%, the same conditions used in Section 5.6.1 in the case of initial drying).

Simulation results are presented in Fig. 5.16, which should be compared with Fig. 5.14b for the case of initial drying. Firstly, the two mass loss curves in Fig. 5.16 are very similar to Fig. 5.14b. The total mass losses at the end of the simulation for the non-hysteresis modelling are very close in these two gures. Secondly, saturation proles at the end of drying and wetting after 500, 1000 and 3000 days are quite similar in Figs. 5.14b and 5.16 for both modelling methods. In fact, the dierence between initial drying and initial wetting is more signicant in the rst cycle for the hysteresis modelling. Results simulated with initial wetting show that saturation at the boundary in the rst cycle is much higher than the following cycles (see the red dash-dotted line and the red solid lines at the end of wetting in the central subgure of Fig. 5.16). Accordingly, the scanning loop of the rst cycle is far above scanning loops of the following cycles (see Cycle

Eect of dierent forms of Richards' equations on hysteresis modelling

As it has been mentioned in Section 5.3, S-based and mixed forms of Richards' equations are two more relevant forms from the numerical point of view. These two forms use dierent state variables: S and P c . Let's consider an experimental condition, in which the sample is rstly submitted to drying and then to wetting (see Fig. 5.17). For each process, if the simulation can reach an equilibrium state, the state variable should be uniform throughout the material.

However, for most cases, the duration of drying is limited. This means that nodes are separately located on the main desorption curve as one example shown in Fig. 5.17a.

When the material is switched to wetting, each node will follow dierent scanning curves. If using S-based form, all scanning curves will go to the horizontal line at S = S e (line AB in the gure), while in the case of mixed form, nal points for all scanning curves will be on the vertical line at RH = RH e (line AC in the gure). It is clear that all scanning curves must pass the line AB if they have to reach the line AC. S e is smaller than any point on the line AC so that the nal mass loss in case of using the S-based form equation is denitely smaller than that of using the mixed form equation. As a result, in Fig. 5.17b, the red dashed curve (from S-based form) decreases more slowly than the blue curve (from mixed form). The area ABC in Fig. 5.17a is related to the gap between these curves in Fig. 5.17b. Hence, the mass loss curve simulated by the S-based form of Richards' equation does not match the measured curve well. This is one reason that the mixed form equation is chosen for our simulations in this chapter.

In real conditions, the water transport in porous materials is mainly due to the gradient of capillary pressure. So, there is no wonder why the mass loss curve predicted by the mixed form equation provides better results. Obviously, for the non-hysteresis modelling, there is no this kind of problem about the choice of dierent forms of Richards' equations. This also indicates that, if one wants to take into account hysteretic eects in modelling of moisture transport, choosing which state variable is an important consideration.

Eect of hysteresis in k rl (S)

Generally, no hysteresis in k rl (S) is considered (see Section 5.4.2). Although this is accepted by most moisture transport models for soils, it is still of fundamental interest to study how the simulation results will be aected if hysteresis in k rl (S) is taken into account. The line numbers 1, 3 and 6 correspond nodes in Fig 5 .2.

Among the number of models for k rl (S), according to our knowledge, Mualem's model-b [START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF] which was developed in later 1976 is the one that can consider hysteresis in k rl (S).

k rl,1,d (S) = S 0.5 [KL -(KL 1 -KL) KH] 2 (5.6)
For the rst wetting scanning process (1,w), it is:

k rl,1,w (S) = S 0.5 [KL -(1 -KL) KH 1 ] 2 (5.7)
where KL 1 and KH 1 are KL and KH calculated at the inverse points of the scanning curves.

In a similar way to calculate sorption isotherms by Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF], equations for the N th drying and wetting scanning processes can be written as: where the subscript (N,0) represents the value at the starting point of the N th scanning curve.

Thus, k rl,N -1,w (S N,0 ) (or k rl,N -1,d (S N,0 )) corresponds k rl at the end point of the previous scanning process (order N -1).

In the above equations, the current saturation S has to be known to further calculate the relative permeability. Thus, for a scanning process, the scanning curve must be computed rstly.

Theoretically, any hysteresis model can be used to predict the scanning loops. But, only conceptual models such as dependent or independent models based on the domain theory always provide closed form of k rl (S) in the scanning loops. If RH-S scanning loops are not closed (e.g., results predicted by empirical models with PEs), it will generate errors at the end of each process and these errors will increase with the number of loops due to the pumping eect (see the explanation in Section 5.5.2). Hence, Mualem Model II [START_REF] Mualem | A conceptual model of hysteresis[END_REF] (see Section 3.6.1), which was also developed based on the independent domain theory, is used here. [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF][START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF] provide smaller k rl values than VGMa model [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF][START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF], either for the main desorption process or the scanning processes. So, when using VGMb model to inversely determine K l , the K l value is much larger (several orders of magnitude) than that from VGMa model (see Table 2.4).

However, the dierences of k rl values between Fig. 5.18 and Fig. 5.8a are much smaller than the dierence in K l . Moreover, when it is switched to wetting, the k rl (S) scanning curves increase and these k rl values are larger than these for the main desorption process. Totally, the case of using VGMb model has larger liquid permeability K l k rl in wetting than using VGMa model. As a result, the mass loss curve drops sharply when hysteresis in k rl (S) is taken into account. The prediction in this case cannot match the measured mass loss curve well as shown in Fig. 5.19.

Analysis of the moisture penetration depth

As mentioned in the introduction, the penetration depth x p is an important factor to evaluate the durability of the concrete structures since x p is related to the depth that ions can penetrate into the material. Beyond x p , it can be considered that the concentration of ions has no eects by the external ions concentration variations.

To determine this depth, two restrictions have been proposed by Arfvidsson [START_REF] Arfvidsson | A new algorithm to calculate the isothermal moisture penetration for periodically varying relative humidity at the boundary[END_REF].

(1) In a simulation after a number of drying and wetting cycles, there must be a part of the material with a constant saturation that cannot be disturbed by RH variations at the boundary. Let's name this saturation as the undisturbed saturation S und (see Fig. 5.20).

(2) When the undisturbed saturation is reached, this means that the saturation prole at the end of drying (and wetting) is the same for the following cycles. This also implies that the mass loss curve reach a mass equilibrium state for each cycle. The moisture penetration depth x p is dened as the depth where the saturation dierence (∆S ) between the end of drying and the end of wetting is equal to 1% (see Fig. To investigate x p , the boundary condition, including the cycle duration and the RH amplitude, and the initial condition have to be determined. In this study, the cycle duration varies between 1 day and 90 days. The RH amplitude is between 44% (RH max = 97%, RH min = 53%) and 14% (RH max = 82%, RH min = 68%). The initial condition should be determined carefully.

Arfvidsson [START_REF] Arfvidsson | A new algorithm to calculate the isothermal moisture penetration for periodically varying relative humidity at the boundary[END_REF] reported that S und is not related to the initial condition, but the initial condition can aect the number of cycles when S und is reached. In the simulations, it has been conrmed that this assumption is only valid in the case of non-hysteresis modelling as it will be presented below. Thus, for all simulations, the initial condition is xed at RH 0 = 75% as the previous calculations (see Sections 5.6.1 and 5.6.2). This value is the mean value of the maximum RH and the minimum RH at the boundary. An initial drying step is chosen here.

Simulation results show that the trends of x p for non-hysteresis and hysteresis modellings are quite similar (see Fig. 5.21). The moisture penetration depth x p increases with the cycle duration and the RH amplitude. For the small cycle durations, such as 1, 5 and 10 days, even for 20 and 30 days, the dierences in term of x p calculated by both modelling methods are very small (less than 1 mm dierence), regardless the RH amplitude. Notice that one example of using the 30-day cycle duration has been shown in Fig. 5.14b, which presents the similar x p for hysteresis and non-hysteresis modellings (see the positions of the gray dashed lines in the Nevertheless, the dierences of x p increase with the cycle duration and the RH amplitude (compare positions of gray dashed lines in Fig. 5.14b for a 30-day cycle duration and in Fig. 5.14c for a 90-day cycle duration). For the simulations with a 90-day cycle duration and 44% RH amplitude, Fig. 5.21 shows that dierence in x p between both modellings rises to 4 mm (24 mm for the hysteresis modelling and 28 mm for the non-hysteresis modelling). Overall, x p increases more sharply with the cycle duration and the RH amplitude for the non-hysteresis modelling than for the hysteresis modelling. This implies that the non-hysteresis modelling may overestimate x p for large cycle durations and large RH amplitudes. This conclusion can help for the design of durability models used to predict service life in the case of cyclic boundary conditions. To analyse the inuence of the initial condition on x p for the hysteresis modelling, simulations are performed for ve dierent initial conditions, and results are provided in Fig. 5.22. For the high RH 0 values (= 85% and 80%), the predicted penetration depth is almost the same with the one obtained with the initial condition RH 0 = 75% (used previously for all simulations), while for lower RH 0 values (= 65% and 70%), x p is lower, regardless the cycle duration. Figure 5.22 reveals that the proposed method (considering hysteresis) to predict x p in valid for the case initial RH 0 > 75%. If RH 0 75%, this method may underestimate x p .

In practice, the initial condition should be accessed according to the composition of the material and the curing duration. For most cases (maybe except high performance materials), the internal RH after self-desiccation seems to be higher than 75% [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF]. Therefore, there is no need to consider the eect of the initial condition on the calculated x p .

Conclusion

In this chapter, hysteresis models have been coupled with a moisture transport model to simulate drying and wetting cycles. Simulations in the case without hysteresis have also been performed.

Based on experimental verications and comparisons for dierent simulation conditions, the following conclusions can be drawn.

1) The proposed method to implement hysteresis models into moisture transport models has a high applicability. The method was used for both conceptual and empirical models. Two hysteresis indexes U and V have been introduced. The hysteresis index V was used to indicate if the node will change its state. The hysteresis index U recorded the drying and wetting history of each node and was very useful for hysteresis modelling because of its strong sorption history-dependence.

2) Comparisons with measured mass loss curves and saturation proles for cement pastes showed that modelling with hysteresis explicitly provides better results than non-hysteresis modelling. It is principally because hysteresis models is able to provide more relevant RH-S relation, especially at the boundary.

3) Among hysteresis models, results simulated by using Mualem Model II (conceptual model) matched experimental data much better than improved Rubin's model (empirical model). This is largely because improved Rubin's model provided non-physical scanning loops, characterised by the pumping errors which is generally observed for empirical models. Thus, Mualem Model II could be the one recommended for modelling of moisture transport under varying boundary conditions. 4) Comparisons with dierent cycle durations revealed that smaller cycle durations cause smaller dierences between non-hysteresis and hysteresis modellings. The dierences increase with the cycle duration.

At the same time, it also was shown that long term simulations cause larger errors than short term simulations. The errors simulated by the non-hysteresis modelling increase with the RH amplitude; therefore, for the situation with small cycle duration and small RH amplitude, the non-hysteresis modelling is still applicable.

5) For long term simulations, dierent initial steps (initial drying or initial wetting) did no show signicant dierences in mass loss curves and saturation proles. The dierences only occurred for the rst cycle in the case of the hysteresis modelling.

6) Comparison of two (S -based and mixed) forms of Richards' equation revealed that the mixed form is a better choice, not only because it can provide good simulation results, but also because moisture transport is mainly due to the gradient of capillary pressure in the real condition. Although S-based form Richards' equation has higher speed of convergence for the numerical simulation, it limits the evolution of sorption isotherms, leading simulated results away from the measured data.

7)

Verifying by the limited experimental data showed that it is not necessary to consider hysteresis in k rl (S). Further investigations are needed to check this conclusion.

8) The moisture penetration depth x p , which may also indicate the penetration depth of ions such as chlorides, has been carefully analysed. For the non-hysteresis modelling, this study conrmed that x p is related to the cycle duration and the RH amplitude, apart from the initial condition, as this has already been pointed out in the reference [START_REF] Arfvidsson | A new algorithm to calculate the isothermal moisture penetration for periodically varying relative humidity at the boundary[END_REF]. But for hysteresis modelling, the initial condition has clear inuence on x p if RH 0 is below 75%. Hence, the use of the hysteresis modelling to estimate x p should be more relevant for the case RH 0 75%. 9) Comparisons of x p indicated that the non-hysteresis modelling overestimated x p in the case of long cycle durations and large RH amplitudes.

According to comparisons provided in the current chapter, the cumulative errors associated with simulation time can become signicant and unavoidable if the simulation cannot take hysteretic eects into account. The above conclusions are only based on the case of one ordinary cement paste (Paste3, see Table 3.2 in Chapter 3). In the next step research, more cementitious materials (e.g., mortars and concretes) will be used to verify the proposed modelling method and to rene the above conclusions. In these kinds of measurements, the sample must be sealed so that liquid-water is only injected from one face by applying extra pressure and leak from the opposite face, where the rate of outow must be measured after the ow is stable [START_REF] El-Dieb | A high pressure triaxial cell with improved measurement sensitivity for saturated water permeability of high performance concrete[END_REF][START_REF] El-Dieb | Water-permeability measurement of high performance concrete using a high-pressure triaxial cell[END_REF]. However, the disadvantages of the conventional methods are obvious. First, the extra pressure has to be very high. For modern cementitious materials with low water-to-cement ratios and ne pore structures, such as high performance concretes, moisture transport is much slower than the traditional ordinary cementitious materials. For such weakly permeable porous media, the order of MPa pressure may be used. This incredibly increases the risk of physical damage of the pore structure. Second, under the high pressure, the leakage is unpredictable. Third, the essentially saturated state is a strict requirement, especially for the low water-to-cement ratio material, it is very dicult to reach. More recently, an improved device using a controlled oedometer seems to be able to avoid the leakage problem, but it still needs a high extra pressure [START_REF] Picandet | Permeability measurement of fresh cement paste[END_REF].

D l (S) = -k rl K l φη l dP c dS (6.1) D v (S) = - M ρ l RT 2 D v0 φ x D (1 -S)
Owing to these reasons, the indirect method becomes prior to the direct methods. The beam bending method is one of rapid and indirect methods [START_REF] Scherer | Bending of gel beams: method of characterizing mechanical properties and permeability[END_REF][START_REF] Scherer | Measuring permeability of rigid materials by a beam-bending method: I. theory[END_REF][START_REF] Scherer | Measuring permeability of rigid materials by a beam-bending method: Iv. transversely isotropic plate[END_REF], which was originally developed for soft gels and later was applied to cementitious materials. It exerts a certain strain to a long and slender sample to obtain a relaxation curve, which is considered including both hydrodynamic and viscoelatic eects. By tting this relaxation curve, K l can be determined.

This method has very clear requirements for the geometry of the sample. It is less applicable for concretes due to needing very big samples to avoid the inuences of aggregates. A method socalled thermopermeametry was also introduced to cementitious materials research from gels [START_REF] Scherer | Thermal expansion of gels: a novel method for measuring permeability[END_REF][START_REF] Scherer | Measuring permeability by the thermal expansion method for rigid or highly permeable gels[END_REF][START_REF] Scherer | Thermal expansion kinetics: method to measure permeability of cementitious materials: I, theory[END_REF]. It could be used to supplement the shortcomings of the beam bending method. If assuming that the thermal expansion of liquid is always much greater than that from the solid phase, this method can estimate K l by measuring the amount of liquid-water which ows out of the material due to the thermal expansion. The main problem is that this method is more sensitive to saturation condition. If there are some air voids or trapped air because of non-fully saturated materials, it will have signicant inuence on measured results. Comparatively, the dynamic pressurization method is much easier to control, which keeps the sample in a sealed vessel and gradually increases or decreases pressure [START_REF] Scherer | Dynamic pressurization method for measuring permeability and modulus: I. theory[END_REF][START_REF] Grasley | Dynamic pressurization method for measuring permeability and modulus: II. cementitious materials[END_REF]. If repeating the pressure cycles, the residual air in the material can be dissipated. The latest literature revealed that the NMR technique is able to provide very good data for determination of liquid permeability [START_REF] Zamani | The relative humidity dependence of the permeability of cement paste measured using gareld nmr proling[END_REF].

Because of so many diculties in experiments, using theoretical equations based on pore size distribution, pore geometry and connectivity to predict K l become more vital for cementitious materials. Based on pore size distribution measured by mercury intrusion porosimetry, Katz-Thompson model [START_REF] Thompson | The microgeometry and transport properties of sedimentary rock[END_REF] has been proven more suitable for estimating the permeability of sedimentary rocks. This method was also used for cementitious materials [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF]. However, it is only valid for materials with large pores, in which the surface eects can be neglected compared to the pore size. In Chapters 2 and 5, K l was inversely determined by tting drying kinetics based on numerical simulation results. Actually, this kind of method needs to solve one or more partial dierential equations and require a plenty of experimental data. These requirements limit its application. Besides, this method is sensitive to the length of the drying period and the predicted K l may vary according to the external RH during drying.

In Eq. (6.2), for the description of vapour diusion, x D is a very important factor. Millington [START_REF] Millington | Gas diusion in porous media[END_REF] suggested x D = 4/3 for variably saturated porous materials such as soils and this value has been used in cementitious materials research [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Bary | Coupled moisturecarbon dioxidecalcium transfer model for carbonation of concrete[END_REF]. However, soils are more porous materials ans their microstructures as less complex. The calculated values of K l for the four studied materials in the desorption process are provided in Table 6.1. Some K l values estimated by the Katz-Thompson model (taken from [START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF]) and by inverse analysis using the ux boundary condition (taken from Table 2.5) are added in this table for comparison. Values validated by the current proposed method are much smaller.

The dierence is mainly due to the low accurate tting. But Figs. 6.2, 6.3 and 6.4 show that several points are far away from the tting curves. The tting precision needs to be improved by using more relevant equation for D a . It may be also due to the problem of experimental data. Figs. 6.3 and 6.4 show that some measured data at around RH = 40% are very low. As it has been shown in the literature [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF], this range of RH corresponds to the threshold pore size separating C-S-H and capillary porosities (see Section 3.8.5). Therefore, unstable moisture transport in this RH range easily causes low accurate measurement results. Several observations can be seen in Figs. 6.3 and 6.4:

• Hysteresis in D a (S) curves is negligible, either for the measured data or the tted curves. This is in agreement with the conclusion about hysteresis in k rl (S) which has been conrmed for soils and other porous materials [START_REF] Topp | Hysteretic moisture characteristics and hydraulic conductivities for glass-bead media[END_REF][START_REF] Topp | Soil-water hysteresis: The domain model theory extended to pore interaction conditions[END_REF][START_REF] Poulovassilis | The hysteresis of pore water: the non-independence of domains[END_REF]. show smaller hysteresis as well.

• Hysteresis in D a (RH) show that for each material there is a threshold RH value which divides hysteresis into two parts: below this threshold value, D a (RH) in adsorption is larger; above this threshold value, D a (RH) in desorption is larger. Besides of reasons of the dierences in tted K l and x D (see Tables 6.1 and 6.3), this trend is mainly due to the dierent shapes of the main desorption and adsorption isotherms.

• The tting results for adsorption are much better than those for desorption. The main reason is that the measured desorption data are more uctuating than the measured adsorption data.

Improved method

Obviously, Eq. (6.2) only takes into account ordinary diusion. As it has been discussed in Chapter 1, ordinary diusion predominantly occurs in pores between 50 nm and 10 µm (the typical range of large capillary pores) which corresponds to the range of RH > 98% according to Kelvin's law (see Eq. 4.8). However, in most conditions, vapour diusion occurs in the small range of pores whose size is denitely below 50 nm. In the pore size range 2-50 nm, Knudsen diusion plays a main role (e.g., [START_REF] Malek | Knudsen self-and Fickian diusion in rough nanoporous media[END_REF]). Hence, to describe vapour diusion in this pore size range, Knudsen eect has to be taken into account. This eect is able to reduce the diusion process due to the collisions between molecules and pore walls [START_REF] Daïan | Condensation and isothermal water transfer in cement mortar. Part I: pore size distribution, equilibrium water condensation and imbibition[END_REF].

Assuming all pores are independent parallel cylinders, Knudsen diusion coecient for a single pore is formulated as [START_REF] Gertis | Isothermer gastrasnport in porosen stoen aus gas kinetischer sicht[END_REF][START_REF] Daïan | Condensation and isothermal water transfer in cement mortar. Part I: pore size distribution, equilibrium water condensation and imbibition[END_REF][START_REF] Baroghel-Bouny | Characterization of cement pastes and concretes-methods, analysis, interpretations[END_REF]:

D K,single = D v0 1 1 + λ 2r K (6.3)
where r K is the Kelvin pore radius (see Fig. 6.5) which is the eective radius for Knudsen diusion. At a room temperature (around 293 K) and atmospheric pressure (1 atm), the mean free path λ of water molecules is about 8×10 -8 m.

Knudsen diusion only takes place in pores unlled by liquid-water. Therefore, the macroscopic diusivity in all parallel pores is obtained by integrating D K,single over unlled pores (pore size > r p ) as:

D K = D v0 ∞ r K dV V v dr p 1 + λ 2 [r p -t a (RH)] dr K (6.4)
where dV V v dr p is inferred from the pore size distribution (PSD), which is usually calculated based on measured sorption isotherms. V v is the pore volume for vapour diusion. The pore radius is given as r p = r K + t a where t a is the thickness of the adsorbed layer. Hagymassy et al. [START_REF] Hagymassy | Pore structure analysis by water vapor adsorption, I. t-curves for water vapor[END_REF] rstly published the water vapour t a curve (t a vs. RH) for non-porous adsorbent based on water vapour adsorption experiments. A generalized expression for the curve is:

t a (RH) = a H [c H -log(RH)] b H (6.5)
where a H , b H and c H are related to the properties of the material. Baroghel-Bouny [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] used

Hagymassy's method to calculate the t a curve for cementitious materials and found that there is a master curve which is suitable for all the studied materials (see Fig. 6.6). Fig. 6.6 shows that the master t a curve is almost the same as that for Paste2 and Paste3, especially for RH< 63.2%. For PSD, a calculation approach called BJH method is adopted, which was proposed by Barrett, Joyne and Halenda [START_REF] Barrett | The determination of pore volume and area distributions in porous substances. I. Computations form nitrogen isotherms[END_REF]. This method assumes that both capillary and adsorbed water coexist in pores which are treated as cylinders. Radii of capillary pores are governed by Kelvin's law, corresponding to r K in Fig. 6.5. The thickness of adsorbed layers varies with dierent RH and can be described by t a curve. If the experimental sorption isotherms span from RH = 0 to 100%, the BJH method can theoretically calculate the pore size distribution in both mesopore (2 to 50 nm) and in micropore ranges (pore openings < 2 nm). The minimum RH used in sorption isotherms measurements is 11% [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] which corresponds to the minimum pore size around r p ≈ 0.5 nm.

According to the BJH method and measured sorption data [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF], the calculated pore size distributions for the studied materials are displayed in Fig. 6.7. It shows that there are two or three peaks depending on the type of the material. The rst peak is observed for all the materials corresponding to around r p = 1.7 nm which is supposed to represent the microstrcture of the denser C-S-H. The second peak is quite dierent according to the studied materials and even the third peak can be observed for some materials such as Paste5. The second and third peaks may represent the microstructure of outer C-S-H. These dierences in pore size distribution are able to signicantly aect the moisture transport behaviour, in particular for the vapour diusion.

The calculated curves with the Knudsen eect for the studied materials are presented in Fig. 6.8. As mentioned above, the Knudsen eect is only calculated for pores larger than 0.5 nm.

For pore size below 0.5 nm, the Knudsen eect is considered as the same as that for r p = 0.5 nm. Figure 6.8 shows that D K /D v0 remains nearly constant when RH is below a certain value depending on the material. In the range above this RH value, D K /D v0 increases quickly with

Fitting results for K l and x D

The tted x D values for the case of taking into account the Knudsen eect are compared with the case of without the Knudsen eect in Tables 6.2 and 6.3 for desorption and adsorption, respectively. Because the Knudsen eect has no inuence on liquid-water transport, the tted K l values for the basic and the improved methods are very similar. Thus, K l values for the improved method are not shown here. But for the vapour diusion, values of x D are much smaller if the Knudsen eect is considered (see Tables 6.2 and 6.3). and the material with low w/c (Paste3), gel pores are predominant as Fig. 6.7 shows that these materials have a similar PSD. are more capillary pores in the material, the continuous liquid-water is not easy to establish in the low RH range so that the main transport mode remains vapour transport. The dierences of threshold values between desorption and adsorption can be explained by the same reasons as hysteresis behaviour such as the ink-bottle eect (see Section 1.2).

Partial conclusion

This section has compared two ways to t K l and x D based on the measured apparent diusivity.

One method considers vapour transport in cementitious materials as ordinary diusion and the other method views it as Knudsen diusion. Four models to calculate k rl have also been compared. Through these comparisons of the dierent methods, it can be concluded:

1) Among the compared k rl models, Mualem model-b [START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF] provides much higher values of K l than the other methods (Katz-Thompson model, inverse analysis using the rst boundary condition, tting D a using Mualem model-a). Moreover, this model is not very easy to use.

By contrast, Mualem model-a [START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF] is an easy-to-use model, which could be the rst choice for numerical purposes. Besides, hysteresis in D a (RH) is caused by hysteresis in sorption isotherms RH(S) or moisture capacity dS/dP c , but is not related to hysteresis in k rl (S)

or k rl (RH). Notice that k rl can be indirectly veried if using the relationship between k rl and k rg provided by the theory of Mualem model-a [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF] (see Eq. 2.10).

Obviously, k rg is much easier to be determined by experiments in case of cementitious materials. Such work has been done by Monlouis-Bonnaire et al. [START_REF] Monlouis-Bonnaire | Prediction of the relative permeability to gas ow of cement-based materials[END_REF] and Thiéry et al. [START_REF] Thiéry | Modelling of isothermal drying process in cementitious materials -Analysis of the moisture transfer and proposal of simplied approaches[END_REF]. Based on the theory of Mualem model-a, they suggested that = 5.5 and 7.15, respectively. These two proposals need to be veried in the future study.

Eect of boundary conditions on the determination of K l by inverse analysis in laboratory condition

Coupling moisture transport between the porous material and its surrounding is a process of signicant scientic and applied interest. It involves several mechanisms that aect the macroscopic behaviour. These include phase change between liquid and vapour, mass transport by diusion and convection, etc. However, most moisture transport models for cementitious materials in the literature only focus on the mass transport taking place within the material. Moisture transport over the material surface is generally represented by a simplied boundary condition, either by the rst type or by the ux boundary conditions (see Eqs. 1.65 and 1.68 in Section 1.5.6). These conditions only consider one-way eects and do not really take into account mutual interactions between the material and its surrounding.

In the early developed 1D-HAM (Heat-Air-Moisture) PC-program [START_REF] Hagentoft | Simplied analysis of combined heat, moisture and air transport for one-dimensional cases[END_REF][START_REF] Hagentoft | PC-program: 1D-HAM[END_REF], it considered that a ctitious surface lm can be used to account for the material surface resistance. This lm is simulated as a material layer without additional air ow resistance. This method is quite similar to the ux boundary condition. In building energy models, such as EnergyPlus [105],

TRNSYS [START_REF] Trnsys | Transient system simulation program[END_REF], HAM and simplied models (e.g., EMPD), they assume that state variables are uniform in the air, so that only one node is considered in the region of surrounding to couple moisture [105,[START_REF] Trnsys | Transient system simulation program[END_REF]. Laurindo and Prat [START_REF] Laurindo | Numerical and experimental network study of evaporation in capillary porous media: Phase distributions[END_REF][START_REF] Laurindo | Numerical and experimental network study of evaporation in capillary porous media: Drying rates[END_REF] considered the presence of a mass boundary region (MBR) over the external surface of the porous media to estimate the drying rates. Thus, the coupled system of materials and their surrounding can be viewed as three parts: a) the porous material, b) the MBR and c) the part beyond the MBR in the environment (see Fig. 6.9).

In the material, moisture transport can be simulated by models mentioned in Chapters 1 and 2.

Actually, for most models in the literature, they do not distinguish the mass boundary region and part (c) [START_REF] Erriguible | Simulation of convective drying of a porous medium with boundary conditions provided by CFD[END_REF][START_REF] Steeman | Conservative modelling of the moisture and heat transfer in building components under atmospheric excitation[END_REF][START_REF] Steeman | Modelling indoor air and hygrothermal wall interaction in building simulation: Comparison between CFD and a well-mixed zonal model[END_REF][START_REF] Steeman | Evaluation of the dierent denitions of the convective mass transfer coecient for water evaporation into air[END_REF], and call them together as an atmospheric boundary layer (e.g., [START_REF] Blocken | CFD simulation of the atmospheric boundary layer: wall function problems[END_REF]). However, modelling results by Laurindo and Prat [START_REF] Laurindo | Numerical and experimental network study of evaporation in capillary porous media: Phase distributions[END_REF][START_REF] Laurindo | Numerical and experimental network study of evaporation in capillary porous media: Drying rates[END_REF] concluded that the thickness of the MBR x e has a great eect on drying rate. Yiotis et al. [START_REF] Yiotis | Coupling between external and internal mass transfer during drying of a porous medium[END_REF] further pointed out that the drying rate is inversely proportional to x e . These conclusions let us take a new look at the conditions used during our experiments which have been used to t K l in Chapters 2 and 5. In experimental conditions, samples were placed in desiccators with constant RH which is controlled by the saturated salt solution. Generally, the distance between the surface of the sample and the surface of the saturated salt solution is 5 cm -15 cm. In this region, only water vapour diusion occurs. If vapour diusion in this region has an inuence on moisture transport in the material, K l determined by inverse analysis in Chapters 2 and 5 can be also aected.

On the whole, the purpose of this section is to check whether there is an inuence on K l determined by inverse analysis if taking into account vapour diusion in the mass boundary region. Let's call this condition as diusive boundary condition since only vapour diusion is considered. A simple 1D model will be introduced to simulate vapour diusion in this region.

Moisture transport in both material and the MBR are simultaneously solved. Some discussions about x e and its eects on mass loss and K l are presented in the last part of this section.

Modelling of water vapour diusion in the MBR

The air in the MBR is viewed as an incompressible ideal gas. Uno [START_REF] Uno | Plastic shrinkage cracking and evapouration formulas[END_REF] analysed several traditional equations for the drying evaporation from the surface of wetted concretes, and he nally concluded that the drying rate is related to the velocity of ambient air and how fast diusion occurring at a given temperature. Since there is no air velocity in our experiments, the mass balance equation for vapour transport in the MBR only contains vapour diusion with a constant diusivity.

∂ρ v ∂t = D v0 ∂ 2 ρ v ∂x 2 (6.6)
At the surface, the continuity of vapour movement is expressed as:

(ρ v ) -= (ρ v ) + (6.7)
whereand + represent the MBR and the material sides as shown in Fig. 6.9.

Moisture transport in the material is simulated by the simplied mode used in Chapter 5

(see Section 5.2). The continuity of moisture ux at the surface is given by:

Modelling validation

The validation will check if the newly developed method coupling internal and external moisture transport can provide simulation results that agree with experimental data. The agreement should be found in both mass loss curves and saturation proles. Here, Paste4 is taken as an example (the case of drying at RH e = 53.5%) and experimental data have been displayed in x e and K l . The value for K l is set as 2.0 × 10 -21 which was calibrated by inverse analysis using the ux boundary condition (see Table 2.4 in Chapter 2).

Thus, the value for x e can be changed to t measured mass loss curve during drying. Eventually,

x e = 3 cm is found to be able to provide relevant simulation results. The same values of x e and K l are also used for modelling of wetting. For the hysteresis modelling, good agreements between simulation results and experimental data can be found for both mass loss curves and saturation proles. For the non-hysteresis modelling, it overestimates the mass changes during wetting. Actually, these results are very similar to results calculated by using the ux boundary condition as shown in Figs. 5.6 and 5.7.

Discussion about eect of the MBR on the tting of K l by inverse analysis

Eect of x e on simulated drying kinetics

The above comparison may indicate that the main factor which can aect the simulation results is K l , rather than x e . To prove this assumption, the mass loss curves simulated by using the same K l value but dierent x e values are shown in Fig. 6.11. After drying around one year, the dierence of the relative mass loss between x e = 1 cm and 15 cm is only very small. This shows that x e is less sensitive than K l in the numerical modelling.

Relation between K l and x e

The value of x e is adjusted between 0 and 15 cm to calibrate K l based on the experimental mass loss curve during a drying experiment at controlled RH. The relation between x e and K l for the three studied materials are shown in Fig. 6.12. Points at x e = 0 corresponds to the rst type boundary condition (see Eq. 1.65). 6.12 shows that a very good relation between x e and K l for all the studied materials.

The K l values at x e = 15 cm are around two times larger than values at x e = 0 cm, but they are still in the same order of magnitude. Hence, the dierences between K l values calibrated by the rst type and the diusive boundary conditions are not so large.

In Fig. 6.12, K l values determined by the ux boundary condition are highlighted. These values are located in the gure between x e = 0 to 6 cm. In our experiments, the distance between the salt solution surface and samples is between 5 to 15 cm. Even if the maximum value of x e = 15 cm is chosen, K l values in Fig. 6.12 are still very close to K l values determined by using the ux boundary condition. The comparison is able to conclude that the inverses analysis method using these three dierent boundary conditions (rst type, ux and diusive) can show good agreements in terms of tted K l values. This may indicate that the inverse determination of K l is quite reliable and shows very small inuence of the type of boundary condition.

Partial conclusion

In this section, a coupled model has been used to simulate moisture transport in the mass boundary region and in the material. This model only considers vapour diusion in the MBR according to our experimental conditions. The main conclusions are:

1) If the same K l combining with dierent values of x e is used, the coupled model provides quite similar mass loss curves as in the case using the ux boundary condition.

2) The relationship between x e and K l reveals that dierent x e values from 0 cm to 15 cm do 6.3. Comparisons of K l values determined by dierent methods not aect moisture transport within the material very much.

3) All results and discussions show that the inverses analysis method using the three usual boundary conditions (rst type, ux and diusive) provide quite similar results of determined K l values on the basis of drying experiments. Hence, there is no necessity to take into account vapour diusion in the MBR for determination of K l in the case of laboratory experimental conditions.

Above observations are restricted to laboratory conditions in which only 1D vapour diusion is considered. If one wants to study natural conditions, which means that eect of air velocity in the MBR have to be taken into account, a 1D coupled model seems to be not so relevant because the air always ows along the surface of the material rather than in the vertical direction.

Although the air does not directly aect moisture transport in the vertical direction, it can certainly increase the drying rates.

Comparisons of K l values determined by dierent methods

Overall, there are nine methods which have been used to determine K l (see Fig. 6.1). All K l values corresponding to these methods are presented in Fig. 6.13 for the four studied materials (see Table 3.2). Values for the Katz-Thompson model are taken from the literature [START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF].

From the second to the fth methods (red bars in Fig. 6.13), K l values are calibrated by inverse analysis by tting drying kinetics. When simulations were performed with the diusive boundary condition (see Section 6.2.1), the thickness boundary region x e was adjusted from 0 cm to 15 cm according to our experiments and a series of K l values has been determined. In Fig. 6 In this study, modelling of drying and wetting cycles in the case of cementitious materials has been described with comprehensive verications by experimental data. This section highlights the main conclusions of this work, which are shown in the sequence of chapters.

Modelling of isothermal drying

For modelling of drying and wetting cycles, models initially developed to simulate moisture transport under periodic boundary conditions were reviewed rstly. Three methods were summarised: using separate transport coecients for drying and wetting, non-hysteresis modelling and hysteresis modelling. The literature [START_REF] Johannesson | A two-phase moisture transport model accounting for sorption hysteresis in layered porous building constructions[END_REF][START_REF] Johannesson | A numerical approach for non-linear moisture ow in porous materials with account to sorption hysteresis[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF] has pointed out that taking into account hysteresis can improve the simulation results. However, there were no further experimental data to consolidate this conclusion, neither in-situ nor laboratory data. Therefore, this study is devoted to modelling moisture transport with hysteretic eects and to providing experimental data to verify the modelling results and relevancy.

The second step is to nd a sucient moisture transport model. On the basis of previous studies [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF][START_REF] Coussy | Mechanics of porous continua[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Coussy | Mechanics and physics of porous solids[END_REF][START_REF] Bear | Modeling Groundwater Flow and Contaminant Transport[END_REF], a sophisticated multiphase model, which is able to describe moisture transport through advection of liquid-water, advection of gas-phase and diusion of water vapour and dry air, was recalled. In experimental verications, two methods to calculate the relative permeabilities of liquid-water and gas-phase were compared. Both methods can provide good simulation results compared with experimental data. The only dierence can be found in the values of the liquid-water intrinsic permeability K l and the gas-phase intrinsic permeability K g validated by these two methods.

Theoretically, K l and K g should be the same because there correspond to intrinsic properties of the material and only dependent on the pore structure. However, measurements of K g show that K g is much larger than K l [START_REF] Hamami | Inuence of mix proportions on microstructure and gas permeability of cement pastes and mortars[END_REF][START_REF] Loosveldt | Experimental study of gas and liquid permeability of a mortar[END_REF][START_REF] Kameche | Measurement of the relative gas permeability of ordinary concrete: inuence of saturation degree and the sample size[END_REF]. Hence, both cases (K l = K g or K l K g ) were considered in this study. According to the asymptotic analysis, it was able to distinguish quite permeable and weakly permeable porous materials. For the case of cementitious materials, the use of K l = K g implies that cementitious material can be considered as weakly permeable media, while materials are apt to show quite permeable behaviour if K l K g . Gas pressure variations, including overpressure and underpressure, were observed for weakly permeable materials. But for quite permeable materials, gas pressure in the material always remains constant (equal to the atmospheric pressure). If further neglecting advection of gas-phase, a diusion-like equation only with liquid-water transport can be obtained for the case K l = K g [START_REF] Mainguy | Modèles de diusion non linéaire en milieu poreux. Application à la dissolution et au séchage des matériaux cimentaires[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF]. But for the case K l K g , if advection of gas-phase is negligible as well, a simplied model with advection of liquid-water and vapour diusion is obtained. This simplied model was used to simulate moisture transport under drying and wetting cycles conditions in Chapter 5.

Dierent transport modes were investigated for transport both at the boundary and in the material. In the high water content range, advection of liquid-water is predominant, while in the low water content range, vapour transport plays a main role. These dierent transport modes are related to the material properties, initial and boundary conditions. For natural conditions, two transport modes generally occur simultaneously.

Modelling of drying and wetting cycles

The third step for modelling is to nd an appropriate hysteresis model. Thus, detailed comparisons of hysteresis models have been conducted, which include models for the tting of the main sorption curves, for the prediction of the main adsorption curve based on the known desorption curve, and for the simulation of scanning loops. Quantitative criteria were used to evaluate the relevancy of the tting. Comparisons showed that most hysteresis models taken from soil science can be used for cementitious materials.

Among the selected main curves tting equations, the three-parameter models showed better results than the two-parameter models. Models selected for predicting one main isotherm based on the other main isotherm seem not relevant. Concerning hysteresis models, the empirical ones 7.1. General remarks showed a slight better prediction for the rst scanning curves than the domain theory models, but the pumping eect remains a big issue for the empirical model. The domain theory models can avoid this non-physical behaviour and can provide closed scanning loops. This is one of the main advantages to use conceptual models.

After comparisons of hysteresis models, a new method to determine the main desorption and main adsorption curves by using the fewest experimental data was proposed. To predict both desorption and adsorption isotherms, ve points with suggested positions (three for the main desorption curve and two for the main adsorption curve) are enough for ordinary cementitious materials. For some high performance materials, the use of one or more points in the low RH range can greatly improve the tting accuracy.

The fourth step for modelling of drying and wetting cycles is to prepare the input data for modelling and experimental data for verication. Experimental investigations about eects of varying RH at the boundary on moisture distribution within cementitious materials were carried out. Measurements of mass loss curves and saturation proles were carried out during drying, wetting and re-drying. Comparisons of mass loss curves calculated based on data from gamma-ray attenuation method and measured by weighing showed that gamma-ray attenuation technique can provide very reliable results to assess the repartition of moisture within the material.

Water vapour sorption isotherms for various cement pastes were measured by using a dynamic vapour sorption instrument (DVS). Weibull's equation [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF] was used to extrapolate the mass loss curve measured by DVS. This equation can yield very good tting results. In DVS measurements, the main adsorption isotherms were not measured. Hence, the method proposed in Chapter 3 was employed to estimate the main adsorption curve thanks to the measured main desorption curve and the rst adsorption scanning curve. Results showed that this method can provide satisfactory shapes of the main adsorption curves.

The fth step for modelling is to couple hysteresis models with a moisture transport model.

The proposed method, involving two hysteresis indexes to implement hysteresis models into moisture transport models, showed a high applicability.

The method can be used for both conceptual and empirical hysteresis models. Comparisons with measured mass loss curves and saturation proles concluded that modelling taking into account hysteresis explicitly provides better results than non-hysteresis modelling.

It is principally because hysteresis models can provide a more relevant RH-S relation at the boundary. Among hysteresis models, because improved Rubin's model (an empirical model) [START_REF] Rubin | Numerical method for analyzing hysteresis-aected, post-inltration redistribution of soil moisture[END_REF][START_REF] Zhang | A review and statistical study of existing hysteresis models for cementitious materials[END_REF] yields non-physical scanning loops, characterised by pumping errors, this model did not provide as good results as Mualem Model II (a conceptual model) [START_REF] Mualem | A conceptual model of hysteresis[END_REF]. Therefore, Mualem Model II is considered as the one recommended for modelling of moisture transport under varying boundary conditions.

Comparison of dierent forms of Richards' equation revealed that the mixed form, using ∇P c as the driving force, is a better choice which provides better simulation results than the form using ∇S as the driving force.

Comparisons with dierent cycle durations showed that smaller cycle durations and boundary RH amplitude cause smaller errors for non-hysteresis modelling. The errors increase with the cycle duration and boundary RH amplitude. For long term simulations, dierent initial steps (initial drying and initial wetting) did no show signicant dierences in mass loss curves and saturation proles. Furthermore, experimental verications showed that it is not necessary to consider hysteresis in relative permeability to liquid-water k rl (S).

From a practical point of view, one important factor is the moisture penetration depth x p which can be dened under periodic boundary conditions. For non-hysteresis modelling, this study conrmed that x p is related to the cycle duration and the RH amplitude, but not to the initial condition. However, for the hysteresis modelling, if the initial saturation is lower than 75% it has a clear inuence on x p . Hence, to estimate x p with hysteretic eects, our model is more applicable for initial RH values higher than 75%. Comparisons of x p revealed that the non-hysteresis modelling may overestimate x p in the case of long cycle durations and large RH amplitudes.

Relevancy of the determination of the liquid-water intrinsic permeability

In the last part of this thesis, two perspectives were introduced to provide more information for the determination of the intrinsic permeability to liquid-water K l . The rst perspective focused on tting measured apparent diusivity values to determine K l . The resistance factor to vapour diusion can also be assessed by this kind of tting through the determination of the parameter

Limitations and perspectives for future research

The work conducted in this thesis represents a contribution to the numerical simulation of durability issues concerning concrete structures. Due to knowledge and time limitations, there are several perspectives, which can be proposed for the future research.

1) Gas pressure variations (overpressure and underpressure) have been found in numerical simulations. However, gas pressure variations have not been veried by experimental data.

The tentative idea is to pre-deploy pressure sensors when the samples are cast. Hence, during the drying and wetting measurements, gas pressure variations can be measured.

2) When comparing dierent hysteresis models, the conclusion was that no appropriate model can be used to predict the main adsorption curve only based on the measured main desorption curve. But it still remains interest to carry out further studies on this topic because the main adsorption curve is not easy to measure compared to the main desorption curve. The discussion about reasons of hysteresis showed that dierent water states can be observed in dierent pore size. Generally, the main desorption and main adsorption curves are very close in the low RH range because adsorbed water in gel is predominant. This water does not cause signicant hysteresis. The ink-bottle eect and dierent contact angles lead to more signicant hysteresis in the capillary water dominated range. The main idea is to infer the main adsorption isotherm based on the theoretical relation between two main curves and the measured main desorption isotherm. The relation between two main curves can be established according to reasons of hysteresis such as the ink-bottle eect [START_REF] Espinosa | Inkbottle pore-method: Prediction of hygroscopic water content in hardened cement paste at variable climatic conditions[END_REF], the assumption of dierent shapes of pores [START_REF] Ranaivomanana | Toward a better comprehension and modeling of hysteresis cycles in the water sorptiondesorption process for cement based materials[END_REF][START_REF] Ranaivomanana | Prediction of relative permeabilities and water vapor diusion reduction factor for cement-based materials[END_REF] and the assumption of dierent levels of connectivity [START_REF] Pinson | Inferring pore size and network structure from sorption hysteresis[END_REF]. The measured main desorption will be used to calibrate parameters in the model.

3) In the current research, the proposed modelling method has been veried by experimental data. However, it still lacks the in-situ measured data for verication. Further research needs to nd data from literature or to carry out measurements in the eld.

4) In this study, the proposed model focuses on moisture transport in the hygroscopic range.

But for natural conditions this model needs to be extended to the liquid-water suction range, in which the material is in direct contact with liquid-water. For modelling of this transport process, the modication to the capillary curve (P c -S) is needed. Firstly, measured sorption isotherms should be extended to retention curve range, which is far above the hygroscopic range due to water in large capillary pores or pores connected by microcracks [START_REF] Ranaivomanana | Toward a better comprehension and modeling of hysteresis cycles in the water sorptiondesorption process for cement based materials[END_REF][START_REF] Ranaivomanana | Prediction of relative permeabilities and water vapor diusion reduction factor for cement-based materials[END_REF]. Secondly, the currently used equations for sorption isotherms (all main curve tting equations in Chapter 3) are not sucient because they provide innite moisture capacity at S = 1. Therefore, a new equation which is able to avoid this drawback is needed such as an equation proposed in latest reference [START_REF] Zhou | Predicting water permeability and relative gas permeability of unsaturated cement-based material from hydraulic diusivity[END_REF]. Then, the extended model can be applied to concrete structures under more severe conditions, such as rain-drying weather conditions, tidal zone, groundwater uctuating zone, etc.

5) It is clear that transport mechanisms in dierent pore size are dierent. In gel pores, vapour is transported by surface diusion, Knudsen diusion, etc. But in capillary pores, Knudsen diusion is mixed with ordinary diusion. Liquid-water can move due to capillary pressure or can condense on pore walls because of adsorption. Thus, the further research has to focus on the integration of these kinds of moisture transport mechanisms for dierent size of pores in a unique model.

6) When comparing dierent models to calculate the liquid-water relative permeability, these models were not veried by experimental data due to the diculty to measure this kind of data. But an indirect method can be used based on the relationship between k rg and k rl indicating in Mualem model-a (see Eq. 2.10) because k rg is much easier to measure.

Further work will carry out this kind of research and apply research results to simulate moisture transport.

7)

As it has been mentioned in the general introduction, rarely pure water is involved in the degradation processes. Moisture transport is always the essential mechanism able to induce other durability issues for concrete structures; therefore, moisture transport has to be coupled with other transport phenomena (e.g., heat, ions) to study the moisture behaviour in cementitious materials under drying and wetting cycles conditions. Moisture transport under temperature gradients has been widely studied in the literature [START_REF] Philip | Moisture movement in porous materials under temperature gradients[END_REF][START_REF] Milly | Moisture and heat transport in hysteretic, inhomogeneous porous media: a matric head-based formulation and a numerical model[END_REF][START_REF] Kerestecioglu | Theoretical and computational investigation of simultaneous heat an d moisture transfer in buildings: Eective penetration depth theory[END_REF][START_REF] Hagentoft | Simplied analysis of combined heat, moisture and air transport for one-dimensional cases[END_REF][START_REF] Derluyn | Hysteretic moisture behavior of concrete: Modeling and analysis[END_REF] and it is not dicult to couple with moisture transport. Furthermore, water in cementitious materials is always mixed with chemical components, which can decrease the chemical activity of the liquid-water and further aect liquid-water transport. Therefore, the liquidwater transport potential such as P c has to be modied according to the change of ions concentration [START_REF] Baroghel-Bouny | Modelling of isothermal coupled moistureion transport in cementitious materials[END_REF][START_REF] Baroghel-Bouny | Prediction of chloride binding isotherms of cementitious materials by analytical model or numerical inverse analysis[END_REF]. In experiments, there were no external sources of ions so it was considered that the chemical eect on moisture transport was fully included in the measured sorption data. For each RH value reached in the material during drying or wetting, the eect of the chemical activity is similar as on the position on the sorption curve for the same RH value. Thus, Kelvin's equation for P c does not need to be modied.

  dénition d'une profondeur pour laquelle le prol hydrique du béton est modié par les variations périodiques d'humidité relative permet de mieux comprendre comment la modélisation de la pénétration des espèces ioniques est inuencée par les cycles d'humidication-séchage. Par ailleurs, notre analyse révèle qu'il est pertinent de considérer l'eet de Knudsen pour la diusion de la vapeur an d'améliorer la prédiction de la diusivité apparente. Mots clefs : Matériaux cimentaires ; Transferts hydriques ; Hystérésis ; Isothermes de sorption de vapeur d'eau ; Cycles d'humidication-séchage ; Simulation numérique ; Investigations expérimentales ; Variation de pression de gaz ; Modes de transports ; eets de Knusen ; Diusion apparente ; Couche limite massique. Fitted desorption curves for the selected materials by using van Genuchten's equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Predicted α(S)τ for three materials calculated by Millington's and Thiéry's proposals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. 18

 18 Ration of liquid-water ux to the total ux in the material during drying of Paste4 (RH 0 = 89% and RH e = 53.5%). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 Simulation results of mass loss curves, including total, liquid-water and vapour, during drying of Paste4 (RH 0 = 89% and RH e = 53.5%). . . . . . . . . . . . . . 2.20 Comparison of the contribution of liquid-water transport to total mass loss during drying of Paste3, Paste4 and Paste5 for initial RH 0 = 90% and varying external relative humidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Schematic representation of WVSIs. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Schematic representation of Mualem's diagram for the main desorption and adsorption processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . 6 . 4

 364 Main curves tting results for Paste3 material. . . . . . . . . . . . . . . . . . . . 3.7 Schematic drawing of Feng and Fredlund's model to predict the adsorption branch from the desorption branch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Prediction results of the main adsorption curves for Paste2 material. . . . . . . . 3.9 Comparison of the predicted main adsorption branch based on the measured main desorption branch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.10 The scanning loops simulated by Mualem's diagram. . . . . . . . . . . . . . . . . 3.11 An example of scanning loops calculated by Mualem Model II. . . . . . . . . . . 3.12 Schematic representation of Mualem's diagram for scanning processes described with the dependent domain model. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 The calculation of p d during a drying process in Mualem dependent model. . . . 3.14 Schematic representation of the calculation of scanning curves by Li's model. . . 3.15 Predicted wetting scanning curves and drying scanning curves for Paste3. . . . . 3.16 Comparison results for wetting scanning isotherm predicted by selected hysteresis models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Predicted scanning loop for Paste3 compared with experimental data. . . . . . . 3.18 Denitions of DPE and WPE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Simulated hysteresis loops by Li's model and improved Rubin's model (taking Paste3 as an example). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 The evolution of PEs calculated based on two RH uctuations. . . . . . . . . . . 3.21 The mean residual for all materials at each measured point. . . . . . . . . . . . . 3.22 Comparison of results calculated by three versions of FX model. . . . . . . . . . . 3.23 Comparison of calculated R 2 and MME for FX model. . . . . . . . . . . . . . . . 3.24 Comparison of calculated R 2 and MME for VG3 model. . . . . . . . . . . . . . . 3.25 Results calculated by FX and VG3 models for Paste2 in the case of adding one more point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The gamma-ray signal intensity at dierent positions inside materials after 1-day curing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Proles of porosity measured by gamma-ray attenuation for cement pastes. . . . 4.3 Examples of saturation proles measured by gamma-ray attenuation. . . . . . . xiv 4Comparisons of mass loss curves measured by gamma-ray attenuation and by global weighing for CEMI + SF pastes. . . . . . . . . . . . . . . . . . . . . . . . 4.5 An example of controlled RH used in measurements (blue line) and measured mass changes (red line) at dierent RH. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 An example showing dierent extrapolation methods. . . . . . . . . . . . . . . . 4.7 One example of the comparison of original and extrapolated desorption curves measured by DVS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Pressure plate extractors used in measurements at Lund University. . . . . . . . 4.9 WVSIs measured by DVS and pressure plate for CEM I materials. . . . . . . . . 4.10 Predicted main adsorption curves. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Implementation of the hysteresis modelling. . . . . . . . . . . . . . . . . . . . . . 5.2 Schematic representation of a sample during the drying and wetting measurements.141 5.3 Simulated sorption isotherms vs. experimental data for Paste3. . . . . . . . . . . 5.4 Comparisons of mass loss curves simulated by non-hysteresis and hysteresis modellings for Paste3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Comparisons of mass loss curves during drying RHe = 63.2% simulated by using two tted K l values for Paste3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Comparisons of simulated mass loss curves with experimental results for Paste4 and Paste5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Saturation proles simulated by non-hysteresis (ω = 1.0) and hysteresis modellings are compared with experimental data for Paste4 and Paste5. . . . . . . . . . . . .

5. 10

 10 Comparisons of simulation results with measured data for PCEMIII06. . . . . . 5.11 Comparisons of simulation results with measured data for PCEMIII06. . . . . . 5.12 The dierences of boundary conditions used in non-hysteresis and hysteresis modellings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Simulation results performed with Mualem Model II and improved Rubin's model to investigate the PEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Simulation results of dierent cycle durations in the cases of non-hysteresis and hysteresis modelling by taking Paste3 as an example. . . . . . . . . . . . . . . . . 5.15 Simulation results in the case of using a 22% RH amplitude (between 64% and 86%) by taking Paste3 as an example. . . . . . . . . . . . . . . . . . . . . . . . . 5.16 Simulation results in the case of initial wetting by taking Paste3 as an example. . 5.17 The dierences of two forms of Richards equations (taking Paste3 as an example). 5.18 Evolutions of k rl (S) calculated by VGMb at dierent depths for Paste3 . . . . . . 5.19 Comparisons of mass loss curves calculated in the cases of with or without hysteresis in relation S-k rl for Paste3 (for the case of drying at 53.5%). . . . . . . . 5.20 Denitions of the undisturbed saturation S und and the moisture penetration depth x p after a number of cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21 Moisture penetration depth in relation with the cycle duration and the RH amplitude for the cases of non-hysteresis and hysteresis modelling by taking Paste3 as an example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.22 Comparisons of x p for dierent initial conditions and cycle durations calculated by using Mualem Model II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Methods to determine K l which are compared in this research. . . . . . . . . . . xv 6.2

4 . 1

 41 Materials and mix designs used in the present study. . . . . . . . . . . . . . . . . 4.2 RH cycles used during drying-wetting experiments. . . . . . . . . . . . . . . . . 4.3 The mass adsorption coecient for studied CEM I pastes. . . . . . . . . . . . . . 4.4 The comparison of global porosity (%) measured by weighing and gamma-ray for cement pastes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ρr 1 γN • m - 1 κ 1 λDF 1 µm 3 • m - 3 θm 3 • m - 3 A 2 C 1 CT 0

 11113333210 Abbreviation DPEDrying pumping error.

m 2 • s - 1

 21

m 2 • s - 1 D S Moisture diusion coecient under saturation gradient. m 2 s - 1 DT 1 E 1 fI

 21111 Moisture diusion coecient under temperature gradient.kg • m -1 s -1 K -1 D v Vapour diusivity.m 2 • s -Mass transport coecient between material and environment (emissivity). kg • Pa -1 • m -2 • s -1 xx E m Rate of mass exchange between liquid-water and vapour. kg • m -3 • s -(r, ρ) Total pore water distribution function. f (φ, S) Resistance factor. -Pore water distribution function related to pore necks. -H(P c ) Cumulative pore water distribution function related to pore necks. -Transmitted photon intensity. -J l , J v , J a Flux of liquid-water, vapour and dry air. kg • m -2 • s -1 k d Material constant for drying in the hysteresis model proposed by Johannesson et al. k m Mass transport coecient in the airside.kg • m -2 • s -1 K pThe slope of a scanning curve.-K v Vapour permeability. kg • Pa -1 • m -1 • s -1 k wMaterial constant for wetting in the hysteresis model proposed by Johannesson et al.-

l

  Length of the material. m l(ρ) Pore water distribution function related to pore bodies. -L(P c ) Cumulative pore water distribution function related to pore bodies. -Weighting factor used for the drying process in Mualem dependent model. -Weighting factor used for the wetting process in Mualem dependent model. -P atm Atmospheric pressure. Pa P cm Capillary pressure related to the medium pore radius.

m • s - 1 v0l

 1 e RH at the boundary. l , v v , v a Velocity of liquid-water, vapour and dry air. m • s -1 x p Moisture penetration depth in the surface layer. m x r Moisture penetration depth in the deeper layer. m Subscripts Initial value or at the starting point of a curve. 105 Parameters for materials dried at 105 • C. Liquid-water.satParameters for materials at the saturated state.

  The corresponding sorption isotherms.

Figure 1 . 2 :

 12 Figure 1.2: Schematic drawing of ink-bottle eect.

Figure 1 . 3 :

 13 Figure 1.3: Schematic drawing of contact eects.

Figure 1 . 6 :

 16 Figure 1.6: An example for the rst wetting scanning curve calculated by the hysteresis model

Figure 1 . 7 :

 17 Figure 1.7: A triangular representation of PM domain for the main desorption and adsorption curves. The main drying process starts from point B (the maximum RH) and decreases gradually to point O (the minimum RH). The main adsorption process is the inverse process.

S con = S c = rc 0 dV dr dr = 1 -

 1 exp(-Br c ) (1.54)

  58) (a) Drying scanning curves. (b) Wetting scanning curves.

Figure 1 . 10 :

 110 Figure 1.10: Scanning curves calculated in the DuCOM [13].

Figure 2 . 1 :

 21 Figure 2.1: Schematic representation of the geometry of the problem. The surface in bold black lines is sealed by aluminium sheets and the ellipse in thin line on the left-hand side represents

Figure 2 .

 2 Figure 2.2 shows quite good tting results, especially for the points in the moderate RH range. Some points in the very high RH range (e.g., RH = 97%) and at RH = 33% are not tted very well.

Figure 2 . 2 :

 22 Figure 2.2: Fitted desorption curves for the selected materials by using Eq. (2.4).

Figure 2 . 3 :

 23 Figure 2.3: Predicted α(S)τ for three materials calculated by Millington's[START_REF] Millington | Gas diusion in porous media[END_REF] and Thiéry's[START_REF] Thiéry | Modélisation du séchage des bétons -Analyse des diérents modes de transfert hydrique[END_REF] proposals.

  2.7) and the combination of van Genuchten's equation + Mualem model-b is denoted by VGMb. Here, only equations to calculate k rl for the main desorption process are introduced.

2

 2

( 2

 2 .11) where KL and KH are two normalized integral functions calculated based on capillary pressure.The expressions to these two functions are:

Figure 2 . 2 Figure 2 . 4 :

 2224 Figure 2.4 shows calculated k rl (RH) and k rg (RH) for Paste3 by using the above two methods. It is clear that Method-2 provides smaller k rl and k rg in the whole range of RH than Method-1.The relative permeability to gas-phase k rg calculated by Method-2 has been veried by

  Method-2 (K0 = 3.5 × 10 -19 m 2 ).

Figure 2 . 5 :

 25 Figure 2.5: Comparisons of simulated mass loss curves based on two dierent methods to calculate k rl and k rg for Paste4.

Figure 2 . 6 :

 26 Figure 2.6: Comparisons of calculated mass loss curves based on two methods to calculate k rl and k rg for Paste5.

  (a) Method-1 (K0 = 1.95 × 10 -21 m 2 ).

  Method-2 (K0 = 3.5 × 10 -19 m 2 ).

Figure 2 . 7 :

 27 Figure 2.7: Comparisons of simulated saturation proles based on two dierent methods to calculate k rl and k rg for Paste4.

  Method-2 (K0 = 160 × 10 -19 m 2 ).

Figure 2 . 8 :

 28 Figure 2.8: Comparisons of simulated saturation proles based on two dierent methods to calculate k rl and k rg for Paste5.

Figure 2 . 9 :

 29 Figure 2.9: Comparisons of simulated mass loss curves based on two dierent methods to calculate k rl and k rg for Concrete3 and Paste3.

  Concrete3 (K0 = 1.8 × 10 -21 m 2 ). Paste3 (K0 = 0.39 × 10 -21 m 2 ).

Figure 2 . 10 :

 210 Figure 2.10: Calculated saturation proles for Concrete3 and Paste3 (Method-1 to calculate k rl and k rg ).

2. 3 . 1

 31 Mass balance equations of gas and moisture By putting Eqs. (1.14)-(1.16) into the mass balance equations of gas-phase Eqs. (1.3) and (1.5),

14 Pg

 14 0.04 0.06 0.08 0.1 0.12 0.Initial S0 = 0.99.

Figure 2 . 11 :

 211 Figure 2.11: Proles of gas pressure during drying of Paste4 with dierent values of initial saturation.

  Initial S0 = 0.99.

Figure 2 . 12 :

 212 Figure 2.12: The maximum and minimum gas pressure during drying of Paste4 under various external RH with two dierent values of initial saturation.

Figure 2 . 13 :

 213 Figure 2.13: Calculated mass loss curve and saturation proles in the case K l K g based on Method-1 to calculate k rl and k rg for Paste5.

Figure 2 . 14 :

 214 Figure 2.14: Proles of gas pressure during drying of Paste4 with initial saturation S 0 = 0.99.

Fig. 2 .Figure 2 . 15 :

 2215 Fig. 2.15 for Paste4 by using VGMa model to compute k rl . Roughly speaking, point at S ≈ 0.42 (RH ≈ 0.56) divides the mass transport into two parts. Below this point, the dominant transport is vapour diusion (represented by f (φ, S)RH), and above this point, it is liquidtransport transport (represented by k rl (S)).

Figure 2 . 16 :

 216 Figure 2.16: Mass loss curves calculated by three versions of models for Paste4, using the same K l = 1.7 × 10 -21 m 2 validated by the multiphase model (seeTable 2.3). The external RH is

  .3). The external RH is 53.5%. a Obtained by drying at RH = 63.2%.

  Transport mode two.

Figure 2 . 17 :

 217 Figure 2.17: Schematic representations of two moisture transport modes.

  The case K l = Kg. The case K l < Kg.

Figure 2 . 18 :

 218 Figure 2.18: Ration of liquid-water ux to the total ux in the material during drying of Paste4 (RH 0 = 89% and RH e = 53.5%).

  The case K l = Kg. The case K l Kg.

Figure 2 . 19 :

 219 Figure 2.19: Simulation results of mass loss curves, including total, liquid-water and vapour, during drying of Paste4 (RH 0 = 89% and RH e = 53.5%).

Figures 2 .Figure 2 . 20 :

 2220 Figures 2.19a and 2.19b clearly show that at the beginning of drying (only several days) the mass loss related to liquid-water movement plays the major role, while the contribution of water vapour increases gradually. This is because the liquid-water is released from the pore network during drying, and more space is available for the transport of water vapour. At the end of simulation, the contribution of liquid-water movement in Figs.2.19a and 2.19b should correspond to the ratios of the bold black curves at the boundary in Figs. 2.18a and Fig.2.18b.The dierences between two cases K l = K g and K l K g are obvious in Fig.2.19. In the former case, advection of liquid-water is always predominant in the total mass loss during

Figure 3 . 1 :

 31 Figure 3.1: Schematic representation of WVSIs. The main desorption curve (BA), the main adsorption curve (AB), the rst scanning curve in drying (DE ), the rst scanning curve in wetting (C B) and the second scanning curve in wetting (E B) are illustrated.

(3. 1 )

 1 A rectangle diagram was proposed by Mualem (see Fig.3.2)[START_REF] Mualem | A conceptual model of hysteresis[END_REF] to depict how the independent domain model works. The horizontal axis represents the water distribution function h(r)

Figure 3 . 2 :

 32 Figure 3.2: Schematic representation of Mualem's diagram [53] for the main desorption and adsorption processes. The gray domain represents completely-lled pores and the blank domain stands for fully-dried pores.

  3.1) by adding two weighting functions (p d ≤ 1 and p w ≤ 1). θ = p d (r, ρ, θ)p w (r, ρ, θ)f (r, ρ)drdρ (3.7)

  data are gathered in Table3.2 from[START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] measured by the saturated salt solution method. These data can meet the above-mentioned comparison requirements very well. All materials are made from the same OPC cement (CEM I-52.5, according to EN 197-1 European standard). They include ordinary and high performance hardened cement pastes and concretes. Water-to-cement ratios range from 0.2 to 0.6. High performance materials are Concrete2 and Paste2, which contain 10% of silica fume to cement in weight. In order to reach chemical stability before the measurements, materials were sealed-cured for at least 6 months for concretes and 2 years for cement pastes.Then, cement pastes were crushed into small specimens with mass of about 8 g. Because the content of aggregates has a small inuence on the measured sorption curves[START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF], concrete specimens were cut into thin slices, 2 or 3 mm a The same cement is used for Concrete1 and Paste1, but it is from dierent supplier to other materials. b Silica fume.thickness with mass between 20 and 100 g. This size still can keep small sands and partial ITZ eects. Measurements started from desorption at RH = 100%, decreasing step-by-step to RH = 3%, which is considered as the dry reference state. Then, RH increased stepwise to 100% to obtain the adsorption isotherm. To reduce the variations of RH during the measurements as well as the carbonation risk, only a small hole on the top of the desiccator is open for weighing.

3 .3 and 3 . 4 .

 334 Results for two materials (Paste2 and Paste3, representing a high performance and an ordinary materials) are illustrated in Figs. 3.6 and 3.5. Overall, all models provide satisfactory tting results (mean R 2 adj > 0.98).

Figure 3 . 3 :

 33 Figure 3.3: Comparison of R 2 adj and NME for desorption isotherms. Each circle (•) in the gure stands for one material (see Table 3.2). Concrete2 and Paste2 are highlighted. The error bars represent the standard deviation for each studied statistical criteria (R 2 adj and NME) and each model.

3 . 5 .

 35 The performance of the tri-modal model, nearly passing through all measured points, is obviously better than the uniand the bi-modal models for both main isotherms in the low RH range. In terms of statistical analysis, using a tri-modal model improves the tting results greatly (see highlighted values of R 2 adj and NME for Concrete2 and Paste2 in Figs. 3.3 and 3.4). In the high RH range, the tri-modal model yields a similar shape of curve as the bi-modal model.Paste3(a) Paste3 material.(b) Enlarged gure for Paste3.

Figure 3 . 6 :

 36 Figure 3.6: Main curves tting results for Paste3 material. To make the display clearly, only three models, VG2, FX and bi-modal models, are chosen to represent uni-and multi-modal models. Fitting results for low RH values are shown in the enlarged gure.

3 . 5 )

 35 . Combining Eq. (3.16) with Eq. (3.6), an equation calculating θ w (P c ) from θ d (P c )

Figure 3 . 7 :

 37 Figure 3.7: Schematic drawing of Feng and Fredlund's model [160] (improved by Pham et al. [163]) to predict the adsorption branch from the desorption branch. Three points A, D and C are used to calculate point B.

1/b d ( 3 . 24 )

 324 Points A, C, D and E should satisfy the relation: |P c,wA -P c,wE | = |P c,dD -P c,dC | (3.25) Capillary pressure of point B on the adsorption curve is obtained by solving Eqs. (3.22)-(3.25). P c,wB = P c,wAtwo parameters (a w and b w ) used in Feng and Fredlund's model [160] are calculated on the basis of capillary pressure at points A and B. a w = S wA P c,wA bw 1 -S wA (3.27)

The subscript ( 1 ,

 1 w) represents the rst wetting scanning curve. The accuracy of this model is sensitive to the position of the starting point of the wetting scanning curve. If the scanning curve starts at a high RH, the model tends to be the same as Mualem Model II-1 [170]. If the scanning curve starts at a low RH, it will only use the wetting scanning curve to predict the adsorption isotherm. According to the position of the starting point in available experimental data, comparisons were performed in three levels: low RH (RH 1 = 12%), medium RH (RH 1 = 33%) and high RH (RH 1 = 53%). Not all materials in the database can reach this requirement. Only ve materials (Concrete1, Concrete2, Concrete3, Paste2 and Paste3) are used for validation and comparisons.

II- 1 (

 1

Figure 3 . 8 :

 38 Figure 3.8: Prediction results of the main adsorption curves for Paste2 material. Two points in the Feng and Fredlund's model[START_REF] Feng | Hysteretic inuence associated with thermal conductivity sensor measurements[END_REF] (improved by Pham et al.[START_REF] Pham | A practical hysteresis model for the soilwater characteristic curve for soils with negligible volume change[END_REF]) are at RH = 67% and RH = 46%.

MII- 1 PFigure 3 . 9 :

 139 Figure 3.9: Comparison of the predicted main adsorption branch based on the measured main desorption branch. Results are from models: Mualem Model II-1 (MII-1) [170], Parlange's model (P) [168] (modied by Braddock et al. [169]), Feng and Fredlund model's (FF) [160] (improved by Pham et al. [163]), xed points version of Feng and Fredlund's model (FF-f ) and Mualem Model

S 1 ,

 1 d (P c ) = S w (P c,1 ) -[L(P c,1 ) -L(P c )] [1 -H(P c )] (3.30)

  is deduced according to Mualem's diagram:S N,w (P c ) = S(P c,N ) + [L(P c ) -L(P c,N )] [1 -H(P c,N )] for L(P c ) ≤ L(P c,N -1 ) (3.32)where S(P c,N ) is the saturation at the starting point of the current scanning curve, which must be the end point of the previous scanning curve. If L(P c ) ≥ L(P c,N -1 ), Mualem's diagram indicates that L(P c,N -3 ) will be used instead of L(P c,N -1 ) until the scanning curve reduces to the main

Figure 3 . 10 :

 310 Figure 3.10: The scanning loops simulated by Mualem's diagram. The rst drying, second wetting and third drying scanning curves are shown in this gure.

S

  N,d (P c ) = S(P c,N ) -[L(P c,N ) -L(P c )] [1 -H(P c )] for L(P c ) ≥ L(P c,N -1 ) (3.33) If L(P c ) ≤ L(P c,N -1 ), L(P c,N -3 ) will be used instead of L(P c,N -1 ) for calculation. Equations (3.31-3.33) comprise a set of formulas to calculate arbitrary scanning loops.

Figure 3 . 11 :

 311 Figure 3.11: An example of scanning loops calculated by Eqs. (3.31)-(3.33).

(3. 34 )Figure 3 . 12 :

 34312 Fig. 3.13).The function for the rst scanning curve in drying (order N = 1), starting on the adsorption

S 1 ,

 1 w (P c ) = S d (P c,1 ) + p d (S 1 ) [1 -S w (P c,1 )] [S w (P c ) -S w (P c,1 )](3.36) where p d (S 1 ) is p d at the starting point (S d (P c,1 ), P c,1 ). Functions for scanning loops can be easily derived based on Mualem's diagram[START_REF] Mualem | Dependent domain model of capillary hysteresis[END_REF][START_REF] Mualem | A hysteresis model based on an explicit domain-dependence function[END_REF][START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF]. For the wetting scanning curves of order N (odd number and > 1), the expression is written as:S N,w (P c ) = S(P c,N ) + p d (S N ) [1 -S w (P c,N )] [S w (P c ) -S w (P c,N )] , S w (P c ) ≤ S w (P c,N -1 ) (3.37)For the drying scanning curves of order N (even number), the expression is: S N,d (P c ) = S(P c,N ) -p d (S) [1 -S w (P c )] [S w (P c,N ) -S w (P c )] , S w (P c ) ≥ S w (P c,N -1 ) (3.38) One can see that wetting scanning curves are provided by explicit formulas, while equations for drying scanning curves are implicit forms with respect to the current capillary pressure P c . Therefore, a drying scanning curve must be calculated by an iterative method. Substitution of Eq. (3.34) into Eq. (3.36) yields the same equation for the rst scanning curve in wetting as the one used by Mualem Model II. It means that these two models will share the same rst wetting scanning curve. An example of calculated p d .

Figure 3 . 13 :

 313 Figure 3.13: The calculation of p d during a drying process in Mualem dependent model [55].

  d ln[P c (S)] = ln[ Pc (S)] -ln(P c,1 ) ln[P c (S)] -ln(P c,1 ) β d ln[ Pc (S)] (3.39) where β is a material parameter. Pc (S) = P c,d (S) is capillary pressure on the main desorption branch for the drying process and Pc (S) = P c,w (S) is on the main adsorption branch for the wetting process (see Fig. 3.14). By integrating Eq. (3.39), one can obtain the expressions for a drying scanning curve: ln[P c (S)] = ln(P c,1 ) + |ln[P c,d (S)] -ln(P c,1 )| β d +1 -|ln(P c,d1 ) -ln(P c,1 )| β d +1 1/(β d +1) (3.40) and for a wetting scanning curve: ln[P c (S)] = ln(P c,1 ) -|ln(P c,1 ) -ln[P c,w (S)]| βw+1 -|ln(P c,1 ) -ln(P c,w1 )| βw+1 1/(βw+1)

Figure 3 . 14 :

 314 Figure 3.14: Schematic representation of the calculation of scanning curves by Li's model [59].

  dP c,w (S) dS + c w |P c -P c,w (S)| P c,d (S) -P c,w (S) -|P c -P c,w (S)| (3.44) where c d and c w are the internal state variables for drying and wetting processes, respectively. In Eqs. (3.43) and (3.44), the slope of the main sorption isotherm (-dP c,d (S) dS ) and the distance between two main isotherms (P c,d (S) -P c,w (S)) represent irreversible eects. The part of |P c -P c,d (S)|, depending on the current capillary pressure, represents the reversible eect. The coecients of c d (or c w ) is used to control the reversible eect and it is a function of the water content. To simplify the application of this model, Wei et al. [61] assumed constant values of c d and c w . Clearly, the reversible eect can be observed at the beginning of the scanning curve due to the large initial slope, like it has been illustrated by Wei et al. [61] and in the model of Hogarth et al. [175, 176] for soils. When Eqs. (3.43) and (3.44) are applied to simulate scanning curves for cementitious materials, the model predicts that the scanning curve crosses the main sorption isotherm because of the signicant reversible eect. However, experimental data of the scanning curves [23, 12] do not show such a reversible eect. Hence, a modication to Eqs. (3.43) and (3.44) is proposed in this thesis to reduce the reversible eect and new relations are yielded for a drying scanning curve: K p = -dP c,d (S) dS + c d |P c -P c,d (S)| P c,d (S) -P c,w (S) (3.45) and for a wetting scanning curve. K p = -dP c,w (S) dS + c w |P c -P c,w (S)| P c,d (S) -P c,w (S) (3.46)

  2), have been studied. One example of a comparison between experimental data and predictions (Paste3) performed by above models are provided in Fig. 3.15 (other comparison results are included in the document supplementary materials). The results obtained by Mualem Model II [53] and Mualem dependent model [55] are illustrated in the same gure. For each empirical model, one parameter should be valid for all measured wetting scanning curves as well as the other one should be valid for all measured drying scanning curves. Thus, during tting, these unknown parameters (β d and β w in Eqs. 3.40 and 3.41, c d and c w in Eqs. 3.45 and 3.46 and γ d and γ w in Eqs. 3.48 and 3.49), are optimized to be suitable for all measured wetting (or drying) scanning curves for each material.

  (a) Mualem's models[START_REF] Mualem | A conceptual model of hysteresis[END_REF][START_REF] Mualem | A modied dependent-domain theory of hysteresis[END_REF].(b) Li's model[START_REF] Li | Modelling of hysteresis response for arbitrary wetting-drying paths[END_REF].(c) Modied Wei's model (proposed). (d) Improved Rubin's model (proposed).

Figure 3 . 15 :

 315 Figure 3.15: Predicted wetting scanning curves (solid lines) and drying scanning curves (dasheddotted lines) for Paste3 compared with experimental data (symbols) from [23].

Figure 3 . 16 :

 316 Figure 3.16: Comparison results for wetting scanning isotherm predicted by Mualem Model II (M-II) [53], Li's model (L) [59], modied Wei's model (W-M) [61] and improved Rubin's model (Im.-Rubin) [177].

  Modied Wei's model (proposed).

  Improved Rubin's model (proposed).

Figure 3 . 17 :

 317 Figure 3.17: Predicted scanning loop (solid lines) for Paste3 compared with experimental data

Figure 3 .

 3 Figure 3.18 implies amplitudes of DPE and WPE can be quantied.

Figure 3 . 18 :

 318 Figure 3.18: Denitions of DPE and WPE.

Figure 3 . 19 :

 319 Figure 3.19: Simulated hysteresis loops by Li's model [59] and improved Rubin's model (taking Paste3 as an example).

  RH fluct. WPE, High RH fluct. DPE, Low RH fluct. WPE, Low RH fluct. (b) Improved Rubin's model.

Figure 3 . 20 :

 320 Figure 3.20: The evolution of PEs calculated based on two RH uctuations.

3 . 21 .

 321 Points with the minimum values of MR are chosen as the more relevant positions to determine the desorption or adsorption isotherm.

Figure 3 . 21 :

 321 Figure 3.21: The mean residual for all materials at each measured point.

Figure 3 . 22 :

 322 Figure 3.22: Comparison of results calculated by three versions of FX model. Points chosen to solve equations are highlighted (solid symbols).

  Figures 3.23 (FX) and 3.24 (VG3) also show that this simplication can provide the similar accuracy to the original method.

Figure 3 . 23 :

 323 Figure 3.23: Comparison of calculated R 2 and MME for FX model. Each diamond ( ) or circle (•) for R 2 or MME stands for one material. + is the mean value of R 2 and × is the mean value of MME.

Figure 3 . 24 :

 324 Figure 3.24: Comparison of calculated R 2 and MME for VG3 model. The other explanations of legend are the same as Fig. 3.23.

Figure 3 . 25 :

 325 Figure 3.25: Results calculated by FX and VG3 models for Paste2 in the case of adding one more point.

  The hardened samples were submitted to gamma-ray attenuation test to check the homogeneity inside the sample after one-day curing. Results for two examples (CEM I with w/c = 0.6 and CEM III with w/c = 0.5) are presented in Fig.4.1. The density of the material is quite uniform throughout samples, implying no severe aggregation occurred.

Figure 4 . 1 :

 41 Figure 4.1: The gamma-ray signal intensity at dierent positions inside materials after 1-day curing.

  The porosity proles measured by gamma-ray attenuation for nine mixtures are shown in

Figure 4 . 2 :

 42 Figure 4.2: Proles of porosity measured by gamma-ray attenuation for cement pastes. Error bars are added.

4. 2 . 3 4 )

 234 Proles of saturation Saturation during experiments at each point is calculated based on measured densities at saturated and dried states. As mentioned above, variations of measured gamma-ray intensities at dierent time are caused by the dierent moisture content in the material. Thus, S is calculated based on measured gamma-ray intensities at saturated and dried states at each depth of the sample. Two examples of measured saturation proles for PSF05 and PSF06 during drying, wetting and re-drying are shown in Fig. 4.3. Because the duration for each process is not long enough, the inuence of varying RH at the boundary on water content has not reached the inside part of the material. Saturation in PSF05 beyong x = 40 mm shows no changes during experiments. For PSF06, there is only a small decrease. Both Figs. 4.

  for 28d Drying at 33% for 91d Wetting at 80% for 84d Re-drying at 53.5% for 58d (a) PSF05 (CEM I+SF, w/b = 0.5) for 28d Drying at 33% for 91d Wetting at 80% for 84d Re-drying at 53.5% for 58d (b) PSF06 (CEM I+SF, w/b = 0.6)

Figure 4 . 3 :

 43 Figure 4.3: Examples of saturation proles measured by gamma-ray attenuation.

  , w/c=0.5 Weighing, w/c=0.6 Gamma-ray, w/c=0.6

Figure 4 . 4 :

 44 Figure 4.4: Comparisons of mass loss curves measured by gamma-ray attenuation and by global weighing for CEMI + SF pastes.

Figure 4 . 5 :

 45 Figure 4.5: An example of controlled RH used in measurements (blue line) and measured mass changes (red line) at dierent RH.

Figure 4 . 6 :

 46 Figure 4.6: An example showing dierent extrapolation methods.

Figure 4 .

 4 Figure 4.7 shows one example of extrapolated mass loss curves and comparison between orig-inal and predicted desorption curves using Eq. (4.7). Since the equilibrium state was almost reached during each measurement process, the extrapolation does not show signicant improvement to the desorption curve, especially in high and low RH ranges. However, in RH = 30-50% range, the extrapolation improves the results. Baroghel-Bouny[START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] has reported that this feature is related to the transitional range between capillary porosity and C-S-H porosity. Sorption

Figure 4 . 7 :

 47 Figure 4.7: One example of the comparison of original and extrapolated desorption curves measured by DVS.

Figure 4 . 8 :

 48 Figure 4.8: Pressure plate extractors used in measurements at Lund University. The circled specimen was blown away from the center due to the suddenly increased gas pressure.

4. 4 .

 4 Determination of the main adsorption curve

Figure 4 . 9 :

 49 Figure 4.9: WVSIs measured by DVS and pressure plate for CEM I materials. Two points measured by pressure plate have been highlighted (solid symbols).

  (f) PSF06 (CEM I+SF, w/b = 0.6).

Figure 4 . 10 :

 410 Figure 4.10: Predicted main adsorption curves. Solid squares represent the points used to calculate the main adsorption curve.

Figure 5 . 1 :

 51 Figure 5.1: Implementation of the hysteresis modelling.

Figure 5 . 2 :

 52 Figure 5.2: Schematic representation of a sample during the drying and wetting measurements. The bold black lines represent the sealed sides of the sample, and the ellipse in the thin line on the right-hand side represents the surface that is exposed to ambient environment. The numbers indicate the positions of nodes that are chosen to show the calculated k rl and f (φ, S) in Fig. 5.8.

Figure 5 .Figure 5 . 3 :

 553 Figure 5.3 displays the comparison of measured and predicted sorption isotherms for Paste3. Main curves are tted by the two-parameter version of van Genuchten's equation (see Eq. 2.4) and scanning curves are predicted by Mualem Model II [53] (solid scanning curves) and improved Rubin's model [177, 144] (dashed scanning curves). The two parameters used in improved Rubin's model are optimized to t all measured scanning curves. They are γ d = 5.8 and γ w = 24.8 (see Eqs. 3.48 and 3.49). Due to the lack of measured scanning curves for Paste4 and Paste5, these parameters are considered as the same as for Paste3.Figure 5.3 shows that both hysteresis models yield similar results for the prediction of the rst scanning curves, either for drying or wetting.

5. 4 . 3

 43 Verication with experimental data from the literature Simulation results of the mass loss curves (mass loss vs. time) for Paste3 are shown in Fig. 5.4.

  5.5). Therefore, in the next sections K l determined by means of drying experiments at RH e = 53.5% will be used for the modelling of the wetting process. In the same way, simulated and measured mass loss curves for Paste4 and Paste5 are compared in Fig. 5.6. Saturation proles at the end of drying and the end of wetting for these two materials are shown in Fig. 5.7. Even though predicted mass loss curves show a slight dierence with the measured curves during the wetting processes (mainly for Paste5), Mualem Model II is still the best one which can provide a high agreement with experimental data. Again, results computed ω = 1.0 Non-hysteretic ω = 0.5 (b) Drying at RH e = 63.2%.

Figure 5 . 4 : 2 Figure 5 . 5 :

 54255 Figure 5.4: Comparisons of mass loss curves simulated by non-hysteresis and hysteresis modellings for Paste3. Drying at RH e = 53.5% or 63.2% and wetting at RH e = 97%.

5 . 7

 57 shows that these three modelling methods (Mualem Model II, improved Rubin's model and non-hysteresis) provide very dierent results at the end of wetting. Curves simulated by the non-hysteresis modelling are far above the measured ones (see the dashed-dotted curves in Fig.5.7), which leads to around 20-30% dierences on the surface of the material compared to the measured proles. The case of using improved Rubin's model

Figure 5 . 6 :

 56 Figure 5.6: Comparisons of simulated mass loss curves with experimental results for Paste4 and Paste5. Drying at RH e = 53.5% and wetting at RH e = 97% .

Figure 5 . 7 :

 57 Figure 5.7: Saturation proles simulated by non-hysteresis (ω = 1.0) and hysteresis modellings are compared with experimental data for Paste4 and Paste5, drying at RH e = 53.5% and wetting at RH e = 97%. Red lines and inverted triangle symbols correspond to simulated and measured proles at the end of drying, respectively. Blue lines and triangle symbols correspond to simulated and measured proles at the end of wetting, respectively.

Figure 5 . 8 :

 58 Figure 5.8: Evolutions of k rl and f (φ, S) at dierent depths simulated by VGMa [90, 89] for Paste3 (drying at RH e = 53.5%). The numbers of curves corresponds to the node positions which have been marked in Fig. 5.2.

Figure 5 . 9 :

 59 Figure 5.9: Comparisons of simulation results with measured data for PCEMI6.

  (a) Mass loss curves.(b) Saturation proles.

Figure 5 . 10 :

 510 Figure 5.10: Comparisons of simulation results with measured data for PCEMIII06.

Figure 5 . 11 :

 511 Figure 5.11: Comparisons of simulation results with measured data for PCEMIII06.

Figure 5 . 12 :

 512 Figure 5.12: The dierences of boundary conditions used in non-hysteresis and hysteresis modellings. Points 3 and 6 correspond to the nodes which have been shown in Fig. 5.2.

Figure 5 . 13 :

 513 Figure 5.13: Simulation results performed with Mualem Model II [53] and improved Rubin's

  The cycle duration is 90 days.

Figure 5 . 14 :

 514 Figure 5.14: Simulation results of dierent cycle durations in the cases of non-hysteresis and hysteresis modelling by taking Paste3 as an example. The subgures on the left-hand side show the mass loss curves. Partially enlarged gures are also provided. The central subgures show saturation proles at the end of drying and at the end of wetting after 500, 1000 and 3000 days. The solid-black lines indicate the initial condition. The gray dashed lines show the moisture penetration depth, calculated by the method provided in Section 5.6.6. The subgures on the right-hand side show simulated sorption isotherms (scanning loops) at dierent depths. The positions of points 1, 3 and 6 in the material have been marked in Fig. 5.2.

  state and only the part in the vicinity of the surface is really aected. But for cycle durations of 30 and 90 days, the dierences between both modelling methods are more obvious. Mass loss curves calculated by the non-hysteresis method reach very low values. These dierences are also illustrated with the simulated saturation proles. The hysteresis modelling shows similar shapes of saturation proles for 30-and 90-day durations. For the non-hysteresis modelling, the saturation prole progressively rises during the simulation. It is due to more moisture moving into the material so that the mass of the material increases gradually (see the left-hand side subgures in Figs. 5.14b and 5.14c).The main reason of this dierence refers to the dierent levels of saturation at the boundary as explained in Section 5.5. The saturation for the wetting boundary conditions is remarkably inconsistent between hysteresis and non-hysteresis modellings. For the non-hysteresis modelling, saturations at the boundary are 0.87 and 0.90 for 30-and 90-day cycle durations, respectively, while for the hysteresis modelling saturation at the boundary is between 0.77 and 0.80. These dierent boundary conditions are also clearly presented in sorption isotherms (see the scanning loops for point 1 in the right-hand side subgures of Fig.5.14).

Figure 5 . 15 :

 515 Figure 5.15: Simulation results in the case of using a RH amplitude = 22% (between 64% and 86%) by taking Paste3 as an example. The explanations of the legend are the same as in Fig. 5.14.

Figure 5 . 16 :

 516 Figure 5.16: Simulation results in the case of initial wetting by taking Paste3 as an example.

1 and

 1 Cycles 2 in the right-hand side subgure of Fig. 5.16).

Figure 5 . 17 :

 517 Figure 5.17: The dierences of two forms of Richards equations (taking Paste3 as an example).

k0.5 2 ( 5 . 8 )

 258 rl,N,d (S) = S 0.5 KL + (KL N,0 -KL) KH + k rl,N -1,w (S N,0 ) S N,0 k rl,N,w (S) = S 0.5 KL -(KL -KL N,0 ) KH N,0 + k rl,N -1,d (S N,0 )

Figure 5 . 18 :

 518 Figure 5.18: Evolutions of k rl (S) calculated by VGMb [90, 128] at dierent depths for Paste3 (for the case of drying at RH e = 53.5% and wetting at RH e = 97%).

Figure 5 . 19 :

 519 Figure 5.19: Comparisons of mass loss curves calculated in the cases of with or without hysteresis in relation S-k rl for Paste3 (for the case of drying at 53.5%).

Figure 5 . 20 :

 520 Figure 5.20: Denitions of the undisturbed saturation S und and the moisture penetration depth x p after a number of cycles.

  m p lit u d e (-) C y c le d u ra ti o n (d ) Penetration depth (m) (a) Non-hysteresis modelling.

  m p lit u d e (-) C y c le d u ra ti o n (d ) Penetration depth (m) (b) Hysteresis modelling.

Figure 5 . 21 :

 521 Figure 5.21: Moisture penetration depth in relation with the cycle duration and the RH amplitude for the cases of non-hysteresis and hysteresis modelling by taking Paste3 as an example.

Figure 5 . 22 :

 522 Figure 5.22: Comparisons of x p for dierent initial conditions and cycle durations calculated by using Mualem Model II[START_REF] Mualem | A conceptual model of hysteresis[END_REF]. The RH amplitude is 44%.

  The desorption processExamples of tted D a (RH) and D a (S) curves in the desorption process for Paste5, Paste2 and Paste3 are presented in Figs. 6.2, 6.3 and 6.4. The shapes of tted curves by all models (VGMa, VGMb, FXMa and FXMb) are very similar. However, VGMb shows the curve slightly lower compared to other models in the high saturation range in Fig.6.2. In the range close to S = 0, all tted curves drop sharply, which are mainly due to RH→ 0 in Eq. (6.2). The same trend is also shown by the experimental data in[START_REF] Zamani | The relative humidity dependence of the permeability of cement paste measured using gareld nmr proling[END_REF].

Figure 6 . 2 :

 62 Figure 6.2: Fitted D a curves by four models for Paste5 in the desorption process.

x

  D value for Paste2 is close to the value proposed by Millington (x D = 4/3) and the value for Paste3 is close to the value proposed by Thiéry et al. (x D = 2.74). However, the values for Paste4 and Paste5 are much larger. Hysteresis in D a (RH) and D a (S) Measured D a data for Paste2 and Paste3 include both adsorption and desorption processes and therefore it is able to compare hysteresis between desorption and adsorption as it has been discussed in the literature [11]. Calculation results based on the four studied models for Paste2 and Paste3 are shown in Figs. 6.3 and 6.4, respectively. Because Mualem model-b [128] provides the same equation for the main adsorption process as Mualem model-a [89] (see Section 5.6.5), in Figs. 6.3 and 6.4 curves for VGMa.Ads. and FXMa.Ads. are the same as curves for VGMb.Ads. and FXMb.Ads., respectively. The tted K l and x D values for the adsorption process are presented in Tables 6.1 and 6.3, respectively.

Figure 6 . 3 :

 63 Figure 6.3: Fitted D a curves for Paste2 during both desorption and adsorption processes.

Figure 6 . 4 :

 64 Figure 6.4: Fitted D a curves for Paste3 during both desorption and adsorption processes.

Figure 6 . 5 :

 65 Figure 6.5: Denition of pore radii.

Figure 6 . 6 :

 66 Figure 6.6: The t a curves.

Figure 6 . 7 :

 67 Figure 6.7: The pore size distribution calculated by BJH method.

Figure 6 . 8 :

 68 Figure 6.8: The calculated Knudsen eect for dierent materials.

2 ) 3 )

 23 K l values determined by tting measured D a data are much lower values provided by inverse analysis. The main reason could be the low accuracy tting. If comparing cementitious materials with soils or sands, the resistance eects to water vapour diusion should be more signicant. Hence, when the Knudsen eect is taken into account, it provides more relevant x D values which appear to separate the studied materials in two groups (high performance and ordinary materials). One group with low x D values represents eects of gel microstructure and the other group with high x D values shows eects of capillary microstructure.In the previous chapter, only VGMa and VGMb model were implemented to simulate moisture transport in cementitious materials. The newly introduced FXMa, FXMb in this chapter have not been implemented in a moisture transport model yet. Further work is needed to implement these models to simulate moisture transport and to check the applicability of them.

8 )

 8 The left boundary condition of the MBR is xed by a constant vapour density and the right boundary condition is controlled by the ux with respect to the material. The transport equation in the MBR is solved simultaneously with moisture transport within the material for each iteration. The same numerical method as in Chapter 5 (see Section 5.3) is used here.

Figure 6 . 9 :

 69 Figure 6.9: Schematic representation of the system in case of our experimental conditions. P v0 shows where vapour pressure is controlled by the saturated salt solution. x e and l represent the thickness of the MBR and the sample. The moisture exchanges occur at the position x = 0.

Figs. 5 .6 and 5 . 7 .

 557 Figs. 5.6 and 5.7. The description of experiments has been given in Chapters 2 and 5. In this coupled model, there are two unknowns: x e and K l . The value for K l is set as 2.0 × 10 -21 which

Figure 6 . 10 :Figure 6 . 11 :

 610611 Figure 6.10: Comparisons of simulated mass loss curves and saturation proles with measured ones for Paste4.

Figure 6 . 12 :

 612 Figure 6.12: Relationship between x e and K l for the studied materials. Paste3(53.5) represents experiment at RH e = 53.5% and Paste3(63.2) represents experiment at RH e = 63.2%. High- lighted (solid) symbols are determined by using the ux boundary condition.

Figure

  Figure 6.12 shows that a very good relation between x e and K l for all the studied materials.

  .13, only K l values obtained at x e = 10 cm are presented, which is considered as the most representative thickness. From the sixth to the ninth methods (blue bars), K l values are determined by tting measured apparent diusivity D a for desorption. Notice that the newly proposed FXMa and FXMb for tting D a seem not as stable as VGMa and VGMb because they present larger K l value variations. More clearly, FXMb illogically provides a smaller K l value for Paste4 than Paste3. This is quite unusual because a more porous material (like Paste4) should have a larger permeability than a less porous material (like Paste3).Thus, methods based on FXMa and FXMb will not be compared below.Roughly speaking, for each material, K l values indicate that there are two groups for all methods.The rst group with higher K l values consists of the Katz-Thompson model and

Figure 6 . 13 :

 613 Figure 6.13: Comparisons of K l values determined by dierent methods for the studied materials.

x

  D . Fitting results showed that in the low RH range the use of Knudsen eect can provide more relevant x D values for cementitious materials. Two groups of materials can be identied corresponding to low (0.3 < x D < 0.92) and high (2.22 < x D < 2.83) x D values representing respectively high performance and ordinary materials. In the high RH range, tted K l values are much lower than those determined by inverse analysis in Chapter 2. This kind of results is mainly due to the low tting accuracy and some problems which provide uctuating experimental data.The second perspective is to investigate the eect of vapour diusion in a mass boundary region on determination of K l by inverse analysis during a drying experiment in laboratory conditions. Coupled simulations have provided the relation between the thickness of the mass boundary region x e and K l . This relationship revealed that dierent x e values from 0 to 15 cm do not aect moisture transport in the material very much. All comparisons and discussions indicated no necessary to take into account vapour diusion in the mass boundary region when determining K l by inverse analysis. It further proved that this kind of inverse determination of K l can present very stable results for three types of boundary conditions (the rst, ux and diusive boundary conditions).

  Comparisons of simulated saturation proles based on two dierent methods to calculate k rl and k rg for Paste4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Comparisons of simulated saturation proles based on two dierent methods to calculate k rl and k rg for Paste5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Comparisons of simulated mass loss curves based on two dierent methods to calculate k rl and k rg for Concrete3 and Paste3. . . . . . . . . . . . . . . . . . . . 2.10 Calculated saturation proles for Concrete3 and Paste3 (Method-1 to calculate k rl and k rg ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Proles of gas pressure during drying of Paste4 with dierent values of initial saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Proles of gas pressure during drying of Paste4 with initial saturation S 0 = 0.99. .15 Relationships of f (φ, S)RH(S) vs. S and k rl (S) vs. S for Paste4. . . . . . . . . 2.16 Mass loss curves calculated by three versions of models for Paste4. . . . . . . . . 2.17 Schematic representations of two moisture transport modes. . . . . . . . . . . .

.5 Comparisons of simulated mass loss curves based on two dierent methods to calculate k rl and k rg for Paste4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Comparisons of calculated mass loss curves based on two methods to calculate k rl and k rg for Paste5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 2.12 The maximum and minimum gas pressure during drying of Paste4 under various external RH with two dierent values of initial saturation. . . . . . . . . . . . . 2.13 Calculated mass loss curve and saturation proles in the case K l K g based on Method-1 to calculate k rl and k rg for Paste5. . . . . . . . . . . . . . . . . . . . . 2xiii List of Figures 2

  6.10 Comparisons of simulated mass loss curves and saturation proles with measured ones for Paste4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Eects of x e on the mass loss curves during drying. . . . . . . . . . . . . . . . . 6.12 Relationship between x e and K l for the studied materials. . . . . . . . . . . . . . 6.13 Comparisons of K l values determined by dierent methods for the studied materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classication of pores and features in concrete . . . . . . . . . . . . . . . . . . . . 2.1 Properties of materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	List of Tables
	1.1
	xvi

Table 1 .

 1 1: Classication of pores and features in concrete (taken from[START_REF] Thomas | The Pore System and Classication of Pores, accessed November[END_REF]).

	Type of pore	Description Size	Water		Properties
	Capillary pores	Large	nm 10 µm -50	Evaporable Bulk water Permeability, strength
		Medium	50 -10 nm	Evaporable Moderate	Permeability, strength,
				menisci		shrinkage (high RH)
	Gel pores	Small	10 -2.5 nm	menisci Evaporable	Strong	Shrinkage (up to 50% RH)
		Micro	2.5 -0.5 nm Non-evaporable,	Shrinkage, creep (35-11%
				no-menisci,	inter-	RH)
				molecular interactions
	Interlayer	Structural < 0.5 nm	Non-evaporable,	Shrinkage, creep (<11%
	spaces			ionic/covalent bond	RH)

Below interlayer water, there is some chemically-bound water which is concerned by the transport. Only in extreme conditions, this kind of water can be lost, such as drying at high temperature (e.g., 105

  1.3a). Hydrophobic or hydrophilic1.3. Mechanisms of moisture transport impurities in pores also cause hysteresis (see Figs. 1.3b and 1.3c) because they make water easily or dicultly to enter pores. Besides, the polarization of water molecules is recognised as one reason. A pre-wetted material is more easily to accept water molecules while a dried material shows strong repulsion. This can also be explained by the dierent energy states during drying and wetting.

Table 1 .

 1 1.3. Mechanisms of moisture transportHence, Knudsen diusion widely exists in gel pores and partially in capillary pores according to 1.

  RH p is RH in the material at the point x p .

	). Note that	dθ dt	=	∂θ ∂RH p	dRH p dt	where	∂θ ∂RH p	is obtained
	based on sorption isotherms and

  ψ ref = ψ(RH ref ).The reference values ψ ref and RH ref can be chosen arbitrarily for each material. For convenience, it is better to let ψ ref = 0.

	variable. Kirchho 's potential ψ(RH) has the same properties as D F (RH) and varies for dier-
	ent materials.
	An important result is that the average value of Kirchho 's potential in the material over
	2 a time period is equal to the average value of Kirchho 's potential at the boundary. Another
	(1.24) important feature of Kirchho 's potential for numerical modelling, reported by Kalagasidis et
	al. in the study about moisture conductivity [107], is that it is very insensitive to the mesh
	renement; thus, the numerical calculation converges much faster. In other words, it can give
	where t p is the cycle period. This equation can be also used to estimate both x p and x r in
	a stable numerical solution with a coarse or moderate mesh. But, using Kirchho 's potential
	Eqs. (1.21), (1.22) and (1.23). Furthermore, these equations can be used to estimate the evolution
	is extremely sensitive to the quality of the input data. It is not surprising because the integral
	of water content or RH in the moisture buering layer.
	form of ψ(RH) is very sensitive to the shape of D F (RH) curve.
	Even though the EMPD model is a very handy method to estimate the moisture behaviour
	in inuenced zones of the material, the drawbacks of this model are also clear. As it can be
	seen, this model highly relies on the moisture penetration depth, which has to be assumed or 1.5.4 Method from Li et al.
	calculated by empirical equation (i.e., Eq. 1.24). That gives the results less accuracy. Another
	diculty is that the EMPD model cannot be used for arbitrary boundary conditions since it is
	more suitable for perfectly cyclic boundary conditions.
	1.5.3 Kirchho's ow potential
	Kirchho 's ow potential used for modelling of moisture transport in porous media was rst
	introduced by Arfvidsson and Claesson [6, 7]. Fick's rst law was chosen to describe the diusion
	process. The potential can be relative humidity or water content. Let's take RH as an example,
	so that a new function ψ can be obtained by Kirchho transformation.
			RH
	ψ(RH) = ψ ref +	D F (RH)dRH	(1.25)
			RH ref
	Thus, the diusion equation can be rewritten as:
	J F = -	dψ dRH	∂RH ∂x	= -	∂ψ ∂x	(1.26)
	Equation (1.26) indicates that moisture diusion is driven by the gradient of ψ and D F (RH) ≡
	1. This certainly is the simplest moisture diusion potential for the case of one independent state

where

  [START_REF] Baroghel-Bouny | Prediction of chloride binding isotherms of cementitious materials by analytical model or numerical inverse analysis[END_REF] where a d , b d , a w , b w and ω l are parameters which need to be determined by experimental data.

	Again, subscripts d and w represent the parameters for the main desorption and main adsorption
	processes, respectively.

  .48) For an adsorption scanning curve, one needs to replace RH d in (1 + a d )RH d , a d RH d , c h RH d , exp(b d RH d ) and d h RH d by RH w . Totally, the H-function itself needs 7 parameters (a d , b d , a w , b w and ω l for two main curves and c h and d h for scanning curves) which have to be experimentally determined.

Table 2 .

 2 1: Properties of materials. were cut into around 10 cm-length cylinders, and then samples were sealed by self-adhesive aluminium foil sheets, but one face was open for moisture exchanges with ambient environment (see Fig.2.1). All samples were stored in a sealed space (e.g., desiccators or dry

	Materials	w/c	Porosity	Initial RH (%)	Drying duration	Sources of data
		(w/b)
	Paste3	0.35	0.31	83	157 days	[23, 24]
	Paste4	0.45	0.41	89	371 days	[23, 24]
	Paste5	0.60	0.49	95	353 days	[23, 22, 42]
	Concrete3	0.43	0.122	93	356 days	[23, 22, 42]
	demoulded, materials

keepers) for measurements at room temperature (T = 23 ± 0.1 • C). The external RH was xed at RH e = 53.5% by using a saturated salt solution (magnesium nitrate, Mg(NO 3 ) 2 ). Besides,

Table 2 .

 2 2: Values of a V and m V for the selected materials tted by Eq. (2.4).

	Materials	Paste3	Paste4	Paste5	Concrete3
	a V (×10 7 Pa)	4.43	3.21	1.35	4.06
	m V	0.48	0.47	0.39	0.49

used to simplify Eq. (2.4); thus, only two parameters need to be determined.

Fitting results by using this two-parameter equation for the selected materials are presented in Fig.

2

.2. Values of these parameters for all the selected materials are presented in Table

2

.2.

Table 2

 2 .3 are too small. In other words, the measured data reveal that K l K g [84, 85]. Hamami et al. [84] measured the gas permeability for 24 compositions of cement pastes and mortars (w/c = 0.4 and 0.6) and results show the measured data are between 10 -16 and 10 -17 m 2 . The same results can be found in the work by Loosveldt et al. for mortars [85]. More recently, Kameche et al. [142] reported K g ≈ 2.5 × 10 -17 m 2 for concrete (w/c = 0.5). It should be acceptable to assume that K g is around 1 × 10 -17 m 2 .

	Table 2.4: The intrinsic permeabilities to liquid-water validated by experimental data for dierent
	materials.
		Methods	Paste3	Paste4	Paste5	Concrete3
	K l	VGMa model (×10 VGMb model (×10 -21 m 2 ) 0.36 (0.24 a ) -19 m 2 ) 1.6 (1.0 a )	1.7 3.2	25 160	1.8 2.3

  .4). Compared to Table 2.3, only some values of K l in Table 2.4 are a little smaller. One example of simulation results for Paste5 is shown in Fig. 2.13. Results provide quite good agreements with measured data. It is clear that both cases K g = K l and K

l K g can be used to t mass loss curves and predict saturation proles.

a

Obtained by drying at RH = 63.2%.

  2.4. Modelling of drying cementitious materials in the caseK l K g -Quite permeable porous materials: t F /t g D 1 leading to K g /(φτ ) 4.5 × 10 -15 m 2 . If putting the values of φ and K g into Eqs. (2.28), (2.29) and (2.30), it yields t F /t g D > 1 forall the selected materials. This means that for the case K l K g the selected materials can be considered as quite permeable materials according to the classication mentioned above. In this

	Because K l	K g , there is t l D /t g D	1;
	-Weakly permeable porous materials: t F /t g D	1 leading to K g /(φτ )	4.5 × 10 -15 m 2 .
	Because K l	K g , there is t F /t l D	1.
			1.05	0 day
			1.04	1 h 32 days
		Pg (10 5 Pa)	1.01 1.02 1.03	79 days 365 days
			1
			0.99
			0.98
			0	0.02 0.04 0.06 0.08 0.1 0.12 0.14
				x (m)

situation, advection of gas-phase is faster than diusion of gas phase. According to Eq. (2.15), the advective ow of gas-phase is able to dissipate any variations of gas pressure, either related to evaporation-diusion phenomena for movement of liquid-water by capillarity. Expectedly, gas pressure variations are not as signicant as the case K l = K g . One example for Paste3 is shown in Fig.

2

.14. Only small gas pressure variations can be seen in the part near the surface of the material. After a certain period of drying, gas pressure in the material is almost equal to the atmospheric pressure. Figure

2

.14 reveals that the main reason causing gas pressure variations is advection of gas-phase. The larger K g is able to dissipate gas pressure quickly.

  37)If adopting T = 293K, with P vs = 2333 Pa, it gives thatM v ρ l RT 2 η l D 0 P vs ≈ 3 × 10 -21 m 2 .

	Equation (2.36) is rewritten as:

Table 2 .

 2 5:The intrinsic permeabilities to liquid-water validated by experimental data for two versions of simplied models.

		Methods	Paste3	Paste4	Paste5	Concrete3
	K l (×10 -21 m 2 )	First simplied model (Eq. 1.17) Second simplied model (Eq. 2.33)	0.5 (0.33 0.46 (0.3 a ) a )	2.2 2.0	34 32	10 8

Equation (1.17) is another form of Richards equation able to describe moisture transport by the sole movement of liquid water. The liquid-water diusivity D l in Eq. (1.17) is a simplied version of Eq. (2.34). Thus, there are two versions of simplied models, Eq. (1.17) and Eq. (2.33). The intrinsic permeabilities inversely obtained by using drying kinetics for these two simplied models are provided in Table

2

.5 for the case using Method-1 to calculate k rl and k rg . Compared to Table

2

.4, those values are larger than that from the full multiphase model, which is because more mass transport is needed to balance neglected advection of the gas-phase in the rst simplied model and the neglected vapour diusion in the second simplied model.

  During experimental verications, two methods to calculate relative permeabilities of liquidwater and gas-phase were also compared. Both methods can provide acceptable simulation results compared with experimental data. The only dierence was found in the values of K l validated by these two methods. Method-1 based on Mualem model-a[START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF] provided K l values slightly lower than other indirect methods such as the Katz-Thompson model or the beam bending method, while values from Method-2 based on Mualem model-b[START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF] are higher. Method-1 seems easier to be implemented because equations in this method are analytical.

Verications of modelling results were conducted through comparing simulated saturation proles with measured ones. Comparisons revealed that the predicted proles match the measured curves quite well. Hence, this multiphase transport model has a high applicability in case of cementitious materials.

Table 3 .

 3 1: Proposed multi-level approach to model hysteresis.

	Levels	Available experimental data	Description
	Level 1	Both main branches and one	One scanning curve is used to deter-
		scanning curve for each drying	mine parameters.
		and wetting process
	Level 2	Both main branches	Predicting scanning loops without
			additional parameters
	Level 3	One main branch	Determining the other main branch

Table 3 .

 3 3: Proposed models for the multi-level approach to model hysteresis.

	Levels	Experimental data	Models combination
	Level 1	Both main isotherms and one	FX (VG3) + L (W-M1)
		scanning isotherm
	Level 2	Both main isotherms	FX (VG3) + M-II
	Level 3	One main isotherm	FX (VG3) + M-II-1 + M-II
	Level 4	No isotherm data	No models available

Comparisons in previous sections show that Level 1 and Level 2 provide very reliable predictions. Because there is no model which is able to infer one main isotherm from another main curve, the prediction accuracy of Level 3 needs to be improved. Level 1 and Level 2 indicate that at least both measured main desorption and adsorption isotherms are required to calculate scanning loops.

Table 4 .

 4 1.

Table 4 .

 4 1: Materials and mix designs used in the present study.

	Binders	Name	w/c (w/b)	Porosity (× 100%) a
		PCEMI04	0.4	0.377
	CEM I-32.5 R	PCEMI05	0.5	0.457
		PCEMI06	0.6	0.511
		PCEMIII04	0.4	0.216
	CEM III/B-42.5 N	PCEMIII05	0.5	0.456
		PCEMIII06	0.6	0.510
		PSF04	0.4	0.344
	CEM I-32.5 R + 10% silica fume	PSF05	0.5	0.407
		PSF06	0.6	0.498

The classication of these binders is according to EN 197-1 European standard. Cements were supplied by HeidelbergCement . There are 9 dierent mix designs (3 w/c × 3 binders).

Table 4 .

 4 .1 in Chapter 2). Experiments were conducted under room temperature (23 • C). The RH cycle used during experiments is presented in Table 4.2. All samples were stored in dry keepers with a certain saturated salt solution which provided a stable and constant RH environment. 2: RH cycles used during drying-wetting experiments.

Table 4 .

 4 3: The mass adsorption coecient for studied CEM I pastes.

	Composition	µ i	w/c = 0.4 w i w i µ i	w/c = 0.5 w i w i µ i	w/c = 0.6 w i w i µ i
	CEM I	0.0774	0.7143	0.0553	0.6667	0.0516	0.6250	0.0484
	Water	0.0857	0.2857	0.0245	0.3333	0.0286	0.3750	0.0321
	w i µ i			0.0798		0.0802		0.0805

The mass absorption coecient is calculated based on the chemical composition of the material. Because calculation results show that cement pastes made by three binders (CEM I, CEM III and CEMI+SF) have very close µ values

[START_REF] Thiéry | Modélisation de la carbonatation atmosphérique des matériaux cimentaires avec prise en compte des eets cinétiques et des modications microstructurales et hydriques[END_REF][START_REF] Wang | Modélisation du transport multi-espèces dans les matériaux cimentaires saturés ou non saturés et éventuellement carbonatés[END_REF]

; only calculated coecients for CEM I pastes are presented in Table

4

.3 (w i is the weight ratio of each composition).

If applying Eq. (4.1) to the saturated state and the dried state, it gives the equation to calculate porosity at each depth of the sample

[START_REF] Thiéry | Modélisation de la carbonatation atmosphérique des matériaux cimentaires avec prise en compte des eets cinétiques et des modications microstructurales et hydriques[END_REF]

:

Table 4 .

 4

	4: The comparison of global porosity (%) measured by weighing and gamma-ray for
	cement pastes.
	Materials	Methods	w/c = 0.4	w/c = 0.5	w/c = 0.6
	CEM I	φ weigh φ γ	37.7 ± 0.6 45.7 ± 0.2 51.1 ± 0.0 37.6 ± 0.1 46.1 ± 0.4 49.8 ± 0.9
	CEM III	φ weigh φ γ	31.6 ± 1.5 45.6 ± 2.2 51.0 ± 1.0 31.6 ± 1.4 43.5 ± 1.9 50.7 ± 0.7
	CEM	φ weigh	34.4 ± 0.3 40.7 ± 0.0 49.8 ± 0.1
	I+SF	φ γ	34.3 ± 0.2 40.5 ± 0.1 49.7 ± 0.2

Comparisons of global porosities measured by these two methods are given in Table

4

.4.

It shows a high agreement between porosities measured by these two methods. The biggest dierence which is found for paste CEM III with w/c = 0.5 is only around 2.2%.

Table 5 .

 5

		1: Validated liquid intrinsic permeabilities (×10	-21 m 2 ) for dierent materials.
		PCEMI05	PCEMI06	PCEMIII05	PCEMIII06	PSF05	PSF06
	K l	28	73	0.14	0.25	0.65	0.95

  5.13, improved Rubin's model clearly presents signicant PEs for both surface point (e.g., point 1) and inner point (e.g., point 3). For point 1, during the rst cycle it appears the largest PEs, which can be quantied by S dierence between point B and point A (see Fig.5.13b).If there is no PE, point B should return to point A on the main desorption curve after the rst cycle. The later cycles seem to have very small relative errors for point 1. However, for point 3, the PEs constantly grow with the number of cycles, which causes that S becomes lower and lower. The dierent evolutions of PEs between point 1 and point 3 are because the surface point has larger RH uctuations than the inner point. As it has been explained in[START_REF] Zhang | A review and statistical study of existing hysteresis models for cementitious materials[END_REF], small RH uctuations yield larger PEs than large RH uctuations. Consequently, improved Rubin's model leads to a global increase of mass loss curve according to the number of cycles due to the lower saturation for each node (see Fig.5.13c). The comparison results demonstrate that if the empirical model fails to eliminate the PEs, the cumulative errors associated with uctuations of hysteresis loops become signicant and cause non-physically simulated results.

  Two basic equations KL and KH (see Eq. 2.12) are used to calculate k rl during the main drying, the main wetting and scanning processes. Equations for the main desorption process have been given in Section 2.2.3 (see Eqs. 2.11-2.13). According to the principle of independent domain theory, k rl

	for the main adsorption process is expressed as [128]:
	k rl,w (S) = S 0.5 KL 1/2	(5.5)
	Notice that if putting KL equation in Eq. (2.13) into Eq. (5.5) one can have the same equation
	to calculate k rl for the main adsorption process as that in Mualem model-b (see Eq. 2.9). It is
	easy to deduce the equation for the rst drying scanning process (1,d) according to Mualem's
	diagram (see Fig. 3.10).

  x D +2 P vs RH φ -S) is the available space for vapour transport and x D is a parameter related to material properties. Terms dP c /dS, k rl and P v depend on saturation S. Eq. (6.1), the liquid permeability K l k rl (S) is one of crucial factors for liquid transport.Generally, k rl (S) is considered as a function of S and calculated based on sorption isotherms (see Section 2.2.3). To perform a moisture transport modelling, the liquid intrinsic permeability has to be determined. It can be measured by experiments or estimated by theoretical models. The conventional methods to measure K l were carried out on fully saturated materials. Therefore, if vapour diusion is negligible during the measurements, the ux of liquid-water is expressed by Darcy's law solely.

	dP c dS	(6.2)
	where φ(1 In

  For cementitious materials, x D value may be larger than 4/3 to present the stronger tortuous eects. Calibrating with O 2 and CO 2 diusion coecient measurements for partially saturated mortars (performed by Papadakis et al.[START_REF] Papadakis | Physical and chemical characteristics aecting the durability of concrete[END_REF]), Thiéry et al.[START_REF] Thiéry | Modélisation du séchage des bétons -Analyse des diérents modes de transfert hydrique[END_REF] proposed x D = 2.74. However, diusion of water molecules is dierent to O 2

or CO 2 . A recent literature

[START_REF] Zamani | The relative humidity dependence of the permeability of cement paste measured using gareld nmr proling[END_REF]

adopted the moisture transport model from

[START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Baroghel-Bouny | Modelling of isothermal coupled moistureion transport in cementitious materials[END_REF][START_REF] Baroghel-Bouny | Prediction of chloride binding isotherms of cementitious materials by analytical model or numerical inverse analysis[END_REF]

to t water content proles measured by NMR at the steady state moisture ow conditions. In this

Table 6 .

 6 [START_REF] Scrivener | Innovation in use and research on cementitious material[END_REF] shows that FXMb model unexpectedly provides a smaller K l for Paste4 than Paste3.It may mean that FXMb is a less stable model.

Table 6 .

 6 1: Fitted K l (×10 -21 m 2 ) for the desorption and adsorption processes.For a given material, the x D values in Table6.2 provided by dierent models are very close so that the D a curves computed by dierent models are quite similar in the low RH range. The

	Materials	Katz-	Desorption Flux BC a VGMa VGMb FXMa FXMb VGMa Adsorption FXMa
	Paste2 Paste3 Paste4 Paste5	Thompson <10 -3 a 37 c 71 c 231 c	-0.46(0.3 d ) 0.171 0.026 2.0 0.296 320 3.010	0.219 23.60 28.40 751.0	(VGMb) 0.039 0.02 5.650 4.66 1.000 -290.0 46300 -0.022 0.272 0.362	(FXMb) 3.01 3390 --

•

  Experimental data for Paste2 show very clear hysteresis in D a (RH) curves between adsorption and desorption. All tted curves by four models captured this trend. Hysteresis in D a (RH) for Paste3 is not as clear as for Paste2 and the tted curves also

		10 -9
		10 -10
	)
	(m 2 s -1	10 -11
	D a	10 -12
		10 -13	Exp. Ads. Exp. Des.	FXMa.Ads. FXMa.Des.
			VGMa.Ads.	FXMb.Des.
		0 -14 10	0.2 VGMa.Des.	0.4 VGMb.Des. 0.6	0.8	1
				RH (-)

a Taken from research results by Baroghel-Bouny

[START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF]

. b Taken from Table

2

.5. c Taken from research results by Nguyen

[START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF]

. d Obtained by drying at RH e = 63.2%.

  VGMb and FXMb do not present signicantly dierent results to other models. Thus, hysteresis in k rl (S) or k rl (RH) cannot be used to explain hysteresis in D a (S) or D a (RH). In other words, hysteresis in D a (RH) observed with measured data (see Fig.6.3a) may be not caused by hysteresis in k rl (S) or k rl (RH). It may be due to hysteresis in RH(S) or dP c /dS (see Eqs. 6.1 and 6.1).The above discussions reveal that x D is clearly larger than the theoretical one. Besides, the tting for desorption is not very good especially in the low RH range. In the next subsection, an improved method taking into account the Knudsen eect will be used to describe vapour

		10 -9
		10 -10
	s -1 )	10 -11
	(m 2
	D a	10 -12
				Exp. Ads.	FXMa.Ads.
		10 -13		Exp. Des.	FXMa.Des.
				VGMa.Ads.	FXMb.Des.
				VGMa.Des.	VGMb.Des.
		0 -14 10	0.2	0.4	0.6	0.8	1
				RH (-)
		diusion.

• Although Mualem model-a provides hysteresis in both relations k rl (S) and k rl (RH),

Table 6 .

 6 2: The tted x D for the desorption process. and x D values for Paste4 and Paste5 are not so dierent (2.22 < x D < 2.83). The phenomenon may indicate that Paste2 and Paste3 have some similarities in gel microstructures, as well Paste5 and Paste4 in capillary microstructures. This is rather reasonable because in the high performance material (Paste2)

	Materials	Millington [93]	Thiéry et al. [120]	Without Knudsen eect VGMa VGMb FXMa FXMb VGMa VGMb FXMa FXMb With Knudsen eect
	Paste2			1.54 1.56 1.59 1.61 0.34 0.36 0.37 0.40
	Paste3 Paste4	4/3	2.74	2.23 2.17 2.25 2.21 0.90 0.89 0.92 0.91 4.26 4.10 4.34 4.24 2.39 2.22 2.47 2.38
	Paste5			4.47 4.32 4.42 4.33 2.83 2.67 2.77 2.68

An important observation in

Table 6.2 which should be pointed out is that if considering the Knudsen eect x D values for the four studied materials appear in two groups: Paste2 and Paste3 have very close x D values which are below 1 (0.3 < x D < 0.92),

Table 6 .

 6 3: The tted x D for the adsorption process.Figures 6.2, 6.3 and 6.4 show that for each D a curve there is a threshold RH value which divides the curve into two parts. One part is related to D l in the high RH range and the other part is from D v in the low RH range. The same trend is also shown in Fig. 2.15. The threshold RH values are presented in Table6.4 for all the materials in the case of using VGMa model and considering Knudsen eect.

	Materials	Without Knudsen eect VGMa (VGMb) FXMa (FXMb) VGMa (VGMb) FXMa (FXMb) With Knudsen eect
	Paste2	1.70	1.66	0.97	0.93
	Paste3	2.54	2.38	1.79	1.63
	Threshold values of RH for vapour and liquid transport

Table 6 .

 6 4: The calculated RH threshold values for using VGMa model with Knudsen eect.Two trends can be seen in this table: the threshold values increase with w/c and values in adsorption are larger than that in desorption. A higher threshold value means that vapour diusion is more signicant. This is the same as the conclusion provided in Chapter 2 (see Section 2.5) when discussing the dierent transport modes. In adsorption, adsorbed layers are formed at low RH and more water molecules are condensed with the increase of RH. If there

		Paste2	Paste3	Paste4	Paste5
	Desorption	0.41	0.66	0.67	0.74
	Adsorption	0.87	0.92	-	-

Acknowledgements

Acknowledgements

was conducted at IFSTTAR, within the research group FM 2 D (Formulation, Saiedpour Mahsa conducted the DVS measurements for me. Fei. Her persistent encouragement has enabled me to complete this dissertation. She provided the necessary strength and motivation to nish this thesis. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7 / 2007-2013) under grant agreement 264448.

Concrete2 Paste2

and each model.

these three-parameter models can be used for a variety of cementitious materials. Moreover, one can remark that the widely-used VG2 model [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF] does not perform better than other models (see Figs. 3.3 and 3.4). Results also show that no selected model can simulate the sudden drop of the desorption isotherms from 44% to 33% RH (see Fig. 3.6). This drop may be because there is a threshold pore size (between RH = 44% to 33%) for capillary pores and gel pores.

When measured water content changes from capillary pores to C-S-H pores as emphasised in reference [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF], the drop on measured sorption curves can be seen. Mean values of R 2 adj for the bi-modal model are higher than 0.99 for both desorption and Chapter 4

Unreal fitting

Experimental investigations

Summary

As stated in the previous chapters, the ob jective of this work is to study the internal moisture behaviour responding to external RH changes, not only by numerical modelling but also by experimental investigations. A simplied model for moisture transport during drying has been introduced in Chapter 2. In Chapter 5, this model will be used to simulate moisture transport under drying and wetting cycles. Experimental data will be needed to feed the model, to calibrate undetermined parameters and to verify the modelling results. This chapter will provide these experimental data. They are complementary to the data taken from the literature or data which have been measured at IFSTTAR in the previous studies [START_REF] Baroghel-Bouny | Characterization of cement pastes and concretes-methods, analysis, interpretations[END_REF][START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF][START_REF] Nguyen | Modélisation des couplages entre hydratation et dessiccation des matériaux cimentaires à l'issue du décorage[END_REF].

Two main experiments, drying-wetting cycles measurements and sorption isotherms measurements, will be introduced. Sorption isotherms measured by DVS and pressure plate are the input data for modelling. DVS and pressure plate measurements were conducted at Lund University which is one research partner in the TRANSCEND pro ject. Measured mass loss curves during drying will be used to determine the liquid intrinsic permeability by inverse analysis which has been adopted in Chapter 2. Mass loss curves in latter processes (wetting and re-drying) and saturation proles measured by gamma-ray attenuation technique will be used to verify the dierent modelling methods in Chapter 5.

Chapter 5

Modelling of moisture transport under drying-wetting cycles

Summary

In this chapter, a method coupling a moisture transport model with hysteresis models is proposed.

Theoretically, this approach is applicable for most types of hysteresis models. Two hysteresis models, including a conceptual model and an empirical one, have been implemented and compared. Experimental data from the literature and measured in this thesis (see Chapter 4) are used to verify modelling results. Verications show that the conceptual model yields better results

than the empirical one. The main reason is that the empirical model provides a non-physical hysteretic behaviour, so-called pumping eect. Meanwhile, comparisons of non-hysteresis and hysteresis modellings are carried out for dierent cycle durations, RH amplitudes, initial states, etc. All comparisons and investigations enhance the necessity of considering hysteresis to model moisture transport under cyclic drying and wetting conditions. The moisture penetration depth is also compared in cases of modelling with and without hysteretic eects. It has real valuable meaning for engineering.

Most simulation results in this chapter have been included in the paper that has been submitted to Cement and Concrete Research journal [START_REF] Zhang | Eect of cyclic boundary conditions on moisture transport within cementitious materials[END_REF].

Chapter 6

Investigation of the reliability of K l determination Summary

In the simplied model used in Chapter 5, the key transport coecient is the liquid intrinsic permeability K l which is rarely veried by experimental data. In Chapters 2 and 5, K l was determined by inverse analysis through tting drying kinetics. This chapter focuses on comparisons of dierent methods to determine K l based on experimental data. These methods include tting measured apparent diusivity values and inverse analysis with dierent boundary conditions.

First of all, let's have a global overview of methods which are compared in this study (see Fig. 6.1). Totally, there are nine methods belonging to three main types. The Katz-Thompson model [START_REF] Thompson | The microgeometry and transport properties of sedimentary rock[END_REF] is generally considered to estimate and compare K l in case of cementitious materials in references [START_REF] Baroghel-Bouny | Characterization and identication of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials[END_REF][START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF].

Values of K l estimated by this model for the selected materials will be further compared with the other methods based on inverse analysis of drying kinetics and apparent diusivity measurements in this chapter.

The rst part of the current chapter will indirectly determine K l by tting the apparent diusivity D a data measured by Baroghel-Bouny [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF]. In the tting, the resistance factor f (φ, S)

to vapour diusion is also assessed. Four dierent models are used to calculate the liquid-water relative permeability k rl . Results show that K l values determined by this method are much lower than that provided by inverse analysis in Chapter 2.

Actually, the boundary conditions (the rst type and the ux boundary conditions) used in Chapters 2 and 5 may not be able to describe moisture transport for an experiment condition which uses the saturated salt solution to control a constant RH in a desiccator. Therefore, the second part of this chapter provides an analysis for the impact of the experiment condition on the result of determination of K l by inverse analysis. The air part in a desiccator is described as the mass boundary region. A model is needed to simulate moisture diusion in this region, and then this model will be coupled with the moisture transport model used in Chapter 5. The relationship between the thickness of this region and K l will be discussed. Simulations and comparisons conclude that K l values provided by the proposed method and by using the ux boundary condition do not have signicant dierence.

Determination of K l by tting measured apparent diusivity values

As the conclusion in Chapter 2 indicates, if assuming that gas pressure remains constantly equal to the atmospheric pressure [START_REF] Mainguy | Role of air pressure in drying of weakly permeable materials[END_REF][START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF] and Mualem's model-b [START_REF] Mualem | Hysteretical models for prediction of the hydraulic conductivity of unsaturated porous media[END_REF]. These two combinations are denoted by FXMa and FXMb models.

Besides, dierent main curve tting equations provide dierent shapes of sorption isotherms; thus the validated K l values will be also dependent on main curve tting equations. However, the above discussions also revealed that if using other main curve tting equations (beside of VG) k rl values can be aected. In the study by Poyet [217], it has shown that using dierent main curve tting equations to evaluate K l is not a fully relevant to the eective permeability K l k rl . Hence, this study does not consider the inuence of dierent main curve tting equations on K l .

Overall, four versions of k rl models are used: VGMa, VGMb, FXMa and FXMb. Note that Mualem model-b uses dierent equations for the main desorption and adsorption processes so VGMb and FXMb need both main desorption and main adsorption isotherms. Besides,

VGMb and FXMb present hysteresis in k rl (S). Therefore, through tting the measured apparent diusivity it will be possible to indirectly verify these k rl models.

Experimental data and tting results

Experimental data for D a

For measurements of moisture diusivity, Nilsson has compared dierent methods [START_REF] Nilsson | A comparison between dierent methods to determine moisture transport properties of cementitious materials[END_REF], including the cup tests, steady-state, non-steady-state measurements, etc. All methods seem timeconsuming. Hence, it is better to use experimental data which are available in the literature.

Baroghel-Bouny [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] provided measured D a data for a variety of cementitious materials. These data were indirectly obtained during sorption isotherms measurements by using a theoretical method, in which moisture transport in the material between two small RH steps was described by Fick's diusion with a constant D a . Therefore, at each measured point for sorption isotherms, the value of D a can be determined by tting the mass loss curve. Actually, this is the inverse analysis method that we used in Chapters 2 and 5. But the dierence is that Baroghel-Bouny [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part I: Essential tool for analysis of hygral behaviour and its relation to pore structure[END_REF] used small RH interval around 10%, while in Chapters 2 and 5 the RH intervals are much larger such 95% to 53.5% for Paste5 (see Table 2.1). The small RH interval allows to obtain accurate results, which showed good agreements with the cup test results [START_REF] Baroghel-Bouny | Water vapour sorption experiments on hardened cementitious materials. Part II: Essential tool for assessment of transport properties and for durability prediction[END_REF]. Totally, experimental D a data are available in the main desorption process for: Paste2, Paste3, Paste4 and Paste5, and in the main adsorption process for: Paste2 and Paste3. According to the method described above, all available measured curves of D a (RH) are tted.