K Sultan

G Gintzburger

M Obaton

C Robin

H Touchane

A Guckert

Nitrogen fixation of annual Medicago-Rhizobium associations during cool season in Mediterranean dry areas

Keywords: nitrogen fixation, 15 N dilution, annual medics, low temperature, yield

Introduction

Annual Medicago (medics) represent a great potential of forage production in areas suffering shortage of livestock food; however, low winter temperature delays the growth and reduces the nitrogen fixation of Medics in Mediterranean dry areas. In this context, the identification of Medics able to fix nitrogen and produce dry matter at low temperature is necessary in order to extend the grazing period.

Materials and methods

The experiment was carried out at ICARDA in Syria (Aleppo) in 1999. Three accessions were used, two of M. aculeata, accessions 80 (A) and 5099 (B) and one of M. rigidula, accession 716 (R). Each accession was inoculated with Sinorhizobium strains appropriate for nodulation under low temperature (S. meliloti strain 620 with accession A, strain 508 with B and strain BZI with R). The percentage of fixation-derived shoot nitrogen was quantified at 5 growth stages using 15 N dilution technique with barley as reference plant.

Results and discussion

Results indicate that nitrogen fixation initiated at low temperature, as early as growth stage 5-7 leaves. Accession A had the lowest yield and N fixation activity among the accessions tested. Percentages of N derived from fixation increased regularly during growth and reach 62, 85 and 87% at maturity stage for A, B and R accessions respectively. Extrapolated to a field surface, the N fixed amounted to 41, 109 and 113 kg.ha -1 for A, B and R accessions respectively.

Conclusion

The use of accessions more tolerant to low temperature should contribute to satisfy forage needs during cool season.

------------------- ------------------- ------------------- ------------------- Total N in plant (kg ha -

Table 1 .

 1 Production of dry matter (kg ha -1 ; at the end of cold period and at maturity)

		End of cold period	End of spring
		Inoculated	control	Inoculated	control
	A (M. aculeata 80)	476 de	377 e	2531 e	2096 e
	B (M. aculeata 5099)	757 e	566 cd	3110 cd	2730 de
	R (M. rigidula 716)	715 b	504 d	3706 b	2810 de
	For a given period, data with similar superscript letters do not differ at p<0.05 level,
	LDS test
	Table 2. Nitrogen fixation (total N in plants, amounts of N derived from fixation; Ndfa and
	percentages of N derived from fixation ; %Ndfa) at different growth stages
	5-7 leaves	16-18 leaves	End flowering	Maturity

  Data in same column with similar superscript letters do not differ at p<0.05 level, LSD test

		1 )
	A	2 bc	7 b	30 b	66 b
	B	3 a	14 a	87 a	129 a
	R	2 c	13 ab	104 a	129 a
	Ndfa (kg ha -1 )
	A	0.2 a	1 a	19 b	41 b
	B	1.6 b	8 a	63 a	109 a
	R	0.6 c	7.8 a	86 a	113 a
	Ndfa (%)
	A	10 c	14 b	62 c	62 b
	B	56 a	58 a	71 b	85 a
	R	35 b	63 a	83 a	87 a

Acknowledgements

The authors thank ICARDA for the field experiment, the french Embassy in Damascus for the PhD grant allocated to KS.


est destinée au dépôt

Reviewed by Thomas Saucède

Diversity of Antarctic marine life is an endless source of fascination for biologists who work in the Southern Ocean. Beside emblematic marine mammals, sea birds, and fish, and all tenants of the water column, the Antarctic seafloor is home to a profusion of living forms dominated by macro-invertebrates, the macrobenthos. The Antarctic macrobenthos shows a remarkable variety of shapes and colours, as well as singular assemblages with no counterpart worldwide. Over the last 40 million years, these organisms have evolved in partial isolation from the rest of the World's Ocean and have acquired many original features. They have adapted to singular environmental conditions, established in the Southern Ocean in stages since the end of the Eocene, among which near-freezing sea waters, ice disturbance, and extreme seasonality in primary production are key factors. Formerly unexpected in polar seas, in which prevail harsh conditions for human perception, such a diversity is now revealed by underwater images and scientific catches to biologists who work in the Southern Ocean aboard research vessels or at established nearshore stations.

It is yet concealed to other visitors of the Antarctic and Museum specimens preserved in ethanol and formalin do not render anymore the vivid colours of living forms. Martin Rauschert's and Wolf Arntz's book aims to portray the startling diversity of the Antarctic macrobenthos as the authors observed it in the field: by illustrating the original shapes and colours of the living forms.

The book is the product of a rich collaboration between two experts in benthic research who have led various expeditions to the Antarctic. During two overwinterings and a seasonal stay at the Soviet station Bellingshausen in the eighties Martin Rauschert explored the shallows of King George Island (South Shetland Islands) by scuba diving to observe and photograph benthic communities in their natural habitats. At the same time, Wolf Arntz led the first West German expedition of the polar research vessel Polarstern to the Weddell Sea and around the Antarctic Peninsula where he collected macrobenthos on the Antarctic continental shelf and upper slope by means of towed gears. After reunification of Germany, Rauschert joined several expeditions of RV Polarstern upon Arntz's invitation and photographed many organisms sampled in the catches. He understood the importance of colouration for identification of individual species, while it is rarely used by taxonomists, as species diagnostic features are based on morphological characters that are distinguishable on museum specimens.

After a short preface, an instructive introduction exposes the main singularities of Antarctic benthic communities: their "Palaeozoic-like" composition dominated by sessile filter feeders, the eurybathy of many species, endemism, original life strategies such as the prevalence of brooding species, and adaptation to freezing conditions. The composition of communities is the result of both in situ evolution and immigration of species coming from the north and the deep sea. The authors highlight the influence of tectonic, climatic, and oceanographic histories on marine evolution, such as the parts played by past glaciations and by the Antarctic Circumpolar Current in the isolation of populations. They also emphasize the importance of adaptation to the characteristics of Antarctic environments, namely cold waters, ice disturbance, low sedimentation rates, extreme seasonality of primary production, and lack of durophagous predators otherwise common in the oceans.

Underwater images and picture plates constitute the central part of the guide. It is composed of a first collection devoted to living communities of the seafloor with pictures taken in the shallows of King George Island and in deeper areas of the Scotia Arc and the Weddell Sea. Representative catches were also photographed upon their arrival on deck to show sampling work onboard RV Polarstern. More than 1,100 species are illustrated individually in a second set of 80 colourful plates organized according to the systematics. This impressive catalogue documents about 25% of the macrobenthic species ever described in the investigated area. The secret for the vivid colours and nice shapes of the shots is that specimens were photographed alive or fresh at very low temperatures immediately after sampling. All taxonomic groups are depicted but some of them are treated in more details depending on their prevalence in Antarctic ecosystems and on the authors' expertise. Plates only mention taxon names, while supplementary information on location, size, and systematics of specimens is condensed in the following section. For each taxonomic group, short paragraphs written in a simple style give the main systematic clues. The guide ends with a short suggestion of publications.

Documenting such a diversity of living forms is unavoidably open to taxonomic uncertainties, omissions, and mistakes. A certain proportion of Antarctic species is not named yet, others are being revised, and many taxa include cryptic species that by definition cannot be distinguished based on morphological features alone. The reader is repeatedly notified about these taxonomic limitations, and taxonomists are invited to suggest corrections when needed. The book could also benefit from more consistency in the naming of taxa, and the section devoted to systematics, although written for non-specialists, could be refined. Despite the impressive number of species illustrated and the enormous task it represents, uncompleteness remains a real weakness for a field guide. Here, it is partially compensated by the wide taxonomic coverage of the catalogue and by the fact that common taxa are illustrated. More than a mere field guide, the book is a captivating catalogue for the reader curious to learn about diversity of life. Whatever the imperfections, the fascination of all the readers who will open the book, be they taxonomists or not, is the best evidence that the authors have reached their goal.

Thomas Saucède

Université de Bourgogne Franche-Comté, Dijon, France Email: Thomas.Saucede@u-bourgogne.fr

Martin

  Rauschert and Wolf E. Arntz: Antarctic macrobenthos: a field guide of the invertebrates living at the Antarctic seafloor. Arntz & Rauschert Selbstverlag, Wurster Nordseekueste, Germany, 2015, 143 pp Thomas Saucède To cite this version: Thomas Saucède. Martin Rauschert and Wolf E. Arntz: Antarctic macrobenthos: a field guide of the invertebrates living at the Antarctic seafloor. Arntz & Rauschert Selbstverlag, Wurster Nordseekueste, Germany, 2015, 143 pp. Polar Biology, 2016, 39 (4), pp.759-760. 10.1007/s00300-016-1905-2. hal-


Zhan Yu

Jianfu Shao
email: jian-fu.shao@polytech-lille.fr

Gilles Duveau

Minh-Ngoc Vu

Gilles Armand

Numerical modeling of deformation and damage around underground excavation by phase-field method with hydromechanical coupling

Keywords: Radioactive waste disposal, COx claystone, damage, time-dependent deformation, hydromechanical coupling, phase-field method

come   L'archive ouverte pluridisciplinaire

Introduction

Geological disposal is one of solutions envisaged in many countries for radioactive waste management. In France, an underground research laboratory (URL) has been constructed at Bure by the French national agency for radioactive waste management (Andra). The Andra URL is located in the Callovo-Oxfordian (COx) claystone layer at an average depth of 490 m. One of the missions of the URL is to carry out a series of in-situ experiments to investigate thermo-hydromechanical responses of the host rock (Armand et al., 2017a). Among those experiments, a number of so-called mine-by tests have been performed, for example the GCS gallery [START_REF] Vu | Excavation induced over pore pressure around drifts in the callovo-oxfordian claystone[END_REF]. The principle of those tests is monitoring hydromechanical responses of the COx claystone around a gallery (pore pressure, displacement, damaged zones, crack distributions) during the excavation phase and post-excavation phases. Based on the experimental data obtained from the GCS experiment, a numerical modeling benchmark has been organized with the purpose to validate and compare different constitutive models and numerical methods [START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF]Armand et al., 2017b). For instance, elastic-plastic and viscoplastic models have been used to describe the mechanical behavior of the COx claystone [START_REF] Mánica | A time-dependent anisotropic model for argillaceous rocks. application to an underground excavation in callovo-oxfordian claystone[END_REF][START_REF] Souley | Hydro-elasto-viscoplastic modeling of a drift at the meuse/haute-marne underground research laboratoratory (url)[END_REF][START_REF] Cuvilliez | An elastoviscoplastic constitutive model for geomaterials: Application to hydromechanical modelling of claystone response to drift excavation[END_REF]. The initiation and growth of localized cracks have been considered as a strain localization process which was described by using high gradient plastic models [START_REF] Van Den Eijnden | Modeling the strain localization around an underground gallery with a hydro-mechanical double scale model; effect of anisotropy[END_REF][START_REF] Pardoen | Modelling the influence of strain localisation and viscosity on the behaviour of underground drifts drilled in claystone[END_REF]. In additional to those continuum mechanics approaches, a discrete block-spring method has also been developed to capture excavation induced multiple under both static and dynamic loads (Miehe et al., 2010a;[START_REF] Borden | A phase-field description of dynamic brittle fracture[END_REF][START_REF] Ambati | A review on phase-field models of brittle fracture and a new fast hybrid formulation[END_REF]. The phase-field method can be easily implemented in standard finite element codes and has successfully been applied to various engineering problems including multi-physics coupling [START_REF] Miehe | Phase field modeling of fracture in multiphysics problems. part ii. coupled brittle-to-ductile failure criteria and crack propagation in thermoelastic-plastic solids[END_REF], finite deformation [START_REF] Borden | A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects[END_REF]. It has also been extended to plastic materials [START_REF] Fang | Phase field fracture in elasto-plastic solids: variational formulation for multi-surface plasticity and effects of plastic yield surfaces and hardening[END_REF][START_REF] Choo | Coupled phase-field and plasticity modeling of geological materials: From brittle fracture to ductile flow[END_REF]. However, in most previous studies, only tensile cracks have been described for materials subjected to tensile stresses. In most underground engineering applications such as the geological disposal of radioactive waste, host rocks are generally subjected to initial compressive stresses. In that situation, both tensile and shear cracks can be generated. Therefore, it is essential to take into account those complex cracking processes. For this purpose, a novel phase-field method is proposed in this paper by introducing two independent damage variables, respectively representing tensile and shear cracks. Further, the phase-field method is extended to hydromechanical coupling in saturated porous media in order to account for the effect of pore fluid pressure on damage processes. The time-dependent deformation of the COx claystone is also taken into account by using a viscoplastic model, which is coupled with the shear damage evolution. The proposed model is then implemented in a finite element code and applied to studying the hydromechanical responses and damage processes of the COx claystone around the GCS experimental gallery.

Phase-field method for saturated porous media

We consider here a saturated porous medium occupying the volume Ω. It is subjected to the body force f b in Ω, the surface force t N on the external boundary ∂Ω f and the prescribed displacement u on the external boundary ∂Ω u . Meanwhile, the porous medium is also subjected to the fluid flux q on the external boundary ∂Ω q and to the prescribed fluid pressure p on the external boundary ∂Ω p . We shall determine the fields of displacement (strains and stresses) and of pore fluid pressure, the initiation and propagation of cracks inside Ω during the whole loading history. The isothermal conditions are adopted throughout the present study.

Regularized crack fields

Failure of most brittle rocks is driven by the transition from diffuse damage to localized cracks or fractures [START_REF] Paterson | Experimental rock deformation-the brittle field[END_REF]. There are sharp displacement discontinuities across those localized cracks. Further, the strain localization leads to the pathological mesh dependency in the classical continuum damage mechanics solutions. Suitable regularization methods should be adopted. In the framework of phase-field method (Miehe et al., 2010a), a sharp crack is replaced by a regularized damage band whose width is controlled by a characteristic length. On the other hand, differently with metal materials under tensile loads, complex cracking modes are observed in rock-like materials under compression-dominating stresses [START_REF] Wong | Micromechanics of faulting in westerly granite[END_REF][START_REF] Wong | Analysis of crack coalescence in rock-like materials containing three flawspart i: experimental approach[END_REF]. In general, both tensile and shear cracks are created and coupled. Therefore, in order to easily deal such mixed cracks, two phase fields are considered in the present study, respectively representing the tensile and shear cracks.

Therefore, the total regularized crack surface area Γ l d is expressed as follows:

Γ l d = Γ t l d (d t ) + Γ s l d (d s ) = Ω {γ t (d t , ∇d t ) + γ s (d s , ∇d s )}dV (1)
where Γ α l d (α = t, s) are the contributions of tensile and shear cracks to the total crack surface area. Two scalar-valued functions γ α (d α , ∇d α ) denote the tensile and shear crack density (surface area per unit volume). Different forms can be used to define the regularized crack surface density γ α . A common form was introduced in [START_REF] Ambrosio | Approximation of functional depending on jumps by elliptic functional via t-convergence[END_REF] and it is adopted here:

γ α (d α , ∇d α ) = 1 2 { 1 l d (d α ) 2 + l d ∇d α .∇d α } ; α = t, s (2)
In this function, a scalar-valued variable d α (x) (α = t, s) is introduced for each type of crack and conventionally called the phase field (or damage) variable. Physically, the value of d α (x) (α = t, s) indicates the damage state of material, more precisely d α (x) = 0 being the sound state while d α (x) = 1 the fully cracked one. It is worth noticing that the crack surface density γ α is a function of both d α and its gradient ∇d α so that its value at a material point is influenced by the damage distribution insider a neighboring zone. l d is a length scale parameter controlling the width of regularized damage bands.

On the other hand, following the spirit of Griffith theory in fracture mechanics, the energy needed for the propagation of cracks can be related to the crack surface area created.

Thus, the energy density per unit volume requested to create the tensile and shear cracks can be defined as:

w c (d t , d s , ∇d t , ∇d s ) = g t c γ t (d t , ∇d t ) + g s c γ s (d s , ∇d s ) (3)
Two coefficients g t c and g s c define the material toughness for the tensile and shear crack respectively.

Variational framework

The variational approach proposed in [START_REF] Francfort | Revisiting brittle fracture as an energy minimization problem[END_REF] is here adopted and generalized to saturated porous media, which exhibit an elastic behavior, viscoplastic flow and crack growth. Further, it is assumed that the viscoplastic dissipation is influenced by the evolution of crack fields. Thus, The total energy functional E of Ω can be expressed in the following general form:

E(ε e , ε vp , m e , m vp , d t , d s ) = Ω w e (ε e , m e , d t , d s )dV + Ω h vp (d t , d s )w vp (ε vp , m vp )dV + Ω w c dV (4)
ε e is the elastic strain tensor while ε vp the viscoplastic one. m e and m vp are the elastic and viscoplastic fluid mass change per unit initial volume. w f , w c and w vp represent respectively the density of elastic free energy, viscoplastic dissipation and that needed for the crack growth as indicated in (3). The function h vp (d t , d s ) defines the influence of crack fields on the viscoplastic dissipation. Its form will be given later. It is worth noticing that for the sake of completeness, the energy functional is here written in a general form. It is assumed that the viscoplastic flow should induce an irreversible fluid mass change m vp . This is not the case for a purely deviatoric viscoplastic flow in an isotropic material.

Elastic free energy

According to the poroelastic theory [START_REF] Coussy | Poromechanics[END_REF]Cheng, 2016), the constitutive relations of an undamaged saturated porous medium can be expressed as:

σ -σ 0 = C b0 : ε e -B(p -p 0 ) (5) p -p 0 = M (-B : ε e + m e ρ 0 f ) (6)
σ and p are respectively Cauchy stress tensor and fluid pressure while σ 0 and p 0 their initial values at the reference state. C b0 denotes the drained elastic stiffness tensor. ρ 0 f defines the volumetric mass of fluid at the reference state. B and M are respectively the Biot's coefficients tensor and modulus, which are functions of volumetric compressibility of porous medium, solid matrix and fluid as well as porosity. In the case of isotropic porous media, B is reduced to a scalar coefficient b so that B = bI with I being the second order unit tensor.

The Biot's modulus is accordingly given by 1/M = (b -φ)/K m + φ/K f , with K m being the bulk modulus of solid matrix, K f that of fluid and φ porosity.

In order to easily introduce the impact of damage on the poroelastic behavior, the elastic free energy of undamaged saturated porous medium (without the contribution of fluid convection) can be conveniently written in the following form:

w 0 e (ε e , p) = 1 2 σ b : ε e + 1 2 (p -p 0 ) 2 M , σ b = (σ -σ 0 ) + B(p -p 0 ) (7)
where σ b denotes the Biot's (elastic) effective stress tensor, acting as the thermodynamic force conjugated with the elastic strain tensor. On the other hand, it is noticed that the evolution of tensile crack is generally driven by tensile stress while that of shear crack by differential compression stresses. Thus, in order to define physically based criteria for the growth of tensile and shear cracks, the Biot's effective stress tensor is decomposed into a positive (tensile) part and a negative (compression) part such as σ b = σ b+ + σ b-, so that the elastic free energy function is rewritten as follows:

               w 0 e (ε e , p) = w 0 e+ (ε e ) + w 0 e-(ε e ) + 1 2 (p-p 0 ) 2 M w 0 e+ = 1 2 σ b+ : ε e w 0 e-= 1 2 σ b-: ε e (8)
The two parts of Biot's effective stress tensor are calculated by using the classical fourth order projection operators such as:

       σ b+ = P + σ : σ b σ b-= P - σ : σ b (9)
The operators P ± σ are further constructed from the spectral decomposition of the stress tensor such suggested in [START_REF] Lubarda | Damage model for brittle elastic solids with unequal tensile and compressive strengths[END_REF][START_REF] Murakami | Continuum damage mechanics: a continuum mechanics approach to the analysis of damage and fracture[END_REF]:

       P + σ = 3 β=1 σ b β + n β ⊗ n β ⊗ n β ⊗ n β P - σ = I -P + σ ( 10
)
I denotes the fourth-order identity tensor. n β (β = 1, 2, 3) define the three orthogonal principal directions of the stress tensor while σ b β are the three principal stresses. The bracket

. + indicates:        a + = a, a ≥ 0 a + = 0, a < 0 (11)
The projection operators are widely used for the decomposition of stress or strain tensor into positive and negative parts [START_REF] Miehe | Comparison of two algorithms for the computation of fourth-order isotropic tensor functions[END_REF][START_REF] Miehe | Algorithms for computation of stresses and elasticity moduli in terms of seth-hill's family of generalized strain tensors[END_REF][START_REF] Zhang | A new phase field fracture model for brittle materials that accounts for elastic anisotropy[END_REF]. In general, the projection operators for stress P σ and for strain P ε are different. In this study, the decomposition of stress tensor is adopted. Compared with the strain tensor decomposition as that used in (Miehe et al., 2010a), the stress decomposition avoids the implication of elastic parameters. It is particularly convenient for anisotropic materials as that considered in this work.

In cracked materials, the poroelastic properties are affected by the damage evolution.

Different forms of degradation functions have been proposed. In the present study, two distinct functions are introduced to describe the impact of tensile and shear cracks on the positive and negative parts of elastic energy. However, for the sake of simplicity, the variations of Biot's coefficient b and modulus M with induced damages are neglected. Their values are kept constant in the present study. Therefore, the elastic free energy of damage porous medium is expressed as follows:

w e (ε e , p, d t , d s ) = h t (d t )w 0 e+ + h s (d s )w 0 e-+ 1 2 (p -p 0 ) 2 M ( 12
)
The common form widely used in previous studies [START_REF] Miehe | Comparison of two algorithms for the computation of fourth-order isotropic tensor functions[END_REF][START_REF] Miehe | Algorithms for computation of stresses and elasticity moduli in terms of seth-hill's family of generalized strain tensors[END_REF][START_REF] Zhang | A new phase field fracture model for brittle materials that accounts for elastic anisotropy[END_REF] is here adopted for two degradation functions:

h α (d α ) = (1 -k)(1 -d α ) 2 + k ; α = t, s (13)
k is a small positive value introduced to ensure the positive definiteness of elastic stiffness tensor after the material is fully broken (d α = 1).

Viscoplastic deformation

Viscoplastic theory is here used for the description of time-dependent deformation. As discussed above, the elastic strain of saturated porous medium is driven by the Biot's effective stress. However, plastic or viscoplastic deformation is generally driven independently by stress tensor and fluid pressure. The existence and validity of effective stress concept has not so far been fully proven. However, in order to simply the formulation of plastic and viscoplastic models, some micro-mechanical and experimental studies have been performed

on the determination of plastic yield or strength criteria of saturated rock-like materials [START_REF] De Buhan | On the validity of the effective stress concept for assessing the strength of saturated porous materials: a homogenization approach[END_REF][START_REF] Lydzba | Stress equivalence principle for saturated porous media[END_REF][START_REF] Xie | Experimental investigation and poroplastic modeling of saturated porous geomaterials[END_REF]. It was found that the commonly used Terzaghi's effective stress concept provided a good approximation for the description of fluid pressure effect. It is defined as σ t = σ + pI, with I being the second order unit tensor. Therefore, in this study, it is assumed that the viscoplastic flow is driven by the Terzaghi's effective stress. Further, as the emphasis is here put on the description of damage fields, a simple viscoplastic model based on the so-called Lemaitres law is adopted, as in some previous studies for clayey rocks [START_REF] Souley | Hydro-elasto-viscoplastic modeling of a drift at the meuse/haute-marne underground research laboratoratory (url)[END_REF]. With the assumption of small strains, the total strain increment dε is the sum of elastic and viscoplastic components:

dε = dε e + dε vp (14)
The increment of viscoplastic strain dε vp is calculated from its flow rate εvp :

dε vp = εvp dt (15)
Furthermore, the viscoplastic flow rate tensor is assumed to be co-axial with the deviatoric stress tensor, that is:

εvp = 2 3 εvp q s t (16)
where q = 3s t : s t /2 is the conventional deviatoric stress, with s t being the deviatoric part of the Terzaghi's stress tensor σ t . The magnitude of viscoplastic strain rate is defined by the scalar variable εvp . It depends on the current value of deviatoric stress q and the state of accumulated viscoplastic strain:

εvp = η q -σ s σ ref n + (1 -ε eq vp ) m (17)
η is a viscosity parameter controlling the initial rate of viscoplastic flow. σ s defines the threshold stress for viscoplastic flow. σ ref is a reference stress (unit of stresses adopted, here 1 Pa) in order to keep the term under the power law dimensionless. n and m are two material parameters controlling the evolution of viscoplastic flow rate. The scalar variable ε eq vp denotes the equivalent viscoplastic strain which is defined by:

ε eq vp (t) = t 0 2 3 εvp (τ ) : εvp (τ )dτ (18)
The viscoplastic energy dissipation density can be calculated by:

w vp (ε vp ) = t 0 σ t (τ ) : εvp (τ )dτ (19)
With this simple model, only the shear viscoplastic strains are produced. Therefore, it is reasonable to assume that the viscoplastic dissipation is influenced by the shear crack growth only. In this study, the following simple form is used for the viscoplastic-damage coupling function:

h vp (d t , d s ) ≡ h vp (d s ) = χh s (d s ) , χ ∈ [0, 1] (20)
The parameter χ defines the effect of shear crack on the viscoplastic dissipation. h s (d s ) is the same as the degradation function given in (13).

Evolution of crack fields

Based on the different energy parts defined above, the total energy functional can be rewritten as follows:

E = Ω w e (ε e , p, d t , d s )dV + Ω g t c γ t (d t , ∇d t ) + g s c γ s (d s , ∇d s ) dV + Ω h vp (d s )w vp (ε vp )dV + Ω (1 -χ)w vp (ε vp )dV (21)
It is assumed that the growth of tensile and shear cracks as well as the viscoplastic flow are three independent dissipation processes. Each of them verifies the positiveness requirement independently. Further, based on previous studies [START_REF] Nguyen | A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography[END_REF][START_REF] Wu | Chapiter one -phase field modeling of fracture[END_REF], each of two crack fields can be determined by solving the minimization problem of E. By calculating the first order variation of E and using the unilateral stationary condition for the total energy functional with respect to each phase-field variable, i.e. δE = 0 for δd α > 0 and δE > 0 for δd α = 0 (α = t, s), one gets the following governing equations for the evolution of two crack fields (α = t, s) : et al., 2016;[START_REF] Wu | Chapiter one -phase field modeling of fracture[END_REF]. By considering the specific forms adopted above for w e , h vp and γ α , the governing equations for two crack fields become:

                               -∂we ∂d α -∂hvp ∂d α w vp -g α c δ d α γ α = 0 , ḋα > 0 , in Ω -∂we ∂d α -∂hvp ∂d α w vp -g α c δ d α γ α ≤ 0 , ḋα = 0 , in Ω ∂γ ∂∇d α .n = 0 , on δΩ (22) with δ d α γ α = ∂γ α ∂d α -div ∂γ α ∂∇d α being the variational derivative of crack field d α (Nguyen
               -h t (d t )w 0 e+ -g t c 1 l d d t -l d div(∇d t ) = 0 , ḋt > 0 -h s (d s )(w 0 e-+ χw vp ) -g s c 1 l d d s -l d div(∇d s ) = 0 , ḋs > 0 (23) where h t (d t ) = -2(1 -k)(1 -d t ) and h s (d s ) = -2(1 -k)(1 -d s )
are the derivatives of two degradation functions with respect to two damage variables.

According to the second relation of ( 23), the evolution of shear crack d s is driven by the compressive part of the elastic strain energy associated with the intact material state w 0 e-. This is physically not fully justified. In practice, the shear cracking of rock-like materials subjected to compression stresses is essentially driven by the maximum shear stress along a critical plane. By considering the classical Mohr-Coulomb criterion, the maximum shear stress can be conveniently represented by the difference between the major and minor principal stresses for a given mean value of these ones. Inspired by some previous studies [START_REF] Li | A damage model for hard rock under stressinduced failure mode[END_REF][START_REF] Zhou | Phase field modeling of brittle compressive-shear fractures in rocklike materials: A new driving force and a hybrid formulation[END_REF], an alternative driving energy (force) based on the Mohr-Coulomb-based maximum shear stress is here introduced for the shear crack growth.

Further, in order to take into account the effect of fluid pressure, the maximum shear stress is calculated from the Terzaghi effective stress tensor. Therefore, the modified driving force for the shear crack w s -is expressed as:

w s -= 1 2G σ t 1 --σ t 3 - 2cosϕ + σ t 1 -+ σ t 3 - 2 tanϕ -c 2 + (24)
with the bracket . -such as:

       a -= 0, a ≥ 0 a -= a, a < 0 (25)
σ t 1 and σ t 3 are the major and minor Terzaghi effective principal stress. c and ϕ denote the cohesion and frictional angle of material. By substituting the physically-based driving energy into (23), the criteria for two crack fields are now expressed as:

               -h t (d t )w 0 e+ -g t c 1 l d d t -l d div(∇d t ) = 0 , ḋt > 0 -h s (d s )(w s -+ χw vp ) -g s c 1 l d d s -l d div(∇d s ) = 0 , ḋs > 0 (26)
On the other hand, in order to describe the irreversible process of crack growth under general loading paths including unloading and reloading cycles, the concept of loading history functional has been introduced in (Miehe et al., 2010b). Based on that concept, three energy history functionals are defined as follows:

                               H t (t) = max[w 0 e+ (τ )] τ ∈[0,t] H s -(t) = max[w s -(τ )] τ ∈[0,t] H vp (t) = max[w vp (τ )] τ ∈[0,t] (27)
The time variable τ designates the loading history from the initial state to the current time step t. Using these energy history functionals, the evolution of both tensile and shear cracks is determined by the following relations:

               -h t (d t )H t -g t c 1 l d d t -l d div(∇d t ) = 0 -h s (d s )(H s -+ χH vp ) -g s c 1 l d d s -l d div(∇d s ) = 0 (28)

Numerical implementation in finite element method

In the present study, four unknown fields, namely the displacement u(x, t), pore fluid pressure p(x, t), tensile damage d t (x, t) and shear damage d s (x, t), should be determined inside Ω and at any time instance t. They are determined by solving four boundary values problems by using the framework of finite element method.

The mechanical problem is governed by the equilibrium equations, cinematic relations and constitutive law:

                                               divσ + f b = 0 in Ω σ • n = t N on ∂Ω f σ -σ 0 = C b (d t , d s ) : ε e -B(p -p 0 ) ε e = ε -ε vp , ε = ∇ s u (29)
The drained elastic stiffness tensor of damaged material C b is given by:

C b (d t , d s ) = h t (d t )P + σ + h s (d s )P - σ : C b0 (30)
As mentioned above, when d t = 1 (or d s = 1), one has h t = k (or h s = k ) to preserve the positiveness of C b . However, the effect of k on all other quantities is not essential. Therefore, for the sake of simplicity, in the subsequent calculations, one takes k = 0.

The fluid flow problem is described by the hydraulic diffusion equation including Darcy's conduction law, fluid mass balance equation and poroelastic relation equation ( 6). One has:

               div( k(d t ) µ .∇p) = 1 M ∂p ∂t + B : ∂ e ∂t in Ω q • n = q on ∂Ω q (31)
where µ is the dynamic viscosity of fluid. k denotes the permeability tensor of cracked porous medium. It is assumed that the permeability is affected by the tensile crack d t by using the following exponential law:

k(d t ) = k 0 exp(βd t ) (32)
where the coefficient β controls the evolution rate of permeability. k 0 is the initial permeability tensor of intact porous medium.

On the other hand, in this study, we shall consider the structural anisotropy of elastic properties and of permeability of the studied COx claystone. However, due to the difficulty to get feasible experimental data and for the sake of simplicity, the Biot's coefficients tensor B is assumed to be isotropic and a scalar Biot's coefficient b is used.

With the help of growth criteria (28), the evolution problems of two crack fields are respectively governed by the following equations:

                       h t (d t )H t + g t c 1 l d d t -l d div(∇d t ) = 0 in Ω ḋt ≥ 0 in Ω d t = 1 on Γ t ∇d t • n = 0 on ∂Ω (33)
for the tensile crack field and:

                       h s (d s )(H s -+ χH vp ) + g s c 1 l d d s -l d div(∇d s ) = 0 in Ω ḋs ≥ 0 in Ω d s = 1 on Γ s ∇d s • n = 0 on ∂Ω (34)
for the shear crack field. In the above relations, Γ t and Γ s respectively denotes the fully cracked domain by the tensile and shear damage.

By using the classical weak formulation and geometrical meshing, and based on the suitable shape functions for the approximation of four fields (see Appendix), one obtains the systems of discrete equations to be solved at each time step. In the present work, the displacement and pore pressure fields are solved by a fully coupled method. The corresponding system of equations is given as follows (see Appendix for details):

               R uu ∆U + C up ∆P = ∆F e (∆tR pp + M pp )∆P + C pu ∆U = ∆t(-R pp P + ∆F ω ) (35)
where U and F e respectively denote the column matrix of increment nodal displacements and forces at the current loading step (for the sake of simplicity, the step index i + 1 is omitted).

P and F ω are respectively the column matrix of increment nodal pore pressures and hydraulic loads. The expressions of all matrices involved in the coupled hydromechanical equations are presented in Appendix.

Similarly to the hydromechanical problem, two damage variable and their gradients inside each element are also approximated by the nodal values by using appropriate shape functions and related derivatives. The local governing equations are transformed into the following discrete systems of equations to be solved. Namely, for the tensile damage field, one gets:

                               K d t d t = F d t K d t = Ω {(g t c /l d + 2H t )N T p N p + g t c l d B T p B p }dV F d t = Ω 2H t N T p dV (36)
and for the shear damage field:

                               K d s d s = F d s K d s = Ω {(g s c /l d + 2(H s -+ χH vp ))N T p N p + g s c l d B T p B p }dV F d s = Ω 2(H s -+ χH vp )N T p dV (37) K d α (α = t, s)
is the global stiffness matrix related to the tensile or shear crack field. F d α denotes the column matrix of nodal driving forces while d α is that of nodal damage variables at the current loading step. It is worth noticing that with the help of three energy history functionals defined in ( 27), the values of damage variables at the current loading step are directly determined as functions of the accumulated driving energy for each cracking mechanism.

It is obvious that four fields are coupled. Indeed, the elastic stiffness and permeability are dependent on the damage variables which are inversely driven by the values of displacement, stress and pore pressure. Based on previous studies [START_REF] Nguyen | A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography[END_REF][START_REF] Yu | Numerical study of thermo-hydro-mechanical responses of in situ heating test with phase-field model[END_REF] and preliminary calculations, a staged numerical scheme is used in this study. At each loading increment, the hydromechanical solutions are first determined by solving the coupled system (35) and by using the elastic stiffness and permeability calculated with the values of damage variables stored at the end of previous step. Then, the obtained values of displacement, pore pressure and stress are used for the updating of damage variables by solving the crack systems ( 36) and ( 37). These updated values are then stored and used as input data for the hydromechanical problem at the next loading step. Furthermore, in the calculations presented below, an explicit time integration scheme with relatively small time increment is adopted for the calculation of viscoplastic strains. At each time step t, the accumulated equivalent viscoplastic strain defined in (18) obtained at the end of previous step ε eq vp (t -∆t) (its initial value is 0) is used to calculate the scalar viscoplastic strain rate εvp according to (17). It is then used to calculate the rates of viscoplastic strain components with (16). The increments of viscoplastic strains are finally given by (15).

Short and long term responses of GCS gallery

In this section, the proposed phase-field model is applied to investigate short and long term hydromechanical responses of a gallery excavated in the COx claystone layer of the 20 underground research laboratory (URL) of Andra [START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF].

General presentation and input data

A partial view of the Andra URL is shown in Figures 1 and2. The so-called GCS gallery is excavated along the direction of the major horizontal stress at a depth of -490 m. This is an experimental gallery in order to capture the hydromechanical responses of COx claystone during and after excavation phases. A series of sensors have been placed at different locations inside various boreholes, for instance OHZ1521 and OHZ1522 for monitoring pore pressure, and OHZ1501 and OHZ1707 for measuring displacement. Damaged zones around the gallery and induced variations of permeability have also been identified [START_REF] Armand | Geometry and properties of the excavation induced fractures at the meuse-haute-marne url drifts[END_REF]. A detailed presentation can be found in [START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF]. In this study, two-dimensional simulations are performed on a representative cross section of gallery with the assumption of plane strain. Taking into account the symmetrical conditions, only a quarter of section is considered. Based on the choices made in the previous benchmark [START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF], the selected geometrical domain and the related boundary conditions are presented in Figure 3. In particular, the impermeable condition is assumed on the external boundaries, except on the gallery wall. The finite element mesh adopted is also illustrated, containing 2080 elements and 2173 nodes. A finer mesh is used in the zone close to the gallery wall. The excavation operation is modeled by the progressive vanishing of initial stress vector and pore pressure on the gallery wall to desired values. In this study and according to the benchmark [START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF], the initial stress vector is step by step decreased during 28 days in order to reflect the effect of excavation front. This process is conventionally represented by a so-called deconfinement ratio (1 indicates the initial state and 0 the completed excavated one). On the other hand, the initial pore pressure is dropped rapidly in 24 hours.

The evolution curves of deconfinement ratio and pore pressure are presented in Figure 4. According to previous studies (Armand et al., 2017b), the COx claystone exhibits a structural (initial) anisotropy of elastic modulus, failure strength and hydraulic properties.

However, coupling between the structural anisotropy and the induced one has not been clearly identified by experimental data. In the present study, for the sake of simplicity and clarity, only the initial anisotropy of elastic properties and permeability is considered while the other parameters are assumed to be isotropic. The values of basic elastic parameters and permeability are based on those defined in the previous benchmark simulations coordinated by Andra [START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF]. These values were selected as average ones issued from a large number of laboratory tests (Armand et al., 2017b). However, in the present study, the phase-field method is adopted to describe the cracking process in terms of localized damaged zones. Both tensile and shear damages are taken into account. The evolution of tensile damage zone is controlled by the material toughness coefficient g t c . Its value cannot be experimentally measured (not like the classical energy release rate used in fracture mechanics theory) but can be directly connected to the uniaxial tensile strength. For the COx claystone, the value of g t c is identified with an average value of uniaxial tensile strength of 3 MPa. The shear damage evolution is controlled by three parameters, say the toughness coefficient g s c , frictional angle ϕ and cohesion c. In the present study, the following method is adopted for their identification. Firstly, from the peak deviatoric stresses obtained from three conventional triaxial compression tests under confining pressures of 2, 6 and 12 MPa, reported in (Armand et al., 2017b), the trial values of ϕ and cohesion c at the failure (peak) state are calculated by using the classical Mohr-Coulomb criterion. Then, it is assumed that the frictional angle remains the same at the shear damage initiation threshold as at the failure state. The value of cohesion c is decreased to a desired value to define the initiation state of shear damage, for example the transition point from the initial linear elastic part to nonlinear stage in stress-strain curves. However, such a damage threshold point is generally not easy to capture in the case of clayey rocks. In practice, for the sake of simplicity, a small value of deviatoric stress is taken. This leads to consider that the value of c at the damage threshold is relatively small. In this study, c=0.1 MPa is used. The value of g s c is finally identified by numerical fitting of peak values of deviatoric stress. The simulations of three triaxial compression tests by using the elastic-damage (phase-field) model are shown in Figure 5. It is observed that the peak deviatoric stresses are correctly reproduced for three values of confining pressure. There is a good agreement between the numerical stressstrain curves and experimental data for the confining pressure of 2 MPa. However, for the tests under 6 and 12 MPa, the numerical model clearly under-estimate the strains before the peak stress. This is due to the fact that the instantaneous plastic deformation of claystone before peak strength is not neglected in the elastic-damage modeling in order to keep the constitutive model as simple as possible. However, it will be seen that this omission of plastic deformation in the triaxial tests does not affect the good prediction of convergence displacement around the gallery. The latter is strongly influenced by the growth of damaged zones around the gallery wall. This interesting feature will be discussed in the next section. Moreover, it is also useful to mention that the failure and post-peak responses of tested samples are generally driven the initiation and propagation of localized cracks. Under prescribed uniform stresses or strains, the localization of induced cracks can be related to material heterogeneity or weakness zones, for instance the spatial variation of calcite content. Therefore, each sample should be considered as a small structure composed of a heterogeneous medium. This feature is considered in our ongoing works. The parameters involved in the viscoplastic model are calibrated from triaxial compression creep tests reported in (Armand et al., 2017b), by using an optimal numerical fitting.

Three tests are used with a confining pressure of 12 MPa and three deviatoric stresses such as q =17, 25 and 30 MPa. In Figure 6, one shows the comparisons between numerical results and experimental data. In spite of the simplicity, the viscoplastic model still captures the main features of creep deformation of the Cox claystone under these loading conditions. (17,25,30 MPa): comparisons between numerical results (continuous line) and experimental data (dotted lines with triangles) reported in (Armand et al., 2017b) Moreover, in the framework of phase-field method, the scale length l d controls the width of localized damage bands. For the sake of numerical efficiency, the value commonly used is the twice of the smallest element size. This choice is adopted in this study so that l d =0.1m is taken here. A sensitivity study on the impact of l d has been presented in our previous paper [START_REF] Yu | Numerical study of thermo-hydro-mechanical responses of in situ heating test with phase-field model[END_REF]. It has been observed that for a given value of l d , the distribution of damaged zones was indeed not affected by mesh size. Therefore, the phase-field method actually allows avoiding the pathological mesh dependency related to strain and damage localization.

The permeability variation coefficient β can be identified from suitable experimental data showing the relationship between permeability variation and tensile damage growth.

Unfortunately, such data are not easy to obtain. In this study, its value is estimated from in-situ measurements of permeability variation around excavated cavity in COx claystone reported in [START_REF] De La Vaissiere | Gas and water flow in an excavation-induced fracture network around an underground drift: A case study for a radioactive waste repository in clay rock[END_REF]. Finally, it is worth noticing that the triaxial compression tests were performed on samples which are not fully saturated. On contrary, the COx claystone is initially fully saturated with an initial pore water pressure. Therefore, the values of frictional angle and material toughness for shear damage evolution identified from laboratory samples can be higher than those of in-situ claystone. These ones can be adequately readjusted for gallery modeling. The selected set of parameters for numerical simulations of GCS gallery is presented in Table 1.

two principal stresses in the cross sections are quasi-isotropic (see Figure 3). Therefore, the anisotropic damage distribution is essentially related to the anisotropic elastic properties and permeability. More precisely, tensile cracks (with the tensile damage value close to 1) can be observed along the horizontal axis in a zone close to the gallery wall. At the same time, a larger shear damage zone is obtained also along the horizontal axis. The maximum shear damage value is not found at the wall but in a zone inside the ground formation. In Figure 8, one presents the synthetic diagram of tensile and shear fractures observed around the GCS gallery reported in [START_REF] Armand | Geometry and properties of the excavation induced fractures at the meuse-haute-marne url drifts[END_REF][START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF]. One can see that tensile fractures (green lines) are localized closely to the wall while shear fractures are distributed a little away from the wall but extended to a larger region. It seems that the numerical damage zones are qualitatively in agreement with the experimental observations. In order to make a quantitative evaluation, the distributions of tensile and shear damage along the horizontal axis at the end of excavation are plotted in Figure 9. According to the observations on the induced fracturing around the excavated structures oriented in the direction of the principal horizontal major stress including the GCS gallery reported in [START_REF] Armand | Geometry and properties of the excavation induced fractures at the meuse-haute-marne url drifts[END_REF][START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF], the average extensional fracture extent is 0.15×D, 0.4×D and 0.1×D, respectively at roof, sides and floor (D being the diameter of borehole). Regarding the shear fracture zones, the observed maximum extent would be 0.5×D, 1×D and 0.5×D, respectively at roof, sides and floor. The numerical values of damages along the horizontal axis can be compared with the observed extents at the gallery sides. Though the numerical damaged zones cover the observed fractured ones, the extents of strongly damaged zones seem to be smaller than those of observed fractured ones. More precisely, at 0.4×D, the corresponding tensile damage is about 0.3, which is much lower than 1. At 1×D, the value of shear damage becomes even less that 0.1. Therefore, in a general way, the predicted strongly damaged zones seem to under-estimate the observed fractured zones, in particular for the shear cracks. A sensitivity study is presented in the next sub-section on the dependency of damaged zones on the shear toughness coefficient g s c .

Furthermore, in the numerical results, only diffuse damage zones are obtained while sharp fractures are observed around the excavated structures including the GCS gallery [START_REF] Armand | Geometry and properties of the excavation induced fractures at the meuse-haute-marne url drifts[END_REF][START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF]. The reproduction of such fractures is a delicate task. Their occurrence can be influenced by several factors including material heterogeneity or weakness zones enhancing the localization of fractures. This important feature is not taken into account in the present work but will be investigated in future by considering for example the spatial heterogeneity of mineralogy (for instance calcite content) in the COx claystone in a micro-mechanics based constitutive model. In Figure 10, one shows the numerical result of pore pressure distribution at the end of excavation. It is clearly seen that the pore pressure distribution is strongly anisotropic between horizontal and vertical directions. Due to the short duration (28 days) and the small values of initial permeability, the pore pressure variation is resulted essentially from the volumetric strain induced by the poroelastic coupling rather than by fluid flow. As a consequence, due to the anisotropic elastic tensor, for instance the Young's modulus in the perpendicular orientation is smaller than that in the parallel one, we obtain a zone of over-pressure along the horizontal direction at some distance from the wall. But the pore pressure distribution is also affected by the excavation induced damage growth modifying the volumetric strain [START_REF] Vu | Excavation induced over pore pressure around drifts in the callovo-oxfordian claystone[END_REF]. However, the effect of permeability increase as that presented above is not observed yet. In Figure 11 and 12, one shows the evolution of pore pressure during the excavation phase at different selected measuring points. It is seen that the general trends observed in the pressure mesurements are quite correctly reproduced by the numerical results in spite of scatters at some positions, in particular the vertical direction. In Figures 13 and 14, we compare the numerical predictions and experimental measurements of the permeability variation [START_REF] De La Vaissiere | Gas and water flow in an excavation-induced fracture network around an underground drift: A case study for a radioactive waste repository in clay rock[END_REF] along the vertical and horizontal axes at the end of excavation. One can clearly see that the permeability is sig-nificantly enhanced by the induced cracks around the GCS gallery. In consistency with the crack distribution presented above, the increase of permeability along the horizontal direction is clearly more important than the vertical one. The numerical predictions of permeability obtained by using the empirical relation (32) reproduce well the general trends observed in the experimental measurements, but they are systematically higher than the measured ones. It is worth noticing that the permeability variation is inherently correlated with the pore pressure evolution. By choosing other values of the parameter β introduced in (32), it is possible to get some better predictions of the permeability variation. But by doing this, very large scatters are obtained on the prediction of pore pressure given in Figure 10. It appears that it is not possible to get an appropriate value of β which gives a good prediction of permeability and pore pressure simultaneously. In this study, the value of β is selected in order to get a quite good description of pore pressure variation. Furthermore, the self-sealing process of fractured zone which can take place in reality is not taken into account in the present model. This can also lead to an over-estimation of permeability variation in numerical modeling with respect to the measurements. Further studies are needed on this issue by searching for a finer description of the permeability variation with induced damage. Furthermore, it is useful to mention that in the present work, the values of Biot's coefficient and modulus are taken constant. Only the permeability values are functions of tensile damage. This is a strong simplifying assumption, which can induce important errors in the pore pressure prediction. In particular, in the zones with high values of tensile damage, open cracks are created. The value of Biot's coefficient should be close to 1. It is therefore needed to properly describe the transition from the initial values of Biot's coefficient and porosity to 1 in such cracked zones. However, suitable experimental data are still needed to calibrate such transitions laws. This important feature is not considered here and will be investigated in future studies.

Long term responses

In Figures 15 and16, the variations of pore pressure with time are presented at four measuring points respectively inside the horizontal borehole OHZ1521 and vertical one OHZ1522.

From a quantitative point of view, the numerical predictions are not fully in agreement with the experimental measurements. For instance, the measured pore pressure decreases more quickly than the simulated one. One of the reasons can be that the real configuration is 3D where the fluid flow also take place in the longitudinal direction. However, in a global way, the predictions provided in this study are little better than those obtained in the previous benchmark modeling cited above [START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF]. More precisely, it seems that the predictions of pore pressure along the horizontal axis are better than those along the vertical one. In particular, the pore pressure variation inside the over-pressure zone (points PRE04 and PRE05 in Figure 15) is well reproduced, including the over-pressure regime during the early period after excavation. Along the vertical axis (see Figure 16), there is no over-pressure zone and one essentially observes the reduction of fluid pressure at all four points. The sharp reduction of pore pressure at the closet point to the wall (point PRE03) is quite well described by the numerical modeling. For both directions, the pore pressure decreases with time more and less quickly for all measuring points. This general trend is also well reproduced. However, further studies are still needed in order to improve the prediction of pore pressure variation. In Figure 17, we present the variations of convergence displacement on the gallery wall along the horizontal and vertical directions. The numerical results agree quite well with the measured values. It is obvious that the horizontal convergence is much larger than the vertical one. It is recalled that the elastic modulus in the horizontal direction (E =6000MPa) is higher than that in the vertical one (E ⊥ =4500MPa). Therefore, the large horizontal displacement is directly related to the onset of important damaged zone in this direction.

However, the anisotropy ratio of convergence between the horizontal and vertical direction of the numerical results is rather close to the lower bound of that observed in the measurements, say between 1.5 and 2. This difference can be due to the fact that the horizontal convergence is a bit under-estimated by the numerical modeling because the strong displacement discontinuities in the sharp tensile fractures in this direction are not explicitly taken into account. This shows some limitation of phase-field method. As mentioned above, the value of shear toughness coefficient g s c is not easy to determine.

A reference value of g s c =4100N/m is chosen and used for the calculations presented above. In order to investigate its effects on hydromechanical responses of the GCS gallery, two comparative calculations are performed respectively using g s c = 3700N/m and 4500N/m. In Figure 18, the fields of tensile and shear damages around the gallery are presented and compared between three cases while the comparisons of damage variation along the horizontal axis are given in Figure 19. One can first observe that the damage fields are clearly affected by the value of g s c . Both the intensity and extent of shear damage zone increases (or decreases)

when the value of g s c is lower (higher). Such a result seems to be quite logic as the parameter g s c represents the material resistance to shear damage growth. But another interesting result is that the distribution of tensile damage is also significantly affected by g s c even if the value of tensile toughness g t c is not modified. This clearly indicates that two damage mechanisms are inherently coupled. Indeed, the driving energy controlling the tensile damage w 0 e+ is dependent on stress-strain fields which are effected by the shear damage evolution. With the results obtained with g s c = 3700N/m, the tensile damage value obtained at 0.4×D is higher than 0.4. The observed tensile cracked zone is thus better described [START_REF] Armand | Geometry and properties of the excavation induced fractures at the meuse-haute-marne url drifts[END_REF][START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF]. The intensity and extent of shear damage zone are also more close to the observed ones than the reference case. However, with this lower value of g s c , the convergence displacements are over-estimated by the numerical predictions, as shown in Figure 21, in particular for the vertical direction. In Figure 22, one shows the evolution of pore pressure at the horizontal sensor PRE04. One can see that the pressure evolution at this measuring point is significantly affected by the value of g s c . In particular, when the toughness parameter is reduced to 3700N/m, there is a rapid decrease of pore pressure which becomes significantly lower than the measurement. This is due to the large increase of permeability in the damaged zone as the tensile damage is significantly enhanced. According to these comparisons, it seems that the choice of g s c = 4100N/m is a good compromise. Case

g s c =3700N/m g s c =4100N/m (reference) g s c =4500N/m d t d s
1: E = E ⊥ =5250MPa, ν ⊥ =ν =0.24, G = E /2/(1 + ν ⊥ )=2117MPa
Case 2: E =6000MPa and E ⊥ =4500MPa, ν ⊥ =0.24 and ν =0.35, G=2400MPa

Case 3: E =6500MPa and E ⊥ =4000MPa, ν ⊥ =0.24 and ν =0.35, G=2400MPa This is due to the fact that along the horizontal axis, the damage evolution is already quite important without considering the viscoplastic energy. Its contribution is not high enough to significantly affect the damage evolution. But in the vertical direction, as the amplitude of damage due to poroelastic effects is small, the contribution of viscoplastic flow can enhance quite significantly the damage evolution. In Figure 35, we compare the horizontal and vertical convergences between four cases. In consistency with the shear damage field, the viscoplastic effect on the horizontal convergence is smaller than that on the vertical one.

E = E ⊥ = 5250MPa E = 1.3E ⊥ = 6000MPa (reference) E = 1.6E ⊥ = 6500MPa d t d s
k h =2k v =6×10 -20 m 2 (reference) k h =k v =4.5×10 -20 m 2 d t d s
When the value of χ increases, the time-dependent variation of convergence is decreased and quickly tends towards a stationary value. But at the same time, the shear damage is enhanced in the vertical direction. Such a particular result is directly related to the conception of viscoplastic model adopted here. Indeed, the viscoplastic flow rate depends on the deviatoric stress only. Due to the growth of damages, the values of stress in the damaged zones are decreased due to the degradation of elastic stiffness. Therefore, the viscoplastic flow rate is reduced. More precisely, as the damage values are higher in the horizontal direction than in the vertical one, the viscoplastic flow rate is higher in the vertical direction and more sensitive to the value of χ. This phenomenon can also explain that the shear damage is more significantly affected by the viscoplastic flow in the vertical direction than in the horizontal one, as already mentioned in Figure 34. Obviously, the interaction between damage evolution and viscoplastic deformation depends on the type of viscoplastic model used. In some types of rocks, the viscoplastic deformation may be enhanced by the induced damage. In order to take into account this type of effect, it is needed to consider that the viscoplastic flow rate depends explicitly on damage value, for instance through the variation

Conclusion

In this paper, a new phase-field method has first been proposed for saturated porous media. Two independent damage variables have been introduced in order to easily describe complex cracking process including tensile, shear and mixed cracks. In particular, the tensile crack is driven by the tensile elastic strain energy while the shear crack is governed by the generalized Mohr-Coulomb effective shear stress. Further, the interaction between the shear crack and viscoplastic flow is also taken into account.

The proposed method was implemented in the framework of a finite element code. It was applied to the numerical study of hydromechanical responses and damage processes around an underground gallery excavated in the COx claystone layer. The structural anisotropy of elastic properties and permeability of the claystone has been taken into account. It is found that the damage and cracking processes around the gallery were strongly coupled with anisotropic hydromechanical responses. A strong tensile damage zone is localized close to the gallery wall. The shear damage has a smaller amplitude than the tensile one but extends to a larger zone. The numerical predictions of damaged zones are in good agreement with the experimental observations. Distributions of pore pressure and displacement have also been investigated. The numerical results are globally in consistency with the experimental measurements. Some improvements are still needed in order to better describe the permeability evolution and to incorporate anisotropic plastic deformation and spatial variability of claystone properties. Other types of viscoplastic models can also be envisaged in order to investigate different interactions with the growth of damage fields.

As a conclusion, the present phase-field model allows describing many features of the COx claystone during and after the excavation, such as damaged zones with distinct distinctions of tensile and shear cracks, pore pressure distribution, permeability change, and gallery convergence. However, the efficiency of the proposed method needs to be verified through modeling of other in-stu experiments, for instance gallery excavation along the mi-nor horizontal stress (GED). Three dimensional simulations should also be performed in order to investigate geomtrical effects. After that, the proposed method can be used as efficient tool to assess the robustness and safety of the designed tunnel configuration of the industrial disposal (CIGEO) project. Some improvements of viscoplastic model can also be envisaged, for instance by considering the enhancing effect of induced damage on viscoplastic flow. Further studies are also needed to better describe the permeability variation with induced cracks and self-sealing process.

Figure 1 :Figure 2 :

 12 Figure 1: Partial view of Andra URL and principal stress directions[START_REF] Seyedi | transverse action-a model benchmark exercise for numerical analysis of the callovo-oxfordian claystone hydromechanical response to excavation operations[END_REF]

Figure 3 :

 3 Figure 3: Geometrical domain and boundary conditions (left) and finite element mesh (right)

Figure 4 :

 4 Figure 4: Mechanical and hydraulic deconfinement curves

Figure 5 :

 5 Figure 5: Axial and lateral strains versus deviatoric stress in triaxial compression tests with different confining pressures (2, 6, 12 MPa): comparisons between numerical results (continuous line with ϕ=27.5 • and

Figure 6 :

 6 Figure 6: Creep tests on COx claystone with a confining pressure of 12 MPa and three different deviatoric

Figure 7 :Figure 9 :

 79 Figure 7: Distribution of tensile (left) and shear damage (right) after excavation

Figure 10 :

 10 Figure 10: Distribution of pore pressure at the end of excavation, together with the selected points for

Figure 13 :Figure 14 :

 1314 Figure 13: Variation of permeability along the horizontal direction around GCS gallery at the end of

Figure 15 :

 15 Figure 15: Pore pressure evolution with time at four measuring points of the horizontal borehole OHZ1521

Figure 17 :

 17 Figure 17: Evolution of horizontal and vertical convergence displacements on the gallery wall

Figure 18 :Figure 19 :Figure 20 :Figure 21 :Figure 22 :

 1819202122 Figure 18: Distributions of tensile and shear damage at t=1228 days for three values of g s c (3700, 4100, and 4500N/m)

Figure 23 :Figure 24 :Figure 25 :Figure 26 :Figure 27 :

 2324252627 Figure 23: Distributions of tensile and shear damage at t=1228 days for three groups of E /E ⊥ (1, 1.3 and 1.6)

Figure 28 :Figure 29 :Figure 30 :Figure 31 :

 28293031 Figure 28: Distributions of tensile and shear damage at t=1228 days for isotropic and anisotropic perme-

Figure 34 :Figure 35 :

 3435 Figure 34: Variation of shear damage with distance to the gallery along horizontal and vertical directions

Acknowledgement:

This work is jointly supported by the French National Agency for radioactive waste management (ANDRA) and University of Lille through the i-site project MULTIFRAC. In particular, the three first authors are grateful to Dr. DM Seyedi who has initiated this work through ZY's doctoral thesis and provided many valuable suggestions before he left Andra to CEA.

Viscoplastic cracking coupling χ=0

Viscoplastic threshold stress

Viscoplastic flow rate parameter n = 0.63

Viscoplastic flow rate parameter m = 530

Short term responses due to excavation

In Figure 7, one shows the zones of tensile and shear damage induced by the excavation. It is found that the damaged zones are clearly asymmetrical between the horizontal and vertical axes. It is recalled that the GCS gallery is parallel to the major horizontal stress and the Following the classical Galerkin method, the weak forms of static equilibrium and hydraulic diffusion equations can be written as:

Similarly, the weak forms of governing equations of the tensile and shear crack fields are expressed as follows:

In the framework of finite element method, the fields of displacement, pore pressure and two damage variables inside each element are approximated in terms of their nodal values U, P and d α with the help of selected shape functions N u and N p . In this study, the same shape functions are used for the pore pressure and each damage variable. The gradients of these fields are accordingly approximated with the help of derivatives of shape functions B u and B p . Further, the shape functions are not modified during the loading history. Therefore, for each loading step, one gets:

The same approximations are used for the trial fields, that is:

By substituting these approximations into the integral forms, one obtains the following discrete system of equations for the nodal displacements at the current loading step:

The discrete equations for the nodal values of pore pressure are given by:

(∆tR pp + M pp )∆P + C pu ∆U = ∆t(-R pp P + ∆F ω ) 47)


X Saloppé

; Pham

& Oswald

Quality of Life (QOL): Which contribution to the Good Lives Model (GLM) ?

Introduction

Ø The Good Lives Model (GLM) is a humanist approach to rehabilitating forensic people that inspires international care teams. According to this model, the improvement of psychological well-being is major element to consider in order to reduce the risk of recidivism [START_REF] Van Damme | Quality of life in relation to future mental health problems and offending: Testing the good lives model among detained girls[END_REF][START_REF] Ward | The good lives model of offender rehabilitation: Clinical implications[END_REF].

Ø In addition, the primary needs (physical health, autonomy, emotional balance, social, affective and family relations, happiness, etc.) are at the heart of the GLM and cover important areas of quality of life (QOL, physical health, Psychological, social relationship, environment).

Ø The perceived QOL can thus be a possible operationalization of the GLM.

Ø The participants (n=55) are patients from the Forensic Psychiatric Hospital (FPC) in Centre Regional Psychiatrique "Les Marronniers" in Tournai, Belgium.

Ø Participants completed :

ü Two measures of QOL : ü Four risk measures :

• Sex Offender Risk Appraisal Guide (SORAG; Quinsey, Rice, & Harris, 1998).

• Risk for Sexual Violence Protocol (RSVP; Hart, Kropp, & Laws, 2003).

• Violence Risk Appraisal Guide (VRAG; Harris, Rice, & Quinsey, 1993).

• Historical Clinical Risk-20 (HCR-20; Webster, Douglas, Eaves et Hart, 1997).

Method Discussion and Conclusion

Aims of the Study

To test the hypothesis of a negative association between the generic and specific measures of QOL and the risk of sexual and violent recidivism The results generally support the contribution of QOL measures as operationalization of the GLM in relation to the instruments of structured clinical judgment that integrate the dynamic variables but also the historical domain.

Ø No significant correlations for static risk assessments.

In line with the idea that the GLM is a dynamic model. An association has been found between psychopathy and this domain [START_REF] Pham | Influence of psychopathy on self-perceived quality of life in forensic patients: A cohort study in Belgium[END_REF].

It is important to outline what we want to improve : Prosocial Expectations Antisocial Outcome

Importance of contextualizing the results by a double evaluation and an accurate assessment of the patient's relational environment an it interactions.

Results

Contact : xavier.saloppe@crds.be View publication stats

  r-: QI/WHOQOL-BREF and MQL total score (r=-.30*; r=-.33*) r + : Social Desirability total score / MQPL Total score (r=.27*) Partial correlations (MPQL and n=24) * p < .05. ** p < .01. (Past Mental Disorder) and MQPL (Fairness) (r=.47*)Partial correlations n=29)

  Therefore, it's important to systematize longitudinal assessments.The Social relationship domain requires special attention ! Social relationship domain is positively associated with the Risk of sexual recidivism and offence measure[START_REF] Van Damme | Quality of life in relation to future mental health problems and offending: Testing the good lives model among detained girls[END_REF].

Ø

  These two aspects can be overcome via this evaluation model :