Mariane Peter-Borie

Arnold Blaisonneau

Sylvie Gentier

Théophile Guillon

Xavier Rachez

Study of Thermo-Mechanical Damage around Deep Geothermal Wells: from the Micro-Processes to Macroscopic Effects in the Near Well

Keywords: bonded particle model, breakout, deep geothermal well, DEM, granite, induced tensile fractures, Melleray, near well, PFC, sandstone, Soultz, thermo-mechanical damage

The different processes involved in the life of a geothermal well, from drilling to exploitation, can damage the rock mass in the near well area. In this paper, we propose to study the potential damage linked to the mechanical and thermo-mechanical effects of the well drilling, the well development and the well exploitation. The cooling of the rock mass of the near well pre-damaged by drilling process is a complex phenomenon with the superimposition of different kind of loadings at different scale that lead us to use modeling with a micro-macro approach. To confront the results of the modeling with the reality, we propose to base our study on real cases. For studying mechanical and thermo-mechanical loadings due to drilling and development of the well, we focus our study on the granitic reservoir exploited in the framework of the enhanced geothermal system (EGS) of Soultz-sous-Forêts (France). The study of the thermo-mechanical loading due to well exploitation is performed for a sandstone in the conventional heat exploitation of Melleray (Loiret, France). These simulations highlight the thermo-mechanical damage of a geothermal well linked to the different steps of its life.

INTRODUCTION

Numerous references to damaging of well walls under mechanical and thermo-mechanical solicitations can be found in the literature, particularly in the context of oil extraction. For a long time now, this phenomenon has been classically studied to determine the stress state by analyzing the shape of breakouts and of drilling-induced tensile fractures (e.g. [START_REF] Bell | Northeast-southwest compressive stress in Alberta evidence from oil wells[END_REF][START_REF] Zoback | Well Bore Breakouts and in Situ Stress[END_REF][START_REF] Stephens | Hydraulic Fracturing Theory for Conditions of Thermal Stress[END_REF]. In sandstones, it can lead to sand production problems ("sanding", e.g. [START_REF] Morita | Typical Sand Production Problems: Case Studies and Strategies for Sand Control[END_REF]. Similar damaging is observed in deep geothermal wells: for example in Soultz-sous-Forêts enhanced geothermal system, ultrasonic borehole imager performed after drilling operation show breakouts and induced fractures (Bérard, 2003, Valley and[START_REF] Valley | Stress state at Soultz to 5km depth from wellbore failure and hydraulic observations[END_REF]; in Melleray deep geothermal exploitation, a large phenomenon of sanding occurred in the injection well [START_REF] Boisdet | Projet Trias : expérimentation en vue de la réinjection de saumures géothermales dans les grès du Trias profond[END_REF]. In order to have a better understanding of these phenomena in deep geothermal wells, and bearing in mind the aim of optimizing the reservoir connection to the well and to limit the rock pervasive damaging, we propose to study the effects of mechanical and thermo-mechanical loadings associated with the drillings, well development and exploitation.

The complexity and the superimposition of the physical processes involved in these operations lead us to have a numerical modeling approach. The numerical method is chosen according to the physical processes we wish to study. The thermal solicitation induces differential strains at the origin of thermo-mechanical stresses. When these stresses exceed the mechanical resistance of the rock, microcracks and failures can appear. Strains at the origin of this process can be mainly due to two causes: on the one hand a difference of temperature in the rock mass, on the other hand the heterogeneity of the grains contraction in the rock. Because of this heterogeneity, two adjacent mineral phases can contract at different rate and thus generate uneven strains at the grain boundaries [START_REF] Wanne | Bonded-particle modeling of thermally fractured granite[END_REF]. Discrete Element Method (DEM) based on the micro-macro approach looks fairly adapted to catch these thermo-mechanical processes. The DEM allows to take into account the physical phenomena at the granular phase level (micro scale), as well as their impact on the mechanical behavior of the near well zone (macro scale).

After a brief presentation of the DEM used in our work, this paper first deals with how the rocks involved in the studied cases are modeled as assemblies of bonded particles. It then presents the results of the modeling of the well drilling, the well development and the well exploitation.

NUMERICAL APPROACH: BONDED PARTICLE MODEL

Bonded particle models are carried out with the PFC2D code (© Itasca), whose principle is the modeling of the movement and interaction of the non-uniform-sized circular particles using discrete element method (DEM Cundall, 1971). The definition of the microscopic parameters, i.e. particles and bonds geometrical, physical, mechanical and thermo-mechanical characteristics, is crucial to reproduce the behavior of the rock at a macro scale [START_REF] Potyondy | A Bonded-Particle Model for Rock[END_REF]. After a brief presentation of the used DEM, this section mainly deals with the building of the bonded particle model for rocks with grained structure such as granite or sandstone.

Discrete element method

PFC2D models simulate the movement and interaction of stressed assemblies of rigid circular particles using the Distinct Element Method (DEM). The model is composed of non-uniform-sized distinct particles that displace independently from one another, and interact only at contacts or interfaces between them (Itasca, 2008a). The behavior of the contacts is characterized using a soft contact approach, wherein the particles are allowed to overlap one another at contact points (magnitude of the overlap is related to the contact force via the force-displacement law, and all overlaps are small in relation to particle sizes). Bonds can exist at contacts between particles. The mechanical behavior of such a system is described in terms of the movement of each particle and the interparticle forces acting at each contact point (figure 1; Itasca, 2008a). In the modeling cases presented in this paper, the particles are initially bound by a contact Parallel Bond (PB). The macroscopic behavior of the model is governed by the density of the arrangement of the particles, their size distribution and particle properties and contacts [START_REF] Potyondy | A Bonded-Particle Model for Rock[END_REF]. The thermal capabilities of PFC allow the simulation of transient heat conduction and storage, with each particle being seen as a heat reservoir. For each particle, the temperature value changes with respect of its own specific heat coefficient and thanks to a thermal pipe at each contact with the neighboring particles, if the two particles at the contact are overlapping or if a bond is present (active contact); a pipe length and thermal resistance are assigned to each active contact. When a thermal pipe exists between two particles of two different temperatures, a heat flux takes place in the thermal pipe. Due to their temperature evolutions, a volumetric response (contraction/dilation) of the particles is induced by the integration of radii increment, depending on their respective thermal expansion (Itasca, 2008b). All these thermal volumetric strains are at the origin of a new mechanical equilibrium of the whole model and so of a new stress/strain state. When the model is composed by numerous particles (from about half a million particles), this thermo-mechanical coupling requires a prohibitive computational time. In that case, a one-way coupling only is taken into account to capture the effect of temperature on the mechanical behavior: a homogeneous analytical solution gives the temperature fields, which are then patched on the numerical model particles.

Bonded particle rock models

The first step of the bonded particle rock building deals with the conceptualization of the real rock. This step is important as it defines the scale of the discretization of the rock according to the scale of the physical phenomenon involved in the problem we have to solve. The particles of the numerical model can represent either the grains of a grained-structured rock, either a homogenized volume of rock (that should be representative within the meaning of representative volume element (RVE) concept).

In the problem involved in the studied cases in this paper, a part of the strain generated by the thermal loading applied to the rock mass occurs because of the heterogeneous thermo-mechanical behavior of the mineral phases. The scale of the involved physical phenomena in the studied problem leads to discretizing the sandstone and the granite at the grain scale. The correspondence of the different mineral phases of the rocks with the numerical elements is performed according to their role in the behavior of the rock [START_REF] Peter-Borie | A particulate rock model to simulate thermomechanical cracks induced in the near well by supercritical CO 2 injection[END_REF][START_REF] Peter-Borie | De la roche à son modèle numérique par approche particulaire : conceptualisation et applications[END_REF]: while the particles are predominantly associated with the elastic properties (e.g., Young's modulus) and volumetric characteristics (density) of the material, contacts are mainly associated to the failure. Thus, the mineral phases that compose the granular phases of the sedimentary rocks or the crystals of plutonic rocks, are represented by particles. The contacts between the grains or crystals (for sedimentary rocks: grain-to-grain contacts or grain-to-cement-to-grain contacts, for plutonic rocks: crystal joints) are modeled by Parallel Bonds (PB) as this kind of bond approximates the mechanical behavior of brittle elastic cements. The choice of relative value of PB radius (defined as the PB length reported to the particles diameter and expressed between 0 and 1) should be estimated according to observations of rock thin sections. It could be different for the various kinds of contact (grain-to-grain or grain-to-cement-to-grain contact) for a same sedimentary rock. For plutonic rocks, the crystals are completely jointed by rock fabric processes. However, in specific cases such as hydrothermal alteration, we can make the assumption that the degree of interlocking of crystals decrease locally because of dissolution processes. In this case, the PB length should be also reduced.

Granite bonded particle model

The Soultz granite is a Hercynian monzogranite characterized by phenocrystals of alkali-feldspars in a matrix of quartz, plagioclase, biotite and minor amphibole. In the studied areas, this rock has been hydrothermally altered. As a consequence, most of the plagioclase has been altered in clay. Following the conceptualization methodology, all of the mineral phases will be modeled by particles. Each particle family has the concentration and the crystal size distribution characterizing the corresponding mineral phase in the rocks and that have been determined by image analyses of rocks section by [START_REF] Riss | Granitic core cross section: numerical modelling[END_REF]. The table 1 summarizes these data. In this bonded-particle model of the hydrothermally altered granite, we assume that 85% of the grains joint have been altered. The length of PB modeling such altered joint is considered to be equal to 99% of the diameter of the smallest particle of the couple of bonded particles. 3.0 10 -3 2.0 10 -3 1.0 10 -3 1.5 10 -3 Maximal radius (m) 12.0 10 -3 6.0 10 -3 2.0 10 -3 3.5 10 -3 Young modulus (MPa) 53 30 72 65

Table 1: Composition of the hydrothermally altered granite of Soultz and properties of the particles modeling the crystals

At the particle scale (micro scale), the properties must be physically consistent with the mineral phase characteristics, and must enable the model to reproduce the macroscopic mechanical and thermo-mechanical behavior of the rock (macro scale). In this way, the mechanical properties of the particles are chosen consistent with the mineral phase of the crystal it models (see table 1 for Young Modulus of each mineral phase). The elastic properties of each PB are equal to the average of elastic properties of the two adjacent particles that constitute the bond. The bonds strengths are defined by the friction angle, cohesion and normal strength in table 2. We can note that the altered crystal joint is characterized by a cohesion slightly lower than that of the non-altered joint.

Sandstone bonded particle model

The clastic rock involved in the conventional heat exploitation of Melleray is a Triassic sandstone. Few data is available for this specific facies involved in the Melleray well. In this paper, we focus on the sandstone called white sandstone with dolomitic cement. From textural definition (from Cautru and Robelin, 1984), we know that it is a quartzarenite packstone to grainstone with low content of feldspars grains and few lithic fragments contents. The grain size distribution is known heterometric around a min value of 100 10 -6 m. For cementing the grain, a major dolomitic cement, lower kaolinite cement and few quartz overgrowth are also noticed. Following the definition of the quartzarenite (i.e. composed of greater than 90% detrital quartz) and the packstone to grainstone concept (frame grain supported), we make the assumption that the quartz concentration in the rock is about 95%, feldspars concentration about 4% and lithic fragments concentration about 1%. These rates are reported for particle family and a grain size distribution for the particle is chosen uniform and heterometric between a minimum radius of 50 10 -6 m and 1 10 -3 m (table 4). The PB are considered to model mainly grain-dolomitic cement-grain contacts (60%), then grain-clayey cement-grain contacts (30%) and finally grain-grain contacts (10%). The rock structure defined as packstone to grainstone associated with a porosity of 10% leads to using a mean value of PB length of 94% of the diameter of the smallest particle of the couple of bonded particles (table 5).

Mineral phases feldspars

Lithic fragments Quartz Concentration 4% 1% 95% Minimal radius (m) 0.05 10 -3 0.05 10 -3 0.05 10 -3 Maximal radius (m) 1.0 10 -3 1.0 10 -3 1.0 10 -3 Young modulus (MPa) 30 25 60

Table 4: Composition of the sandstones and properties of the particles modeling the grains

The microproperties of particles in the sandstone (table 4) model are based on those used for the granite model: they should be physically consistent with the mineral phase characteristics. The quartz particles Young's modulus is slightly lower in the sandstone than in the granite: it is equal to 60 GPa (65 GPa in the granite). For the Feldspars, the Young's moduli of the alkali and of the altered plagioclases in the granite are respectively 53 GPa and 30 GPa. A value of 30 GPa, equal to those of the altered plagioclase in granite, have been chosen for the feldspars in sandstone to take into account the weathering of this mineral during the transport phase of the sedimentary process.

The petrofabric differences between granite and sandstone are mainly in the grains joint characteristics and properties. In the case of granite, the cooling and solidification of magma with mineral crystallization results in strong joints between crystals. The sandstone is the result of cementation of pre-compacted grains that creates globally weaker links between grains. The three models of sandstone, called G2, G2a

STUDY OF MECHANICAL AND THERMO-MECHANICAL DAMAGE AROUND DEEP GEOTHERMAL WELL

The effects of drillings, well development and exploitation on the near well area are studied by modeling the corresponding mechanical and thermo-mechanical loadings on the rock modeled with the bonded-particle approach. Simulations are based on two real cases in order to compare and to discuss their results. The study of the impacts of drillings and development of the well is based on the case of the enhanced geothermal system of Soultz-sous-Forêts where data is available thanks to well loggings. The effects of geothermal exploitation (fluid injection) are treated in the case of Melleray where an important sanding phenomenon has been observed after injection tests [START_REF] Boisdet | Projet Trias : expérimentation en vue de la réinjection de saumures géothermales dans les grès du Trias profond[END_REF]. Prior to the simulation of a cold fluid injection, the mechanical loading resulting from the well drilling is modeled, as it is a starting point of any operation in a well.

Drilling and development of a well: case of Soultz-sous-Forêts

The Soultz EGS site is organized around three main deep boreholes, GPK2, GPK3 and GPK4, which are more than 5,000 m deep.

The ultrasonic borehole imager (UBI) logs performed in the wells have revealed the presence of local damage around the wells. It results in breakouts and/or drilling-induced tensile fractures (figure 2). Depending on the depth, it appears that one damage type is predominant. An example from [START_REF] Valley | Stress state at Soultz to 5km depth from wellbore failure and hydraulic observations[END_REF] for the well GPK3 is given on the figure 3a andb. The breakouts occur primarily below 3,000 m while the induced tensile fractures mainly appear above this depth. Most of the breakouts develop along the east-west axis. The breakouts have mechanical origin and are associated with the drilling process in an anisotropic stress context: in the simple case of a vertical borehole penetrating a rock mass in which one principal stress is vertical, breakouts indicate the orientation of the minimal horizontal stress [START_REF] Bell | Northeast-southwest compressive stress in Alberta evidence from oil wells[END_REF]. However, locally north-south-oriented breakouts appear (Figure 3c), i.e. in the direction of the major horizontal stress. According to [START_REF] Th | Estimation du champ de contrainte dans le massif granitique de Soultz-sous-Forêts : implication sur la rhéologie de la croûte fragile[END_REF], this kind of breakout could be of thermal origin: a stable stage of pervasive tensile micro-cracking may occur before failure localizes and the resulting damage may take the shape of breakout. In order to study the development of near well rock mass damaging, and the conditions under which the mechanical and thermal damaging occurs, we propose to model the mechanical effect related to drilling followed by the thermo-mechanical effects induced by cold fluid injection without overpressure (case of cooling by mud during the drilling or of cooling by fluid during injection following the drilling) and with overpressure (case of development of the well). A parametric study on the difference of temperature between the rock and the injected fluid and on the applied overpressure is performed.

Model definition

The model geometry is a 2D horizontal section of the hydrothermally altered granite centered around the vertical well. To compare the results of the modeling with the observations performed by [START_REF] Valley | Stress state at Soultz to 5km depth from wellbore failure and hydraulic observations[END_REF] and Bérard ( 2003), two depths are considered: 2,000 m, where induced tensile fractures are mainly observed, and 5,000 m where breakouts are dominant. The magnitudes of major stresses are calculated from [START_REF] Valley | Stress state at Soultz to 5km depth from wellbore failure and hydraulic observations[END_REF] characterizations of stress magnitudes.

As the DEM is very CPU time and memory size consuming, the number of needed particles of the models is limited. However, the boundaries of the model must be far enough to avoid boundary effects. That's why we chose to embed the granular near-well model (PFC) within a continuous model (FLAC, Itasca, 2002). The coupling method between the continuous and discontinuous models is developed in [START_REF] Shiu | Discrete modeling of near-well thermo-mechanical behavior during CO 2 injection[END_REF]. For a well diameter of 21.6 cm, the bonded particle model size is a square of 0.65x0.65 m included in a continuous model FLAC2D of 2x2 m.

The modeling of the drilling effect is performed in two main steps. First, the stress state measured or calculated at the depth of the near well model is applied on the intact rock mass model (no well). It represents the initial state of the rock mass. Then, the mechanical effects of the well drilling are modeled by removing the particles located on the well perimeter. In order to reproduce the progressive discharge of the well linked to mud presence, a force reduction procedure is used at this step to release progressively the unbalanced force of particles lying on the well surface [START_REF] Shiu | Discrete modeling of near-well thermo-mechanical behavior during CO 2 injection[END_REF].

The thermo-mechanical effects due to rock cooling either during the drilling process (cooling by mud injection) or during fluid injection following the drilling are studied by applying a very low over-pressure (0.1 MPa) and a heat flux corresponding to various temperature differences between fluid and rock on the particles forming the well wall (table 7). The thermo-mechanical effects of the rock cooling associated with an overpressure is modeled to study the impact of the well development on the near well rock. The over-pressure linked to the fluid injection is applied on the well boundary (5 MPa, 10 MPa and/or 15 MPa). Then, the thermal loading due to the colder fluid injection is modeled (table 7).

Results

Numerical results for the rock mass discharge following the well drilling are shown in Figure 4. PB cracks highlight the damage of the near well rock mass. The damage intensity is closely linked to the depth: barely no damage is observed at 2,000 m depth (only few cracks -figure 4a) whereas numerous cracks appear at 5,000 m depth (figure 4b) mainly concentrated in the direction of minor horizontal stress, and induce a changing of the well outline (figure 4c). In fact, at 5,000 m deep, the coalescence of the cracks leads to a local increase in the well radius from 10.8 cm initially to a maximum of 14.0 cm (30% increase). Beyond this increase of the well radius (interpreted as breakouts), preferably aligned cracks highlight the presence of a damaged area. Numerical results of the well development are shown in figure 5 (2,000 m depth) and figure 6 (5,000 m depth). At 2,000 m deep, we can observe (figure 5):

 An increasing number of the diffused cracks around the well with the increase of temperature difference and overpressure; 

The coalescence of cracks from the well wall in the direction of the major horizontal stress. It results in an initiation of induced tensile fractures for: o A 120°C temperature difference between the rock mass and the fluid with an overpressure of 5 MPa, o A 100°C temperature difference between the rock mass and the fluid with an overpressure of 10 MPa;  As in the cooling by the mud case study, secondary tensile fractures initiation is observed (figure 5B on zoomed views); their development is quickly aborted (before 20 minutes of simulated time).

At 5,000 m depth, we can observe (figure 6):

 A number of the diffused cracks more important around the well than at a 2,000 m depth;  For a 150°C temperature difference, the coalescence of cracks from the well wall in the direction of the major horizontal stress results in an initiation of induced tensile fractures. The fractures propagation stops before 2 minutes of thermal simulation. Globally, simulating drilling of the well highlights the formation of breakouts at a 5,000 m depth, whereas only few cracks are observed at a 2,000 m depth. The tensile fractures develop under thermal loading at 2,000 m depth and their initiation and development is facilitated by the fluid overpressure: the higher the overpressure, the lower the temperature difference required to develop tensile fractures. These results are consistent with the observations performed by [START_REF] Valley | Stress state at Soultz to 5km depth from wellbore failure and hydraulic observations[END_REF] on UBI images in the wells. Moreover, the initiation of several tensile fractures around the well can be the source of rock ruptures that may take the shape of breakouts as suggested by [START_REF] Th | Estimation du champ de contrainte dans le massif granitique de Soultz-sous-Forêts : implication sur la rhéologie de la croûte fragile[END_REF]. The results also highlight the rock damaged zones beyond the breakouts, which can thus be taken into account in the subsequent thermal loading.

Exploitation of the well: case of Melleray

In the early 1980s, one of the first sites to exploit the deep Triassic aquifer in the Bassin de Paris (France) was set up in Melleray (Loiret). However, a significant and rapid decrease of the injectivity rate during injection tests leads the geothermal doublet to be abandoned [START_REF] Boisdet | Projet Trias : expérimentation en vue de la réinjection de saumures géothermales dans les grès du Trias profond[END_REF]. It was associated with a large sanding phenomenon in the injection well (partially filled with 11 m of sand). A numerical model is proposed to investigate for a potential rock damaging that can lead to a sand production. The well drilling and the first hours of its exploitation are modeled.

Model definition

The model is similar to that presented in section 3.2.1 for the Soultz case. The model geometry is a 2x2 m rock mass horizontal section, with the near well region (0.54x0.54 m) being modeled using the discontinuous code PFC2D and being embedded in a continuous code (FLAC2D). The considered depth corresponds to the mean depth of the injection well open-hole, i.e., 1,500 m, where the rock temperature is estimated to be equal to 80°C (from [START_REF] Dewertz | GMY2 Forage géothermique de Melleray[END_REF]. A parametric study is conducted to estimate the ultimate sandstone brittleness that would correspond to the onset of sanding. Three models of sandstone, called G2, G2a and G2b from strongest to the weakest, have been built to create different levels of sandstone strength under shear and tensile stresses.

The initial stress state is assumed to be anisotropic: σ H =σ v /1.4 and σ h =σ H /1.25= σ v /1.75. The principal vertical stress σ v is taken as the overburden weight, where an average density of 2.6 is considered. The pore pressure at 1,500 m is set equal to 15 MPa (hydrostatic pressure). These assumptions lead to the following effective stress state: an effective vertical stress magnitude of 24 MPa, an effective major horizontal stress magnitude of 13 MPa, and an effective minor horizontal stress magnitude of 7 MPa.

The discharge resulting from the drilling of the 18-cm diameter well is modeled in the first place. Then, the colder temperature (35°C; BRGM, 1983) is applied on the well boundary (no overpressure is applied). Given the fine-grained structure of the sandstone, a large amount of particles has to be considered in the numerical model. To spare computational time, a one-way coupling only is taken into account to capture the effect of temperature on the mechanical behavior: a homogeneous analytical solution gives the temperature fields, which are then patched on the numerical model particles.

Results

The results of the mechanical effect due to the 18-cm diameter well drilling in the three sandstone models are shown in figure 7. Two different behavior of the near well rock mass are observed. The sandstones G2 and G2a, characterized by the highest strengths, are slightly damaged (figure 7a andb) whereas the important damaging of the sandstone G2b (figure 7c) leads to a local increase of the well radius in the direction of the minor horizontal stress (breakout formation): initially equal to 9 cm, it can locally reach 10.4 cm (figure 7d). A statistical analysis performed on the cracks that appeared during both the drilling and thermal phases highlight that PB representing grain-dolomitic cements-grain preferably break, although the clayey cement are the weakest. Further investigation should be done to understand this observation. However, this result is consistent with the in situ observations (BRGM, 1983): the analysis of the sand deposed in the well reveals that dolomite is frequent while clay is qualified as a trace element. The observed sanding process is only potentially reproduced by the weaker bonded particle sandstone model. A very low cohesion of the sandstone could produce breakout, associated with sand release in the well, given the stress state of Melleray well. The breakout formation is associated with an important diffuse damaging of the rock. We can make the assumption that this damaged rock area may be mobilized the force linked to the potential water circulation in this rock (not taken into account in the presented model). The thermal loading increases the breakout size and the diffuse rock damaging. Moreover, the tensile micro-cracking observed in the direction of the maximal horizontal stress may result in damage that takes the shape of breakout and leads to sand release in the well.

DISCUSSION AND CONCLUSION

In this paper, we study the effects of the mechanical and thermo-mechanical loadings due to the different processes involved in the life of a geothermal well, from drilling to exploitation, through a numerical modeling using a micro-macro approach (bonded particle models). The study is based on two real cases. The study of the drilling and the development of the well is performed on the base of the granitic reservoir of Soultz-sous-Forêts (France) and the study of the well exploitation is performed for a sandstone in the conventional heat exploitation of Melleray (Loiret, France).

In addition to being a relevant approach for thermo-mechanical processes study, the bonded particle models at grain scale allow the integration of the rock petrofabric specificity. Sandstone and granite, which are globally composed by the same mineral phases, result from different petrofabric process. As a consequence, they have very different mechanical properties. The grain size and mainly the contact properties that take into account the bond type of the rock allow the building of different mechanical behavior with similar particles properties. In the numerical models, the bonds specificities have an impact in the cracks apparition and propagation. In the case of the sandstone, we can notice that the dolomitic cements were more likely to break than the other kinds of cement. This kind of observation can be the base for a detailed petrographic study of the nature of mineral released in the well. But to validate the interpretations from this modeling, the setting of the mechanical and thermo-mechanical properties of the particles and bonds should be performed with a great care in accordance with the modeled rock element.

In both the studied cases (Soultz and Melleray), modeling at grains scale allows to reproduce the damage highlighted in the wells by direct (sand particles) or indirect (UBI) observations. The drilling process generates breakout initiation in the direction of the minimum horizontal stress. But the depth at which this phenomenon occurs depends on the stress state and the rock compressive strength (the lesser the rock compressive strength is, the shallower the breakout appears). In addition, the modeling allows the quantification of the damaging beyond the area where the rock failed. Thus, the diffuse damage developed beyond the breakout can be taken into account when the subsequent thermal loading is applied. The thermal loading associated with a large enough temperature difference and without any fluid overpressure (during drilling process or well exploitation) induces the development of thermo-mechanical tensile fractures in the direction of the maximum horizontal stress. The development of the main induced tensile fractures is sometimes associated to the initiation of secondary tensile fractures close to the well. The propagation of these secondary fractures starts at the beginning of the thermal loading but stops at an early stage of the modeling. These tensile fractures may result in rock ruptures that may take the shape of breakouts (thermal breakouts) as suggested by [START_REF] Th | Estimation du champ de contrainte dans le massif granitique de Soultz-sous-Forêts : implication sur la rhéologie de la croûte fragile[END_REF]. During the simulation of well development, the overpressure applied at the well wall makes the tensile fractures propagation easier. It may enhance and accelerate the rock mass damage linked to the thermal loading initiated during the well drilling in the direction of the maximal horizontal stress.

These simulations highlight the thermo-mechanical damage of a geothermal well linked to the different steps of its life. The analytical solution cannot take into account all the complexity of the processes and the pre-damaging of the rock. In perspective of this work, we plan to take into account all the history of the well to have a complete overview of the damage resulting from the

Figure 1 :

 1 Figure 1: Force-displacement behavior of the particles (grain behavior) and of the parallel bonds (cement behavior; Potyondy and Cundall, 2004)

Figure 2 :

 2 Figure 2: Schematic cross-section of a wellbore showing breakouts and induced fractures ( H : major horizontal stress,  h minor horizontal stress).

Figure 3 :

 3 Figure 3: Orientation of breakouts (a) and drilling-induced tensile fractures (b) observed in GPK3 (Valley and Evans, 2006); c: UBI in GPK3 between 4,726 m and 4,733 m depth: breakout oriented NS.

Figure 4 :

 4 Figure 4: Cracks in the hydrothermally altered granite following the well drilling-induced discharge at 2,000 m deep (a) and at 5,000 m deep (b). c: Change in the well outline at 5,000 m resulting from the cracks coalescence.The results of the thermo-mechanical effects due to rock cooling either during the drilling process (cooling by mud injection) or during fluid injection at 2,000 m and 5,000 m depths are shown in figure5and figure6, respectively. Cooling of the rock at 80°C and 100°C generates little damage at 2,000 m deep (figure5). A 120°C temperature difference is necessary to observe the development of thermo-mechanically induced tensile fractures in the direction of the maximum horizontal stress. The development of the main induced tensile fractures (in the direction of the major horizontal stress, figure 5A on zoomed view) is associated to the initiation of secondary tensile fractures close to the well (figure 5B on zoomed view). The propagation of these secondary fractures starts at the beginning of the thermal loading but stops at an early stage of the simulation (before 1 minute of simulated time). At a depth of 5,000 m, the thermal loading generates diffuse cracks around the well mainly in the minor horizontal stress direction.

Figure 5 :

 5 Figure 5: Simulated cracks in the hydrothermally altered granite resulting from the thermal loading at 2,000 m deep. On the zoomed views, A refers to the main tensile fracture and B to aborted tensile fractures developed from the wall of the well.

Figure 6 :

 6 Figure 6: Simulated cracks in the hydrothermally altered granite resulting from the thermal loading at 5,000 m deep.

Figure 7 :

 7 Figure 7: Simulated cracks in the sandstones resulting from the well drilling (a: G2, b: G2a, c1: G2b). d. Change in the well outline from the cracks coalescence for the model G2b.Results of the thermal loading after first six hours of exploitation are plotted in figure8. As for the drilling results, two behaviors are observed. On the one hand, G2 and G2a have similar damaging: a slight diffuse damaging in the direction of the minimum horizontal stress initiated at the early stage of thermal loading and associated with the development of thermo-mechanical induced tensile fractures. On the other hand, G2b has the most important damaging in the direction of the minimal horizontal stress expressed by numerous and pervasive cracks in this area. In the direction of the maximal horizontal stress, a thermo-mechanical induced tensile fracture similar to those observed for G2 and G2b is initiated. For the three sandstones, as observed in the case of Soultz, the development of the main induced tensile fractures (in the direction of the major horizontal stress, see figure8Aon zoomed view) is associated to the initiation of secondary tensile fractures close to the well (figure 8B on zoomed view).

Figure 8 :

 8 Figure 8: Simulated cracks in the sandstones after 6 hours of thermal loading over-imposed to cracks due to drilling process. A and B on zoomed view: A is the major tensile fracture and B localizes secondary aborted tensile fractures

Table 2 : Mechanical microproperties of parallel bonds for the granite bonded particle model (mean values for a bond between two particles modeling quartz crystals)

 2

	Friction angle	Cohesion (MPa -mean value)	Normal strength (MPa -mean value)
	Crystal joint	40	270	127
	Altered crystal joint	40	267	126
		Granite bonded particle	Theoretical properties of the hydrothermally altered
			model	granite from Gentier et al. (2002)
	Unconfined compressive strength (MPa)		128
	Cohesion (MPa)		33.5		34.0
	Friction angle (°)		39.2		40.0
	Young modulus (GPa)		33.4		32.7

The resulting macroproperties, estimated through simulations of uniaxial and triaxial tests on bonded particles rocks model, are presented in table 3. The macroproperties of the particles assembly and associated bonds are globally those, theoretically estimated by

[START_REF] Gentier | Modélisation de l'interface puits-échangeur (site de Soultz-sous-Forêts)[END_REF]

.

Table 3 : Properties of the bonded particles granite model and of the theoretical properties of the hydrothermally altered granite (from Gentier et al., 2002)

 3

Table 5 : Mechanical microproperties of parallel bonds for the sandstone bonded particle models (mean values for a bond between two particles modeling quartz crystals)

 5 modeling uniaxial and triaxial tests on bonded particles sandstone models. From G2 to G2a, the cohesion values of the bonds have been reduced -the contacts normal strength remains the same (table2). It results in a decrease of the unconfined compressive strength and of the cohesion of the rock at macroscale. From G2 to G2b, both the cohesion values and the contacts normal strength have been reduced. It results in a decrease of the unconfined compressive strength, of the cohesion and of the Young modulus of the rock at macroscale.

		PB proportion Friction angle	Cohesion (MPa -mean value)	Normal strength (MPa -mean value)
	Quartz grain-	G2		38.5	74		20
	dolomitic cement-	G2a	60%	38.5	37		20
	quartz grain	G2b		38.5	19		11
	Quartz grain-	G2		38.5	68		19
	clayey cement-	G2a	30%	38.5	34		19
	quartz grain	G2b		38.5	17		10
		G2		38.5	90		25
	Quartz grain-quartz grain	G2a	10%	38.5	45		25
		G2b		38.5	23		13
				Sandstone models
			G2		G2a	G2b
	Unconfined compressive strength (MPa)	59		48	29
	Cohesion (MPa)	19		15	7
	Friction angle (°)	41		39	41
	Young modulus (GPa)	40		39	22

and G2b from the strongest to the weakest model, have been built to create different levels of sandstone strength under shear and tensile stress. The table 5 synthetizes the used microproperties and the table 6, the resulting macroproperties estimated by

Table 6 : Properties of the bonded particles sandstone models

 6

Table 7 : Cases of the parametric study of the development of near well rock mass damaging at Soultz due to well drilling and to well development. Two depths are considered (2,000 m and 5,000 m) -stress magnitudes are calculated from Valley and Evans (2006).

 7

			Rock				Hydrothermally altered granite
			Depth		m		2,000	5,000
		Initial state	Major horizontal stress magnitude  H	MPa		48	123
			Minor horizontal stress magnitude  h	MPa		26	68
			Vertical stress magnitude  v	MPa		52	128
		Well drilling	Discharge	Well diameter	inch cm		8.5 21.6	8.5 21.6
		Well cooling	Fluid overpressure mud/rock temperature difference	MPa °C	0.1 80/100/120	0.1 100/150
		Well development	Fluid overpressure fluid/rock temperature difference	MPa °C	5 80/120	10 80/100/120	10 150	15 100/150

  superimposition of the different steps: drilling, cooling by the mud, cooling by fluid injection (injection test) with or without overpressure, cooling and overpressure apply during well development if needed, and finally well exploitation.ACKNOWLEDGEMENTThis research was carried out in the framework of both the GEISER (Geothermal Engineering Integrating Mitigation of Induced Seismicity in Reservoirs) European project funded by the European Commission and the CLASTIQ2 project funded by the BRGM and the French Environment and Energy Management Agency. The authors are grateful to Fabian Dedecker and Wenjie Shiu from Itasca Consultants S.A.S. for technical support for these modeling.


Cathy Philippe
email: cathy.philippe@gustaveroussy.fr

Arthur Tenenhaus

Vincent Guillemot

Jacques Grill

Vincent Frouin

Multiblock analysis of omics and imaging data with variable selection

Keywords: Bioinformatics, Genetic Imaging, Medical Informatics

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Malignant cerebral tumors represent 20% of the pediatric neoplasms. Among them 60% are malignant glial tumors. Different locations in the brain may denote different types of tumor, in terms of cell of origin and/or microenvironment. In the present work, we want to integrate gene expression (GE) data and comparative genomic hybridization (CGH) data, in order to determine the molecular profiles of pediatric high grade glioma (pHGG) according to their location in the brain, and then confirm these hypotheses and characterize each tumor type. Regularized generalized canonical correlation analysis (RGCCA) [START_REF] Tenenhaus | Regularized Generalized Canonical Correlation Analysis[END_REF] is a general framework for multiblock analysis covering and unifying a large panel of existing multiblock methods. In such a context as medical genomics, where p ≫ n settings are routinely encountered, it is often needed to provide short lists of the most relevant features in the predictive model, in order to facilitate its biomedical interpretation. RGCCA has been extended in [START_REF] Tenenhaus | Variable selection for generalized canonical correlation analysis[END_REF] to address the variable selection issue by adding an ℓ 1 -penalty to the RGCCA optimization problem.

Section II describes SGCCA's criterion, and the cohort studied in this work. We applied SGCCA to select GE features and CGH segments, according to the location of the tumor in three classes: hemispheres, pons, midline. The results are presented in section III and discussed in section IV.

II. MATERIALS AND METHODS

II.1. Method

In RGCCA [START_REF] Tenenhaus | Regularized Generalized Canonical Correlation Analysis[END_REF], J blocks of p 1 , . . . , p J variables measured on the same set of n individuals are considered. The J × J matrix C = (c jk ) denotes the graph of connections between blocks : c jk = 1 if blocks j and k are connected and c jk = 0 otherwise. RGCCA's regularization parameter τ j is set to 1 for all the blocks. As presented in equation ( 1), SGCCA's criterion is the same as RGCCA's but its constraints are modified to take into account an ℓ 1 penalty on the a 1 , . . . , a J in order to induce sparse weight vectors.

         max a 1 ,a 2 ,...,a J J j,k=1;j =k c jk g(cov(X j a j , X k a k )) s.t.
a j 2 = 1 and a j 1 ≤ s j , j = 1, . . . , J

where g is a function called the scheme. In this work, we chose to limit g to the three most used functions in the multiblock literature :

• g(x) = x, called the Horst scheme,

• g(x) = |x|, called the centroïd scheme and

• g(x) = x 2 , called the factorial scheme.

The sparsity of the weight vectors a 1 , . . . , a J is induced by the use of a soft-thresholding operator [START_REF] Tenenhaus | Variable selection for generalized canonical correlation analysis[END_REF].

II.2. Material

All children, between 0 and 18 year old, with a confirmed high grade glioma, biopsied at Necker-Enfants-Malades hospital (Paris, France), have been first included in the cohort. DNA and RNA extractions were performed on the snap-frozen biopsies. We kept all patients with enough material in both DNA and RNA to perform CGH and GE microarrays. The final cohort of 53 patients is composed of 22 tumors located in the hemispheres (HEMI), 11 in the midline (MIDL) and 20 in the pons (DIPG for diffuse intrinsic pontine glioma). The GE block consists in 15702 genes and the CGH block in 1229 segments. The experimental setting is shown in Figure 1. The design, named "cascade", denotes the central dogma of genetics: DNA (CGH block) gives RNA (GE block) that gives the phenotype (LOCation block). Two other designs were tested: "complete" connecting all the blocks and "hierarchical" without connection between CGH and GE.

SESSION IMAGERIE GÉNÉTIQUE

III. RESULTS

We compared 9 versions of SGCCA (3 designs × 3 schemes) to 3 methods: ℓ 1 -LDA [START_REF] Witten | A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis[END_REF], performing sparse linear discriminant analysis in a monoblock setting, and supervised CCA [START_REF] Witten | Extensions of sparse canonical correlation analysis, with applications to genomic data[END_REF], performing generalized sparse canonical correlation analysis with the complete scheme, and RGCCA. Test error rates, shown on Figure 2, were computed in a nested cross-validation procedure. The inner 5-fold CV loop carries out the sparsity parameter s j selection and the outer 10-fold CV loop performs the evaluation of the best model. The best performances are provided by SGCCA with the hierarchical scheme, ℓ 1 -LDA and RGCCA being very closed.

We assessed the stability of the different obtained signatures over the 10 CV-folds for the best methods in terms of test error rates (see Table 1) by computing the Fleiss' κ index ( [START_REF] Fleiss | Measuring nominal scale agreement among many raters[END_REF]). For each block, we have an equivalent Fleiss' κ index between the two methods but when considering the length of the corresponding signatures, we notice that SGCCA provides much shorter lists of features compared to ℓ 1 -LDA.

IV. DISCUSSION-CONCLUSION

As a conclusion, SGCCA performs better than the competing methods, in terms of cross-validated test error rates and moreover provides shorter lists of relevant features, which are more interpretable for final users such as biologists or clinicians. Functional analysis of GE and CGH signatures, provided by SGCCA with a hierarchical design and the centroïd scheme, showed implication of selected features in regionalization and processes of brain development, which confirms the initial hypothesis.

Figure 1 :

 1 Figure 1: Three different design matrices C to analyze the pHGG dataset.

Figure 2 :

 2 Figure 2: Cross-validated test error rates comparisons.

Table 1 :

 1 Stability and lengths of signatures for the most efficient methods according to the test error rates.

	GE Fleiss' κ Average length of GE signatures	ℓ 1 -LDA SGCCA hierarchical centroïd 0.476 0.478 9 790 41
	CGH Fleiss' κ Average length of CGH signatures	0.322 481	0.317 35

ACKNOWLEDGMENTS

This work was supported by grants from the French National Research Agency (ANR GENIM; grant ANR-10-BLAN-0128) and (ANR Investissement d'Avenir BRAIN-OMICS; grant ANR-10-BINF-04).