Virginie Hamm
email: v.hamm@brgm.fr

Camille Maurel

Frederik Bugarel

Jean-Yves Hervé

Good practice guidelines on deep geothermal drilling and exploitation from experience in the Paris Basin (Dogger and Albian aquifers)

Keywords: Paris Basin, Dogger, Albian, deep geothermal, good practice guide

This paper presents the development of a good practices guide and recommendations regarding deep geothermal projects in the Paris Basin from the experience gained from the drilling and operation in the Dogger limestone and the Albian-Neocomian sands aquifers.

INTRODUCTION

With the relaunch of deep geothermal projects in the Paris Basin since 2007, new techniques in drilling, well architectures, materials and reservoir development have been performed compared to the first wells drilled between the 70's and the 80's. In this context, it is important to gather the lessons learned in order to secure the upcoming drilling operations. Since 2014, BRGM has produced a good practice guide on behalf of the French Technical Committee on Deep Geothermal Energy (which brings together French public and private organisations involved in the development of deep geothermal energy). This research work has consisted in detailed technical analysis of drilling reports, which resulted in the production of synthetic documents or "fact sheets" summarising the main best practices related to the construction of geothermal wells targeting mainly three deep aquifers in the Paris sedimentary Basin. The aquifers are the Albian and Neocomian sands (around 700-850 m bsl, with five doublets and one single well drilled since 2012 and one doublet in 1989), and the Dogger limestone reservoir (around 1700 m bsl, which was targeted by 180 wells since the 1970s -67 wells since 2007 -and has, in 2021, 46 doublets or triplets operating).

In addition, BRGM with the support of ADEME and geothermal operators, has produced in 2020 a thorough feedback specific of geothermal operations in the Albian and Neocomian silico-clastic reservoirs in the Paris Basin and in other comparable formations in Europe. The French operations are indeed currently facing injectivity deteriorations, which compromises future geothermal development in those type of aquifers. Following the analysis of information collected for the concerned operations (i.e. well completion data, production and injection tests, flow rate logging and exploitation chronicles), recommendations regarding the different phases of the project were formulated. They concern the well design, testing and development, operations, monitoring and propose remediation solutions.

LESSONS LEARNED FROM DEEP GEOTHERMAL DRILLING IN THE PARIS BASIN

Over the last 15 years, the number of geothermal operations has gradually increased in the Ile-de-France region, allowing the accumulation of a significant amount of information and experience, sometimes positive and worthy of being leveraged as best practices and sometimes negative, either immediately or in the medium term, and for which fact sheets were elaborated and propose possibilities for correction or improvement as well as recommendations. The idea is not to make deep geothermal well construction principles mandatory or normative but rather to provide a reminder of the main best practices to be applied. Finally, the aim is to provide, wherever possible, indicators to appraise or assess the quality of the work carried out.

Table 2 lists all the new operations carried out between 2007 and 2020, which, whenever possible, have given rise to lessons learned as presented in the guide. Those operations represent 73 boreholes, including 61 targeting the Dogger aquifer, 8 in the Albian aquifer, 2 in the Neocomian aquifer of the Paris Basin and 2 targeting the Cretaceous of the Aquitaine Basin (see Figure 1).

The guide (Hamm et al, 2019, updated The reminders of the main procedures relate to:

surface constraints (drilling platform and wellhead access); drilling tools (diameter, type, IADC code ); casings (characteristics of casings and threads, storage, handling, inspection and screwing); drilling fluids (manufacturing, parameter checks and fluid treatment); loggings (delayed, instantaneous, mudlogging, production); wellheads; completion schedule; checking of work (completion method, technical checks, IADC daily reports).

Geothermal drilling

The geometry and structure of deep geothermal drilling differ significantly from the ones most commonly found in the oil industry. Sheet n°21 defines the specific constraints determining the structure of geothermal boreholes and provides recommendations to improve the final quality of the wells: well architecture, overlap between casings, positioning of diverting valves (DV), maximum allowable flow rate during operation. Sheet n°22 summarises the specificities caused by the restricted annular spaces in the architecture of new or rehabilitated geothermal boreholes. It mainly provides a reminder of the importance of the sealing integrity (vertical and horizontal) of these annular spaces, regarding cementing quality and protection of casings. In this context, recommendations are given regarding the minimum permissible annular space and standoff.

The sheet also provides the precautions to be taken to ensure proper quality cementing in a restricted annular space and which relate to the geometry of the drilled hole, the efficiency of the centering and the quality of the slurry placed in the annular space (filling and bonding).

Sheet n°23 on "Drilling and completion fluids" provides lessons learned on the use of drilling and completion fluids in deep geothermal drilling in the Paris Basin. Its primary purpose is to provide an understanding of the fundamental role played by mud in drilling operations. The types of mud used in each drilling phase are described together with the main difficulties that may be encountered (drilled hole geometry, partial or total fluid losses, jamming of the drill string due to differential pressure, bacterial proliferation, bubbling and destabilisation of clays). A few qualitative recommendations are made, with particular emphasis on the precautions to be taken in the specific circumstances related to cementing operations, logging phase and reservoir drilling.

Finally, sheet n°24 on "New well architectures" presents and discuss some new or innovative architectures that have been proposed for tapping geothermal resources in the Paris Basin. The architectures discussed are:

the sub-horizontal drain (example of the Cachan works in the Dogger aquifer); the concept of anti-corrosion wells (example of the works at Melun l'Almont and Bonneuilsur-Marne); the oversizing of production liners (case of Bobigny-Drancy doublets).

In each case, the specificities, advantages and disadvantages of these unusual architectures are analysed and discussed.

Geothermal casings

Sheet n°33 on "Composite material" describes the rare lessons learned from operations using glass fiber casings (Villeneuve-La-Garenne, Courneuve-Sud, Melun l'Almont) and more recently, in 2015, Chevilly-Larue and L'Hay-les-Roses where glass fibre re-lined liners were installed. Similar operations or projects for new wells made of composite materials are also being considered. The purpose of this memo is to give an overview of the advantages, disadvantages and constraints involved in the use of this type of material (corrosion resistance, mechanical aspects, ageing of composite casings, constraints regarding use and work, cementing, removable casings).

The efficient centering of casings is one of the aspects that determine the cementing quality of the casings concerned. The general quality and durability of wells is directly related to proper cementing. Unlike oil drilling, geothermal wells use technical casings as production liners in which corrosive fluids flow at high flow rates. The quality of the cement lining protecting the outer surface of these casings is therefore particularly crucial. The fact sheet 24 on "casing centering" provides a reminder of the rules proposed by the oil industry regarding the centering of liners and determines the recommendations best suited to the specific conditions of geothermal energy.

Geothermal cementing

Sheet n°41 describes the two main cementing methods that can be used to avoid the risk of hydraulic fracturing (multi-stage cementing or cementing with light slurry or mixed cementing of light and heavy slurry in a single stage), with their reciprocal advantages and disadvantages.

One of the most important qualitative factors for casing cementing is the wait on cement to dry (WOC). This is particularly true in deep geothermal energy field where the casings are used both to maintain the well and to extract the fluid. Sheet n°42 provides a reminder of the rules most generally applied in the oil industry and ways to adjust these rules to the specific conditions of geothermal energy. The importance of allowing a drying waiting time appropriate to the slurry used is pointed out. The rules defining the "effective" waiting time are defined. A calculation of the minimum or global times (conservative values) to be observed is provided on the basis of the available data on the mechanical strength of the slurry used.

Sheet n°44 determines qualitative criteria for analysing sonic logs (CBL-VDL) and ultrasonic logs (USIT, URS, CAST, IBC, etc.) used for the inspection of casing cements. These criteria are based on the analysis of the transit time of the CBL log which determines the logs quality, the analysis of the quality of the cement bonding, based on the CBL log, using the so-called Bond Index (BI) method proposed by Schlumberger, the quality of the cement placed in the annular space using the impedance values when the corresponding ultrasonic logs (USIT, URS, CAST, etc.) are available.

The vertical isolation integrity of the annular space will depend on the filling and channelling levels. The sheet provides a ranking (CBL and USIT indices) in order to obtain quantified, less suggestive and more homogeneous interpretations of cementing quality, and to allow comparative analysis between wells.

Sheet n°43 summarises the procedures and rules for abandonment of geothermal wells and provides recommendations for some specific cases. It also describes the role of cement plugs and the ageing processes of casings and cements.

Geothermal deviation

Sheet n°51 summarises the information that must appear in deviation programmes together with the different possible formats for such provision (geographical coordinates, definition of targets, deviation calculation parameter, means used, and inclination measurements in the reservoir).

As a supplement, sheet n°52 provides a reminder of the factors governing the quality of deviation operations and the inspections methods that can be used to assess them. Criteria for the deviation qualitative analysis are provided (Dog-Leg Severity value or DLS value and Tortuosity Index or TI). Quantitative indicators are provided, along with recommendations for defining the geometry of targets under the specific conditions of deep geothermal wells.

Geothermal completion

Sandy and sandstone aquifers (Albian, Neocomian and Triassic) require specific completion. The equipment used should be adjusted to the reservoir's operating conditions in order to achieve the required pumping rate while maintaining effective filtering and a minimum pressure drop level. In addition to proper follow-up of procedures, a qualitative and suitable choice of the completion material, strainer slot and granulometry of the gravel pack is essential. Sheet n°51 provides a reminder of the basic principles to be applied in this regard (completion structure, anchor positioning, dielectric connection, filtering characteristics, entering velocity in the screen).

Geothermal logging

Sonic measurements are used to assess the quality of the cement bonding to the borehole casing walls. While the CBL-VDL (Cement Bond Log -Variable Density Log) is a simple and efficient tool, it is important to be aware of its limitations. It may prove to be inaccurate or unsuitable for use in highly deviated wells, on large diameter casings or with low density cements. Ultrasonic measurements have been developed and finalised in an attempt to compensate for the inherent limitations of sonic measurements and to improve, supplement and facilitate their interpretation. The data collected with ultrasonic tools allow better discrimination and interpretation of the discontinuities that may appear in a cemented annular space, with a resolution much higher than the sole axial measurement provided by a CBL log. In some cases, including the analysis of light slurry, they are even the only means available to try to identify and understand the quality of cements. Sheet n°72 provides a reminder of the way these tools work, their limits and the conditions allowing correct interpretation of cementing quality.

As a supplement, sheet n°73 describes the various logging tools available in geothermal industry to monitor the production and injection borehole casings, and thus enable monitoring of their changes during operation (mechanical tools, ultrasonic tools and electromagnetic tools). As the vast majority of deep geothermal boreholes are build using steel casings, the tools developed for the needs of the oil industry are adjusted to the geothermal domain. Recent developments also make it possible to troubleshoot specific well architectures, in particular those made of composite material casings (like ultrasonic tools). In addition, an analysis method is provided to quantify the damage of a steel casing, based on data from inspection logs.

Sheet n°74 describes the measures applicable to the characterisation of deep geothermal reservoirs. This reservoir characterisation phase, carried out after drilling, is crucial for exploration, knowledge and optimal operation of the reservoir. The memo first discusses the logs used for the geological characterisation of reservoirs and then the logs used for the hydrogeological characterisation. A summary table of the available logs is provided at the end of the sheet, with the measurement principle associated, the type of information obtained, and the disadvantages and advantages of each method.

Geothermal development and testing

The proper performance of development operations is one of the fundamental factors that determine the quality and success of the implementation of geothermal installations. This is particularly true in the case of clastic (sandy) aquifers where the filtering quality directly affects production capacity and costs. Sheet n°81 provides the development procedures and the means that can be used to assess the results and quality of development operations. The term "development" used covers operations ranging from the supplying of water to the well but also the start-up of production and hydraulic or chemical stimulation. Formation testing, or "long term testing", are implemented after development operations characterised by "step testing". They are a final step to drilling operation in order to determine the local characteristics of the geothermal reservoir. The conditions under which this testing is carried out, the acquisition of data and the final interpretation of the tests are essential assessment factors, which should be remembered in order to understand the operation of the reservoir and its changes. Sheet n°82 determines a common framework for carrying out tests in order to harmonise practices in the field of low-enthalpy geothermal operations and enrich the databases shared by the industry in a relevant way.

Geothermal operation

In addition to the standards and applicable regulations in the operating permits, compliance with the operating conditions of wells and the nominal characteristics of the associated equipment should ensure the longevity of geothermal facilities. The physical and chemical restrictions related to the use of the geothermal fluid mainly apply to wells, pumping equipment, pipes and valves and heat exchangers. Sheet n° 91 only deals with the operating conditions of the producer and injector wells and is based primarily on the lessons learned from the Dogger operations in the Paris Basin. It provides information and recommendations regarding the operating constraints related to the operating flow rate, the temperature of the reinjected fluid, the bubble point pressure , the risks of corrosion-deposits and the pumping equipment's. It also provides recommendations for triplet-related operations. Corrosion-deposit phenomena represent a major risk (technical, environmental and financial) for geothermal operations against which it is essential to take precautions. The materials used for the design of wells and surface facilities are mostly steels (API K55 unalloyed carbon steel for well casings, 316 L stainless steel or duplex steel for equipment). The use of noble metals (titanium, nickel-based alloys) is reserved for sensitive equipment (heat exchangers, pumps, instrumentation, process tubes, etc.). Corrosion can generate holes in well casings that can cause interconnection between aquifers, especially between high salinity aquifers such as the Dogger and shallower aquifers used for drinking water (e.g. Albian). Hole repairs involve costly well rehabilitation operations (relining of casings) and increased electrical pumping costs due to pressure drops caused by the reduction in the diameter of new casings. In addition, deposits accumulate on casing walls, thereby narrowing the fluid passage and increasing pressure losses in casings, which can decrease the flow rate, increase pumping and/or injection energy, or increase the drawdown and/or injection pressure. Particles from corrosiondeposition processes can clog filters and/or heat exchangers on the surface, requiring costly dismantling and cleaning operations. These deposits can also accumulate in the open end of the reinjection well, but can also cause clogging of the nearby injector well. Well and/or open-end cleaning operations with a drilling rig can have a significant financial impact on operations and are always associated with technical risks, as is any work in a well. Sheet n°91 addresses the type of corrosion phenomena and the main influencing parameters, the types of deposits, the treatment system, the types of corrosion inhibitors used in geothermal energy in the Dogger, the lessons learned from new operations carried out since 2007 and the resulting recommendations.

The choice of an optimal filtering solution contributes decisively to limiting disturbances and improving the availability of surface geothermal facilities, thereby significantly increasing productivity. In some hydrogeological contexts (e.g. sandy and/or unconsolidated aquifers such as the Albian, Neocomian, Triassic), geothermal water filtering is required to reduce the risk of physical clogging of the reinjection borehole and the aquifer near the injection works. The choice of filter technology will affect operator cleaning frequency and duration and the pumping power consumption caused by hydraulic pressure drops. Malfunctions generally occur when particles and debris are as large as the operating clearances of moving parts (in the case of dewatering pumps, re-injection pumps and valves) or when their size causes clogging (in the case of plate heat exchangers and small diameter tappings in surface pipes). Sheet n°92 provides a reminder of filter selection and performance indicator definition criteria, the type and origins of solid particles and debris found in the geothermal fluid, existing filtering devices and the development of new generations of filters. It also describes the advantages and disadvantages of devices. The more or less rapid clogging of these wells is mostly due to the transfer of fine particles from the producing wells. For all the operations carried out in Ile-de-France (5 in the Albian, 1 in the Neocomian, see Figure 1), injection difficulties were encountered during the initial development and pumping test phases of the works, and continue, or even deteriorate, during the operation of the doublets. Damage to the reservoir, within a radius of the injector well (internal filter cake), can be more significant in some cases where remediation operations (backwashing in particular) seem to have a limited effect.

The recommendations formulated are intended to propose preventive solutions (in particular for new projects) and curative solutions (for existing operations). They aim in particular to:

limit the entry of fine particles into the production well and their transfer to the reinjection works; avoid clogging of the injector structures; restore the performance of the doublets.

The recommendations [START_REF] Hamm | with the cooperation of Hervé J-Y -Forage Assistance -Good practice guide for lessons learned from deep geothermal drilling[END_REF] include a reminder of the good practices to be implemented for the realisation of geothermal projects targeting sandy aquifers (clastic formations), from the design phase of the doublets and the realisation of the drilling works (in particular concerning completions), the characterisation of the reservoir, the development of the wells and the pumping tests, up to the operation phase (see Table 3).

For geothermal installations already in operation, the main recommendations are:

adapting the operating flows to the admissible flows (based on the dimensions of the well, critical speeds and strainer inlet speeds); the installation of sufficiently fine surface filtration; the implementation of backwashing of the injection wells (if the sizing of the screens allows it).

The purpose of the backwash operation is to encourage the mobilisation of particles from the production well or endogenous particles which have progressively clogged the injection well's screens and the reservoir's pores in the near-well environment. It should be done at a flow rate equivalent to or preferably higher than the operating flow rate (a margin of +10% is generally accepted) and it is essential to avoid sudden variations in pressure (overpressure or underpressure), in particular sudden stops which can destabilise the formations and cause additional clogging.

Backwashing can be carried out by air-lift assisted pumping or by simple pumping. Particular care should be taken at the start of air-lift assisted production to avoid oxygenation of the geothermal loop.

The positive effect of backwashing on the performance of the Albian and Neocomian doublets, and more generally on the exploitation of sandy aquifers, remains to be demonstrated, particularly over time. On the other hand, this measure seems to be quick to implement and inexpensive, in terms of the equipment required and the time needed to interrupt operations. It has proved its worth on some doublets in the Albian aquifer, and operational test phases could be continued on the different sites and developed on others to verify the effectiveness of this technique in the long term.

Continued monitoring of hydraulic, physico-chemical, bacteriological and particulate parameters at the level of the operations is also essential for assessing the risk of clogging of the wells.

For future operations, in addition to the recommendations listed before, the main recommendations concern the need to define the role of the well (producer or injector), only after development and testing phase and estimates of injectivity and productivity indices (pumping tests, flowmetry); the fact of sizing the well screens and gravel pack accordingly so as to be compatible with the injection and production in each well; compliance with good practice in terms of the design of wells and the sizing of their components (diameter, casing, completions) according to the characteristics of the formations (grain size) and the operating conditions envisaged, as well as compliance with good practice in terms of development and pumping tests, giving priority to gradual increases in flow; the possibility of fitting out the injector structure and the surface installations to facilitate cleaning and backwashing operations.

Exploitation

Set up a surface filtration system that is adapted to the particle size of the fines, the operating rates, the pressures considered, the equipment of the doublet, the hydraulic properties of the reservoir at the injection well and the admissible pressure drops Filter the production water before reinjection into the injector well to avoid clogging by fines of 10 µm or less in diameter. Limit pressure losses due to filtration

CONCLUSIONS

The relaunch of deep geothermal energy since 2007 has provided exhaustive feedbacks of practices on geothermal deep well drilling, mostly over the Paris Basin. It has led to the edition of a "good practice" guide, comprising 22 fact sheets covering 9 areas: drilling, casings, cementing, deviations, completions, logs, development and testing, and well operations. Each area includes one or more sheets addressing specific topics and providing reminders of the rules and procedures to be followed to avoid failures and, whenever possible, providing more quantitative recommendations and indicators. The idea is not to make deep geothermal well construction principles mandatory or normative but rather to provide a reminder of the main best practices to be followed. Finally, the aim is to provide, whenever possible, indicators to appraise or assess the quality of the work carried out.

It should be noted that some rules, procedures and recommendations referred to in these sheets are taken into consideration in the analysis of ADEME's fund application within the Heat Fund and in the analysis by the technical committee for the application of the guarantee fund (short and long-term risks).

The origin of the clogging in the Albian and Neocomian operations seems to be mainly due to the presence of particles coming from the production well and progressively clogging the injection well which can reach sizes smaller than 1 µm. Injection difficulties were encountered during the development and testing phases and continue, or even deteriorate, during the operation of the doublets. Recommendations related to the design of the wells, the execution, the development, the production and injection tests and the operation of the doublets have been formulated, following the analysis of the operations. Recent research works carried out by IFPEN (publication to come) have also identified the potential role of the heat exchanger (hydrodynamic and thermal processes) that could have an impact on particles aggregation. Some operations are now testing the impact of filtering before and after the heat exchanger.

Figure 1 :

 1 Figure 1: Localisation of the geothermal doublets taping the Albian or Neocomian sands (pink) or the Dogger Limestones (green and blue) in Paris area

  of the work and the placement of the well screens and gravel pack in the well by video inspection log Ensure the filtration of particles and allow the implementation of well cleaning and development End of drilling Acquire additional data at the development boreholes by well loggings, sampling and formation logging prior to well development Improve geological characterisation of the reservoir, suspended fines, estimate reservoir productivity (at low flow), reservoir temperatures and pressures, nature of fluids on each well Developpement Respect good practice in well development and favour long development phases with a gradual increase in flowrate and increasingly long stages Avoid clogging of the injection well by discharging fines-laden water into the well Developpement Characterise the evolution of the mobilised particules load, measure turbidity and its evolution as a function of production flows in the development and test phases Improve the characterisation of fines (nature, quantity, origin) and adapt flowrates to minimise their production Developpement/ Exploitation Discharge pumped water at the surface during development, testing and commissioning of the loop or during well rehabilitation and cleaning operations Avoid clogging of the injection well by discharging fines-laden water into the well Developpement Check the efficiency of the development operations by camera inspection (existing gravel behind screens, limit traces of fines as much as possible) and by turbidity analysis or particle counting of the water at the end of the development Maximise the chances of successful filtration and characterise the water and its suspended particle content Developpement Carry out a flowmeter log at nominal flow rate at the end of the development Characterise the intervals contributing to the flows and estimate the entrainment velocities at the inlet of the wells under operating conditions Testing Applicate good practice in long term formation testing including re-pressurisation times after development phase, pumping/injection and pressurisation times and flow rates achieved Characterise the hydrodynamic parameters in the vicinity of the well Testing Monitor the turbidity of the water produced during the longterm tests and characterise the particle load at several stages of the tests Ensure the effectiveness of the development of the wells Testing Filter water before re-injection during injection trials Avoid clogging of the injection well by discharging fines-laden water into the well Testing Estimate the entrainment velocities at the inlet of the well screens for the measured productive thickness (ideally close to the design flow rate) Avoid destabilisation of the reservoir formations in the vicinity of the wells and adapt the exploitation rates Exploitation Clean the connecting pipes with clean water before commissioning the geothermal loop and discharge to the surface outlet Remove any fines that may be present and avoid the discharge of fines-laden water into the reinjection well Exploitation Adapt the maximum operating flowrates according to the estimated drawdown, critical velocities and screen entry velocities for each well Limit destabilization and erosive forces in the production well environment and mobilization of suspended fines. Limit drawdown in the production well Exploitation Define the initial state for the physico-chemical and bacteriological parameters of the water, the particulate load, the hydrodynamic parameters of the reservoir and characteristics of the connection between the wells and the reservoir Establish the performance of the well and have a reference to identify possible drifts linked to the progressive clogging of the wells Exploitation Monitor the wells and operations (physico-chemical, particle load, bacteriological, hydrodynamic) Identify potential drifts and impacts of rehabilitation (cleaning, flow reversal) on performance by comparing measurements with the initial state of the wells Exploitation Set up backwash or flow reversal phases at the injection wells and adapt the installations in the well and on the surface (electrical cabinet, pump or pistoning, dewatering column, water discharge) to repeat the operations while limiting operating interruptions Maintain the performance of the doublet by mobilising the fines present at the level of the screens, gravel pack and in the reservoir at the injection well (preventive and corrective)

Table 1 : Fact sheets including lessons learned from 2015 to 2020 Theme Fact sheet n° Specification sheet topic

 1

	on 2021)

Table 2 : Deep geothermal drilling carried out between 2007 and 2020 Geothermal operations Aquifer targeted

 2

	Year of comple-tion	Number of wells	Underground manager project	Boreholes

Table 3 : Recommendations and good practices for sandy formations Phases Recommandations or good practices Objectives

 3

	Exploration	Acquire new data to increase reservoir knowledge (first exploration well)	Caracterize the reservoir before drilling and choose adapted well design and completion
		Respect good practice in well design and dimensionning of
	Design	well elements (well diameter, casings, completion)	Improve geothermal well quality
		depending the formation characteristics
	Design	Take into account avantages and disavantages of deviated wells (setting up the gravel pack) versus vertical wells	Minimize operational risks of completion during works
	Design	Determine the role of the well (producer or injector) after devlopment of wells and testing phase	Maximise the chances of success for reinjection
	Design	Use preferably same well architecture between both well (producer and injector)	Maximise the chances of success for reinjection
	Design	Dimension and choose well screens and gravel pack compatible with injection and production	Minimize risks of clogging and reservoir destabilization during backwashing operation. Allow reversible flows
	Design	Dimension well screens and gravel pack in accordance to grain size and operational conditions	Obtain an efficient filtration for the achieved flowrate. Avoid destalization near well.
	Design	Anticipate the possibility of specific equipments at the surface and in the injection well to facilitate cleaning and backwashing	Clean or unclog wells and near-well formations with minimal disruption
	Drilling	Realize drilling work with an experimented compagny in deep geothermal drilling	Minimize operational failures and improve wells quality
	Drilling	Anticipate the involvment of an experimented engineering compagny for fluid management and reservoir testing	Minimize operational failures and improve wells quality

Acknowledgements

This work was supported by ADEME (French Agency on Environment and Energy management) and was possible thanks to the participation of geothermal operators and organisations involved in the development of deep geothermal energy projects.


Charles Maragna
email: c.maragna@brgm.fr

Virginie Hamm

Camille Maurel

Deployment of 5 th Generation District Heating and Cooling grids (5GDHC) in France: two case studies in Orleans and Strasbourg metropolises

Keywords: 5GDHC, shallow geothermal, waste heat, Orleans metropolis, Strasbourg Eurometropolis

BRGM is carrying out two opportunity studies on the Metropolises of Orléans and Strasbourg for the implementation of 5 th Generation District Heating and Cooling (5GDHC) grids. In Orléans, one considers supplying the extension of an economic activity zone with the 5GDHC powered by a combination of waste heat and shallow geothermal energy. A detailed energy model is being built in TRNSYS, and preliminary results related to the consumer stations are reported in this paper. In Strasbourg, the 5GDHC will supply new residential and tertiary buildings as well as a shopping centre by using the Rhine alluvial aquifer as the main energy source.

INTRODUCTION

The Interreg North Western Europe D2Grids project (2018)(2019)(2020)(2021)(2022)(2023) aims at developing 5GDHC grids, involving several demo sites in Europe (Netherlands, Germany, France, United Kingdom). A 5GDHC grid exchanges thermal energy between buildings with different needs [START_REF] Buffa | 5th generation district heating and cooling systems: A review of existing cases in Europe[END_REF]Interreg NWE D2Grids, n.d.). The grid carries a fluid at low temperature (typically 10-40 °C) to stations that can upgrade the temperature, if needed, to the required level through Heat Pump (HP). Distributed thermal storages can buffer the fluctuation in supply and demand, avoiding oversizing the installation. The 5GDHC architecture allows maximizing the share of low-grade renewables (e.g. shallow geothermal, surficial water bodies, etc.) and waste energy.

The D2Grids project assesses the opportunities to roll out the 5GDHC concept throughout North-western Europe. In a first step, BRGM carried out a regional analysis in three dense urban areas in France, namely Orléans Metropolis, Greater-Paris Metropolis and Strasbourg Eurometropolis respectively located in Centre Val-de-Loire, Grand-Est and Ile-de-France regions.

These regional analyses investigated the existing heating regimes, the existing District Heating Networks (DHN), the available energy sources with a focus on the shallow geothermal potential on each metropolis. For each region, a SWOT (Strength Weakness Opportunity Threat) analysis was built to assess the possibility to implement 5GDHC networks. In a second step, according to the results from the regional analysis and first positive contacts with energy planners and local authority, prefeasibility studies are being carried out in Orléans and Strasbourg metropolises to define local action plans. Orléans metropolis and Strasbourg Eurometropolis mainly use natural gas to provide heating systems. Some high temperature DHN provide heat through the combustion of biomass and wastes. Nevertheless, the objectives fixed in their respective climate and energy plans intend to decrease the overall energy consumption and to increase the share of renewable energy sources in their heating mix. Both metropolises have a proven high shallow geothermal potential, especially for open loop systems, in the Beauce limestones in Orleans Metropolis and in the Rhine alluvial aquifer in Strasbourg Eurometropolis. This potential represents a good opportunity and a relevant energy source for 5GDHC. In accordance to the new environmental regulation of buildings in France (RE2020), 5GDHC represents a key concept to low carbon renewable heating and cooling systems for the residential and tertiary sectors.

ORLEANS METROPOLIS CASE STUDY

The first case study is located in the South of Orléans Metropolis. It is a 45 ha extension of the business park "Parc de la Saussaye" towards the East (cf. Figure 1). The type, volume and thermal needs of the buildings have not been investigated yet, only the surfaces of the plots are available.

The hypothesized 5GDHC grid could valorise the shallow geothermal aquifer of the "Beauce limestones". The aquifer productive layers are typically in the depth range 30 to 50 m, ant its transmissivity is very high (between 3.10 -3 m²/s and 3.10 -1 m²/s). Besides, right in the middle of the existing business park, a factory freezes and processes large amounts of food. As a result, large amount of heat is emitted to the atmosphere at about 35 °C through cooling towers. Due to secrecy reasons, detailed figures cannot be disclosed at this stage. Warming up the 5GDHC grid would be a way to valorise this so-far wasted heat. Here we focus on bi-pipe, bidirectional 5GDHC grids. Such grids have a cold line and a warm line. Stations retrieve fluid from the relevant line, e.g. for heating applications the water is retrieved from the warm line and sent back 4-6 °C colder to the cold line. There is no "departure" and "return" lines as in conventional, high temperature DHC: the flow in each line connecting adjacent stations is bidirectional. If the cooling and heating needs do not compensate each over, flow-rate can be sent to "production" stations (geothermal, waste heat …) which temper the fluid temperature, avoiding the temperature to drift beyond acceptable thresholds.

Questions being addressed

We aim at estimating the cost and CO2 content of the thermal energy produced and delivered by the 5GDHC grid. More specifically, we want to address the following questions:

-What is the "optimal" share of shallow geothermal energy vs. waste heat? -Is it worth insulating the line connecting the industrial site to the new cluster of buildings? Can we expect the warm line to deliver temperature in the range 30-35 °C, which may enable direct heating if low-temperature emitters are deployed? -If so, to which extent is it acceptable to increase the 5GDHN grid temperature without impeding the cooling efficiency? -Indeed, the thermal consumption of the future buildings is barely known. To which extent is a design flexible? For instance, if more thermal energy is required, how can we restrain the flow rates increases and subsequent pump consumptions? -Is it worth extending the lines towards the existing business park, and residential zones located in the village of Saint-Cyr-en-Val (not shown in Figure 1)?

Simulation process

To answer these questions, we use several simulation tools:

- The combination of these tools enable investigating the spatial, thermal, and underground aspects of 5GDHC grid development.

Preliminary results: Thermal behavior of the consumer station

Here we give a short overview of a "consumer" station developed in TRNSYS. Such station can provide heating, cooling and Domestic Hot Water (DHW) to a building. We investigate how design and operational parameters affect the shares of non-renewables (electricity for the Heat Pump and backups) in the thermal energy delivered to a hypothesized building. We first define a "baseline scenario" and we perform one-at-a-time sensitivity analysis. The cold line and warm line temperatures are fixed to constant values.

The thermal loads considered in this study originates from the thermal model of a well-insulated hotel located in Turin, Italy modelled in the framework of the GRETA project (GRETA project, 2017). Overall heating and cooling requirements reaches 249.8 MWh.y -1 (139.2 kW) and 68.1 MWh.y -1 (84.4 kW) respectively. Orleans and Turin climate are indeed similar, with Heating Degrees Days (HDD) and Cooling Degrees Days (CDD) differences being in the range 10-20 %. The average DHW need was estimated following the methodology developed in (ADEME and COSTIC, 2016). It is estimated to be 125 L.day -1 at 40 °C per "standard housing", i.e. 2-bedrooms apartment of 65 m 2 , with hourly, weekly and monthly corrections.

Here we assume that the building is equivalent to 80 "standard housings", the average DHW flow rate is 〈𝑚̇𝐷 𝐻𝑊 〉 = 125×80/24 = 416.7 kg.h -1 . The energy required to warm up the tap water to the delivery temperature 40 °C is 118.2 MWh.y -1 . DHW need reaches about a half (47 %) of the heating need, and is far from being negligible. The main components of the station are a Heat Pump (HP), two heat exchangers, one for direct (passive) cooling and one for direct (passive) heating, and three tanks to store the heating, the cooling and the DHW before it is distributed to the building (cf. Figure 3). Thermal energy is produced according to eight modes of operation:

- Fluid is retrieved from the 5GDHC warm line in modes 1, 3, 7 and from the cold line in modes 2 and 6. We assume a constant temperature difference between inlets and outlets of -4 °C (modes 1, 3, 7) and +4 °C (modes 2, 6). For the baseline scenario, we consider T warm line = 16 °C and T cold line = 12 °C.

The departure temperature from the heating tanks into the heat emitters depends of the outside air temperature as follows:

𝑇 𝑑𝑒𝑝,ℎ𝑒𝑎𝑡𝑖𝑛𝑔 = 𝑎 -𝑏 × (𝑇 𝑎𝑖𝑟,𝑒𝑥𝑡 -16 [°𝐶]) (1)
We considered low-temperature emitters with 𝑎 = 31.1 °C and 𝑏 = 0.56. Note that only lowtemperature heat emitters (typically operating in the range 25-35 °C) could take benefit of modes 7 and 8, bypassing the HP and therefore reducing electricity consumption. Cooling is assumed to be provided at 𝑇 𝑑𝑒𝑝,𝑐𝑜𝑜𝑙𝑖𝑛𝑔 = 15 °C. The difference between departure and return temperatures is 5°C, both for heating and cooling. The HP calorific nominal power is 150 kW.

The HP efficiency depends upon the evaporator and condenser inlet temperatures and originates from data provided by manufacturers. Every heat exchanger is sized so that its pinch is +2 °C.

The HP produced DHW at 60 -65 °C. The French regulation enforces operational constraints for any DHW storage exceeding 400 Litres to avoid the development of legionella [START_REF] Santé | agences régionales de santé dans la mise en oeuvre de l'arrêté du 1er février 2010 relatif à la surveillance des légionelles dans les installations de production[END_REF]. Every 24 h, a controller checks if the temperature at the bottom of tank (the coldest place in the tank) reached 65 °C at least once during the last 24 hours. If not, an internal backup heater is turned on to reach this temperature.

Backups downstream the tanks ensure that the set point temperatures (eq. 1 for heating, 40 °C for DHW) are met at any time.

The The station is modelled is TRNSYS v18 and executed with a time step of 5 minutes.

Figure 44 shows the origin of the thermal energy delivered to the buildings on monthly basis. One can notice that:

-The Heat Pump produces the whole heating. No backup is needed. There is no direct heating, since the warm line temperature is much too low (16 °C). -The Heat Pump produces the whole DHW. No backup is needed. In summer (July and August and too a smaller extent in June and September), most of the DHW is produced simultaneously with cooling (mode 5). However, the warm line feeds the HP most of the year (88 % of the DHW is produced in mode 3). -The cold line is cold (12 °C) enough to provide geocooling all along the summer, though a small part (22.0 %) is provided by the HP through mode 5. We assess the "efficiency" of the station as the ratio of the electricity (for HP and backups) to the total provided thermal energy (249.8+118.5+68.1 = 300.2 MWh). This efficiency factor is 𝐹 = 75.3 300.2 = 17.2 %, meaning that 100-17.2 = 82.8 % of the thermal energy comes from the 5GDHC grid. Indeed, it would be difficult to define efficiency per use, since electricity can be used simultaneously for two uses through modes 4 and 5.

Next, we focus on the influence of the line temperatures, maintaining ΔT line = 4 K between the warm and the cold line in January, April and August.

The share of electricity in the thermal energy F delivered is strongly affected by the line temperature (cf. Erreur ! Source du renvoi introuvable.). Both in January and April, F decreases when the grid temperature increases, as expected. Until c.a. 25 °C, this slight decrease is due to an improved efficiency of the Heat Pump for heating and DHW production, the HP being provided with warmer fluid at its evaporator.

Beyond 25 °C, the dramatic decrease is due to some heating being provided "for free" by bypassing the HP (modes 7 and 8). This effect is more significant in January than in April, since the relative share of DHW, not affected by direct production, is lower. In August, below 17 °C on the warm line (13 °C on the cold line), most cooling is provided "for free" (mode 6), except the cooling produced simultaneously with DHW which consumes some electricity (mode 5). Beyond 13 °C on the cold line, direct cooling is no more possible, the whole cooling is produced through the HP (modes 2 and 5).

Preliminary results: Subsurface modelling

The heat source for the 5GDHC system is the shallow Beauce limestone aquifer. Over the Parc de la Saussaye, the Beauce formation is estimated to be 45 m thick and the top of the formation is located at 15 meters below the ground. The stratigraphy series schematic over the Orléans metropolis underground is presented in Figure 6. In order to study the feasibility of an open geothermal loop in this area, the first step was to build a model of the aquifer at local scale (9 km x 9 km grid). A configurable hydrodynamic and thermal numerical model of a doublet was built in the ComPass software [START_REF] Xing | Parallel numerical modeling of hybrid-dimensional compositional non-isothermal Darcy flows in fractured porous media[END_REF] to study the influence of the installations over the thermal plume and the impact on the pressure field. In the vicinity of the site, several drinking water wells are producing important amount of water over the Beauce and Eocene aquifers. They are located on either side of the foreseen implantation site and have thus been included in the numerical modeling to understand the impact and potential interferences between the existing and foreseen installations. The volumes produced are estimated between 7,000 m 3 /day over one water well and 500 m 3 /day over another one.

The model built is composed of 8 layers, 4 for the Beauce limestones formation, 2 for the marl and sand formations located above the limestones and two for the underlying formations.

The permeability of the limestones (targeted resource) is defined at 3.10 -11 m². A pressure gradient from the south toward the north has been implemented (estimated according to the piezometric network around 0.045 bars per km). The model grid is composed of prismatic cells from 25 m side near the wells to 300 m side at far field limits. An illustration of the grid and locations of water and geothermal wells is presented in Figure 7. The illustration also presents the initial state of pressure in the model with 0.045% hydraulic gradient along X, from right to left. A simplified geometry was used for the simulation and the reservoir is considered to be homogeneous.

In first simulations, the production well has been placed downstream of the injection well, with an angle from the hydraulic gradient, as the impact of high water flow rates in the water well n°2 (located on the right side of 7) is expected to impact the thermal plume around the injection well. First simulations of water well production along with the exploitation of a geothermal doublet have been conducted. The chronicle used for production and reinjection flow rates derives from energy model of office building with heating demand not covered by geothermal energy source since priority is given to the waste heat collection. The cooling is however supplied by a geothermal doublet (consumption per year estimated to 721MWh (1.288 MW). The wells are assumed to be operated with a temperature difference from resource temperature of +6°C and the HP efficiency is estimated at 4. The maximum flow rate to be pumped is thus 64 kg/s. The distance between the injection and production wells considered is 200 m and the distance between the two water well is 3,000 meters. The simulations are run over a period of 10 years with launch of geothermal doublet after 2 years. When observing the pressure field after 10 years of simulation, the impact of water wells productions seems to be important near the doublet and tend to impact the fluid flows. Additional runs will be carried out to identify the impact over the cold or hot plume around the geothermal injection well. On-going simulations also plan to study wells placement to identify optimal positioning of geothermal wells to reduce interferences with drinking water infrastructures.

STRASBOURG EUROMETROPOLIS CASE STUDY

A second pilot site under study is located in Strasbourg, where the department of Renewable Energies and Heat Networks showed interest in the 5GDHC concept and is currently engaging with operational urban planners to identify whether or not an opportunity to develop projects compliant with 5GDHC technology will present itself in the near future.

Strasbourg Eurometropolis is located in Grand-Est Region along the German border and is composed of 33 municipalities with a population of 487 303 inhabitants on a geographic area of 338 km2 (i.e. 1442 hab/km2 in 2019). 60% of the population of the metropolis is concentrated in Strasbourg and the population of the metropolis represents one quarter of the total population in Alsace. The regional analysis has showed that the heating needs are very high (2,5 TWh/y for residential needs and 1,5 TWh/y for tertiary needs) and only 0,2 TWh/y for cooling demand in the tertiary sector. The tertiary sector seems to be the more adapted to 5GDHC future development, as there is in most cases a need of heating and cooling (even if unbalanced) which is not the case for the residential sector.

In order to define a more specific local action plan, contact has been established with the city of Strasbourg and the persons in charge of the energy development and district heating. After several exchanges, a location under development was proposed in the area of the Baggersee pond (see Figure ) to study the possibility for hosting a 5GDHC system. The site will be restructured and is characterized by different usages and needs. Four main types of usages have been identified: residential housings with a surface of 42 000 m2, a shopping center with a surface of 24 000 m2, tertiary needs with a surface of 10 000 m2 and public equipments with a surface of 8000 m2.

5GDHC energy modelling system

In progress (will be completed after the congress)

Subsurface modelling

The heat source for the 5GDHC system is the shallow Rhine alluvial aquifer. The alluvial deposits of the Rhine are the seat of a powerful aquifer (up to 150 m thickness) at the location of the Eurometropolis of Strasbourg. In order to study the feasability of an open geothermal loop in the Baggersee area, the first step was to build a model of the Rhine aquifer at local scale.

For this, we used the data from a regional model of the alluvial aquifer that was done during the Interreg IV Upper Rhine LOGAR "Liaison Opérationnelle pour la Gestion de l'aquifère Rhénan" project (2009Rhénan" project ( -2012)). The primary objective of this regional model was to simulate the global flow and transport of nitrates in groundwater over several decades and assess the impact of measures to improve water quality. The regional model covers an area of 4289 km 2 with 160 km northsouth direction and 90 km east-west direction (Figure 10a). The limits of the Rhine alluvial aquifer are the Vosges at west, the Forêt Noire at east, Karlsruhe at north and Grenzach-Wylen at south. The lithology of the alluvial aquifer is a mixture of fine and coarse sediments. It is composed of two main units: the Neuenburg formation in the upper part and the Bresgau formation in the lower part. These Rhine alluviums lie on the Oligocene marls (Figure 10b).

The regional model is composed of 10 layers, 5 for the Neuenburg formation and 5 for the Bresgau formation.

The choice of this 10-layer discretization was made in order to best represent the vertical variability of the permeability induced by the presence of thicker or thinner clayey interlayers within these two formations. The Highest permeability are in the first 20 meters of the Neuenburg formation. The model grid is composed of 100x100 m square cells which represents more than 4 millions meshes. An extraction of the model was done in the area of Strasbourg (Figure 10a and Figure 10c). The vertical discretization was then simplified in the local model as the permeability in the area of Strasbourg were similar for most of the vertical layers. Thus, in the local model, 3 layers are represented: one layer that represents the Breisgau formation and two layers for the Neuenburg formation with higher permeability in the first layer.

The thermo-hydrodynamic model was built by using Marthe code developed at BRGM for 3D modelling of flows and transfers. Once the local geometrical model is built, a second step was to calibrate the average groundwater flow in order to simulate correctly the hydraulic gradient and also the natural temperature of the aquifer. The third step was to include also the existing geothermal loop in the area as the Rhine aquifer is already largely exploited for heating and cooling needs at the location of the Eurometropolis (see Figure 10c).

Preliminary results

Will be completed after the Congress.

CONCLUSIONS

5GDHCN grids are probably a key for the further development of shallow geothermal energy in most European countries. They offer the possibility to valorise local sources of energies and to couple thermal and electric grids.

Numerical tools are being combined to investigate the spatial, thermal, and underground aspects of 5GDHC grid development on two French conurbations. Preliminary results show that the efficiency of a station is strongly affected by operating temperatures of cold and warm lines. The 5GDHC operating temperature must therefore be optimized, given the available sources (shallow geothermal and waste heat).

Numerical procedures are being developed to automatically create TRNSYS models of the whole system including multiple stations, 5GDHC segments and co-simulate the evolution of the grid and the doublets with ComPASS.

Figure 1 :

 1 Figure 1: Orléans: Map of the area, including the areas to be urbanized (coloured polygons), the hypothetical 5GDHC layout, stations and geothermal wells.

Figure 2 :

 2 Figure 2: Monthly thermal energy demand

  Mode 1: HP produces heating: The warm line feds the HP evaporator and the HP condenser warms up the heating tank. It is triggered before the heating tank temperature falls below the heating set point. -Mode 2: HP produces cooling: The cold line feds the HP condenser, while the HP evaporator cools down the heating tank. It is triggered before the heating tank temperature exceeds the cooling set point. -Mode 3: HP produces DHW. -Mode 4: Simultaneous production of heating and cooling by the HP: the HP evaporator cools down the cooling tank while the tank gets warmed up by the condenser. -Mode 5: HP produces DHW and cooling simultaneously. to mode 4. -Mode 6: Direct cooling: The cold line cools down the cooling tank through a dedicated heat exchanger. HP is bypassed. -Mode 7: Direct heating. Similar to mode 6 with connection to the warm line to feed the heating tank. -Mode 8: Simultaneous direct heating and DHW production.

Figure 3 :

 3 Figure 3: Most elements of the consumer stationA "differential controller" selects the mode of operation by comparing the temperatures of the components,

  volumes of the heating, cooling and DHW tanks are respectively 𝑉 𝑐𝑜𝑜𝑙𝑖𝑛𝑔 = 5.83 m 3 , 𝑉 ℎ𝑒𝑎𝑡𝑖𝑛𝑔 = 8.00 m 3 , 𝑉 𝐷𝐻𝑊 = 0.83 m 3 . The DHW tank can store for 𝑡 𝐷𝐻𝑊 = 𝑉 𝐷𝐻𝑊 〈𝑚 ̇𝐷𝐻𝑊 〉 × 1000 [kg.m -3 ] = 2 h of the average HW consumption.

Figure 4 :

 4 Figure 4: Baseline scenario: Monthly balances on the heating, DHW and cooling tanks, from January to December.

Figure 5 :

 5 Figure 5: Share of electricity in the thermal energy delivered to the building as a function the line temperature, for January and August.

Figure 6 :

 6 Figure 6: Stratigraphy of the underground over the Orleans metropolis area (BRGM, 2007)

Figure 7 :

 7 Figure 7: ComPASS grid structure, location of wells and initial pressure condition from a 0.045% hydraulic gradient along x (pressure in Pa).

Figure 8 :

 8 Figure 8: Pressure in the Beauce limestones reservoir after 10 years of simulation

Figure 9 :

 9 Figure 9: Location of the site for the pre-feasibility study in Strasbourg.

Figure 10 :

 10 Figure 10: a) Extension of the LOGAR model. b) Schematic log of the Rhine alluvial aquifer. c) Case study extension, regulation zones for open loop systems and local hydraulic gradient.

Acknowledgements

The authors would like to thanks Orléans Métropole and Eurométropole de Strasbourg for the data they provided related to the case study. This research work is supported by the Interreg NWE 795 D2Grids project and the BRGM.


Marie-Madeleine Bertucci

et en contexte plurilingue dans l'espace francophone

Éléments de sociolinguistique pour la didactique du français en milieu ordinaire et en contexte plurilingue dans l'espace francophone J'aborderai conjointement l'observation de faits de langue(s) et de pratiques langagières spécifiques aux contextes plurilingues et des éléments de sociolinguistique utiles à la didactique du français en milieu ordinaire et en contexte plurilingue dans l'espace francophone, à savoir celles de processus variationnel, de diglossie, de continuum linguistique, de contacts de langue(s), de situations interlectales, de normes endogènes, d'interactions en discours et de processus identitaires.

I. Le lexique. Les apports de la lexicographie francophone à la didactique du français

Le lexique est un élément problématique pour l'appropriation du français dans la situation qui nous occupe. La question du lexique est généralement peu abordée. Or, des difficultés dans l'appropriation du lexique peuvent générer chez les élèves concernés une situation d'insécurité linguistique, qui peut passer inaperçue dans le contexte scolaire, du fait de sa spécificité. En effet, les élèves peuvent être confrontés à des mots dont la forme est identique et dont le sens diffère. Ces faits de langue(s) s'observent lorsqu'on confronte les français d'Afrique, de l'Océan Indien ou des Caraïbes [START_REF] Bertucci | Les français régionaux dans l'espace francophone[END_REF].

C'est la raison pour laquelle le recours à la lexicographie francophone est précieux dans ce type de contextes où existe d'une part un fort processus variationnel et ce qu'on appelle les norme(s) endogène(s).

Avant d'aborder les faits de langue(s), je vais procéder à un bref récapitulatif théorique.

Je verrai les notions de variation et de processus variationnel, de variétés et de normes endogènes.

La notion de variation

On s'appuiera sur l'approche de la variation retenue par la sociolinguistique depuis les travaux de William Labov, à savoir l'approche l'approche « variationniste » (Gadet, 2007 : 26).

Pour définir la variation, on part généralement de la notion de variétés. On pose en principe qu'il existe, pour une même langue, des traits, qui sont variables et qui relèvent de facteurs externes, à la langue (Gadet, 2007 : 23) (Bavoux, 2000 : 12).

Une polysémie problématique

Je voudrais justifier maintenant la raison de ce détour par la théorie sociolinguistique en faisant maintenant une incursion par La Réunion. Michel Carayol a observé que les manuels scolaires utilisés à La Réunion ne mentionnent pas les particularités lexicales régionales propres au français régional de La Réunion, qu'il définit comme « la langue dans laquelle les locuteurs réunionnais produisent leurs discours quand ils choisissent de le faire en français et non en créole » (1984 : 6). Ainsi, des termes du lexique créole, d'un usage courant en français régional, sont par là même écartés de l'usage scolaire, alors qu'ils renvoient à un quotidien qu'il est difficile voire impossible de désigner autrement (1984 : 5-6). Il insiste clairement sur l'existence de termes ayant un signifiant identique en créole et en français mais offrant dans les deux langues, une opposition sémantique plus ou moins forte. Ces termes ne sont pas identifiés par les élèves et très souvent par les maîtres, ce qui conduit à pérenniser et à consolider des confusions sémantiques qui perturbent l'acquisition et la maîtrise des deux langues. On nuancera cependant ce point de vue. L'idéal serait que les élèves perçoivent la différence entre les registres, entre ceux où le mélange de langues est stigmatisant et ceux où les formes pures ont un coefficient d'étrangeté.

La monosémie est loin d'être dominante, comme on le voit avec l'exemple du verbe couler largement polysémique en français régional : -Le soleil coule derrière la montagne (1). / La case coule quand il pleut fort (2). / Coulez vous-même votre enveloppe dans l'urne (3). / Elle coule le café dans une vieille grègue (filtre) (4). On retrouve ici les sens de disparaître, glisser derrière en 1 ; de ne pas être étanche, fuir en 2 ; d'introduire en 3 ; et de faire passer à travers un filtre en 4 - (Carayol, 1984 : 345).

Couler constitue-t-il un seul élément polysémique avec un noyau commun ou chaque entrée estelle un homonyme des trois autres ? Autrement dit, y a-t-il un seul terme doté de différentes acceptions ou quatre mots distincts à la fois homonymes et homophones (mots de prononciation identique et de sens différents), et de plus homographes, (dotés de la même graphie) les uns des autres.

La question se pose pour un grand nombre de mots formellement identiques, et qui pourtant ont des sens différents.

Je termine sur la variation diatopique avec un dernier exemple. La notion a été élaborée par Allsop puis par Bickerton à propos de l'anglais et du créole de la Guyane. Il a été montré que les faits linguistiques s'ordonnaient en un continuum allant du niveau de créole le plus proche de l'anglais au plus éloigné, le classement correspondant à la position sociale des locuteurs. Le concept de continuum permet de décrire les situations qui opposent deux langues différenciées sous leur forme H (acrolecte) et L (basilecte) entre lesquelles existent des variétés intermédiaires et pour lesquelles, il est difficile de dire où commence L et où finit H. On empruntera à M. Carayol et à R. Chaudenson leur définition du continuum. Il serait « un dia-système bipolaire allant d'un acrolecte caractérisé par des formes socialement valorisées à un basilecte correspondant à l'état de langue dévalorisé socialement ». L'acrolecte et le basilecte partagent de nombreux traits linguistiques et l'opposition porte sur un nombre relativement limité d'éléments. L'intercompréhension est donc relativement aisée entre les deux pôles du continuum, l'objectif n'étant pas de décrire de façon exhaustive tous les idiolectes mais de déterminer les faits linguistiques qui présentent des éléments de variation au sein du continuum. On peut donc figurer le continuum de la façon suivante:

A

Les limites de la notion d'alternance codique

On reprendra d'abord l'analyse que propose Sylvie Wharton (SW) de cet exemple (2003 : 79-88), qui traduit une volonté de dépasser l'analyse classique de ces énoncés en termes de diglossie, et de recourir à la notion d'interlecte (Prudent : 1981). SW souligne que dans le cas de cette phrase extraite d'un corpus, recueilli auprès « d'élèves réunionnais de la maternelle au lycée âgés de 5 à 18 ans » dans une « situation de communication orale » (2003 : 81), précisément une « conversation » (Ibid.), nous sommes confrontés à un « matériau linguistique indissociable » (Ibid. 80). SW fait l'hypothèse de l'existence dans les énoncés de ce type « d'une perméabilité systémique entre les langues » (Ibid. : 80). Cette « perméabilité » se manifeste « au plan formel par une réduction des contraintes pesant sur les traits supposés être distinctifs de l'une ou l'autre langue ». Il convient de prendre en compte également le contexte de l'énonciation et les choix énonciatifs que font les locuteurs en fonction de ce contexte (Ibid.). Cet ensemble de faits conduit SW à postuler que ces « discours » sont « par essence hétérogènes » (Ibid. : 81). Selon SW, l'énoncé « marque une alternance codique (français puis créole) intraphrastique » (Ibid. :82

Conclusion

Il paraît nécessaire d'intégrer dans la pratique scolaire du français ces particularités lexicales et syntaxiques. La difficulté de la description se trouve néanmoins dans l'identification des formes d'une part et dans la détermination de leur appartenance à une langue ou à une autre, en situation de contacts de langues d'autre part. Ceci nous conduit, en conclusion, à aborder les processus identitaires à l'oeuvre dans ces énoncés. En effet, en situation de contact de langues, le comportement langagier devient un véritable acte d'identité par lequel dans la négociation interactionnelle, au contact de l'autre, le choix de la langue, des marques transcodiques, manifestent des facettes identitaires, qui composent une identité complexe.

La notion de norme(s) endogène(s)

  Il existe d'autres types de variation qu'on n'abordera pas ici, car ils sont moins pertinents pour nous dans le cadre de cette formation. Cette terminologie présente des limites. Elle ne permet pas de prendre suffisamment de distance à l'égard de la perception de la langue, fondée sur l'idée que celle-ci est un ensemble « homogène » (Ibid. : 26). Elle conduit, on l'a vu, à stabiliser les variétés et à les cloisonner. Ce faisant, elle occulte le processus variationnel, essentiellement dynamique et instable (Ibid. : 26). L'approche sociolinguistique est fondée sur l'étude de l'oral et sur l'observation de pratiques langagières présentes en discours, et donc potentiellement instables, hétérogènes et en mouvement.Ceci suppose qu'il existe pour un certain nombre de référents des termes distincts selon les endroits, loup et bar par exemple pour le poisson. C'est un exemple classique de variation

	linguistiques ne se confondent pas avec les classes sociales à l'oeuvre dans une société donnée.
	En matière de lexique, la norme endogène, ou norme locale, contextuellement située se
	concrétise, comme on le voit par exemple dans le dictionnaire du français de Madagascar, réalisé
	par Claudine Bavoux, à travers les particularités lexicales du français de Madagascar, qui se
	traduisent en particulier par une néologie, qui s'exerce notamment sur le sens, par
	recontextualisation, comme on l'a vu dans les exemples précédents. L'ensemble de ces termes
	constitue un répertoire qui vient s'ajouter au stock constitué par le français standard, lequel entre
	également dans la compétence du locuteur, mais n'est par nature pas destiné à figurer dans
	l'inventaire
	« En schématisant, [...], on pourrait dire que [...] un français régional, c'est du français commun
	avec quelque chose en plus et, sans doute, aussi, quelque chose en moins »
	de la
	famille » (T. 2),
	Piétiner ou gratter : « aller à pied »,
	Individu : « imbécile ou personne peu recommandable »,
	Être riénard : « être sans le sou » (Id.).
	On trouve des exemples comparables dans le français camerounais. J'emprunte mes exemples à
	Biloa, qui est un linguiste camerounais, qui enseigne à l'université de Yaounde :
	Français du Cameroun
	Politicien « un intrigant ou un menteur »,
	Les bancs : « l'école »,
	Défavori : « quelqu'un qui n'est pas reconnu comme favori » (Biloa, 2003 : 126-127).
	Les exemples de français régional de Côte d'Ivoire ou du Cameroun, que je viens de donner
	renvoient à ce qu'on appelle les normes endogènes, notion que je vais définir maintenant en
	partant de la notion de français régional et en restant dans le cadre de la variation diatopique ou
	géographique, qui va nous conduire progressivement à la variation diastratique ou sociale.
	Les français régionaux s'inscrivent dans le cadre du processus variationnel. Ils sont parlés par une
	communauté linguistique donnée, qui partage les mêmes normes langagières. Les communautés

. On distingue des faits de variation temporelle, c'est la variation diachronique ; de variation spatiale ou géographique, ou régionale, ou locale, c'est la variation diatopique ; sociale, c'est la variation diastratique (Ibid. : 23). diatopique, qui pose relativement peu de problèmes aux élèves, car loup / bar sont deux termes différents. Moins connue est le phénomène lexicologique que j'identifierai comme la polysémie problématique, à savoir des faits de variation diatopique, pouvant être à l'origine de difficultés d'appropriation du français standard par les élèves. Ce phénomène a été décrit comme un processus d'appropriation et de créativité, pour les variétés des français d'Afrique Subsaharienne, pour reprendre le sous-titre du dictionnaire de Suzanne Lafage, Le lexique français de Côte d

'Ivoire, appropriation et créativité (2003)

.

Suzanne Lafage montre comment se met en place un mouvement d'appropriation d'un terme et pour celui-ci la création d'un sens nouveau.

Français de Côte d'Ivoire

Ration signifie « la somme d'argent destinée à l'achat de la nourriture de base journalière

  Garer (V. tr., oral, écrit) : 1. Se garer « J'ai garé devant Monoprix » […] ; 2. Avec un complément désignant une personne, déposer à un endroit donné quelqu'un que l'on a transporté en voiture, « Tu peux me garer à la rue [de/ du + toponyme] » ; 3. Ranger pour conserver, quel que soit l'objet dont il est question, « J'ai là une machine [à coudre] que j'avais louée à quelqu'un, […], je l'ai reprise pour la garer à la maison » ; 4. garé, (être---), loc. verb., argot estudiantin, oral, fam. […] être « paré » contre toute éventualité, « Lui, tu sais, il est garé avec sa famille, mais moi ? ».(Lafage, 2003 : 422-423) On rencontre la même difficulté si on considère l'évolution historique du sens des mots dans une perspective de variation diachronique. C'est un exemple emprunté à un auteur, Jacques Laurent, qui s'appuie sur le Littré. L'exemple est celui du verbe dévisager.Le premier sens de dévisager donné par Littré est celui de « déchirer le visage avec les ongles ou les griffes ».Jacques Laurent aborde le problème de ce qu'il définit comme l'inéluctable évolution de la langue et les moments où elle devient comme étrangère à elle-même, semblable et pourtant autre. C'est ce que je désigne comme la polysémie problématique, laquelle découle d'une part de l'existence du processus variationnel, d'autre part de l'existence du continuum linguistique.

Sens propre : « Prenez garde de vous faire dévisager par ce chat » ; « Je vais chercher le chevalier, madame, et je le dévisage si je le trouve, Dancourt, Chevalier à la mode. » Sens figuré : « Je ne suis point du tout pour ces prudes sauvages. Dont l'honneur est armé de griffes et de dents.

Et veut au moindre mot dévisager les gens », Molière, Tartuffe.

Mais, le sens populaire donné par Littré est le sens moderne : « dévisager quelqu'un, faire effort pour reconnaître les traits de quelqu'un. ».

Le sens indiqué par Littré, note J. Laurent, nous est devenu étranger, malgré une forme identique, ce qui interroge notre faculté de compréhension des textes antérieurs au XX e siècle d'une part et d'autre part pose le problème de la variation en l'élargissant à la variété littéraire, qui se confond avec la variété de prestige sur le continuum linguistique.

  Le macrosystème est défini comme « la formule qui désigne l'ensemble des lectes qui assument la communication dans ce qu'on appelait précédemment la diglossie ou le continuum »(Lebon- Eyquem, 2013 : 26). Les oppositions tranchées entre basilecte et acrolecte que fait apparaître la notion de continuum sont très nettement atténuées si on se place dans le cadre de l'interlecte. On peut se demander si la hiérarchie des langues impliquée par la notion de diglossie ne doit pas être critiquée pour laisser place à de nouvelles formes de minorisation. En effet, les reconfigurations du paysage social font apparaître un « métissage massif », qui remet en question les catégories tant ethniques que linguistiques élaborées entre les années 70 et 90(Lebon-Eyquem, 2013 : Ibid.). Les spécialistes de la question s'accordent à propos de la période contemporaine pour reconnaître l'émergence d'un parler réunionnais caractérisé par des formes labiles et mouvantes difficiles à décrire avec les outils dont dispose la linguistique(Wharton 2003 ; Ledegen, 2010) 2 . On soulignera que ces formes sont différentes de celles qu'offrent les corpus qui ont pu être recueillis à la fin des années 70.Si les notions de continuum linguistique et de diglossie doivent être critiquées dans le cadre de la situation linguistique réunionnaise, elles n'en demeurent pas moins très fécondes d'un point de vue heuristique. Elles continuent d'offrir un cadre théorique pertinent, selon nous, pour l'approche des situations minoritaires et des conflits linguistiques.Cette phrase appartient à la catégorie des phrases clivées construites selon le type emphatique dit par extraction 6 . Ce type de phrase est facultatif. Il ne correspond pas à « un acte de langage » mais possède une structure syntaxique propre (GMF 2016 : 666). L' « emphase désigne […] tout procédé d'insistance ou de mise en relief » (Ibid. : 667). L'emphase par extraction permet la mise en relief d'un constituant « traité comme propos » en encadrant le constituant en question avec le présentatif c'est et le pronom relatif qui ou que (Ibid. : 667). L'élément extrait est désigné comme 7 « focus » ou « foyer » (Ibid. : 726). L'extraction est également analysée comme « une opération de focalisation d'un constituant » (Ibid.). Dans l'occurrence proposée, le présentatif est représenté par « c'est », le constituant mis en relief est « la route ». L'extraction affecte ici une phrase déclarative, le constituant extrait est le sujet : la route amène à Saint-Paul / c'est la route qui amène à Saint-Paul. Cette construction est ambiguë, elle peut aussi être analysée comme « une séquence [présentatif + groupe nominal] incluant une relative, utilisée pour présenter le référent » (GMF 2016 : 726). Deux analyses sont donc possibles. On retient la première. La proposition introduite par qui et qu'on désignera comme une relative, faute d'autres outils, pose trois problèmes 7 . Le premier est syntaxique et apparaît dans la construction sans préposition du verbe amener, le second est sémantique et renvoie au sens du verbe amener, le troisième est celui de la graphie du nom propre « Sinpol ».

	Ainsi Gudrun Ledegen pouvait écrire en 2010 à ce sujet :
	Les pratiques mélangeantes alternantes ont pris de l'ampleur : la comparaison
	d'enregistrements établis en 1978-1980 dans le cadre de l'Atlas linguistique de La Réunion
	avec des corpus d'aujourd'hui (2000-2008 ; Valirun) montre qu'on mélange davantage et de
	façon plus diverse (2010 : 114 citée par Lebon-Eyquem, 2013 : 26).
	français standard	A' français régional	B français créolisé	C créole
	A français littéraire A' français standard A'' français régional : A'' français non normé
			(populaire)
		B français créolisé C créole

C' créole acrolectal C'' créole mésolectal C''' créole basilectal II. La syntaxe -« C'est la route qui amen Sinpol 1 » L'énoncé proposé est un exemple de langues en contact, à savoir le français et le créole réunionnais. On l'analysera comme un exemple de situation interlectale en tenant compte de la proximité des deux langues et de leur parenté génétique. Cette approche nous conduit à dépasser le cadre classique de la diglossie et de la notion de continuum linguistique pour adopter le concept d'interlecte, élaboré par Lambert-Félix Prudent (1981) pour les locuteurs créolophones des Antilles. L'interlecte est adossé aux notions de « sociogenèse, de macrosystème ».

La description que nous allons proposer s'appuie sur les outils linguistiques disponibles dans la grammaire de référence recommandée par les jurys des concours de recrutement des enseignants du second degré 3 . On précisera qu'ils ont été élaborés pour décrire des formes écrites 4 alors qu'il s'agit dans cet énoncé, de formes orales produites en discours, sur lequel pèsent les spécificités d'un contexte 5 , d'un « ensemble de paroles qui ne peuvent être prédites par une grammaire de l'acrolecte ou du basilecte »

(Prudent, 1981 : 31)

. Ces outils sont cependant les seuls dont disposent les enseignants en contexte scolaire. Il paraît donc important de ne pas négliger la description des formes langagières qu'ils permettent. On tentera de proposer une description de ces formes sans perdre de vue le caractère limité des outils utilisés, qui n'ont pas été conçus à cette intention, sans perdre de vue que les propositions, qui vont être faites, sont susceptibles d'être discutées et critiquées. L'étude suivra l'ordre suivant. On analysera d'abord la structure syntaxique de l'énoncé dans son ensemble. On étudiera ensuite les faits d'alternance codique et les critiques qui ont été formulées à l'égard de ce type d'approche.

Les énoncés mélangés. Une critique de l'approche descriptive

  ). Il relève de ce qu'elle appelle les « mayaj 8 » (2003 : 82), à savoir les énoncés mélangés. Cet énoncé n'est donc ni du français standard ni du créole. Cet énoncé en créole serait formulé de la manière suivante : « semin la i amen (anou) Sinpol » (Ibid.). Le français standard proposera « c'est la route qui amène (conduit, va) à Saint-Paul ». SW propose une autre variante plus proche du français régional que de la forme standard : « c'est la route qui amène Ø Saint-Paul », qu'elle nomme une « variante de contact » (Ibid.). Le fait marquant est ici l'omission de la préposition à. À cet égard, on notera que les prépositions ont un emploi spécifique en créole qui se distingue de celui du français, ce qui ne va pas sans causer des difficultés aux locuteurs créolophones.(manguier) ; sabrakane (sabre à canne) » (1985 : 106). Il ajoute que « ne connaissant pas les emplois français de « à » et de « de », […] les créolophones font de nombreuses erreurs dans les structures où entrent ces prépositions » (Ibid. :106). De ce fait, les constructions prépositionnelles du français sont une des catégories grammaticales les plus complexes à maîtriser pour un créolophone (Ibid. : 107) en raison, pour les constructions en « à » et « de » de leur inexistence en créole (Ibid.). PC relève l'absence de « à » dans la liste d'erreurs sur les prépositions qu'il propose dans son ouvrage (Ibid. : 108). Logambal Souprayen-Cavery confirme l'existence d'une « très forte variation » et de « phénomènes d'interlecte » à propos de l'usage de la préposition à (2010 : 172).Au plan sémantique, on peut ajouter qu'un locuteur du français standard utiliserait peut-être plutôt le verbe mener dans l'acception « permettre d'aller d'un lieu à un autre ; ce sentier mène au refuge » (Le Petit Robert 2015) que le verbe amener. Mener est utilisé pour les inanimés, amener pour les êtres animés (Ibid.). AA dans son dictionnaire propose une entrée ménèr, qui n'est pas développée, elle renvoie à amén. Les deux termes paraissent être des synonymes. Sinpol, comme l'indiqueSW (2003 : 82), est la traduction en créole de Saint-Paul, ville située à l'Ouest de l'île. On formulera l'hypothèse que cette graphie semble rejoindre les options de Lékritir 77, formulées par le groupe de recherche et de réflexion sur les graphies du créole réunionnais Oktob 77(Armand, 1987 : LVI).Mylène Eyquem-Lebon, dans un article de 2013, reprend l'analyse proposée en 2003 par SW, et qui a été citée dans ces lignes, pour souligner qu'il est discutable de tenter de décrire et d'ordonner les faits langagiers relevant du code mixing en leur appliquant les structures mises en place par la linguistique française(2013 : 30-31). On notera cependant que c'est faute d'avoir un autre modèle linguistique, une grammaire de l'oral par exemple.On citera à ce sujet les lignes fort éclairantes de MEL qui explicitent fort bien les tensions en présence et les contradictions auxquelles sont confrontés les chercheurs concernés : Ainsi, si depuis quelques années l'interlecte est souvent mentionné dans la littérature par beaucoup de chercheurs souhaitant mettre en avant leur volonté de considérer l'hétérogénéité linguistique et d'étudier la langue dans toute sa variété et sa complexité, sa mise en oeuvre dans les analyses reste problématique. En effet, lors des descriptions des pratiques langagières, son utilisation ne s'est pas accompagnée du virage épistémologique pourtant nécessaire. Bien qu'ils s'en défendent, certains linguistes (et notamment les tenants de l'approche plurilingue) continuent à rechercher des structures et des règles dans de la parole traversée par des conflits sociaux, raciaux et symboliques, à définir des segments, à vouloir les étiqueter, révélant leur nostalgie pour un modèle de langue pur, clair et précis. Cette orientation stabilisante et prédictibiliste, ces découpages de signifiants en tranches fines prennent pour référence le français parce qu'il est plus facilement identifiable. Le basilecte est alors caractérisé de façon soustractive et différentielle, ce qui ne manque pas d'asseoir le pouvoir de la langue dominante (2013 : 30).

	La graphie Sinpol illustrerait, si on suit AA, le principe de simplicité caractéristique d'une
	écriture phonologique énoncé supra.
	Le verbe amèn en créole est transitif comme l'indique le Dictionnaire Kréol Rénioné Français
	d'Alain Armand (1987) et ignore la préposition. Pierre Cellier dans sa Comparaison syntaxique
	du créole réunionnais et du français (1985) note que les prépositions « à » et « de » sont rares en
	créole et surtout observables dans des « structures lexicales figées » du type « piedmang 9

On l'analysera en suivant les remarques formulées par AA à propos de l'orthographe et des graphies. Il s'agit d'abord d'une « proposition orthographique, une parmi d'autres possibles » (Ibid. : LVI). Elle émane de principes qui sont fondés sur la nécessité de distinguer graphiquement deux codes « compte tenu de la parenté génétique et structurelle entre le créole et le français » (Ibid. : LIII). Ces principes sont les suivants : « écriture phonologique […] ; recherche d'une simplicité maximale, […] un signe pour un son et toujours le même ; recherche de la commodité : système dactylographiable sur clavier normalisé, commercialisé ; recherche d'un système qui sans trop se rapprocher du français ne va pas contre les habitudes de lecture des Réunionnais déjà alphabétisés en français ; normalisation sur les variantes basilectales qui sont les plus fréquemment utilisées […] ; intégration systématique des variantes acrolectales portant sur les oppositions i/u , s/ch, z/j et prise en compte des variantes sourdes et sonores de voyelles nasales èn/inn (amèn/aminn), om/onm (rom/ronm) et à un degré moindre de ane/ann (kane/kann) et one/onn (zanone/zanonn) » (1987 : LVI-LVII) .

Cet exemple est extrait de l'article de Sylvie Wharton (2003) « Plurilinguisme et acquisition de la norme scolaire. La situation du créole et du français à La Réunion » in M.-M. Bertucci, J. David (coord.), Les langues des élèves, Le Français aujourd'hui, 143, pp. 82. Il s'agit de la ville de Saint-Paul située sur l'île de La Réunion.

On se reportera à l'article de Mylène Lebon-Eyquem pour une bibliographie plus complète.

Essentiellement la Grammaire méthodique du français (GMF) de M. Riegel, J.-C. Pellat, et R. Rioul, dans l'édition de 2016.

On renvoie à la remarque de F.Gadet (1992 : 85) citée p. 4 de ce document sur ce point.

Non précisé dans l'article d'où il est extrait et qui ne peut être pris en compte de ce fait.

Il existe un type fondé sur le détachement d'un constituant(GMF 2016 : 667).

Ces faits ont été mentionnés dans les copies des étudiants.

« Du créole « « mayaj » (mayaz) » : mélange, emmêlement » (Wharton : 2003 : 82).

Moins courant que « pié mang » d'après AA (1987 : XXXIX).


Quentin Le Lidec

Louis Montaut

Cordelia Schmid

Ivan Laptev

Justin Carpentier

Augmenting differentiable physics with randomized smoothing

In the past few years, following the differentiable programming paradigm, there has been a growing interest in computing the gradient information of physical processes (e.g., physical simulation, image rendering). However, such processes may be non-differentiable or yield uninformative gradients (i.d., null almost everywhere). When faced with the former pitfalls, gradients estimated via analytical expression or numerical techniques such as automatic differentiation and finite differences, make classical optimization schemes converge towards poor quality solutions. Thus, relying only on the local information provided by these gradients is often not sufficient to solve advanced optimization problems involving such physical processes, notably when they are subject to non-smoothness and non-convexity issues. In this work, inspired by the field of zero-th order optimization, we leverage randomized smoothing to augment differentiable physics by estimating gradients in a neighborhood. Our experiments suggest that integrating this approach inside optimization algorithms may be fruitful for tasks as varied as mesh reconstruction from images or optimal control of robotic systems subject to contact and friction issues.

I. INTRODUCTION

A. Motivations

Various physical processes such as image rendering [START_REF] Loper | Opendr: An approximate differentiable renderer[END_REF][START_REF] Liu | Soft rasterizer: A differentiable renderer for image-based 3d reasoning[END_REF] or physical simulation [START_REF] Todorov | Mujoco: A physics engine for model-based control[END_REF][START_REF] Coumans | Pybullet, a python module for physics simulation for games, robotics and machine learning[END_REF], are subject to non-smooth phenomena. These result in badly conditioned gradients which may be uninformative (i.d., null almost everywhere) or even undefined. Typically, rendering [START_REF] Loper | Opendr: An approximate differentiable renderer[END_REF][START_REF] Liu | Soft rasterizer: A differentiable renderer for image-based 3d reasoning[END_REF] is naturally non-differentiable and yields either null or undefined derivatives. Similarly, differentiable simulators exhibits null gradients over entire regions of the trajectory space [START_REF] Werling | Fast and Feature-Complete Differentiable Physics Engine for Articulated Rigid Bodies with Contact Constraints[END_REF]. In both cases, these physical gradients present some discontinuities or lack of regularity, which, in the end, may drastically hinder gradient-based optimization algorithms. In this presentation, we show how randomized smoothing appears as a natural solution to avoid the pitfalls induced by a lack of regularity originating from classic gradient estimators. We notably show its applications in the context of differentiable rendering [START_REF] Le Lidec | Differentiable rendering with perturbed optimizers[END_REF] and differentiable simulation for control [START_REF] Le Lidec | Leveraging randomized smoothing for optimal control of nonsmooth dynamical systems[END_REF].

B. Background on randomized smoothing

Originally used in black-box optimization algorithms [START_REF] Matyas | Random optimization[END_REF][START_REF] Polyak | Introduction to Optimization[END_REF], randomized smoothing was recently introduced in the machine learning community [START_REF] John C Duchi | Randomized smoothing for stochastic optimization[END_REF][START_REF] Berthet | Learning with differentiable pertubed optimizers[END_REF] in order to integrate non-differentiable operations inside neural networks.

In more details, a function g can be approximated by convolving it with a probability distribution µ:

g ϵ (x) = E Z∼µ [g(x + ϵZ)] (1)
which corresponds to the randomly smoothed counterpart of g and can be estimated with a Monte-Carlo estimator as follows:

g ϵ (x) ≈ 1 M M i=0 g(x + ϵz (i) ) (2)
where {z (1) , . . . , z (M ) } are i.i.d. samples and M is the number of samples. Intuitively, the convolution makes g ϵ smoother than its original counterpart g and, thus, yields better conditioned gradients. Using an integration by part yields a zero-th order estimator:

∇ (0) x g ϵ (x) = 1 M M i=0 -g(x + ϵz (i) ) ∇ log µ(z (i) ) ⊤ ϵ . (3)
This zero-th order estimator can be used even when g is non differentiable to obtain first-order information usable in a gradient-based optimization scheme, as in Sec. II. Alternatively, a direct way to estimate the gradients whenever g is differentiable, is to use the first-order estimator:

∇ (1) x g ϵ (x) = 1 M M i=0 ∇g(x + ϵz (i) ). (4)
Such a first-order estimator averages gradients over a noise distribution and, thus, mechanically augments the quantity of captured information in the gradients of g even when they are uninformative e.g null in a region of the optimization space, as in Sec. III.

II. DIFFERENTIABLE RENDERING

An example of a non-differentiable process which can be turned into a differentiable one via zero-th order randomized smoothing is rendering. Rendering is the process which, given a 3D scene and a camera parameterization, Fig. 1: Top: Overview of the rendering process: both rasterization and aggregation steps induce non-smoothness in the computational flow. Bottom: Illustration of the differentiable perturbed aggregation process. The rasterization step is made differentiable in a similar way. outputs the corresponding RGB image. As illustrated in Fig. 1, rasterization-based renderers need to go through two discontinuous steps, called rasterization and aggregation. These discontinuities seem natural: if an object moves on a plane parallel to the camera, some pixels will immediately change color at the moment the object enters the camera view or becomes unocluded by another object. Mathematically, the operations occurring during these steps can be written as argmin operators of Linear Programming problems:

y * (θ) = argmin y∈C ⟨θ, y⟩ (5)
Such operators have null gradients almost everywhere and undefined otherwise, hence making them unusable for optimization. Differentiable rendering approximates the underlying process by a smoother one by adding blur and transparency effects [START_REF] Liu | Soft rasterizer: A differentiable renderer for image-based 3d reasoning[END_REF][START_REF] Le Lidec | Differentiable rendering with perturbed optimizers[END_REF][START_REF] Murthy | gradsim: Differentiable simulation for system identification and visuomotor control[END_REF]. In particular, in [START_REF] Le Lidec | Differentiable rendering with perturbed optimizers[END_REF], we propose to do it by replacing the original y * (θ) operators from rasterization and aggregation by their randomly smoothed counterparts (Fig. 1). If randomized smoothing alleviates the burden of badly conditioned gradients by working with an approximate but smoother renderer, it is necessary to consider a way to reduce the smoothing during optimization in order to solve the original problem. In [START_REF] Le Lidec | Differentiable rendering with perturbed optimizers[END_REF], we introduce a criterion allowing to automatically reduce the injected noise across iterations. This approach is applied to tackle pose optimization and mesh reconstruction tasks and reaches results competitive with the state of the art [START_REF] Le Lidec | Differentiable rendering with perturbed optimizers[END_REF].

III. DIFFERENTIABLE SIMULATION FOR CONTROL

An example of a differentiable process for which first-order gradient estimation via randomized smoothing can be applied to enrich gradients' information is physical simulation. Simulating a physical system subject to contacts and frictions requires to solve a problem formed by the Euler-Lagrange equations of motion, the complementarity constraint accounting for contacts, the Maximum Dissipation Principle and Coulomb's law which governs friction. Solutions of this problem are typically obtained by solving an associated optimization problem [START_REF] Todorov | Mujoco: A physics engine for model-based control[END_REF][START_REF] De | End-to-end differentiable physics for learning and control[END_REF][START_REF] Le Lidec | Differentiable simulation for physical system identification[END_REF] and, then, their gradients can be computed via implicit differentiation. However, as illustrated by Fig. 2 and detailed in [START_REF] Werling | Fast and Feature-Complete Differentiable Physics Engine for Articulated Rigid Bodies with Contact Constraints[END_REF], contacts and friction cause the simulation to be non-smooth and its gradients to remain null over large regions of the trajectory space.

In such a setting, randomized smoothing allows to approximate the underlying physics by a smoother one as shown in [START_REF] Le Lidec | Leveraging randomized smoothing for optimal control of nonsmooth dynamical systems[END_REF][START_REF] Ju | Bundled gradients through contact via randomized smoothing[END_REF] and illustrated by Fig. 2. Interestingly, this approach also allows to frame the stochasticity of exploration as a key element of the success of RL on solving physical problems with contacts and frictions [START_REF] Le Lidec | Leveraging randomized smoothing for optimal control of nonsmooth dynamical systems[END_REF]. If randomized smoothing allows to augment the quantity of information contained in the gradient estimator, it also introduces stochasticity and it is necessary to design accordingly an optimization procedure that preserve the convergence guarantees. To enforce the convergence towards an optimal (local) solution, it remains crucial to reduce the noise injected via the randomized smoothing across the iterations. In [START_REF] Le Lidec | Leveraging randomized smoothing for optimal control of nonsmooth dynamical systems[END_REF], we propose to decrease it in a way that adapts to the problem and avoids the smoothing being reduced too quickly, which would lead to performances similar to classical DDP, or too slowly, which would induce an unnecessary large number of iterations.

Finally, we apply our algorithm to solve a task on the Solo robot [START_REF] Grimminger | An open torque-controlled modular robot architecture for legged locomotion research[END_REF], an 18-DoF robotics system, with frictional contacts. Here, the goal is to reach a final target pose (lifting of the tip of one leg while keeping the three others on the ground), Fig. 4: Randomized DDP makes it possible to achieve complex tasks requiring breaking contacts with the floor on Solo robot such as lifting a leg (see Fig. 3) which requires breaking some initial existing contacts, as illustrated in Fig. 3. Randomized smoothing coupled with Differential Dynamic Programming (R-DDP) allows to solve this task while classical DDP algorithm is not able to precisely apprehend contacts because of non-informative gradients of the dynamics (Fig. 4). Typically, the control obtained from DDP will only approach this pose while maintaining the feet on the ground which results in Solo bending its leg instead of lifting it.

IV. DISCUSSION

Gradients from differentiable physical processes should not be thought about separately from the actual problem and the associated optimization algorithm. It is often the case that exact gradients estimated from analytical expressions or automatic differentiation are likely to provide insufficient information for classical optimization to get satisfying solutions. Randomized smoothing proposes to capture additional information by using a noise distribution to evaluate gradients in the neighborhood of the current optimization stage. Doing so, it moves from a deterministic to a stochastic setting which may hinder performance of optimization algorithms (convergence and reproducibility) and thus requires some adjustments. We propose an approach which decreases the noise adaptively to improve robustness. However, detecting when to increase it is left as future work. Alternatively, analysing the respective stability properties of the zero-th and first order estimators represents another promising research direction [START_REF] Suh | Do differentiable simulators give better policy gradients?[END_REF].

Fig. 2 :

 2 Fig. 2: Top: A slight vertical force is not able to break the unilateral contact, leaving cube on the floor and, thus, the state unchanged. Bottom: The non-smoothness of physics induces null gradients which results in the failure of classical optimization techniques

Fig. 3 :

 3 Fig. 3: Randomized-DDP leverages the smooth dynamics to precisely schedule movements requiring to break contacts.

ACKNOWLEDGMENTS

This work was supported in part by the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute), the European Regional Development Fund under the project IMPACT (reg. no. CZ.02.1.01/0.0/0.0/15 003/0000468), the Grant Agency of the Czech Technical University in Prague, grant No. SGS21/178/OHK3/3T/17, and Louis Vuitton ENS Chair on Artificial Intelligence.