Sophie Joimel

Baptiste Grard

Claire Chenu

Pénélope Cheval

Samuel Mondy

Mélanie Lelièvre

Apolline Auclerc

Laure Vieublé Gonod

One green roof type, one Technosol, one ecological community

Keywords: urban agriculture, extensive green roofs, productive green roofs, soil biodiversity, plants Graphical

Green infrastructures play a key role in the functionality and resilience of urban ecosystems.

The physical, including thickness, chemical and biological properties of the Technosols of green infrastructures on rooftops are highly variable, leading to more or less favourable conditions for soil biodiversity. The aim of this study was to investigate the abundance and diversity of bacteria, fungi, nematodes, collembola and plants communities in relation with abiotic parameters of Technosols on 12 productive and extensive green roofs of the Paris region (France).

Results showed that green roofs harboured a high level of abundance and diversity from microorganisms to micro and mesofauna. Microbial biomass ranged between 16.3 and 419.8 µg DNA g -1 , with a predominance of bacteria, nematodes represented 820-60 700 individuals per kg of soil and between 1000 and 60 700 collembolan were present per m 2 of soil. A total of 13 986 bacterial OTU (Operational Taxonomic Unit), 33 559 fungal OTU, 47 Collembola species, 28 nematodes families, 16 cultivated plant species and 48 spontaneous plant species was identified on all the green roofs studied. Microbial, animals and plants communities were significantly different between the two types of green roofs. Productive and extensive rooftops represent contrasted habitats, which can strongly influence the soil biota. Any voluntary action to enhance soil biodiversity in cities would need to take-into-account both soil properties and the landscape around.

Introduction

Due to the development of urban areas and related environmental issues (e.g. soil sealing, air, water and soil pollution, habitat fragmentation, heat island), the urban ecosystem has been the subject of new societal expectations in the last years with an increasing necessity to reintegrate nature into the city [START_REF] Bratman | Nature and mental health: An ecosystem service perspective[END_REF][START_REF] Cheverry | Sous les pavés la terre[END_REF]. Nature in the city has so far been mostly associated with vegetation and with its recognized role in the landscape and as a component of the green belt network [START_REF] Arnould | La nature en ville : l'improbable biodiversité[END_REF]. The evolution of knowledge and awareness allows nowadays considering green infrastructures as a whole, considering both the soil and above-and below-ground biodiversity and to highlight their importance in the functioning of urban ecosystems. Indeed, green infrastructures play a key role in the functionality and resilience of urban ecosystems, in particular through the ecosystem services they provide (e.g. food production, regulation of the water cycle and floods, carbon storage and climate regulation, biodiversity support) [START_REF] Luederitz | A review of urban ecosystem services: six key challenges for future research[END_REF][START_REF] Tzoulas | Promoting ecosystem and human health in urban areas using Green Infrastructure: A literature review[END_REF]. Thus green infrastructures should temper negative environmental impacts of rapid urbanization [START_REF] Pataki | Coupling biogeochemical cycles in urban environments: ecosystem services, green solutions, and misconceptions[END_REF] and impact positively the biodiversity in urban areas [START_REF] Uy | Application of land suitability analysis and landscape ecology to urban greenspace planning in Hanoi, Vietnam. Urban For[END_REF]. Urban ecosystems, from simple supports for buildings, will be recognized as supports for several ecosystem services.

Because of the scarcity of space and land pressure in the city and the availability of large roof surfaces, the development of green infrastructures on the roofs of buildings is a promising development. Green infrastructures on buildings offer additional space and corridors to enhance biodiversity in cities [START_REF] Mayrand | Green Roofs and Green Walls for Biodiversity Conservation: A Contribution to Urban Connectivity?[END_REF] as island habitats for urban biodiversity, and thus participate to the conservation of ecological networks, the so-called "green and blue infrastructures". Green infrastructures on rooftops can fulfill different functions, from food production (productive green roofs) to ornamental (extensive green roofs) purposes. Green roofs have been also recognized for their better energy efficiency for building and reduced rain runoff over conventional roofing [START_REF] Li | A comprehensive study of green roof performance from environmental perspective[END_REF]. Initiatives have gain local supports and institutional supports and cities authorities. Since 2014, the city of Paris, through its "Parisculteurs" call for initiatives, aims to develop the place of nature in the city, by attracting new farmers, creating social links and employment, and raising awareness of good eating habits, notably through the implementation of rooftop urban agriculture. The role of green roofs in biodiversity supporting, and its drivers, are very poorly known nowadays, particularly in the case of productive green roofs [START_REF] Lin | The future of urban agriculture and biodiversityecosystem services: Challenges and next steps[END_REF].

Substrates of green roofs, that we can consider as soils, are composite and man-made, that involve in some cases the use of urban waste [START_REF] Molineux | Characterising alternative recycled waste materials for use as green roof growing media in the U[END_REF] and can be named Isolatic Technosols (IUSS Working Group WRB, 2014). The soils of green infrastructures on rooftops -more or less thick-are characterized by a high variability in their physical, chemical and biological properties, which varies greatly according to the aesthetic and/or productive functions of the rooftop that could impact positively or negatively the soil biodiversity [START_REF] Rumble | Soil microarthropod community dynamics in extensive green roofs[END_REF]. For example, extensive green roofs have mostly esthetic values and a role in water regulation whereas productive green roofs have moreover functions of food productions or educational services. We need to better address the direct and indirect roles of these soils and their above-and below-ground biodiversity in ecosystem functioning and services in order to be better considered in the management of nature areas into cities. In a recent review, [START_REF] Guilland | Biodiversity of urban soils for sustainable cities[END_REF] outlined the importance of making new studies as urban soils are still very little studied with only a few studies characterizing the biodiversity of urban soils and theirs ecological and functional roles. Studies on productive and extensive green roofs are even scarcer. Thus, biodiversity databases and repositories need to be enriched with urban ecosystems information. A few studies described macroarthropods [START_REF] Madre | Building biodiversity: Vegetated façades as habitats for spider and beetle assemblages[END_REF][START_REF] Madre | A comparison of 3 types of green roof as habitats for arthropods[END_REF] and Collembola [START_REF] Rumble | Soil microarthropod community dynamics in extensive green roofs[END_REF][START_REF] Schrader | Soil formation on green roofs and its contribution to urban biodiversity with emphasis on Collembolans[END_REF] in green roofs, with a high abundance of soil organisms and low species diversity but they did not address simultaneously different groups of soil organisms nor interactions between the aboveand below-ground biodiversity. Recent studies showed for example an influence of soil microbial communities on plant-pollinator interactions (Barber and Gorden, 2014).

However, without multi-taxon studies, we cannot be sure that all groups of soil organisms respond in the same way to these infrastructures. Their different characteristics (e.g. size, behaviour, colonisation pathways) may induce different responses to the roof type.

The objectives of this study were to i) describe the biodiversity of productive and extensive green roofs through the study of soil and floristic biodiversity, ii) determine the key soil parameters that condition the microbial, faunal and floristic biodiversity of green roofs, and especially the water retention capacity, trophic and nutritional (e.g. content of organic matter) as well as chemical (e.g. pH) characteristics of the Technosols and iii) study the cooccurrence between the different soil organisms on the one hand and between soil organisms and plants on the other hand. Understanding the drivers of the abundance and diversity of microbial and faunal communities should make it possible to propose how to optimize biodiversity in cities through the design and implementation of management of green roofs.

We hypothesised that the Technosols characteristics of productive green roofs will favour soil biodiversity but not vegetation diversity compared to extensive green roofs due to differences in edaphic constraints and management. Productive greens roofs being usually deeper and richer in organic matter due to organic waste used in their construction [START_REF] Grard | Recycling urban waste as possible use for rooftop vegetable garden[END_REF][START_REF] Grard | Rooftop farming on urban waste provides many ecosystem services[END_REF] should favour soil biodiversity, especially detritivorous organisms. In contrary, we supposed that, as both green roof types have different purposes, plant diversity could be influenced by practices, as previously reported in gardens such as plants removed by gardeners [START_REF] Kendal | A cultivated environment: Exploring the global distribution of plants in gardens, parks and streetscapes[END_REF]. Indeed, plant diversity is controlled in productive green roofs by targeted planting and by weeding control. It is also necessary to keep in mind that, if plants are mostly choosen for food production or ornamental values, no voluntary introduction of soil organisms are conducted.

2.

Materials and Methods

Studied sites

We selected 12 rooftops (ca 35 m above ground) located in the dense urban environment of the Paris region (Ile-de-France, France), six extensive green roofs and six productive green roofs (Table 1). According to the "Green Roofing Guideline", these green roofs were classified according to three criteria: Technosol's depth, main function of the roof and presence/absence of an irrigation systems (The Roof Greening Working Group, 2002).

Extensive green roofs were characterised by a thin layer (8 to 10 cm-depth) and coarse substrate (typically pozzolan), were not irrigated and mostly covered with ornamental succulent species (Sedum sp. mostly). These species were introduced initially at green roofs construction. Others plants species can thus colonize the green roofs. Productive green roofs were characterized by grown vegetables covering more than 50% of their surface area, a thick layer of Technosol (20 to 30 cm of depth) and high organic matter contents (presence in the Technosols of compost, spent mushroom substrate, shredded wood, potting soil).

Extensive and productive green roof surface areas were circa. 390 m² and 270 m 2 .

Sampling methods

Soil biodiversity samplings and vegetation identification were carried out in April 2017. In each green roof, six plots of 25x25 cm were randomly located in the area covered by vegetation (i.e. excluding paths or impervious surface), at least 1 m from the roof border to avoid edge effects for vegetation identification and for invertebrates sampling.

For each green roof, final representative soil samples (600 g) were composed of 5 soils subsamples (8 to 15 cm depth according to soil depth). This composite soil sample was used for soil, nematodes and microbial analyses.

Microbiological sampling and analysis

From the composite soil sample of each plot, an homogenized aliquot of soil was sieved at 4 mm and 50 g of soil were sent to the Genosol platform (Dijon) to be lyophilized and stored at -40°C for further molecular analyses.

Molecular microbial biomass

Microbial DNA was extracted from 1 g of lyophilized soil subsamples using the procedure described in [START_REF] Plassart | Evaluation of the ISO Standard 11063 DNA Extraction Procedure for Assessing Soil Microbial Abundance and Community Structure[END_REF]. Briefly, soil samples were homogenized in 1 ml of extraction buffer [100 mM Tris, pH8; 100 mM EDTA; 100 mM NaCl; 1% (w/v) polyvinylpyrrolidone; 2% (w/v) sodium dodecyl sulphate] and glass beads for 3×30 s at 4 m.sec -1 in a FastPrep -24 (MPBiomedicals, NY, USA). The samples were then incubated for 30 min at 70°C, and centrifuged at 7 000 g for 5 min at 20°C. After removing the supernatant, proteins were precipitated with 1/10 volume of 3M sodium acetate prior to centrifugation (14 000 g for 5 min at 4°C). Finally, nucleic acids were precipitated by adding one volume of icecold isopropanol. The DNA pellets obtained after centrifugation (14 000 g for 5 min at 4°C) were washed with 70% ethanol. Soil DNA extracts were purified with a polyvinylpyrrolidone spin column and a Sepharose 4B spin column. The quality and the integrity of the soil DNA were checked by electrophoresis on 1% agarose gel. DNA was quantified by fluorescence measurement of the Picogreen dye using a Quant-iT™ dsDNA Assay Kit (Invitrogen, USA).

Molecular analyses of soil bacterial and fungal communities

Quantitative real-time PCR were performed on extracted DNA to quantify 16S and 18S rDNA gene sequences [START_REF] Plassart | Evaluation of the ISO Standard 11063 DNA Extraction Procedure for Assessing Soil Microbial Abundance and Community Structure[END_REF], which led to the estimation of the fungal to bacterial (18S/16S) ratios. For microbial diversity and composition analyses, banks of 16S CGATAACGAACGAGACCT 3′ for 18S primers). Bioinformatics analysis of the sequences was performed using the GnS-PIPE v1.17 (Tardy et al., 2015). In order to compare the data sets efficiently and avoid biased community comparisons, the sample reads were reduced by random selection closed to the lowest data sets (10 000 reads for 16S and 18S-rRNA gene sequences respectively for each soil sample). The retained high-quality reads were used for taxonomy-based analysis using similarity approaches.

Nematode sampling and identification

Aliquots of the composite soil samples (one per plot) from each green roof were pooled in a single sample of 300 g of wet soil for the analysis of nematodes (no replicate and no sieving).

Nematodes were analysed by extraction, quantification and taxonomic identification at the level of the nematode family according to NF/ISO 23611-41 by the ELISOL laboratory.

Collembola sampling methods and identification

For each of the six plots by green roofs, undisturbed soil cores (5 cm depth, 6 cm diameter)

were sampled from the topsoil. Collembola were then extracted from soil samples using a high-gradient McFadyen extractor for one week according to NF/ISO 23611-2 and identified at the species level using available dichotomous keys [START_REF] Gisin | Ökologie und Lebensgemeinschaften der Collembolen im Schweizerischen Exkursionsgebiet Basels[END_REF][START_REF] Hopkin | A Key to the Collembola (springtails) of Britain and Ireland[END_REF].

Plants identification

All spontaneous vascular plants, i.e. those growing and reproducing without human assistance [START_REF] Bossu | Effects of built landscape on taxonomic homogenization: Two case studies of private gardens in the French Mediterranean[END_REF] and cultivated plants were identified at the species level using identification keys [START_REF] Machon | Sauvages de ma rue: guide des plantes sauvages des villes de France[END_REF][START_REF] Mamarot | Mauvaises herbes des cultures -4eme edition[END_REF] in each of the six plots of 25x25 cm per roof type. Their relative abundance was estimated using the Braun-Blanquet cover-abundance scale and then transformed to the ordinal scale. The discrimination of species between cultivated and spontaneous plants was difficult due to the lack of information about plantations during the green roofs installation. We assumed that the species that had resisted over time were species able to colonize extensive roofs. Even if some species of Sedums sp. could have been initially planted, we considered all Sedums sp. as spontaneous species in this study due to the high age of extensive green roofs.

Soil sampling and physicochemical analyses

From the composite soil sample, an homogenized aliquot of soil from each green roof was airdried, disaggregated and sieved at 2 mm. Soil analyses were performed by the laboratory of Paris city (AgiLIMS, France), according to the following standard methods: pH (NF ISO 10390), organic carbon (NF ISO 10694) and total metals (Cd, Cu, Pb and Zn; NF ISO 11466).

In addition, three undisturbed soil cores were sampled for bulk density determination (NF EN 13041). They were saturated with water for 24h and then placed on a suction table to equilibrate at 20kPa corresponding to field capacity (pF2.3). The samples were then dried at 105°C to determine bulk density and soil water content at field capacity, i.e. the water holding capacity (WHC).

Organic C and pH were selected as indicators of soil chemical fertility, bulk density and water holding capacity as indicators of soil physical fertility and total concentrations of Cu, Ni, Pb and Zn as indicators of metal contamination. These metals are often associated with human activities in urban agriculture [START_REF] Joimel | Physico-chemical characteristics of topsoil for contrasted forest, agricultural, urban and industrial land uses in France[END_REF].

Community indices

Microorganisms, nematodes, Collembola and plants communities were described by their biomass, abundance or density (except for plants) and the total number of species per green roof type. Additionally, we computed community indices for structure and composition based on taxonomy and their relative abundance (species-based indices). We calculated the Shannon and evenness indices for each green roof.

Statistical analyses

When indices did not match the basic assumptions of normality and homoscedasticity required for parametric statistics (Wilk-Shapiro test at rejection level of α = 0.05), we used non parametrical one-way kruskal test to assess differences between the two types of green roofs.

We analysed the effects of green roof type on species composition by performing a nonmetric multidimensional scaling (NMDS) for each taxonomic groups, using the Bray-Curtis index, to explore dissimilarities between communities. Differences in species compositions between green roof types were tested by a permutational analysis of similarities (ANOSIM) [START_REF] Clarke | Non-parametric multivariate analyses of changes in community structure[END_REF].

To study the co-variation between soil biological and selected soil abiotic parameters, a coinertia analysis was performed on two PCA: (i) one on soil chemical and physical parameters (C, pH, bulk density, WHC, Cu, Cd, Pb, Zn) and (ii) one on biodiversity parameters (abundances and species richness) of each taxonomic group. Co-inertia was chosen in order to look on the co-structure of biological and physicochemical parameters.

All statistics were performed using R software version 3.2.2 (R Development Core Team, 2015), with "vegan" [START_REF] Oksanen | Vegan : Community Ecology Package[END_REF] and "ade4" [START_REF] Dray | The ade4 Package: Implementing the Duality Diagram for Ecologists[END_REF] packages.

Results

Chemical and physical soil characteristics

Mean organic C contents were significantly lower in extensive roofs (7.8 g kg -1 ) compared to productive roofs (21.0 g kg -1 ) (Table 2). pH was slightly alkaline with values ranged from 7.3 (extensive roofs) to 7.6 (productive roofs). No significant difference was observed between green roof types for metal concentrations. Extensive green roofs retained significantly less water (0.14 g H 2 O g -1 of dry soil) than productive ones (0.75 g H 2 O g -1 of dry soil) at pF 2.3

and their bulk density was very low, but higher in extensive roofs (0.8 g cm -3 ) compared to productive roofs (0.4 g cm -3 ).

Community structures and composition

Microorganisms

Microbial biomass ranged between 16.3 and 419.8 µg DNA g -1 soil (Figure 1) and was significantly higher in productive roofs (279.9 ± 29.3 µg DNA.g -1 soil) than in extensive roofs (111.3 ± 27.7 µg DNA g -1 soil). Fungi/bacteria ratios varied from 3.2 to 13.9 and were lower for the productive roofs (6.4 ± 1.3) than the extensive ones (8.8 ± 2.3). No significant difference was observed between the two types of roofs concerning specific richness, Shannon and evenness indices for bacteria and fungi (Table 3).

Microbial diversity was studied at the phylum, genera and OTU (Operational Taxonomic Unit) scales. Twenty-eight bacterial phyla (apart from the categories "unknown" and "unclassified") were identified and 16 among them were present in all soil samples. Thirteen represented more than 0.5% of the total but 2 phyla were largely dominant: Proteobacteria (39.9 and 38.8% in productive and extensive roofs respectively) and Bacteroidetes (35.3 and 31.1% in productive and extensive roofs respectively). Some phyla were only present in rare samples (Fusobacteria, TM6 et Lentisphaerae).

The same phyla were observed on both types of roofs but the hierarchy of phyla representing less than 10% of the total was slightly different (Figure S1).

A total of 2031 bacterial genera was detected in all soil samples. Focusing on the 50 most abundant genera in each sample (147 genera in total), the 10 000 sequences analysed represented on average 89.8% of the diversity (87. S1).

The factorial map of the NMDS revealed marked differences between bacterial communities between green roof types, along both axes 1 and 2 (Figure 2), and these differences were significant (ANOSIM, P<0.001).

A high proportion of fungi belonged to unknown or unclassified phyla (44.4% in productive roofs and 33.5% in extensive roofs) (Figure S4). Ascomycota was the dominant phylum (36.9% in productive roofs and 50.1% in extensive ones). Few differences were observed between productive and extensive roofs, and mainly for the 20% minority phyla.

Sixty-five fungal genera were present at least once in the soil samples and 34 were present in all samples. More than 50% of fungi were unknown or unclassified (Figure S4). The most prevalent fungal genus was Lojkania (11.5 in productive roofs and 24.4% in extensive ones).

Batrachochytrium represented 8.9 and 7.0% in productive and extensive roofs respectively.

Many genera were poorly represented (42<0.1%).

A total of 33 559 different fungal OTU was quantified considering all soil samples, corresponding to means of 1359 ± 164 fungal OTUs in productive roofs and 1434 ± 170 in extensive ones. Thirteen and nine OTU were dominant (>1%) in productive and extensive roofs (Table S2). Some of them correspond to unknown fungi. Others were Ascomycota (Hypocrea, Verticillium, Leptosphaeria, Lojkania, or unclassified), Basidiomycota (unclassified) and Chytridiomycota (Batrachochytrium).

The factorial map of the NMDS revealed marked differences between fungal communities between green roof types, along both axes 1 and 2 (Figure 2), and these differences were significant (ANOSIM, P<0.001).

Nematodes

A total of 28 nematodes families was present in the Technosol of the six productive and six extensive green roofs (Table 3, Table S3). The number of families varied from 5 to 18 taxa per roof. Total nematodes abundances in soil samples ranged from circa 820 to 60 700 individuals.kg -1 and were significantly different between extensive and productive green roofs. No significant difference was showed for Shannon and Eveness indices.

The factorial map of the NMDS revealed marked significant differences in nematodes communities between green roof types, along both axes 1 and 2 (Figure 3, ANOSIM, P<0.001). On average, trophic groups of nematodes were mostly composed by bacterivorous (51% in productive green roofs and 80% in extensive green roofs) followed by phytophagous in productive green roofs (35%) or omnivorous and carnivorous in extensive green roofs (12%). Two productive green roofs presented phytoparasites. In terms of frequency, the families Plectidae (100%), Tylenchidae (100%), and Cephalobidae (100%) were also always present on both types of roofs. The most frequent families specific on productive roofs were Rhabditidae (100%), Anguinidaeditylenchus (83%), Apelenchoidi (83%) and

Neodiplogasteridae (83%). The species most frequently observed on extensive roofs was Qudsianematidae (67%).

Collembola

A total of 47 Collembola species was identified in the topsoil of the six productive and six extensive green roofs (Table S4). The number of species varied from 9 to 20 species per roof.

The total Collembola density in topsoil ranged from circa 1 000 to 235 600 individuals.m -2 .

No significant difference was observed for Shannon and Eveness indices for Collembola between green roof types.

The factorial map of the NMDS revealed marked differences in Collembolan communities between the two green roof types, along both axes 1 and 2 (Figure 3), and these differences were highly significant (ANOSIM, P<0.001). Productive and extensive roofs had similar frequencies of Proisotoma notabilis (Schaeffer, 1896) (100%) and Protaphorura armata (Tullberg, 1869) (71%). Some species were specific to green roof types. Megalothorax minimus (Willem, 1900) and Proisotoma minuta (Tullberg, 1871) were the most frequently observed species on productive green roofs (71% for both) while Bourletiella hortensis (Fitch, 1863) and Sminthurinus aureus (Lubbock, 1862) were present on extensive green roofs (100% and 86% respectively).

Plants

A total of 16 cultivated plant species and 48 spontaneous plant species was identified in the six productive and six extensive green roofs (Tables 3,S5, S6). The total number of plant species was not significantly different between green roofs on average. Cultivated plant species were mostly vegetables such as lettuce (present on 50% of productive green roofs) or onion (33%), cucumber, radish, chard (17%) and herbs. Parsley and other herbs were also present exceptionally in extensive green roofs. Concerning spontaneous species, Cerastium glomeratum Thuill was present in 50% of productive green roofs but was absent in extensive green roofs. The most frequent species, belonged to genus Sedum, were present in all extensive green roofs (6 different species), especially Sedum album L. (83%) and Sedum reflexum L. (67%). Due to low frequency, no NMDS was performed on plants communities.

Relationship with soil parameters

In order to study the co-variation between soil chemical, physical and biodiversity parameters in the 12 green roofs, we carried out a co-inertia analysis (Figure 4). The analysis showed a high co-variation between biotic and abiotic parameters (RV = 0.62; Monte-Carlo permutation test, P = 0.003). The two first axes explained 94% of the total inertia. Green roofs were separated in two groups corresponding mainly to the two different types of green roofs with the exception for the extensive roof n°6. Fertility parameters were higher in productive green roofs whereas extensive green roofs presented a higher bulk density. Fertility parameters and metal contents co-varied positively with biodiversity indices except for spontaneous plants species richness. Abundances of microbial communities (bacteria and fungi) and species richness and abundance of nematodes co-varied along the axis 1. Along the axis 2, the species richness of bacteria, fungi and Collembola co-varied positively but negatively with abundance of Collembola and species richness of cultivated plants.

Discussion

Green roofs: a possible refuge for soil biodiversity, less for plants

The first objective of this work was to have a better view of the biological patrimony of urban soils, especially of green roofs by comparing productive and extensive roofs. Despite their nature and artificial composition, our results showed that green roofs harbour a high level of abundance and diversity from microorganisms to micro and mesofauna.

Microorganisms

Microbial molecular biomass in green roofs is higher compare to the average values observed from more than 2000 French soils under grasslands (81 µg DNA g -1 soil), forests (77 µg DNA g -1 soil), crops (38 µg DNA g -1 soil), vineyards and orchards (27 µg DNA g -1 soil) and other soils (gathering natural environments, urban parks…) (94 µg DNA g -1 soil) ( platform GenoSol data -Gis Sol, 2015 in [START_REF] Briat | Les sols et la vie souterraine: des enjeux majeurs en agroécologie[END_REF]. These results are similar to those described in the review by [START_REF] Guilland | Biodiversity of urban soils for sustainable cities[END_REF] where urban soils exhibit a higher microbial biomass than agricultural soils [START_REF] Chen | Key nitrogen cycling processes in pine plantations along a short urban-rural gradient in Nanchang, China[END_REF][START_REF] Enloe | Urbanization effects on soil nitrogen transformations and microbial biomass in the subtropics[END_REF][START_REF] Pavao-Zuckerman | Decomposition of chestnut oak (Quercus prinus) leaves and nitrogen mineralization in an urban environment[END_REF]. However, Mc Guire et al ( 2013) observed higher microbial biomasses in urban parks than in green roofs in New York City. These contradictory results could be explain by the fact that urban soils include a wide variety of types of green spaces (parks, gardens, sports fields, etc.) with contrasted characteristics. Our results showed a higher microbial biomass on productive roofs than on extensive roofs These results could be explained by higher organic carbon and clay contents and higher pH in productive roofs (Table 2) as [START_REF] Horrigue | Predictive model of soil molecular microbial biomass[END_REF], showed that soil organic carbon and clay contents, altitude and pH were the best explanatory variables of soil microbial biomass and explained 60% of the variance for microbial molecular biomass by developing a statistical model.

Unlike fungi, bacteria were significantly more abundant in the soils of productive roofs than in extensive roofs and led to 1.4 times lower fungi/bacteria ratios in productive roofs compared to extensive ones. That indicated that productive roofs have a positive effect on the bacterial populations. For comparison, in the study of Mc Guire et al. ( 2013), fungi/bacteria ratios were of the same order and were lower in urban parks than in green roofs (5.3% vs 8.7% respectively).

Bacterial diversity is influenced by soil pH, soil texture and the quality of organic carbon and is more important in alkaline, coarse-textured soils with low C/N [START_REF] Constancias | Contrasting spatial patterns and ecological attributes of soil bacterial and archaeal taxa across a landscape[END_REF][START_REF] Terrat | La caractérisation des communautés microbiennes du sol à l'échelle de la France pour évaluer l'effet de l'usage des sols[END_REF]. Bacterial diversity is also influenced by land use, with lower levels of biodiversity in grasslands and forests (2103 and 1897 taxa, respectively) than in cultivated or vineyard soils (2181 and 2215 taxa, respectively) in soils from the RMQS campaign [START_REF] Terrat | La caractérisation des communautés microbiennes du sol à l'échelle de la France pour évaluer l'effet de l'usage des sols[END_REF]. These trends are the opposite of those observed for microbial molecular biomass, where agricultural soils had a lower microbial molecular biomass compared to natural environments (Dequiedt et al., 2011). Similarly, in green roof Technosols, we did not observe a correlation between the microbial biomass (higher than in soils of the RMQS campaign) and the bacterial diversity (lower than in soils of the RMQS campaign).

Green roofs also harbour a large diversity of fungi with an average number of OTU per roof Although the average level of bacterial and fungal diversity was similar in the two roof types, the structure of the communities at the OTU scale was different, as shown by the NMDS analyses. Soil abiotic characteristics such as concentrations of available nutrients and bulk density may explain differences of communities between roofs (Mc Guire et al., 2013).

It is also noted that knowledge on the origin of microorganisms in green roofs remains low.

Maybe, the origin of substrate could play a major role and explain differences of results between studies.

To summary, microbial characteristics of green roofs are different from soils with other land uses. Furthermore soils of productive and extensive roofs differ both in the quantity of microorganisms they harbour, especially bacteria, and in the type of microorganisms present, with bacterial and fungal diversity specific to a roof type, due to soil characteristics.

Nematodes

The abundance of nematodes in extensive roofs was similar to those observed in cultivated soils (1.10 4 to 1.3 10 4 individuals per kg of soil) [START_REF] Cluzeau | Integration of biodiversity in soil quality monitoring: Baselines for microbial and soil fauna parameters for different land-use types[END_REF][START_REF] Ponge | The impact of agricultural practices on soil biota: A regional study[END_REF]. In productive roofs, it was in contrast, higher than in soils with other land uses. Different studies presented in the review of [START_REF] Guilland | Biodiversity of urban soils for sustainable cities[END_REF] reported a similar [START_REF] Park | Anthropogenic influence of urban development on the soil nitrogen fixing bacteria, nematode community, and nutrient pools[END_REF] or a lower [START_REF] Li | Nematode community structure as bioindicator of soil heavy metal pollution along an urban rural gradient in southern Shenyang, China[END_REF][START_REF] Pavao-Zuckerman | Decomposition of chestnut oak (Quercus prinus) leaves and nitrogen mineralization in an urban environment[END_REF] abundance of nematodes in urban soils compared to agricultural soils in relation to heavy metals concentrations in soil.

Bacteriovorous nematodes dominated communities in green roof soils. Such results are common in young constructed soils [START_REF] Villenave | Rapid Changes in Soil Nematodes in the First Years after Technosol Construction for the Remediation of an Industrial Wasteland[END_REF] such as greens roofs (under the age of 6 in our study). Young industrial Technosols are first dominated by bacteriovorous and then by phytophagous nematodes [START_REF] Cortet | Biodiversité et fonctionnement d'un Technosol construit utilisé dans la restauration de friches industrielles[END_REF] that correlate with the higher bacterial density. [START_REF] Guilland | Biodiversity of urban soils for sustainable cities[END_REF] reported that abundance of carnivorous and omnivorous nematodes was low in urban soils and that the ratio between fungivorous and microbivorous nematodes was significantly lower in urban environments than non-urban land uses, indicating that organic matter decomposition was mainly ensured by bacteria [START_REF] Ferris | A framework for soil food web diagnostics: extension of the nematode faunal analysis concept[END_REF].

In brief, nematodes communities are higher and different in productive compared to extensive soils.

Collembola

The density of Collembola in green roofs was higher to those observed in agricultural (9810 ind.m -2 ) [START_REF] Cluzeau | Integration of biodiversity in soil quality monitoring: Baselines for microbial and soil fauna parameters for different land-use types[END_REF] but also in some urban and industrial soils [START_REF] Joimel | Urban and industrial land uses have a higher soil biological quality than expected from physicochemical quality[END_REF]. For comparison, Rumble and Gange (2013) quantified a mean of 19.10 3 individuals per m² in extensive roofs. [START_REF] Guilland | Biodiversity of urban soils for sustainable cities[END_REF] explained the high abundance of Collembola in green roofs by the absence of earthworms and the low abundance of predatory macroarthropods such as spiders or ground beetles.

Despite their greater abundance in green roofs, several authors observed less diversity of Collembola in green roofs compared to other urban soils such as parks, residential gardens, lawns [START_REF] Joimel | Urban and industrial land uses have a higher soil biological quality than expected from physicochemical quality[END_REF][START_REF] Rzeszowski | The effect of soil nutrient gradients on Collembola communities inhabiting typical urban green spaces[END_REF][START_REF] Santorufo | An assessment of the influence of the urban environment on collembolan communities in soils using taxonomy-and trait-based approaches[END_REF] likely because of habitat fragmentation. Nevertheless, in our study we identified 5 times more species than [START_REF] Rumble | Soil microarthropod community dynamics in extensive green roofs[END_REF] in extensive roofs of London (6 species recorded). It could be due to different origins of substrates in cities, especially compost which have been demonstrated as a way of colonisation for Collembola [START_REF] Joimel | Are Collembola "flying" onto green roofs?[END_REF]. Conversely, mean species richness (12 and 14.5 in productive and extensive roofs respectively) was generally greater than previous studies at regional scales comparing different land uses, ranging from 3.4 to 14 [START_REF] Cluzeau | Integration of biodiversity in soil quality monitoring: Baselines for microbial and soil fauna parameters for different land-use types[END_REF].

While little significant differences in biodiversity levels were demonstrated between green roofs, we observed a strong differentiation of communities as a function of green roof type as shown by NMDS. Parisotoma notabilis is, as frequently reported in literature, the most common species. Some Collembola species are specific to green roof types. Extensive green roofs harbour epiedaphic Collembola as B. hortensis and S. aureus whereas eudaphic Collembola are most present in productive ones, due perhaps to their ecological preferences.

The Collembola species identified in the green roofs were, for most of them, different from those detected in extensive roofs by [START_REF] Rumble | Soil microarthropod community dynamics in extensive green roofs[END_REF] with the exception of Proisotoma minuta but this species was in our case only detected in productive roofs.

To summary, green roofs harbour high level of abundances and diversity of Collembola with strong differentiation of communities between productive and extensive roofs.

Plants

With only 65 species, green roofs were not the most favourable environments for the establishment of native species, compared to poorly anthropised ecosystems which present up to 600 species (Muratet et al., 2008).

In conclusion, few cultivated plant species have colonized the extensive roofs but in contrast, spontaneous plants abundance was limited in productive roofs by the planting of vegetables and weeding practices.

Co-occurrence between organisms and links with green roofs characteristics

Soil physical and chemical characteristics are described to be major drivers of the abundance and composition of plants and soil organisms, including in urban areas (e.g. [START_REF] Huot | Diversity and activity of soil fauna in an industrial settling pond managed by natural attenuation[END_REF][START_REF] Milano | The effect of urban park landscapes on soil Collembola diversity: A Mediterranean case study[END_REF][START_REF] Santorufo | Responses of functional and taxonomic collembolan community structure to site management in Mediterranean urban and surrounding areas[END_REF].

In this study, the abundance and diversity of the different groups of organisms are differently explained by soil parameters, especially for plants and soil organisms. Plants were mostly correlated with bulk density whereas soil biodiversity was linked to metal contents and soil fertility components (organic C, pH, water retention). Yet, previous studies have highlighted the importance of soil characteristics including water availability and organic matter contents for the assemblage of plant communities in urban soils [START_REF] Dana | Urban vegetation of Almera City -a contribution to urban ecology in Spain[END_REF]. In cities, specific constraints such as climatic conditions and human management of productive roofs can have an impact on floristic diversity, in addition to soil characteristics [START_REF] Joimel | Contrasting homogenization patterns of plant and collembolan communities in urban vegetable gardens[END_REF]. We showed that plant diversity was not correlated to soil biodiversity. Previous studies showed contrasted results with absence [START_REF] Mcguire | Digging the New York City Skyline: Soil Fungal Communities in Green Roofs and City Parks[END_REF][START_REF] Rumble | Soil microarthropod community dynamics in extensive green roofs[END_REF] or presence of a correlation in urban parks and green roofs. [START_REF] Rumble | Soil microarthropod community dynamics in extensive green roofs[END_REF] explained these different results by the age of roofs, the diversity of plants, the climate or sampling season.

We may also propose that human management may have less effects on microbial diversity.

Secondly, a strong co-variation between biodiversity and soil characteristics was observed that allowed to distinguish the two types of green roofs. The studied soil characteristics reflected the green roofs functions, either productive or ornamental. Technosols of the studied productive green roofs were rich in organic matter, even higher than other urban agricultural soils such as vegetable gardens [START_REF] Joimel | Physico-chemical characteristics of topsoil for contrasted forest, agricultural, urban and industrial land uses in France[END_REF]. In contrast, Technosols of extensive green roofs were among the urban soils with the lowest fertility. The studied green roofs had low concentrations in metals compared to those classically observed in urban soils [START_REF] Joimel | Physico-chemical characteristics of topsoil for contrasted forest, agricultural, urban and industrial land uses in France[END_REF]. Results showed that some indicators such as abundance of Collembola and species richness of nematodes were rather influenced by soil fertility parameters and others such as abundance of microorganisms and nematodes or diversity of microorganisms and Collembola, by metal contents. Some of these results are consistent with the literature while others are more unexpected. Indeed, many studies reported the effects of soil pH, organic carbon and metal contents on Collembola [START_REF] Joimel | Contrasting homogenization patterns of plant and collembolan communities in urban vegetable gardens[END_REF][START_REF] Santorufo | Responses of functional and taxonomic collembolan community structure to site management in Mediterranean urban and surrounding areas[END_REF] and microorganisms [START_REF] Hui | Lead (Pb) contamination alters richness and diversity of the fungal, but not the bacterial community in pine forest soil[END_REF][START_REF] Huot | Diversity and activity of soil fauna in an industrial settling pond managed by natural attenuation[END_REF][START_REF] Wang | Sensitivities to nitrogen and water addition vary among microbial groups within soil[END_REF] in non-urban soils.

The distribution of molecular microbial biomass at the scale of France was mainly explained by the soil physico-chemical characteristics (Dequiedt et al, 2011). Concerning nematodes, results from [START_REF] Villenave | Rapid Changes in Soil Nematodes in the First Years after Technosol Construction for the Remediation of an Industrial Wasteland[END_REF] showed that the high organic matter content of Technosols and its positive effect on bacterial populations, ensured nematode colonization and progressive diversification. Soil fauna such as Collembola is also correlated to physical parameters related to porosity and aggregation because they preferentially live in air-filled soil pores [START_REF] Wall | Soil Ecology and Ecosystem Services[END_REF]. Moreover, the correlation we observed between metal contents and biodiversity was positive, contrary to previous studies [START_REF] Fountain | A Comparative Study of the Effects of Metal Contamination on Collembola in the Field and in the Laboratory[END_REF]), which we explain by the low concentrations of trace metals in the roof top soils. Finally, no relation between abundance and diversity of the different groups of soil organisms was observed.

The relationships between organic matter and biodiversity, with generally a higher diversity in productive green roofs, which present the higher contents in organic matter, are in line with the conclusion that organic matter content is likely to be the key for successful biodiversity colonisation [START_REF] Villenave | Rapid Changes in Soil Nematodes in the First Years after Technosol Construction for the Remediation of an Industrial Wasteland[END_REF].

Substrate characteristics were also shown to have an influence on the development of the plant cover, especially in extensive green roofs [START_REF] Emilsson | Vegetation development on extensive vegetated green roofs: Influence of substrate composition, establishment method and species mix[END_REF].

Our statistical analyses did not take into account difference of soil depth. It is necessary to note that thickness is different between green roofs: 8 to 10 cm-depth for extensive green roof and 20 to 30 cm-depth for productive green roofs. Previous studies demonstrated that depth soil could influence soil organisms such as nematodes [START_REF] Hafeez | Taxonomic and functional characterization of microbial communities in Technosols constructed for remediation of a contaminated industrial wasteland[END_REF]. However, these conclusions were drawn from deeper Technosols (85 cm) and did not differentiate the first layer (0-15 cm). Moreover, in this study, the soil layers of the green roofs were homogenous and then comparable. So, it is possible that in our case the depth does not play a role in the observed groups. However, the depth may play a role for other taxonomic groups such as earthworms.

To summary on this part, the physical and chemical characteristics of Technosols play a major role in the pattern of soil biodiversity observed in our study.

Synthetic approach

The responses of each studied taxonomic group to the green roofs type were synthetized in a radarplot (Fig. 5). Productive green roofs exhibited, for each group of soil biodiversity, the higher values of abundance and density. They also harboured a higher cultivated plants and nematodes richness. In contrast, extensive green roofs had the higher values of diversity for microorganisms, Collembola and spontaneous plants.

These results highlight the importance of considering different indicators (e.g. abundance vs.

diversity) and different soil organisms to answer a given question. Soil fauna, which usually discriminate less land uses than microorganisms [START_REF] Cluzeau | Integration of biodiversity in soil quality monitoring: Baselines for microbial and soil fauna parameters for different land-use types[END_REF], were very sensitive here to the green roofs type.

In the present study, we only recorded the state of biodiversity at a given point in time.

Moreover, these Technosols are young soils and these findings will need to be re-examined in the long term to investigate the sustainability of these systems. Colonization by organisms from nearby environments may still be limited in young soils and could change over time and contribute to an increase in biodiversity of older green roofs. In future studies, it would also be important to carry out several sampling during the first stages of these young soils in order to assess the biodiversity dynamics.

Conclusion

Green roofs provide habitats for biodiversity instead of impervious surfaces in the cities and host in the first years a high abundance of soil organisms.

  that was 10 times higher than those observed by Mc Guire et al. (2013) in green roofs of New York. The comparison of taxonomic affiliations at the phylum and genera levels highlighted differences in fungal diversity between our study and works of the Mc Guire et al. (2013).

Figure 1 .Figure 2 .

 12 Figure 1. Microbial biomass in the 12 green roofs surveyed. Lower case letters indicate

Figure 3 .

 3 Figure 3. Graphic displaying the first two axes of the non-metric multidimensional scaling

Figure 4 .

 4 Figure 4. Co-inertia analysis of geochemical (circles) and biodiversity (triangles) parameters

Figure 5 .

 5 Figure 5. Radar plot illustrating mean abundance and diversity of microbial, faunal and plant

Table 1 .

 1 But the characteristics of above and below ground biodiversity vary in the different green roof types corresponding to different Technosols with their own characteristics, depending on harsh conditions in extensive ones, and human interventions in productive ones. Characteristics of 6 extensive and 6 productive green roofs

	The contrasted abundance and composition of biotic communities between green roofs type
	may impact on soil functioning.

Table 2 .

 2 Physical and chemical soil characteristics (mean ± standard deviation) of extensive and productive green roofs surveyed. Lower case letters indicate significant differences between green roofs type (alpha =0.05)

	Soil parameters	Unit	Extensive green roofs	Productive green roofs
	pH	-	7.3 b ± 0.25	7.6 a ± 0.10
	C	mg.kg -1	7.8 b ± 6.5	21.0 a ± 1.7
	Density	g.cm -3	0.8 a ± 0.2	0.4 b ± 0.2
	Water holding capacity (pF 2.3)	g H 2 O . g dry soil	0.14 b ± 0.1	0.75 a ± 0.4
	Cd		0.2 a ± 0.3	0.4 a ± 0.1
	Cu Pb	mg.kg -1	28.5 a ± 20.0 55.9 a ± 42.0	41.1 a ± 8.8 56.2 a ± 15.2
	Zn		226.7 a ± 202.2	251.2 a ± 122.8

Table 3 .

 3 Taxonomic diversity indices (mean +/-standard deviation) of microbial, nematodes, collembolan and plant communities in the 12 green roofs surveyed. Lower case letters indicate significant differences among green roofs type (alpha = 0.05)

	Extensive (n=6) Productive (n=6)

Acknowledgements

This work would not have been possible without the involvement of Paris stakeholders in urban agriculture. We are especially thankful to the Espace association, RATP, Culture en Ville, Topager and Veni Verdi, who welcomed us in all their green roofs. The authors wish also to thank Cécile Villenave from ELISOL for nematodes identification, Alain Rakoto for his major contribution on the microarthropods extraction, Valérie Pouteau from technical staff and Quentin Vincent for his help on script R. This work was supported by the Labex BASC ("TROL" project). This work, through the involvement of technical facilities of the GenoSol platform (https://doi.org/10.15454/L7QN45) of the infrastructure ANAEE-Services, received a grant from the French state through the National Agency for Research under the program "Investments for the Future" (reference ANR-11-INBS-0001), as well as a grant from the Regional Council of Bourgogne Franche Comté. The BRC GenoSol is a part of BRC4Env (https://doi.org/10.15454/TRBJTB), the pillar "Environmental Resources" of the Research Infrastructure AgroBRC-RARe.

The raw data of 16S and 18S sequencing were deposited to SRA under the following bioproject PRJNA726534 and analysed results with GnS-PIPE are available (https://nextcloud.inrae.fr/s/8Ttr4sFDyPY3tDA)


R Ravaud

G Lemarquand

V Lemarquand

C Depollier

Torque in PM Couplings: Comparison of Uniform and Radial Magnetization

Keywords: Magnetic coupling, Tile permanent magnet, Torque, Uniform magnetization, Radial magnetization, Three-dimensional calculation. I

We present a three-dimensional study of the torque transmitted between tile permanent magnets uniformly magnetized. All this study is based on the Coulombian model. The torque is calculated semianalytically by considering all the surface densities that appear on the tiles. In addition, no simplifying assumptions are done in the expressions given in this paper. Consequently, the evaluation of the torque is very accurate and allows us to show the drawbacks of using tile permanent magnets uniformly magnetized instead of using tile permanent magnets radially magnetized. Such an approach is useful because it allows us to realize easily parametric studies.

way the torque is transmitted between tile permanent magnets uniformly magnetized. Such a model allows us to realize easily parametric studies and thus to know when the use of tile permanent magnets radially magnetized is necessary or not. Indeed, the angular width has a great influence on the radial field created by a tile permanent magnet uniformly magnetized. As this radial field is not perfectly symmetrical, the torque transmitted is modified. Therefore, the aim of this paper is to determine precisely with a three-dimensional model how the torque transmitted between tiles changes according to the way the tiles are magnetized.

Many authors have studied magnetic couplings [START_REF] Yonnet | Permanent magnet bearings and couplings[END_REF]- [START_REF] Elies | Analytical study of radial stability of permanent magnet synchronous couplings[END_REF] and structures using permanent magnets [START_REF] Yonnet | Passive magnetic bearings with permanent magnets[END_REF]- [START_REF] Moser | Optimization of repulsive passive magnetic bearings[END_REF].

Historically, we can say that the first models used to study magnetic couplings were the two-dimensional models [START_REF] Bancel | Three-dimensional analytical optimization of permanent magnets alternated structure[END_REF]- [START_REF] Abele | Generation of uniform high fields with magnetized wedges[END_REF]. The main reason lies in the fact that a two-dimensional approach is fully analytical and allows us to make an easy parametric optimization of the permanent magnet dimensions. However, this 2D-approach is not valid when we determine the magnetic field far from one magnet [START_REF] Ravaud | Analytical calculation of the magnetic field created by permanent-magnet rings[END_REF].

Tree-dimensional approaches seem to be more difficult to realize parametric studies because they are not fully analytical. In fact, the calculation of the radial and axial magnetic field components created by tile permanent magnets is not strictly analytical but is necessary based on special functions [START_REF] Conway | Inductance calculations for noncoaxial coils using bessel functions[END_REF]- [START_REF] Durand | Electrostatique[END_REF].

More generally, all the semi-analytical or analytical approaches used by several authors allow manufacturers to optimize devices using permanent magnets [START_REF] Ohji | Performance of repulsive type magnetic bearing system under nonuniform magnetization of permanent magnet[END_REF]- [START_REF] Aydin | Minimization of cogging torque in axial-flux permanent-magnet machines: design concepts[END_REF].

This paper presents a semi-analytical approach based on the Coulombian model for calculating precisely the torque transmitted between tile permanent magnets uniformly magnetized. This semi-analytical approach is three-dimensional. We explain in the second section how this problem can be solved. Then, we present a semi-analytical expression of the torque transmitted between two tile permanent magnets uniformly magnetized. Eventually, we present the main drawbacks of tile permanent magnets uniformly magnetized.

II. MODELING TILE PERMANENT MAGNETS WITH THE COULOMBIAN MODEL

This section presents the geometry studied and the model used for the modeling of the torque transmitted between two tile permanent magnets uniformly magnetized.

A. Notation and geometry

The geometry considered and the related parameters are shown in Fig 1 . A two-dimensional representation of the geometry is shown in Fig 2 . We consider here two tile permanent magnets uniformly magnetized. The inner radius of the outer tile is r in1 and its outer one is r out1 . Its height is z bz a and its angular width is θ 2θ 1 . The inner radius of the inner tile is r in2 and its outer one is r out2 . Its height is z dz c and its angular width is θ 4θ 3 . The two magnetic polarization vectors J 1 and J 2 are expressed as follows:

J 1 = -cos( θ 2 + θ 1 2 ) u r -sin( θ 2 + θ 1 2 ) u θ (1)
J 2 = -cos( θ 4 + θ 3 2 ) u r -sin( θ 4 + θ 3 2 ) u θ (2)
The magnetic field created by tile permanent magnets can be obtained by using the Coulombian Model.

Indeed, a permanent magnet can be represented by a magnetic pole surface density that is located on the faces of the magnet and a magnetic pole volume density that is located inside the magnet. In the configuration presented in Fig 1, the magnetic pole volume density is 0 because the magnetic polarizations of the magnets are uniform. Consequently, each tile permanent magnet is modelled by its magnetic pole surface density, which is determined as follows: by denoting n ij , the unit normal vector of the face i of

J 1 . n 1 J 1 cos(θ -θ1+θ2 2 ) σ * 12 J 1 . n 2 J 1 sin( θ2-θ1 2 ) σ * 13 J 1 . n 3 -J 1 cos(θ -θ1+θ2 2 ) σ * 14 J 1 . n 4 J 1 sin( θ2-θ1 2 ) σ * 25 J 2 . n 5 J 2 cos(θ -θ3+θ4 2 ) σ * 26 J 2 . n 6 J 2 sin( θ4-θ3 2 ) σ * 27 J 2 . n 7 -J 2 cos(θ -θ3+θ4 2 ) σ * 28 J 2 . n 8 J 2 sin( θ4-θ3
2 )

TABLE I DEFINITION OF THE MAGNETIC POLE SURFACE DENSITIES LOCATED ON THE MAGNETS

the magnet j, the corresponding surface density σ * ij is determined with (3).

σ * ij = J j . n ij (3)
Thus, four faces of the two magnets are charged with a magnetic pole surface density. All the surface density calculations are represented in Table I. The torque transmitted between two tile permanent magnets can be determined in two steps. The first step consists in calculating the magnetic field created by one tile permanent magnet. The second step consists in integrating the magnetic field created by the first tile on the second one. Let us first consider the tile permanent magnet located on the right of Fig 1

. By using the

Coulombian approach, we can write that the azimuthal field created by this tile is expressed as follows:

H θ (r, θ, z) = J 1 4πµ 0 θ2 θ1 z b za cos(θ - θ 1 + θ 2 2 ) u(r in1 ) | u(r in1 )| 3 r in1 d θdz - J 1 4πµ 0 θ2 θ1 z b za cos(θ - θ 1 + θ 2 2 ) u(r out1 ) | u(r out1 )| 3 r out1 d θdz + J 1 4πµ 0 rout1 rin1 z b za sin( θ 2 -θ 1 2 ) v(θ 2 ) | v(θ 2 )| 3 drdz + J 1 4πµ 0 rout1 rin1 z b za sin( θ 2 -θ 1 2 ) v(θ 1 ) | v(θ 1 )| 3 drdz (4)
where

u(x) = (r -x cos(θ -θ)) u r -x sin(θ -θ) u θ + (z -z) u z (5)
and

v(y) = (r -r cos(θ -y) u r -r sin(θ -y) u θ + (z -z) u z (6)
The next step is thus to express the torque transmitted to the second tile permanent magnet uniformly magnetized (as shown in Fig 1). By using (4), the torque T θ can be determined as follows:

T θ = J 1 J 2 4πµ 0 θ4 θ3 z d zc r in2 H(r in2 , θ, z)r in2 cos( θ - θ 3 + θ 4 2 )d θd z - J 1 J 2 4πµ 0 θ4 θ3 z d zc r out2 H(r out2 , θ, z)r out2 cos( θ - θ 3 + θ 4 2 )d θd z + J 1 J 2 4πµ 0 rout2 rin2 z d zc rH( r, θ 4 , z) sin( θ 4 -θ 3 2 )d rd z + J 1 J 2 4πµ 0 rout2 rin2 z d zc rH( r, θ 3 , z) sin( θ 4 -θ 3 2 )d rd z (7)

B. Semi-analytical Expression of the Torque

The torque transmitted between two tile permanent magnets can be written as follows:

T θ = J 1 J 2 4πµ 0 θ2 θ1 θ4 θ3 dT (1) θ + T (2) θ + z b za dT (3) θ + θ2 θ1 dT (4) θ (8)
θ 4 -θ 3 =θ 2 -θ 1 = π 12 , r in1 = 0.025m, r out1 = 0.028m, r in2 = 0.021m, r out2 = 0.024m, z d -zc = 0.003m, z b -za = 0.003m, za = 0.001m, r = 0.024m, J 1 = J 2 = 1T
where dT

(1) θ , T (2) θ , dT (3)
θ and dT (4) θ are given in the appendix. Strictly speaking, this expression is threedimensional and we take into account all the contributions between the two tiles. However, it is noted that all the contributions have not the same weight and the interaction between the surface densities located on the inner and outer faces of each tile permanent magnet is in fact the most important. Furthermore, it is noted that this expression could probably still simplified and led to incomplete elliptical integrals of the first, second and third kind. In addition, we use the Cauchy principal value for determining the singular cases that appear for example when one tile permanent magnet is exactly in front of the other one.

III. COMPARISON OF THE TORQUE TRANSMITTED BETWEEN TWO TILES RADIALLY MAGNETIZED AND TWO TILES UNIFORMLY MAGNETIZED

Tiles uniformly magnetized are generally less efficient than tiles radially magnetized for magnetic couplings. However, this loss of torque depends greatly on the tile angular width. Therefore, it is interesting to determine this loss of torque for different tile angular widths. For this purpose, we represent in the same figure the torque transmitted between two tiles radially magnetized and the torque transmitted between two tiles uniformly magnetized. On Figure 3, the angular width of each tile is the same ( Fig. 7. Representation of the mean loss of efficiency of the torque transmitted between two tiles uniformly magnetized versus the angular width of the tiles. This loss of torque is calculated in comparison with the torque transmitted between two tiles radially magnetized. The points correspond to the figures 3, 4, 5 and 6. We have : r in1 = 0.025m, r out1 = 0.028m, r in2 = 0.021m,

; θ 4 -θ 3 =θ 2 -θ 1 = π 3 , r in1 = 0.025m, r out1 = 0.028m, r in2 = 0.021m, r out2 = 0.024m, z d -zc = 0.003m, z b -za = 0.003m, za = 0.001m, r = 0.024m, J 1 = J 2 = 1T Π 12
r out2 = 0.024m, z b -za = z d -zc = 0.003m, J 1 = J 2 = 1T.
4, 5 and 6 show that the torque transmitted between two tile permanent magnets radially magnetized does not depend very much on their angular widths whereas it decreases greatly with the increase in the angular width when the tiles are uniformly magnetized.

We can also represent this decrease by calculating its rate of loss versus the angular width of tiles (Fig 7).

Figure 7 is consistent with Figs 3, 4, 5 and 6. Indeed, when the angular width of the tile permanent magnets tends to zero, we can use either tiles radially magnetized or tiles uniformly magnetized. This is the only case in which the choice has not a great importance. Consequently, as tiles uniformly magnetized are cheaper to fabricate, their use is more judicious. However, when the angular width of the tiles used becomes greater, the choice is certainly more difficult and other considerations must be taken into account.

For example, if the first property required in a coupling is really the value of the torque transmitted, the magnetization of the tiles should be radial and not uniform.

IV. APPLICATION : ALTERNATE MAGNET STRUCTURES USING TILES RADIALLY OR UNIFORMLY

MAGNETIZED

We can illustrate the expression established in the previous section by studying the torque transmitted in an alternate magnet structure with 8 tile permanent magnets on each rotor. For this purpose, we use the principle of superposition with [START_REF] Lemarquand | Ironless loudspeakers[END_REF] for the calculation of the torque transmitted between the leading The tiles are either radially magnetized (dashed line) or uniformly magnetized (thick line) r in1 = 0.025m, r out1 = 0.028m,

r in2 = 0.021m, r out2 = 0.024m, z b -za = z d -zc = 0.003m, J 1 = J 2 = 1T
rotor and the lead rotor. We represent this torque versus the shifting angle θ in Fig 8.

Figure 8 is consistent with the previous representations of the torque transmitted between two tiles radially or uniformly magnetized. First, we note that the torque transmitted in the alternate magnet structure is sixteen times greater than the one transmitted between only two tile permanent magnets. Then, we note that the torque transmitted between tiles uniformly magnetized is smaller than the one transmitted between tiles radially magnetized, which is still consistent with Fig 4.

V. CONCLUSION

This paper has presented a three-dimensional expression of the torque transmitted between two tile permanent magnets uniformly magnetized. Then, we have compared the torque transmitted between two tiles uniformly magnetized with two tiles radially magnetized. An alternate magnet structure has been studied to illustrate the three-dimensional expression of the torque transmitted between tiles radially or uniformly magnetized. Some theoretical results have been analytically determined. First, tiles uniformly magnetized are less interesting than tiles radially magnetized for realizing couplings. Indeed, the more the tile angular widths are important, the less the torque transmitted between two tiles uniformly magnetized is important in comparison with the torque transmitted between two tiles radially magnetized. However, the cost of the magnets must be taken into account. Tiles uniformly magnetized are easier to fabricate than tiles radially magnetized (and thus cheaper...).

APPENDIX

We give the expressions of the parameters used for calculating the torque transmitted between two tile permanent magnets uniformly magnetized. The torque can be expressed in terms of semi-analytical expressions using one or two numerical integrations or not.

T θ = J 1 J 2 4πµ 0 θ2 θ1 θ4 θ3 dT (1)
T θ . These expressions are expressed as follows:

A[r 1 , r 2 ] = z b za z d zc -r 2 1 r 2 2 (r 2 1 + r 2 2 -2r 1 r 2 cos( θ -θ) + ( z -z) 2 ) 3 2 dzd z (14) B[θ i , θ j ] = rout2 rin2 z d zc rout1 rin1 z b za -r r sin(θ j -θ i ) (r 2 + r2 -2r r cos(θ j -θ i ) + ( z -z) 2 ) 3 2 drdzd rd z (15)
C[θ i , r i ] = rout1 rin1 z b za z d zc -r 2 i r sin( θ -θ i ) (r 2 + r 2 i -2rr i cos( θ -θ i ) + ( z -z) 2 ) 3 2 drdzd z (16)
D[θ j , r j ] = rout2 rin2 z d zc z b za -r 2 j r sin(θ j -θ i ) ( r2 + r 2 j -2r j ˜r cos(θ j -θ) + ( z -z) 2 ) 3 2 dzd rd z (17)
The torque transmitted between two tile permanent magnets could be calculated directly by numerical means with [START_REF] Samanta | Magnetic bearing configurations: Theoretical and experimental studies[END_REF]. However, the computational cost would be too long. Therefore, we give here four reduced semi-analytical expressions of A[r 1 , r 2 ], B[θ i , θ j ], C[θ i , r i ] and D[θ j , r j ]. We obtain:

A[r 1 , r 2 ] = A (1) [z a , z c ] -A (1) [z b , z c ] -A (1) [z a , z d ] + A (1) [z b , z d ] (18)
with

A (1) [z i , z j ] = r 2 1 r 2 2 r 2 1 + r 2 2 + (z i -z j ) 2 -2r 1 r 2 cos( θ -θ) r 2 1 + r 2 2 -2r 1 r 2 cos( θ -θ) (19)
B[θ i , θ j ] = rout2 rin2 -B (1) [z a -z c , r out1 ] + B (1) [z b -z c , r out1 ] d r + rout2 rin2 +B (1) [z a -z d , r out1 ] -B (1) [z b -z d , r out1 ] d r + rout2 rin2 B (1) [z a -z c , r in1 ] -B (1) [z b -z c , r in1 ] d r + rout2 rin2 -B (1) [z a -z d , r in1 ] + B (1) [z b -z d , r in1 ] d r (20)
with

B[y, r i ] = r r2 -2r rx + y + r2 x log r -rx + r2 -2r rx + y + ri(-x + √ -1 + x 2 r2 -y) 2 √ -1 + x 2 log[A] + ri(x + √ -1 + x 2 ) r2 -y 2 √ -1 + x 2 log[B] (21)
A = 2i(r r(-1 + x 2 ) + r2 (xx 3 + x 2 √ -1 + x 2 )) (-x + √ -1 + x 2 )(r + r(-x + √ -1 + x 2 ))( r2y)

(x + √ -1 + x 2 )(-r + r(x + √ -1 + x 2 ))( r2 -y) 3 2 (23)
x = cos(θ iθ j )

C[θ i , r i ] = rout1 rin1

C (1) [θ i , r i ]dr

C (1) [θ i , r i ] = r i r 2 i + r2 + (z c -z) 2 -2r i r cos(θ 3θ i )

-
D[θ j , r j ] = rout2 rin2 D (2) (z b , z c ) -D (2) (z a , z c ) + D (2) (z a , z d ) -D (2) (z b , z d ) d r

with D (2) (x, y) = sin(θ iθ j )r 2 j r r 2 j + r2 + (xy) 2 -2r j r cos(θ jθ i ) r 2 j + r2 -2r j r cos(θ jθ i )

Fig. 1 .

 1 Fig. 1. Representation of the geometry considered in three-dimensions : two tile permanent magnets uniformly magnetized

Fig. 2 .

 2 Fig. 2. Representation of the geometry considered in two dimensions : two tile permanent magnets uniformly magnetized

Fig. 3 .

 3 Fig.3. Representation of the torque transmitted between two tiles ; thick line: the magnetization is uniform, dashed lines: the magnetization is radial; θ 4 -θ 3 =θ 2 -θ 1 = π 12 , r in1 = 0.025m, r out1 = 0.028m, r in2 = 0.021m, r out2 = 0.024m, z dzc = 0.003m, z bza = 0.003m, za = 0.001m, r = 0.024m, J 1 = J 2 = 1T

π 12 ), π 4 and π 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 1243456 Fig.[START_REF] Elies | Analytical study of radial stability of permanent magnet synchronous couplings[END_REF]. Representation of the torque transmitted between two tiles ; thick line: the magnetization is uniform, dashed lines: the magnetization is radial; θ 4 -θ 3 =θ 2 -θ 1 = π 6 , r in1 = 0.025m, r out1 = 0.028m, r in2 = 0.021m, r out2 = 0.024m, z dzc = 0.003m, z bza = 0.003m, za = 0.001m, r = 0.024m, J 1 = J 2 = 1T

Fig. 8 .

 8 Fig.8. Representation of the total torque transmitted between an eight tile rotor and an eight tile stator versus the angle θ. The tiles are either radially magnetized (dashed line) or uniformly magnetized (thick line) r in1 = 0.025m, r out1 = 0.028m, r in2 = 0.021m, r out2 = 0.024m, z bza = z dzc = 0.003m, J 1 = J 2 = 1T

1 + 2 + 2 √ - 1 +

 1221 x 2 (-y + i r2y r2 -2r rx + y) -1 + x 2 ) -2i r2 x(-1 + x 2 + x √ -1 + x 2 ) (x + √ -1 + x 2 )(-r + r)(x + √ -1 + x 2 )( r2y) 3 x 2 (iy + r2y r2 -2r rx + y)

r i r 2 i

 2 + r2 + (z d -z) 2 -2r i r cos(θ 3θ i ) -r i r 2 i + r2 + (z c -z) 2 -2r i r cos(θ 4θ i ) +r i r 2 i + r2 + (z d -z) 2 -2r i r cos(θ 4θ i ) +r i (z c -z) log -z c + z + r 2 i + r2 + (z c -z) 2 -2r i r cos(θ 3θ i ) -r i (z d -z) log -z d + z + r 2 i + r2 + (z d -z) 2 -2r i r cos(θ 4θ i ) +r i (z d -z) log -z d + z + r 2 i + r2 + (z d -z) 2 -2r i r cos(θ 3θ i ) -r i (z cz) log -z c + z + r 2 i + r2 + (z cz) 2 -2r i r cos(θ 3θ i )


Arnault Lassin
email: a.lassin@brgm.fr

Pedro Felipe

Santos

Marion Ducousso

Pierre Cézac

Laurent André

Experimental measurements and modelling of thermodynamic properties of NaCl-CaCl2 aqueous solutions up to high temperatures and CO2 pressures

Keywords: Brines chemistry, thermodynamics, geochemical modelling, CO2 solubility

Geothermal energy currently uses hot deep brines. However, their exploitation can raise issues specifically due to their high salinity and the variations of temperature and pressure during their pumping. Main disorders are the potential degassing of fluids, including CO2, but also the precipitation of minerals in the facilities (production/injection wells, heat exchangers…) involving expensive maintenance works.

The chemical and physical properties of the geothermal fluids are key control factors, determining the heatcarrying potential for energy transfer over the lifetime of the well. An improved understanding of the properties of these fluids is therefore necessary to avoid exploitation issues as mentioned above, and even optimize site developments and operations. Additionally, the future for exploration, prediction and utilization of novel geothermal technologiesnamely enhanced geothermal systems and supercritical resourcesis intimately tied to the understanding of the physical and chemical properties of the reservoir fluids.

The REFLECT project aims to improve the accuracy and consistency of key thermodynamic and kinetic input data in order to optimize sustainable geothermal reservoir management, power and heat production and reinjection strategies. Geochemical modelling, which relies on the measurement of high quality data, is one of the numerical approaches needed to reach this goal.

In a first step, published laboratory measurements were collected to estimate the properties of saline chloride fluids (NaCl and CaCl2) containing dissolved CO2. A focus was made on properties such as CO2 solubility for medium temperature brines up to 473.15 K.

In parallel to data collection, a modelling work was setup to calculate these fluid properties using the specific tool named PhreeSCALE. Relying on the Pitzer equations, it is able to compute thermal and volumetric properties of aqueous solutions such as heat capacity or density of the geothermal fluids.

The methodology relies on successive steps that include the estimation of Pitzer's interaction parameters for the CaCl2-H2O, CaCl2-NaCl-H2O, and CaCl2-NaCl-CO2-H2O systems to reproduce literature data. The model applies from 298.15 up to 473.15 K, and to elevated pressures.

INTRODUCTION

Geothermal energy currently uses hot deep brines. However, their exploitation can raise issues specifically due to their high salinity and the variations of temperature and pressure during their pumping. Main disorders are the potential degassing of fluids including CO2, but also the development of scaling in the facilities (production/injection wells, heat exchangers…) involving expensive maintenance works [START_REF] Eroini | Kinetic Study of Calcium Carbonate Formation and Inhibition by using an in-situ Flow Cell[END_REF][START_REF] Boch | Scale-fragment formation impairing geothermal energy production: interacting H2S corrosion and CaCO3 crystal growth[END_REF][START_REF] Todd | Chapter 12 -Formation and mitigation of mineral scaling in geothermal power plants[END_REF].

Moreover, these processes are interrelated since the degassing of CO2 is generally concomitant with water evaporation, pressure and temperature decrease during heat extraction operations (Alt-Epping et al., 2013). The chemical disturbances induced by the outgassing (increase of concentrations and of pH) strongly favour mineral precipitation, and thus scaling [START_REF] Thomas | Advances in the study of solids deposition in geothermal systems[END_REF][START_REF] Benoit | Carbonate scaling characteristics in Dixie Valley, Nevada geothermal wellbores[END_REF][START_REF] Simmons | Origins of calcite in a boiling geothermal system[END_REF]. It is therefore of major interest to develop a sound knowledge of CO2 solubility in hot and pressurized brines. The REFLECT project (https://www.reflect-h2020.eu/) aims to improve the accuracy and consistency of key thermodynamic and kinetic input data in order to optimize sustainable geothermal reservoir management, power and heat production and reinjection strategies. More specifically and in line with the scaling issues, one objective of the project is to improve the modelling capacity to predict CO2 degassing in the facility.

Because of the environmental issues related to the climate change resulting (at least partly) from human activity, the exploitation of the underground as a source of decarboned energy, or as a large volume to store permanently waste gases (e.g., CO2) or temporarily energy vectors (natural gas, hydrogen, heat), has regained much interest [START_REF] Kelemen | An Overview of the Status and Challenges of CO2 Storage in Minerals and Geological Formations[END_REF][START_REF] Zivar | Underground hydrogen storage: A comprehensive review[END_REF][START_REF] Mahon | A review of thermal energy storage technologies for seasonal loops[END_REF]. The knowledge developed in these domains can be beneficial to geothermal energy. This is particularly true for experimental CO2 solubility data in brines, acquired in the context of the carbon capture and underground storage technology (CCUS) [START_REF] Carvalho | Carbon dioxide solubility in aqueous solutions of NaCl: Measurements and modeling with electrolyte equations of state[END_REF][START_REF] Mohammadian | Measurement of CO2 Solubility in NaCl Brine Solutions at Different Temperatures and Pressures Using the Potentiometric Titration Method[END_REF][START_REF] Chabab | Thermodynamic study of the CO2-H2O-NaCl system: Measurements of CO2 solubility and modeling of phase equilibria using Soreide and Whitson, electrolyte CPA and SIT models[END_REF].

In this context, various chemical systems have been studied until recently and are now well known (dos Santos et al., 2021a(dos Santos et al., , 2021b(dos Santos et al., , 2020;;Messabeb et al., 2017Messabeb et al., , 2016;;[START_REF] Poulain | Experimental measurements of carbon dioxide solubility in Na-Ca-K-Cl solutions at high temperatures and pressures up to 20 MPa[END_REF]and references there in). However, many chemical systems still need to be investigated or their description improved. As an additional contribution to these developments, the present work focuses on the CaCl2-NaCl-CO2-H2O system. It relies on the collection of bibliographic CO2 solubility data and on their numerical exploitation to develop a robust geochemical model with predictive capabilities.

BACKGROUND AND METHOD

Collection of experimental CO2 solubility data

In order to study the CaCl2-NaCl-CO2-H2O system, focus must be made on the simpler CaCl2-CO2-H2O and NaCl-CO2-H2O systems first. The latter is already well known (see the extensive experimental data collection by dos Santos et al., 2021b) and described with dedicated models (dos Santos et al., 2021b;[START_REF] Duan | An improved model for the calculation of CO2 solubility in aqueous solutions containing Na + , K + , Ca 2+ , Mg 2+ , Cl -, and SO4 2[END_REF][START_REF] Shi | An improved model for CO2 solubility in aqueous electrolyte solution containing Na + , K + , Mg 2+ , Ca 2+ , Cland SO4 2under conditions of CO2 capture and sequestration[END_REF]Zhaoa et al., 2015). The former is a little less known, despite some thermodynamic models have been developed [START_REF] Appelo | Principles, caveats and improvements in databases for calculating hydrogeochemical reactions in saline waters from 0 to 200°C and 1 to 1000 atm[END_REF][START_REF] Bastami | Experimental and modelling study of the solubility of CO2 in various CaCl2 solutions at different temperatures and pressures[END_REF][START_REF] Shi | An improved model for CO2 solubility in aqueous electrolyte solution containing Na + , K + , Mg 2+ , Ca 2+ , Cland SO4 2under conditions of CO2 capture and sequestration[END_REF][START_REF] Springer | A thermodynamic model for predicting mineral reactivity in supercritical carbon dioxide: I. Phase behavior of carbon dioxide-water-chloride salt systems across the H2O-rich to the CO2-rich regions[END_REF][START_REF] Zhao | Measurement and modeling of CO2 solubility in natural and synthetic formation brines for CO2 sequestration[END_REF]. New experimental CO2 solubility data in CaCl2 synthetic brines have been acquired since (Lara Cruz et al., 2021;Messabeb et al., 2017;[START_REF] Poulain | Experimental measurements of carbon dioxide solubility in Na-Ca-K-Cl solutions at high temperatures and pressures up to 20 MPa[END_REF]. They cover ranges of pressures and temperatures that are complementary to previously published data. Indeed, they either extend the pressure and salinity ranges or refine the pressure intervals between measurements (see below). In summary, the temperature, pressure and CaCl2 concentration values covered by all published CO2 solubility data range between 293 and 424 K, 0.1 and 40 MPa, and 0 and 6.3 mol•kg -1 , respectively. Some inconsistencies can be noticed among all these sets of data and, in some occasions, within datasets from same authors. More specifically, [START_REF] Prutton | The solubility of carbon dioxide in calcium chloride-water solutions at 75, 100, 120° and high pressures[END_REF] indicate experimental uncertainties of 3%, but when compared with other datasets, discrepancies of up to 20% can be found. [START_REF] Bastami | Experimental and modelling study of the solubility of CO2 in various CaCl2 solutions at different temperatures and pressures[END_REF] do not provide any indication on their uncertainties, which complicates the comparison with other datasets and models. [START_REF] Tong | Solubility of CO2 in aqueous solutions of CaCl2 or MgCl2 and in a synthetic formation brine at temperatures up to 423 K and pressures up to 40 MPa[END_REF][START_REF] Liborio | Dissolution of carbon dioxide in aqueous electrolyte solutions[END_REF] also reported experimental errors of up to 10 and 15% in their respective datasets. Consequently, a lower weight was given to these datasets while developing the geochemical model for the CaCl2-CO2-H2O system.

Experimental CO2 solubility data for the CaCl2-NaCl-CO2-H2O system were essentially acquired by Lara Cruz et al. (2021), [START_REF] Liu | Solubility of CO2 in aqueous solutions of NaCl, KCl, CaCl2 and their mixed salts at different temperatures and pressures[END_REF], [START_REF] Poulain | Experimental measurements of carbon dioxide solubility in Na-Ca-K-Cl solutions at high temperatures and pressures up to 20 MPa[END_REF], Zhao et al. (2015a), representing almost 70 data points. The temperature, pressure, NaCl and CaCl2 concentration values covered by these data range between 318 and 453 K, 1.2 and 40 MPa, 0.95 and 2.99 mol•kg -1 , and 0.2 and 0.67 mol•kg -1 , respectively.

Numerical modelling

Thermodynamic equilibrium between the CO2-rich phase and the aqueous solution in the autoclave can be described according to the equality of the chemical potential of CO2 in the two phases. This results in the following equation that corresponds to Henry's law [START_REF] Appelo | Equations for calculating hydrogeochemical reactions of minerals and gases such as CO2 at high pressures and temperatures[END_REF]:

𝑚 𝐶𝑂 2 = 𝜑 𝐶𝑂 2 𝑣𝑎𝑝 ×𝑃×𝑦 𝐶𝑂 2 𝛾 𝐶𝑂 2 𝑚 × 𝐾 𝐻 × 𝑒𝑥𝑝 (- 𝑉 𝑚,𝐶𝑂 2 (𝑃-𝑃 0 ) 𝑅𝑇 ) × 𝑚 0 𝑃 0 [1]
𝑚 𝐶𝑂 2 is the molality of aqueous CO2 (mol•kg -1 ), 𝜑 𝐶𝑂 2 𝑣𝑎𝑝 is the fugacity coefficient of CO2 in the gas (or supercritical) phase (unitless), P is the total gas pressure (Pa), 𝑦 𝐶𝑂 2 is the mole fraction of CO2 in the gas phase (unitless), 𝛾 𝐶𝑂 2 𝑚 is the activity coefficient of aqueous CO2 (unitless), KH (unitless) is the Henry's law constant for the dissolution reaction of CO2(g) into CO2 0 , 𝑉 𝑚,𝐶𝑂 2 is the partial molar volume of aqueous CO2 (m 3 •mol -1 ), P0 is the reference pressure (= 10 5 Pa), R is the ideal gas constant (= 8.31446 J•mol -1 •K -1 ), T is the absolute temperature (K) and m0 is the reference molality (=1 mol•kg -1 ).

𝑚 𝐶𝑂 2 and 𝑦 𝐶𝑂 2 are the unknowns of equation [1], P and T are constraints, 𝑉 𝑚,𝐶𝑂 2 is described by the HKF equation of state for aqueous species [START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600°C[END_REF][START_REF] Tanger | Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: Revised equations of state for the standard partial molal properties of ions and electrolytes[END_REF], 𝜑 𝐶𝑂 2 𝑣𝑎𝑝 is determined by the Peng-Robinson equation of state, 𝛾 𝐶𝑂 2 𝑚 is computed according to the Pitzer equations. More details about the latter equations can be found in [START_REF] Pitzer | Activity coefficients in electrolyte solutions[END_REF]. More specifically, the activity coefficient of aqueous CO2 can be expressed as follows (dos Santos et al., 2021b):

ln(𝛾 𝐶𝑂 2 𝑚 ) = 2𝑚 𝐶𝑂 2 𝜆 𝐶𝑂 2 ,𝐶𝑂 2 + ∑ 2𝑚 𝑐 𝜆 𝐶𝑂 2 ,𝑐 𝑛 𝑐 𝑐=1 + ∑ 2𝑚 𝑎 𝜆 𝐶𝑂 2 ,𝑎 𝑛 𝑎 𝑎=1 + ∑ ∑ 𝑚 𝑐 𝑚 𝑎 𝜁 𝐶𝑂 2 ,𝑐,𝑎 𝑛 𝑎 𝑎=1 𝑛 𝑐 𝑐=1 [2]
𝜆 𝐶𝑂 2 ,𝐶𝑂 2 , 𝜆 𝐶𝑂 2 ,𝑐 , 𝜆 𝐶𝑂 2 ,𝑎 are binary Pitzer interaction parameters between two molecules of CO2 0 , one molecule of CO2 0 and a cation c, and one molecule of CO2 0 and an anion a, respectively. 𝜁 𝐶𝑂 2 ,𝑐,𝑎 is a ternary Pitzer interaction parameter between one molecule of CO2 0 , a cation c and an anion a. nc and na are the number of different cations and anions in the solution, respectively. As the Pitzer equation are semi-empirical, these specific interaction parameters cannot be determined a priori and must be fitted using experimental data, like gas solubility in brines amongst others.

In the present work, geochemical calculations are done at thermodynamic equilibrium using the PhreeSCALE software [START_REF] Lach | Thermal and volumetric properties of complex aqueous electrolyte solutions using the Pitzer formalism -The PhreeSCALE code[END_REF]. It results from an evolution of PhreeqC [START_REF] Parkhurst | Description of input and examples for PHREEQC version 3: a computer program for speciation, batch-reaction, onedimensional transport, and inverse geochemical calculations[END_REF], a software that is widely used in the geochemical modelling community. The Pitzer equations that are implemented in these two calculation codes can handle partial dissociation of electrolytes but are limited to ionic strength-dependent viral contributions of second order. Compared to PhreeqC, PhreeSCALE includes a more rigorous treatment of the partial derivatives of the Pitzer equations of the excess Gibbs free energy of the solutes. Consistent thermal and volumetric properties of aqueous solutions such as heat capacity or density of geothermal fluid can be calculated too.

To fit the specific interaction parameters involving CO2 0 , an additional numerical tool named PEST [START_REF] Doherty | PEST. Model-independent parameter estimation[END_REF] is used in combination with PhreeSCALE. Starting from an initial set of interaction parameter values, PEST launches PhreeSCALE, which computes the concentration of dissolved CO2. PEST then compares the results to the experimental values. As long as convergence criterion is not met, PEST suggests a new set of parameter values and launches PhreeSCALE again. When the convergence criterion is met, the fitting procedure stops and the last set of parameters proposed by PEST is saved.

The development of a consistent thermodynamic database is a long term process, and thus necessarily incremental. Therefore, the procedure above must be applied based on pre-existing models for the investigated brines. In the present work, we use the chemical model recently developed by Lassin et al. (submitted) for the NaCl-CaCl2-H2O system, and the one developed by (dos Santos et al., 2021b) for the CO2-NaCl-H2O system. Indeed, the models mentioned previously (section 2.1) mostly apply to the solubility of CO2 in simple binary brines, while the objective here is to consider a mixture of brines whose description is not a straightforward addition of two models for binary brines.

The model of Lassin et al. (submitted) for the NaCl-CaCl2-H2O system is based on the Pitzer equations and includes the partial dissociation of the CaCl2 electrolyte. The standard partial molar properties of the aqueous species involved (namely, Ca 2+ , Cl -, CaCl + , and CaCl2 0 ) are described according to the HKF equation of state [START_REF] Sverjensky | Prediction of the thermodynamic properties of aqueous metal complexes to 1000°C and 5 kb[END_REF]. The model is able to describe the osmotic coefficient of the CaCl2 aqueous brines and salt solubility up to 523 K. It is also able to describe salt solubility in the NaCl-CaCl2-H2O system over the whole range of mixture ratios, at least up to 403 K (experimental solubility data not available at higher temperatures). Moreover, the models of Lassin et al. (submitted) and dos Santos et al. (2021b) are consistent with each other since they both include the interaction parameters of [START_REF] Lach | Experimental data and modeling of solution density and heat capacity in the Na-K-Ca-Mg-Cl-H2O system up to 353.15 K and 5 mol•kg -1 ionic strength[END_REF] for the NaCl-H2O system.

RESULTS AND DISCUSSION

CO2 solubility in the CaCl2-H2O binary system

Only two new specific binary interaction parameters involving the CO2 0 aqueous species were found necessary to describe CO2 solubility in CaCl2 and (NaCl + CaCl2) brines, up to 6 M CaCl2, 40 MPa of CO2 and 423 K. These parameters, namely 𝜆 𝐶𝑂 2 ,𝐶𝑎 2+ and 𝜆 𝐶𝑂 2 ,𝐶𝑎𝐶𝑙 + are temperature dependent, according to the following equation:

𝜆 𝐶𝑂 2 ,𝑐 (𝑇) = 𝐴 + 𝐵𝑇 + 𝐶/𝑇 [3]
The corresponding coefficients are given in Table 1. Results are plotted in Figure 1 for the solubility of CO2 in CaCl2 binary aqueous solution as a function of CO2 pressure, and at various CaCl2 concentrations and temperatures. For each condition, errors  were calculated according to the following equation ( [START_REF] Christov | A chemical equilibrium model of solution behavior and solubility in the H-Na-K-Ca-OH-Cl-HSO4-SO4-H2O system to high concentration and temperature[END_REF], and reported in Table 2:

𝜎 = √ ∑(𝑚 𝐶𝑂 2 ,𝑒𝑥𝑝 -𝑚 𝐶𝑂 2 ,𝑐𝑎𝑙𝑐 )² 𝑁 [4]
where 𝑚 𝐶𝑂 2 ,𝑒𝑥𝑝 and 𝑚 𝐶𝑂 2 ,𝑐𝑎𝑙𝑐 are the experimental a and calculated molalities of dissolved CO2, respectively, at a given condition of CaCl2 concentration, pressure and temperature. The sum covers all conditions of pressure at a given temperature and CaCl2 concentration, represented by N data points.

A) B) C) A) B)
These encouraging results make it possible, by means of geochemical modelling, exploring the solubility of CO2 in extended (NaCl + CaCl2) aqueous mixtures. Thus, a parametric study was done considering the whole range of compositions for the xNaCl + (0.5x/2)CaCl2 mixture, where 0 < x < 1, for three total Cl concentration (i.e., 1.6, 3.6 and 6 mol•kg -1 ), three temperatures (i.e., 323, 373, and 423 K), and two CO2 fugacity values (i.e., 5 and 15 MPa).

The latter constraint is a convenient way to set the concentration of dissolved CO2. Indeed, according to Henry's law and because the chloride content is constant when the mole fraction x of Na varies from 1 to 0, the molality of dissolved CO2 is proportional to its fugacity in the CO2-rich phase. Thus, the effect of the brine mixture composition on the calculated equilibrium pressure can be compared for different salinities. Results are plotted in Figure 3.

They suggest that whatever the temperature and the chloride concentration, the composition of the NaCl+CaCl2 mixture does not affect the pressure required to equilibrate the system with a CO2 fugacity of 5 MPa. At higher CO2 fugacity, namely 15 MPa, the calculated equilibrium pressure varies with the chemical composition of the brine mixture. This variation is more pronounced at high salinity with large Na proportions, and lower temperatures. In these conditions, increasing Ca (at the expense of Na) increases the pressure required to equilibrate with the CO2 fugacity (Figure 3-A). In other words, the presence of Ca in the brine has a salting-out effect on CO2. This effect can be reversed at high temperature and high salinity: the equilibrium pressure slightly decreases when small quantities of Ca replace Na at 423 K (Figure 3-C). At higher Ca content (c.a. 5 mol.%), the saltingin effect of Ca stops and turns to a salting-out effect.

CONCLUSIONS

A new set of specific interaction parameters has been developed to describe the solubility of CO2 in CaCl2 + NaCl aqueous brines according to a geochemical model based on the HKF and Pitzer equations. The model is valid for CaCl2 concentrations up to 6 M, temperatures up to 453 K and pressures up to 40 MPa.

Predictive calculations using this model suggest that, at high pressure, low temperature and high NaCl concentration, the presence of CaCl2 in the mixture increases the salting-out effect of the brine, event at low Ca/Na ratios.

An extension of the model to larger chemical systems may help describing the behaviour of brines that are more representative of the fluids involved in geothermal exploitation plants.

Acknowledgements (optional)

The REFLECT project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement n° 850626. The Carnot ISIFoR and BRGM are also gratefully acknowledged for their support.

  model and suggesting extrapolation capabilities with respect to pressure, temperature and salinity.

Figure

  Figure 1: CO2 solubility in CaCl2 aqueous brines as a function of pressure, for various CaCl2 concentrations, at A) 323 K, B) 373 K, and C) 423 K. Symbols: experimental data (45pru/sav = Prutton and Savage, 1945; 75mal/kur = Malinin and Kurovskaya, 1975; 15zha/dil = Zhao et al., 2015; 13ton/tru = Tong et al., 2013; 17mes/con = Messabeb et al., 2017; 19pou/mes = Poulain et al., 2019; 20lar/ney = Lara Cruz et al., 2021). Lines: model (this work).

Figure

  Figure 2: CO2 solubility in (NaCl + CaCl2) aqueous brines as a function of pressure, at A) 323, 373, and 423 K, and B) 333 and 453 K. Symbols: experimental data, A) Poulain et al. (2019, B) Lara Cruz et al. (2021). Lines: model (this work).

Figure

  Figure 3: Predicted CO2 solubility in (NaCl + CaCl2) aqueous brines as a function of Na mole fraction, for Cl molality of 1.6, 3.6 and 6 M, CO2 fugacity of 5 and 15 MPa, at A) 323, B) 373, and C) 423 K.

Table 1 : Coefficients for the temperature function [3] of the CO2 0 -Ca 2+ and CO2 0 -CaCl + specific interaction parameters.

 1

	Parameter	A	B×10 3	C
	𝜆 𝐶𝑂 2 ,𝐶𝑎 2+	1.45781	-1.44809	-258.144
	𝜆 𝐶𝑂 2 ,𝐶𝑎𝐶𝑙 + -4.05504	5.83521	708.627

Table 2 : Errors  between experimental and calculated CO2 solubility in CaCl2 brines, for each condition of temperature and CaCl2 concentration.

 2

	Temper-		CaCl2	Number of	Error 
	ature (K)	(mol•kg -1 )	data points N
			1	11	0.015
	323		3	4	0.013
			6	4	0.013
			1	30	0.043
	373		3	7	0.035
			6	5	0.009
			1	17	0.022
	423		3	8	0.032
			6	4	0.011
	Figure 1 and Table 2 show that the model satisfactorily
	describes the experimental data within their dispersion
	intervals, between 323 and 423 K, and up to 6 M CaCl2.
	Beyond	the	pressure	values	investigated
	experimentally, the calculated CO2 solubility smoothly
	increases, demonstrating the numerical stability of the

Table 3 : Errors  between experimental and calculated CO2 solubility in (NaCl + CaCl2) brines, for each condition of temperature.

 3

	Temperature (K)	Number of data points N	Error 
	323	8	0.022
	333	5	0.018
	373	8	0.016
	423	8	0.035
	453	5	0.086