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Gas concentration and flow rate measurements as part of methane baseline assessment: Case of the Fontaine Ardente gas seep, Isère, France

Keywords:

Methane and CO 2 gas emissions from a gas seep located in the French Alps, documented over two millenaries, have been quantified along with gas emissions from the miniseepage area contouring the main vent. Several tons per year of both gas phases are released in the atmosphere from the main seep (18 tons of CH 4 and 5.5 tons of CO 2 ) which has been modified by an old borehole. Diffuse seepage brings additional CH 4 (1 ton per year) and CO 2 (2.3 tons per year) which are emitted from a small area of 240 m². Secondary CH 4 oxidation processes are likely to occur near the surface. Biologically produced CO 2 is also found in the soil in the seepage area and the biological component becomes predominant as CO 2 concentrations drop lower than 2% vol. Soil gas CH 4 +CO 2 enrichments are oriented along N45 and N170 pathways that represent the main structural directions of the area. Soil flux anomalies are more scattered because surface clay formations have been strongly reworked at surface. Variety of methods (IRGA, micro-GC, accumulation chamber flux measurements, IR-camera) were tested and are discussed in connection with environmental impact monitoring and the role of sampling conditions, with emphasis on shale gas and carbon storage baseline assessments The role of potential leakage pathways represented by the old borehole structure is also discussed as an analogue with leaky shale gas wells.

Introduction

Natural Earth degassing is a matter of concern and investigation since many years in various fields such as volcanology or geothermics. Often relying on the characterization of the gas species percolating from depth to shallow horizons and subsequently to the atmosphere, these studies have provided better constraints on the sources and amounts of gases released to the atmosphere and on potential migration pathways. Numerous studies have dealt with the evaluation of the amounts of CO 2 and other gases naturally released by volcanic structures (e.g. [START_REF] Greenland | An estimate of gas emissions and magmatic gas content from Kilauea volcano[END_REF][START_REF] Hobbs | Airborne measurements of particle and gas emissions from the 1990 volcanic eruptions of Mount Redoubt[END_REF][START_REF] Brantley | Measured carbon dioxide emissions from Oldoinyo Lengai and the skewed distribution of passive volcanic fluxes[END_REF][START_REF] Halmer | The annual volcanic gas input into the atmosphere, in particular into the stratosphere: a global data set for the past 100 years[END_REF] and the consequences such releases may have on human health (e.g. [START_REF] Sigurdsson | Origin of the lethal gas burst from Lake Monoun, Cameroun[END_REF][START_REF] Baxter | Health hazards and disaster potential of ground gas emissions at Furnas volcano, Sao Miguel, Azores[END_REF]. Gas emissions from geothermal areas and from geothermal exploitation have also received growing interest as contribution to greenhouse gases (GHG) in the atmosphere (e.g. [START_REF] Capaccioni | Light hydrocarbons in gas-emissions from geothermal fields volcanic areas and geothermal fields[END_REF]Cardenelli et al., 2003a;[START_REF] Armannsson | CO 2 emissions from geothermal power plants and natural geothermal activity in Iceland[END_REF]. Furthermore, they induce direct risks for human dwellings (e.g. [START_REF] Durand | Geothermal ground gas emissions and indoor air pollution in Rotorua, New Zealand[END_REF]. Gas releases may also be associated with emissions from hydrothermal resources (e.g. [START_REF] Battani | Geochemical study of the natural CO 2 emissions in the French Massif Central: How to predict origin, processes and evolution of CO 2 leakage[END_REF], from non-volcanic areas (e.g. [START_REF] Rogie | Flux measurements of non-volcanic emission from some vents in central Italy[END_REF] or from other subsoil reservoirs (e.g. [START_REF] Etiope | Migration of carrier and trace gases in the geosphere: an overview[END_REF][START_REF] Voltattorni | Soil gas geochemistry: significance and application in geological prospectings[END_REF]: they are associated with mud volcanoes (e.g. Jud, 2005;Etiope et al., 2011a) and with hydrocarbon plays (e.g. [START_REF] Etiope | Natural Gas Seepage, The Earth's Hydrocarbon Degassing[END_REF]. The budgets of carbon emissions to the atmosphere have been assessed in the 2000's (e.g. [START_REF] Mörner | Carbon degassing from the lithosphere[END_REF] and are regularly refined for CO 2 (e.g. [START_REF] Holloway | Natural emissions of CO 2 from the geosphere and their bearing on the geological storage of carbon dioxide[END_REF] or CH 4 (e.g. [START_REF] Etiope | Natural Gas Seepage, The Earth's Hydrocarbon Degassing[END_REF]. Recently, methane emissions from igneous rocks (abiotic gas) have also been recognized as more widespread and more important than previously thought (e.g. [START_REF] Etiope | Abiotic methane in continental serpentinization sites: an overview, 15th Water-Rock Interaction International Symposium[END_REF].

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Consequently, there is still need to better constrain methane emissions to the atmosphere given its elevated greenhouse power. As pointed out by [START_REF] Etiope | Natural Gas Seepage, The Earth's Hydrocarbon Degassing[END_REF] in his landmark monograph, the importance of hydrocarbon seepage from onshore locations is far from negligible. The refinement of natural gas seepage contributions to the atmospheric methane budget thus contributes to better understanding the impacts on climatic forcing.

Hydrocarbons released by gas seeps, miniseepages, microseepages and emissions from mud volcanoes are phenomena occurring worldwide, in Europe (Germany, Greece, Italy, Romania, Switzerland), Azerbaijan, Turkey, Asia (China, Japan, Taiwan), Australia and USA.

Emissions from ophiolites or igneous intrusions [START_REF] Etiope | Natural Gas Seepage, The Earth's Hydrocarbon Degassing[END_REF] are even more widespread.

Methane seeps are also an important part of the geochemical baseline that needs to be defined when assessing the environmental impact of conventional and unconventional hydrocarbon exploration and exploitation [START_REF] Humez | Redox controls on methane formation, migration and fate in shallow aquifers[END_REF][START_REF] Mayer | Prospects and Limitations of Chemical and Isotopic Groundwater Monitoring to Assess the Potential Environmental Impacts of Unconventional Oil and Gas Development[END_REF]Mayer et al., 2016).

Emissions of alkanes in the alpine foreland are reported in Switzerland (Giswil, Northern Alps; [START_REF] Etiope | Origin and flux of a gas seep in the Northern Alps[END_REF] taking the form of both diffuse degassing area and gas vents with large amounts of thermogenic gas released each year (more than 16 tons per year from 2 main seeps and diffuse degassing). Gas seeps in France are largely under-investigated [START_REF] Etiope | Natural emissions of methane from geological seepage in Europe[END_REF]. Only one site is mentioned in the MAGA database (http://www.magadb.net/global_map/) even though methane point emissions do not seem to be an uncommon phenomenon in the French Subalpine Chains (Figure 1), apparently systematically related to outcropping or underlying Lower Jurassic argillaceous sediments.

We report investigations at the most prominent French alpine gas seep, known for at least two millenaries, the Fontaine Ardente du Gua site, south of Grenoble (hereafter referred to Accepted Manuscript as Fontaine Ardente). First measurements, in June 2015, aimed to define the shape of the gas emissions at surface and preliminary results can be found in [START_REF] Kloppmann | Natural gas seeps in the French Alps: Sources and pathways[END_REF] and in Gal et al. (2017a). Here we present more detailed investigations based on soil gas measurements and on soil gas flux measurements performed in October 2016, allowing a first quantitative assessment of CH 4 and CO 2 fluxes to the atmosphere.

Geological and historical settings

The Fontaine Ardente gas seep is situated 20 km South-West from Grenoble city (Isère department, France), at the foothill of the Urgonian limestone cliffs of the Vercors Massif (Figure 1Error! Reference source not found.). The gas emanates from Callovian claystones and argillaceous limestones (black shales) in a small thalweg in the main bed of the Vernant creek. Gas escapes are thought to be linked with faults but the geometry of these faults remains unclear and so does the origin of the gas phase. There has been a debate between a deep origin, linked to the existence of a deep fault reaching Triassic formations, and a near surface origin, linked to a minor fault only affecting shales [START_REF] Mercier | L'épigraphie et les fontaines ardentes du Dauphiné[END_REF], the region being rich in organic-rich sediments (coal seams of La Mûre or the Motte d'Aveillans coalfield; [START_REF] Berthier | Etude et travaux pour le renforcement en gaz de la Fontaine Ardente -Commune du Gua (Isère)[END_REF].

The Fontaine Ardente seep is known from ancient times (as early as 426 AD) as "burning water" because the seep was initially observed in the stream bed of the Vernant creek (Figure 1). Following the rush in hydrocarbons exploitation in the US (especially in Pennsylvania) and Canada in the XIX th century, pro-parte based on the discovery of natural Accepted Manuscript gas associated with hydrocarbons, attempts to exploit the Fontaine Ardente gas have been made around 1880 [START_REF] Piret | La Fontaine Ardente de Saint-Barthélémy (Isère) -Gaz naturel, charbon, pétrole[END_REF]. It was believed that there must be a considerable gas reservoir together with perennial gas production to allow leakage of methane at ground surface for such a long timea volume of 1.73 x10 3 m 3 (3.1 tons at boiling point) of CH 4 per day was evaluated in the late XIX th century.

The composition of the gas phase was analyzed for the first time at the end of the XIX th century: 98.81% (vol.) of CH 4 , 0.58% of CO 2 , 0.48% of N 2 , 0.10% of O 2 and 0.03% of loss or bias during analysis [START_REF] Piret | La Fontaine Ardente de Saint-Barthélémy (Isère) -Gaz naturel, charbon, pétrole[END_REF]. It was reevaluated in 1960, leading to the sole detection of CH 4 and CO 2 at 91. 16% and 8.84% respectively (cited in AHPG, 2012). Traces of O 2 and N 2 previously described were attributed to atmospheric contamination during sampling.

In view of the exploitation of the gas phase, the primitive site was profoundly modified in the goal of focusing the patchy gas emissions into a unique emitting point [START_REF] Piret | La Fontaine Ardente de Saint-Barthélémy (Isère) -Gaz naturel, charbon, pétrole[END_REF].

Infrastructure was built to deviate stream water and to allow the construction of a gasometer to collect gas without interaction with surface waters. From this gasometer the gas was piped uphill for further use to the plateau surrounding the thalweg where the fountain is located. The exploitation quickly ceased as the gas production could not be enhanced and floods repeatedly destroyed the surface infrastructure. The investigations related to these works concluded that the rock formations near the surface were altered down to a depth of 18 to 20 m below ground surface by the long-term percolation of the gas. Due to flooding in 1956, the site was covered by detrital sediments, the gas emission pathways have changed and the gas flux to the surface decreased.

In order to enhance and better constrain gas emissions, further works were undertaken in the 90's [START_REF] Berthier | Etude et travaux pour le renforcement en gaz de la Fontaine Ardente -Commune du Gua (Isère)[END_REF]. After surveying soil gas (CO 2 soil gas concentration Accepted Manuscript prospecting at 0.5 m depth), 2 boreholes (FA1 and FA2) were drilled down to 35 m below ground level. Only one (FA2; Figure 1) was methane rich, main gas inflow occurring at about 10 m depth. Gas flowrate was measured at 130 l/min at 0.1 bars during 3 hours (borehole opened). Once closed, borehole pressure increased up to 0.6 bars. Exploitation of the gas had been expected to be possible at above 90 l/min but no exploitation was tried. Nowadays the FA2 borehole represents the main seep knowing that the wellhead has been destroyed and that additional reworking led to the building of a concrete platform around the former wellhead and to the flattening of the area close to FA2 borehole as shown by Figure 1Error! Reference source not found..

Methods

Site investigations at Fontaine Ardente used two approaches: soil gas concentrations measurements at 1 m depth and soil gas flux quantification at the soil/atmosphere interface.

Soil gas measurements

Soil gas concentrations were measured according to a well-established methodology used e.g. for monitoring for Carbon Capture and Storage (CCS) sites [START_REF] Gal | The SENTINELLE French research project: subsurface geochemical characterization prior CO 2 Accepted Manuscript injection (baseline survey) -site of Rousse (TOTAL CCS Pilot)[END_REF]. A 1 m square mesh grid was used in 2015 (Figure 1). In 2016, the mesh grid varied from 1 x 1 m to 1 x 2 m. Some measurements were performed outside of this grid and outside of the Vernant river bed. These points serve as reference points for the evaluation of background characteristics of soil gas. Small holes (1 cm in diameter) were drilled down to 1 m depth. Such a small mesh size for systematic sampling is rarely used in similar studies (e.g. [START_REF] Bloomberg | Soil gas flux exploration at the Rotokawa geothermal field and White Island[END_REF] and is a direct consequence of the small size of the area to monitor.

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Using a small mesh is also a way to lower the uncertainty in soil gas flux quantifications that may be related to the use of relatively small flux chambers [START_REF] Parkin | Sampling Protocols -Chapter 3 -Chamber-Based Trace Gas Flux Measurements[END_REF].

Owing to the presence of a concrete platform and surrounding rockwall (gabion wall), the area cannot be monitored totally.

For soil gas investigations, a sampling tube was inserted into the hole of 1 cm in diameter and an Infra-Red Gas Analyzer (IRGA, LFG20 by ADC Gas Analysis Ltd., UK) was plugged, pumping at low flow rate (200 ml.min -1 ) and reaching steady state conditions within tens of seconds. No concentration changes on very short time scale (minutes) were observed. The IRGA allowed a direct estimate of CH 4 , CO 2 and O 2 gas concentrations in soils. Analytical precision for CO 2 and CH 4 was ±0.5% of the reading for low concentration range (0.01-10% vol.), ±3% of the reading for higher concentrations (10 to 50% vol.) and up to ±5% of the reading for the CO 2 sensor and ±16% for the CH 4 sensor for concentrations greater than 50% (a procedure is described in supplementary information to correct the bias for CH 4 concentrations). The precision for oxygen was ±0.4% full scale reading (0-25% vol.).

Complementary characterization of the gas phase was performed in 2015 using gas microchromatography (Agilent 3000) in order to increase the accuracy of CH 4 determinations (IR sensors exhibit non-linear absorption at high gas concentrations). Samples were stored in Tedlar® bags during site monitoring and analyzed by the end of the day. A quantitative assessment of the gas phase composition was then possible for permanent gases (CO 2 , O 2 , N 2 , Ar), alkanes (CH 4 , C 2 H 6 , C 3 H 8 ) and 4 He. Detection limits are 10 ppm for permanent gases, 2 ppm for alkanes and 50 ppm for helium. Precision of the measurement is better than 2% at full scale.

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Soil flux measurements

Soil gas flux measurements were performed in 2016 on the 1 x 1 to 1 x 2 m mesh grid. At some locations, flux monitoring was not technically feasible as the ground was too pebbly, in the bed of the Vernant creek or strongly modified by the concrete platform and the rockwall (Figure 1). The accumulation chamber technique, with external recirculation, was used. Such closed chamber measurements are widely used for CO 2 flux quantifications (e.g. [START_REF] Chiodini | Soil CO 2 flux measurements in volcanic and geothermal areas[END_REF][START_REF] Fridriksson | CO 2 emissions and heat flow through soil, fumaroles, and steam heated mud pools at the Reykjanes geothermal area, SW Iceland[END_REF] and are also applied to CH 4 flux measurements (e.g. Etiope et al., 2013a;[START_REF] Gal | The SENTINELLE French research project: subsurface geochemical characterization prior CO 2 Accepted Manuscript injection (baseline survey) -site of Rousse (TOTAL CCS Pilot)[END_REF]. Reliable and representative estimates of the soil gas flux can be obtained as this technique does not rely on corrections depending on soil characteristics (Chiodini et al., 1998, and references therein). The flux chamber system is a customized version of the soil gas flux system of Echo Instruments (Slovenia). The aluminum chamber has a semi-spherical geometry with a basal area of 0.0289 m 2 and a volume of 0.0018 m 3 (soft ground) to 0.0023 m 3 (hard ground). There is no anchor installation prior to measurement as the sharp-edged base of the chamber is shaped to be easily inserted in the ground. A fan with tunable speed ensures the required mixing of the headspace gas [START_REF] Parkin | Sampling Protocols -Chapter 3 -Chamber-Based Trace Gas Flux Measurements[END_REF]. Pressure, temperature and relative humidity inside the chamber, as well as pumping rate, are continuously monitored thus giving estimates on the possible influence of external parameters on the measurements. The following gas species are monitored: CO 2 and CH 4 by Non-Dispersive Infra-Red (NDIR) detectors (0 to 5000 ppmv ± 2% and 0 to 10000 ppmv ± 5% respectively), O 2 and H 2 S by electrochemical cells (0 to 25% vol. ± 2% of the reading and 0 to 10000 ppmv ± 5% of the reading respectively). The measurements are performed directly on the soil surface without removal of the first centimeters of the soil as it was sometimes reported [START_REF] Fridriksson | CO 2 emissions and heat flow through soil, fumaroles, and steam heated mud pools at the Reykjanes geothermal area, SW Iceland[END_REF]. Around two thirds of the fluxes were measured on bare soil or gravel soil, the rest on vegetated soil Accepted Manuscript (grass). This allowed quantification of the gas emissions to the atmosphere under undisturbed conditions (soil removal may artificially enhance gas emissions if performed shortly before measurement).

The flux measurement was based on the rate of CH 4 and CO 2 increases (positive fluxes), and possible opposite O 2 decrease ("negative" fluxes), in the chamber. Usually positive values indicate fluxes directed from the soil to the atmosphere and negative values indicate flow from the atmosphere into the soil (e.g. D' Alessandro et al., 2018). Details on the procedure used to calculate flux is given as supplementary information.

Results and discussion

Meteorological conditions

The influence of meteorological conditions is a widely reported phenomenon (e.g. [START_REF] Toutain | Gas geochemistry and seismotectonics: a review[END_REF]. The two campaigns were performed under contrasted climatic conditions.

In June 2015, high rainfall amounts were recorded prior to the field acquisitions (40 mm on June 13 and 14). Such amounts can lower the soil gas concentrations in the upper part of the soil (0 to 40 cm depth)and so the gas flux at the soil/atmosphere interfaceas the result of a barrier effect that reduces gas emissions to the surface when water is percolating downwards and/or dissolving part of the soil CO 2 [START_REF] Hinkle | Environmental conditions affecting concentrations of He, CO 2 , O 2 and N 2 in soil gases[END_REF]. At the opposite, increases of the concentrations by a "piston" accumulation effect can be observed deeper in the soil [START_REF] Bouma | On the assessment of root and soil respiration for soils of different textures: interactions with soil moisture contents and soil CO 2 concentrations[END_REF]. Rainfall has less influence on CH 4 as it is less soluble in water than CO 2 [START_REF] Weiss | Carbon dioxide in water and seawater: the solubility of a non-ideal gas[END_REF][START_REF] Wiesenburg | Equilibrium Solubilities of Methane, Carbon Monoxide, and Hydrogen in Water and Sea Water[END_REF]. Gas emission can also be affected by Accepted Manuscript pressure variations with possible anti-correlation of gas with air pressure [START_REF] Toutain | Gas geochemistry and seismotectonics: a review[END_REF] even if this relation has not been observed in all cases (e.g. [START_REF] Gal | The SENTINELLE French research project: subsurface geochemical characterization prior CO 2 Accepted Manuscript injection (baseline survey) -site of Rousse (TOTAL CCS Pilot)[END_REF].

The situation was opposite in October 2016. Only 4 mm of rainfall occurred during the two days preceding the measurements and August and September 2016 were drought months (only 37.8 + 37.1 mm precipitated compared to 67.2 + 105.7 mm on thirty-year average; http://www.meteofrance.com/climat/france/grenoble/38384001/releves). Influence of rainfalls on gas emissions may be of secondary importance in 2016 as water uptake by vegetation and/or evaporation after rainfalls may preclude water to percolate deep in the soil. This period of drought also allowed enlarging the monitoring area by performing some data acquisitions directly in the river bed of the Vernant creek (Figure 1).

Statistical parameters

Main statistical characteristics of soil gas concentrations measured in the field by gas chromatography in 2015 and IRGA in 2016 are reported in Figure 2. The 2016 IRGA data were completed by some gas chromatography measurements (N 2 , 4 He and C 2 H 6 ) in order to cross-check high CH 4 concentrations measured by IRGA. The gas phases monitored in 2015 (Gal et al., 2017a) and the gas phases monitored in 2016 are in good agreement:

-Soil gas concentrations cover a wide range of variation typically linked to the existence of a gas vent. Concentrations of the predominant gas in the main vent (CH 4 ) ranged from zero up to 90% vol.

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-CO 2 exists across all the monitoring area with maximum values around 11% for the main vent and background values in the surrounding forest soils around 0.2% (Figure 1).

-Trace gases are detected less frequently. C 2 H 6 and 4 He appeared to be detectable in half of the samples in 2015 and their concentrations reached respectively 590 and 250 ppm. In a few samples, having a composition similar to that of the main seep (FA2 inFigure 1), very low amounts of C 3 H 8 were measured close to the instrumental detection limit (2 to 3 ppm in the present case). Interestingly, H 2 S has not been detected in 2015 but has been monitored at high concentrations in 2016 in four samples (up to 2.2% vol. in two samples; around 200 ppm in two other samples; below detection limit -50 ppmin the two last samples).

-Oxygen and N 2 concentrations varied from atmospheric concentrations down to near zero values for O 2 and residual content for N 2 . The gas emitted at the main seep is composed by 89% CH 4 , 9 to 9.5% CO 2 , 1% residual N 2 and traces of Ar (less than 0.07%), 4 He (0.018%) and C 2 H 6 (0.0002%). H 2 S measured in 2016 accounts for additional 2% vol. These values well match with the corrected IRGA measurements (respectively 92-93% CH 4 , 9-9.2% CO 2 and 0% O 2 ). By comparison, a smaller gas seep reported locally (Meylan-Rochasson gas seep; Gal et al., 2017a) is richer in CH 4 (94.5%), in C 2 H 6 (1.9%) and C 3 H 8 (0.3%), and has lower N 2 residual amount (0.35%), lower 4 He (0.007%) and CO 2 contents (0.39%).

Main statistical characteristics of soil gas fluxes measured in 2016 are reported in Figure 2.

Oxygen decrease over the duration of flux measurements (not reported in Figure 2) were measured on 9 points (of 69) and ranged between -3.5 and -20 g/m 2 /h. Both CO 2 and CH 4 Accepted Manuscript fluxes reached values close to 16 to 17 g/m 2 /h at maximum. A CO 2 flux was always possible to monitor whereas CH 4 fluxes were measurable only on the four-fifths of the data.

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Among the dataset, about two thirds of the data clearly showed the influence of the gas seep (high CH 4 , presence of 4 He, C 2 H 6 and CO 2 ).

CO 2 -O 2 relationship

One of the most frequently studied relationships in soil gas studies is CO 2 vs. O 2 (e.g. [START_REF] Romanak | Process-based approach to CO 2 leakage detection by vadose zone gas monitoring at geologic CO 2 storage sites[END_REF] Figure 3). Three main mechanisms result in linear correlations of both gases. By order of increasingly negative slopes of O 2 vs. CO 2 lines (Figure 3A), these are i) dilution, occurring when a deep CO 2 endmember dilutes the O 2 +N 2 pool in near surface environments (slope of -0.2); ii) replacement, due to aerobic respiration in soil that consumes 1 mole of O 2 to produce 1 mole of CO 2 (slope of -1) ; iii) CH 4 oxidation in near surface environments where 2 moles of O 2 are needed to produce one mole of CO 2 +H 2 O (slope of -2). The linear regression line calculated using the 2015 and the 2016 measurements has a slope (-2.07) very close to this CH 4 oxidation line albeit data scattering is not negligible for CO 2 concentrations exceeding 3 to 4% vol. This suggests that CH 4 oxidation plays an important role in the appearance of CO 2 in the soil gas even if, in 2016, concentrations up to 4 to 5% of CO 2 (in the absence of CH 4 ) can still be attributed to near surface biological CO 2 production. This 3 to 4% vol. CO 2 concentration range seems to be a threshold from which CH 4 oxidation process has a significant influence onto CO 2 production whatever the sampling period considered. At lower CO 2 concentrations, data are often lying in between the CH 4 oxidation line and the replacement line. This indicates a mixed origin with part of the CO 2 deriving from CH 4 oxidation and part of the CO 2 deriving from biological processes. The latter become predominant for CO 2 concentrations lower than 2% vol. Such concentrations levels can Accepted Manuscript reasonably be assumed as the mean biological background concentrations (e.g. Gal et al., 2014 and references therein). Nonetheless, Figure 4 outlines that even at such low CO 2 concentrations the influence of CH 4 oxidation process may exist thus suggesting a complex behavior of gas emission towards the surface.

The CO 2 vs. O 2 plot allows the identification of another group of points that plots fairly well on the replacement line (Figure 3A). Their CO 2 concentrations reach more than 5% vol. and may represent soils with strong biological activity. Such concentrations are consistent with concentrations up to 8-10% of biologically derived CO 2 reported in the literature (e.g. [START_REF] Gal | Soil gas ( 222 Rn, CO 2 , 4 He) behaviour over a natural CO 2 accumulation, Montmiral area (Drôme, France): geographical, geological and temporal relationships[END_REF]Gal et al., , 2017b)). These points are mainly located at the periphery of the monitoring zone (black dots in Figure 3B). This may suggest the maximum extension of the methane-derived CO 2 -plume does not exceed a dozen of meters.

O 2 -N 2 relationship

Processes are further constraint by O 2 vs. N 2 relationships [START_REF] Romanak | Process-based approach to CO 2 leakage detection by vadose zone gas monitoring at geologic CO 2 storage sites[END_REF][START_REF] Romanak | Towards a method for leakage quantification and remediation monitoring in the near-surface at terrestrial CO 2 geologic storage sites[END_REF]. concentrations. This corresponds to respiration processes and/or the presence of air as defined by [START_REF] Romanak | Process-based approach to CO 2 leakage detection by vadose zone gas monitoring at geologic CO 2 storage sites[END_REF][START_REF] Romanak | Towards a method for leakage quantification and remediation monitoring in the near-surface at terrestrial CO 2 geologic storage sites[END_REF]. O 2 depletion on this line may be linked to biological processes, at least for O 2 concentration ranging from 21 down to 15-16% as suggested by the CO 2 -O 2 plot. At O 2 concentrations close to 15-16%, the gas analyses still indicate that no Accepted Manuscript CH 4 is associated with N 2 , O 2 and CO 2 . When O 2 concentrations are lower than 15%, then the O 2 depletion becomes balanced by CH 4 and CO 2 concentrations increases (up to 12% and up to 9% respectively). Points having constantly high N 2 concentrations are located in the peripheral area surrounding the Fontaine Ardente main seep (black dots in the map; Figure 3C) where the influence of the gas seep becomes weaker. N 2 is only concerned by mixing with seeping gases, whereas O 2 is consumed by oxidation of both organic matter in the forest soil and of methane.

The second fit in Figure 3C (orange fit) has a slope of +0.265, close to the atmospheric O 2 /N 2 ratio of 0.268, and represents the progressive mixing of air with N 2 and O 2 free gas vent (dilution line). The corresponding points are in the vicinity of the main seep. This location also well corresponds to points with a O 2 vs. CO 2 slope of >=2 (Figure 3B) highlighting CH 4oxidation. Some data (7 points, yellow black circles in Figure 3C) fall between these two main lines and they may correspond to some altered gas compositions. The spreading of these data is influenced by the CH 4 content of the soil gas phase (points with the lower N 2 concentrations are the most enriched in CH 4 , up to 74 %). These points are subject to a variable influence of the gas vent with different rates of mixing between the geogenic and the biological endmembers. Plotted on the map (Figure 3C), these data are located at the peripheral of the gas vent highlighting the progressiveand non-linearvanishing of the geogenic gas input, progressively hidden by the biological gas production in soil.

CO 2 -CH 4 and O 2 -CH 4 relationships

As CO 2 and O 2 are closely linked, the relationships between CH 4 concentrations and CO 2 or O 2 concentrations can be investigated in parallel (Figure 4). The CH 4 vs. O 2 relationship Accepted Manuscript shows less dispersion than the CH 4 vs. CO 2 relationship, as it is not affected by biological production of CO 2 in soil that may alter the compositional ratios in between gas phases.

The linear fits indicated in black in Figure 4A and Figure 4B represent the influence of atmospheric dilution. In the gas composition of the Fontaine Ardente main seep, no O 2 is present and the CH 4 concentration is close to 90%. If air is present in the soil (c.a. 78% of N 2 and 21% of O 2 ) then a decrease of the CH 4 concentration by 10% is accompanied by an increase of +2.1% in O 2 and of 7.8% in N 2 which corresponds to the atmospheric N 2 /O 2 ratio.

The linear fits reported in blue in Figure 4A and Figure 4B correspond to rapid O 2 decrease accompanied by CO 2 increase at near constant CH 4 with a slope close to -2.4. The underlying process is oxidation of organic matter in the wood soil and root respiration, predominant in points located at the periphery of the gas seep. Minor contribution of CH 4 oxidation to the O 2 depletion and to the CO 2 enrichment may also exist. In the vicinity of the gas seep the dominant process is the seepage of methane and associated CO 2 . 5A). In the present case study, a CH 4 flux is considered as high when it is greater than 2 g/m²/h. Point 17 has a CH 4 to CO 2 flux ratio close to 1, compared to 2.1 and 2.4 for the two other points ( 28Accepted Manuscript and 31), whereas the ratios between CH 4 and CO 2 concentrations are respectively 6.1 (point 17), 9.4 (point 28) and 8.6 (point 31). In the lower CH 4 flux range (grey dots in Figure 5B) CH 4 flux represents approximately half of the CO 2 flux.

Two conclusions come from these statements. The first is that the emission of CH 4 is highly variable over short distances albeit the higher fluxes are monitored in close vicinity to the main seep. The second is that there is no relationship between the soil concentrations at 1 m depth (Figure 4) and surface fluxes (Figure 5). This is a common observation in non-active areasno volcanic nor geothermal activity (e.g. [START_REF] Gal | The SENTINELLE French research project: subsurface geochemical characterization prior CO 2 Accepted Manuscript injection (baseline survey) -site of Rousse (TOTAL CCS Pilot)[END_REF]. In areas of deep gas fluxes (CO 2 : e.g. [START_REF] Battani | Geochemical study of the natural CO 2 emissions in the French Massif Central: How to predict origin, processes and evolution of CO 2 leakage[END_REF]CH 4 : e.g. Etiope et al., 2011a) or where gas fluxes are intentionally induced (e.g. [START_REF] Schroder | The role of soil flux and soil gas monitoring in the characterisation of a CO 2 surface leak: A case study in Qinghai, China[END_REF], the link with concentrations may be stronger, even though complicated by transport through a thick soil column (e.g. [START_REF] Jones | Continuous monitoring of natural CO 2 emissions near Romelessons for low-level CO 2 leakage detection[END_REF].

The impervious geological substratum of the black shales would favor discrete gas flux on fractures rather than diffuse fluxes. Another preferential pathway might be drillholes that could have concentrated the formerly rather patchy gas leakage. At the surface level, the combination of former reworking and the existence of a clayey cover may lead to poor connection between CH 4 enrichment in the soil and CH 4 emission at the soil surface as the driving force is not sufficient for allowing stronger efflux.

Numerous points in Figure 5B show low to very low CH 4 fluxes (less than 0.1 g/m 2 /h excepted 1 sample at 0.18) and a wider scattering of the CO 2 flux (up to 4.5 g/m 2 /h). The locations of these points are generally randomly distributed all around the gas seep with the exception of some points (light blue and light blue-green dots in Figure 5B) which have CO 2 flux greater than 1 g/m 2 /h and which seem to surround the main seep a few meters away.

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Even if the values of CH 4 fluxes are low, they are still influenced by the gas seep. Methane emissions quantified in forests or in grasslands are indeed frequently 1000 times lower [START_REF] Jones | Greenhouse gas emissions from a managed grassland[END_REF][START_REF] Arnold | Fluxes of CO 2 , CH 4 and N 2 O from drained coniferous forests on organic soils[END_REF] and may rise up to 0.01 g/m 2 /h as a consequence of agricultural practices such as cattle slurry spreading [START_REF] Jones | Greenhouse gas emissions from a managed grassland[END_REF] or in wetlands (Le [START_REF] Mer | Production, oxidation, emission and consumption of methane by soils: A review[END_REF]. Similarly, CO 2 fluxes reaching 4.5 g/m 2 /h are still high compared to fluxes measured in standard environments where they are generally 10 times lower with the exception of agricultural spreading (up to 3.3 g/m 2 /h; [START_REF] Jones | Greenhouse gas emissions from a managed grassland[END_REF] or enhanced emissions in clayey soils (up to 2.3 g/m 2 /h; [START_REF] Gal | The SENTINELLE French research project: subsurface geochemical characterization prior CO 2 Accepted Manuscript injection (baseline survey) -site of Rousse (TOTAL CCS Pilot)[END_REF]. The measured CH 4 and CO 2 fluxes are then still influenced by the existence of the gas seep but there is no clear relation between the strength of the CO 2 flux and the location of the sampling point (Figure 5B). It is then complicated to distinguish between a CO 2 flux mainly linked to the gas seep (CO 2 produced by CH 4 oxidation) and a variable contribution to the CO 2 flux of biological origin.

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The most interesting case is point 73. At this location, the O 2 depletion is high (-10 g/m 2 /h), the CO 2 flux is high (4.5 g/m 2 /h) and the CH 4 flux is zero. In the soil, there is an O 2 depletion that is partially balanced by the presence of CO 2 (CO 2 /O 2 = -1.5) and only traces of CH 4 are found. The classical way for interpreting this O 2 depletion is to consider that an oxygen flux from the atmosphere to the soil exists. Nevertheless, point 73 plots on the CO 2 flux -O 2 flux linear fit that is defined for locations where a CH 4 flux exists. The O 2 depletion monitored at this point thus represents the influence of the CH 4 gas seep and the genesis of O 2 depletion is a consequence of CH 4 oxidation that produces CO 2 that is to some extent mixed with biologically produced CO 2 . The other time scale that can be considered is the 3 min duration of flux integration during chamber measurements. Gas emission of major species can also be variable over few minutes (Figure 7). Point 10 represents the routine situation with linear increase of the CO 2 concentration indicating constant gas emission. The situation was different for some monitoring points. Examples of varying flow are shown for points located close to the gas seep (Figure 7). We observe both examples of sudden and simultaneous increases of CO 2 and CH 4 fluxes (point 13) and decreases (points 15 and 17) that are not necessarily synchronous for both gases (point 17). Very short-term changes have also been observed (point 17). These variations may induce a bias in the estimation of fluxes. Of 69 CO 2 and 54 CH 4 flux measurements, respectively 17 and 12 showed variations, and for two in each case fluxes were potentially underestimated.

We could not perform repeated monitoring at these points so the bias linked to these short to very short term fluctuations of the seepage rate could not be precisely assessed. These fluctuations are also attested by the rapidly changing height of the flame as depicted inFigure 8. An estimation of the bias that may be related to this short term variability has been done by examining each gas accumulation curve during time. On the 69 CO 2 flux measurements, 17 showed temporal variation during the CO 2 accumulation phase among which only 2 were potentially underestimated. On the 56 CH 4 flux measurements, 12 showed temporal variation and by them 2 were also potentially underestimated. Thus, the flux measurements may rather lead to a slight overestimation of the global CO 2 and CH 4 budgets released into the atmosphere so that our estimates can be considered conservative.

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. Spatial distribution

Albeit our dataset is relatively limited compared to those that may be used e.g. in volcanic environments (e.g. [START_REF] Ciotoli | Tiber delta CO 2 -CH 4 degassing: A possible hybrid, tectonically active Sediment-Hosted Geothermal System near Rome[END_REF], such investigations on small areas can nevertheless yield to accurate estimates of both the gas emissions and the pattern of these emissions [START_REF] Etiope | Origin and flux of a gas seep in the Northern Alps[END_REF]. As may be viewed from data available in the supplementary information, our dataset is skewed and neither ranges under normal nor log-normal data distributions. The existing bimodal distribution is a consequence of the existence of two distinct phenomena. The first is the presence of the gas seep and the second are common biological processes in soils. The definition of threshold values is thus done using quantilequantile (Q-Q) plots (e.g. [START_REF] Ciotoli | Geostatistical analysis of soil gas data in a high seismic intermontane basin: Fucino Plain, central Italy[END_REF] further allowing the calculation of the quantity of gas emitted to the atmosphere using Graphical Statistical Analysis (e.g. Bloomberg et al., 2012, and references therein).

Variogram calculations were done for the data falling in the 20x20 m side square around the main seep (see supplementary information). All the data show an anisotropic pattern with the main axis elongated along the N45 direction with a length ranging from 6.5 to 13.7 m (9.2 m as a mean). Such a shape is in good agreement with the spatial extent of the gas vent that may be deduced directly from field measurements: high CH 4 values were found along the axis of the Vernant creek which flows to the NE and these high values were found in a narrow perimeter (less than 20 m of side).

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Data are used to build classed post maps (shown in supplementary information) and interpolated maps (Figure 9). Classed post maps are presented for the entire monitoring area with detail for the vicinity of the main seep whereas contours maps only refer to this area of reduced size. These maps highlight once again that the majority of soil gas anomalies are located within few meters around the gas seep. From the classed post maps it is not obvious to define whether the degassing is structured along pathways inherited from structural or morphological features or if the degassing occurs randomly close to the vent.

A better definition of the spatial distribution of the degassing processes is given by contour maps (Figure 9). The N45 direction, which is the average flow direction of the Vernant creek and which was also highlighted by variogram calculations, suggests the existence of preferential pathways inherited from structural constraints. A second direction of degassing can also be defined using the helium gas contents in soil probably related to the ability of helium to laterally diffuse as a consequence of its low molecular weight (e.g. [START_REF] Toutain | Gas geochemistry and seismotectonics: a review[END_REF]. This second direction is close to N170, a direction that is commonly found in the limestones of the Vercors Massif. As for classed post maps, flux data do not allow defining or refining the structural shape of the gas emissions to the surface. Contour maps are dominated by a few outliers (high fluxes) which induce a peak-like interpolation. This may be a consequence of site alterations in the since the 19 th century: the point where CO 2 and CH 4 emissions to the atmosphere are at their maximum value is indeed located on a break of slope, in front of the gabion wall that has been built to shelter the fountain, as visible on Figure 1 in the foreground of the panoramic photo.

Figure 9

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Gas emissions to the atmosphere

Even if the Fontaine Ardente gas seep has a rather limited spatial extensionother seeps may have considerably larger geographical manifestations (e.g. Etiope et al., 2011b) the evaluation of the CH 4 and CO 2 emissions to the atmosphere is interesting. As shown by the site monitoring, the main emission point is clearly the main seep itself, probably in strong relation with the existence of the former FA2 borehole (Figure 1) that focuses the gas emission by draining. Nonetheless, gas fluxes to the atmosphere are not restricted to the former borehole and exist over a larger area that extends northeastwards from the seep and their contribution has also to be quantified.

Estimations on the gas release at the seep mouth are done using the fire dynamics model of methane flux vs. flame size (Etiope et al., 2011b, and references therein) using the heat of combustion given by Etiope et al. (2013a) for CH 4 . Typology of the gas emission is taken from flame height and diameter at emission point estimated using Figure 8. As the main seep exhibits rapid changes in the flame height, the emission rate ranges between 15 kg per day (flame of 38 cm height and 12 cm on diameter) and 84 kg per day (flame of 77 cm height and 24 cm on diameter). Based on the flame size variations that may be observed at the seep at short time scale, the amount of CH 4 released in the atmosphere can be estimated to be somewhere between 40 and 50 kg per day. The estimated release rate in the 90's, obtained from pressure considerations [START_REF] Berthier | Etude et travaux pour le renforcement en gaz de la Fontaine Ardente -Commune du Gua (Isère)[END_REF], was 130 l/min giving approx. 120 kg of CH 4 per day. Our present estimates, based on the flame height, may then be underestimated e.g. as a consequence of short duration of the measurements. A significant uncertainty on the result has been pointed out by Etiope et al. (2011b) especially when the gas emission is Accepted Manuscript high and turbulent. This new estimate may also represent a real decrease in the feeding rate of the borehole, the gas flux migrating along this preferential pathway changing with time.

The case of the Fontaine Ardente degassing area is also interesting insofar as this gas seep is CO 2 -rich. Using the CH 4 flux estimated in the 90's and the proportion of CO 2 in the gas phase (approx. 11%), the quantity of CO 2 released in the atmosphere through the vent can be roughly estimated at 14 l/min i.e. 36 kg per day. Based on flame height estimates and still considering a proportion of CO 2 in the gas phase of 11%, then the estimate of the CO 2 release at the gas vent is close to 15 kg per day as a mean (range: 4.5 to 25 kg per day).

The CH 4 and CO 2 emissions from the soil are first quantified on a 84 m 2 area included in the larger 20 x 20 m area reported on Figure 9. The area is not a square area as the vicinity of the gas seep cannot be investigated (concrete plugs + gabion) but its geometry is likely to be representative as most of the efflux is located inside this small region (see Figure 9). This approach considers that the flux measured at one point is valid for 1 m 2 ; in case the grid mesh is 2 x 1 m, then the value assigned in between two real measurements is the mean value of these two measurements. The sum of the fluxes is 44.8 g/h for CH 4 and 88.3 g/h for CO 2 leading to releases of 1.07 kg of CH 4 and 2.2 kg of CO 2 per day. A refined estimate is obtained using the interpolated maps. Integration of the anomaly is done using Surfer® software and indicates, for the whole flux monitoring zone (240 m 2 ; Figure 9), a release of 2.83 kg of CH 4 and 6.27 kg of CO 2 per day. These estimates are in perfect agreement with those obtained using the 84 m 2 area [1.07 x (240/84) = 3.05] suggesting that even the basic method could lead to realistic evaluation in the present case. Nonetheless this flux estimate doesn't take into account gas emissions that may occur with the area covered by concrete plugs which may be fissured and permeable to gas.

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More sophisticated methods (e.g. [START_REF] Bloomberg | Soil gas flux exploration at the Rotokawa geothermal field and White Island[END_REF] or other spatial interpolation methods (e.g. [START_REF] Schroder | Evaluating the performance of soil flux surveys and inversion methods for quantification of CO 2 leakage[END_REF] may be used to quantify more precisely the leakage associated to the gas seep. Nonetheless, even well constrained data interpolation methods lead to biases in the emission estimates. During controlled release experiments, [START_REF] Schroder | Evaluating the performance of soil flux surveys and inversion methods for quantification of CO 2 leakage[END_REF] report differences of 20 to 45% using the method that best fits the data. In the present case, the main emission is the gas seep itself, and errors on the soil emissions may be considered as negligible. Methane emission at gas vent ranges from 18 to 44 tons per year (2016 flame estimate / 1990 data) and CO 2 emission ranges from 5.5 to 13 tons per year. Referring to our measurements, the CH 4 emission from soils represents less than 6% of the CH 4 emitted at the seep (1.03 ton per year) whereas the CO 2 emission from soils accounts for 42% of the CO 2 emitted at the seep (2.3 tons per year) including biogenic soil CO 2 . This biological component can be estimated to account for two thirds of the CO 2 release on the basis of volumes calculated using interpolated maps with a biological CO 2 production fixed at 1.25 g.m -2 .h -1 [START_REF] Werner | CO2 emissions from the Yellowstone volcanic system[END_REF].

Conjoint emission of CH 4 and CO 2 in gas seep is not a common phenomenon. In the European database from [START_REF] Etiope | Evidence of subsurface anaerobic biodegradation of hydrocarbons and potential secondary methanogenesis in terrestrial mud volcanoes[END_REF] or the data from other countries (Etiope et al., 2011a), which concern mud volcanoes, CO 2 -rich emissions (8% vol. or more) are rare and linked to thermogenic CH 4 . CH 4 seep rates associated to mini-and micro-seepages range from less than 10 up to 446 g/m 2 /day; associated CO 2 seep rates range from 3 to 535 g/m 2 /day [START_REF] Etiope | Geologic emissions of methane to the atmosphere[END_REF]. Maximum seep rates measured at point 17 are in this range (408 and 374 g/m 2 /day for CH 4 and CO 2 resp.). Abiotic methane emissions from ultrabasic rocks are associated with very low CO 2 emissions (Etiope et al., 2011b). In such cases, CH 4 emissions can reach hundreds of g per day per m 2 but CO 2 emissions only reach 65 g/m 2 /day when soil litter is present and 10 g/m 2 /day when soil is absent. Eternal flames Accepted Manuscript from the USA (Etiope et al., 2013a) or gas seeps reported in Switzerland [START_REF] Etiope | Origin and flux of a gas seep in the Northern Alps[END_REF] are not reported to be CO 2 -rich but CH 4 emissions can reach 40 kg per day and 10 kg per day respectively in the US and in Switzerland (strong miniseepages, from hundreds of g/m 2 /day up to a few kg/m 2 /day). Similarly, studies of the large thermogenic gas seepage of Katakolo (Greece) only report CH 4 emissions (from 5.4 to 98.6 kg per day onshore; Etiope et al., 2013b) albeit 6 to 8% vol. of CO 2 are monitored in the gas phase. Quantification of CO 2 emissions from natural methane seeps is rare, opposite to numerous studies of CO 2 emissions in volcanic or geothermal areas (e.g. Cardinelli et al., 2003b).

The Fontaine Ardente area has thus a specific behavior that may partly be a consequence of the presence of a draining structure (borehole) and related impact onto surface properties of soil. Methane is found at high concentrations in the soil with moderate to weak leakage into the atmosphere. Most of the CH 4 is vented through the borehole that constitutes by itself an important gas seep. According to the terminology of [START_REF] Etiope | Natural Gas Seepage, The Earth's Hydrocarbon Degassing[END_REF], miniseepage occurs around the gas vent but only accounts for 5% of the global budget of CH 4 emitted (compared to the lowest estimate of 18 tons/y). This differs from the gas emission reported in the Giswil by [START_REF] Etiope | Origin and flux of a gas seep in the Northern Alps[END_REF] where two thirds of the CH 4 are vented through two main seeps and one third is vented by miniseepage on a smaller area (115 m² compared to 240 m² at Fontaine Ardente area). The other specific character of the Fontaine Ardente seepage area is the richness of the gas phase in CO 2 . At the scale of the miniseepage area, this leads to a quantity of CO 2 emitted being more than two times that of CH 4 . This is a rather unique situation in the Alpine context and GHG emissions evaluations have to considerer both CH 4 and CO 2 gas phases to be pertinent.

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The other specificity of the Fontaine Ardente main seep is that it represents a unique analogue of a leakage situation. Hydrocarbon resources are rarely located at few meters below ground level. Here, a borehole intersects at c.a. 10 m depth geological formations which act as drain for CH 4 +CO 2 gas mixture. The borehole as preferential pathway focuses gas emissions to the surface, similarly to degassing pipes like those in lakes Nyos and Monoun [START_REF] Halbwachs | Degassing the "killer lakes" Nyos and Monoun, Cameroon, EOS[END_REF]. The gas which is not vented through the borehole diffuses laterally in the subsurface layers through pathways whose geometry is likely to be influenced by the network of local fractures.

Old pictures, previous to drilling, clearly show numerous small vents, whereas at present there is a unique and dominant gas seep with only minor contribution of diffuse seepage. This highlights the impacts of small-scale works on gas flux patterns.

Implications for methane or CO 2 baseline assessment and monitoring

Several tens of tons per year of both CH 4 and CO 2 gas phases are naturally released in the atmosphere by the Fontaine Ardente main seep thus contributing to GHG emissions over at least two millenaries. Apart from gas released at the borehole mouth, the amounts of CO 2 emitted through soil degassing may largely exceed the range of natural CO 2 flux (from 7.6 to 76 g/m²/day; Klusman, 2015, and references therein) with maximum values greater than 350 g/m²/day. Similarly, CH 4 emissions reaching 10 g per m² per day around the main seep are considerably higher than those measured over grasslands during summertime with 4.86±22.8 mg per m² per day (mean + standard deviation) reported by Klusman (2015, and references therein). Apart from this quantification of GHG emissions, a second interest of this study was to test monitoring techniques and strategies to better assess the natural gas Accepted Manuscript baseline for impact studies. The production of unconventional natural gas has profoundly transformed the global energy game over the past ten years. Environmental stakes of shale gas exploration and exploitation show some similarities with the CCS thematic, specifically with the risk of leakage from geological formations from which CH 4 can be produced or in which CO 2 can be stored, albeit the gas species of interest have very different behavior in e.g. their reactivity or dissolution properties.

The example of the Fontaine Ardente seepage area makes it possible to investigate simultaneously methane and CO 2 migration on a unique natural analogue and to learn more about containment monitoring in settings where potential pathways to the surface exist [START_REF] Jenkins | The state of the art in monitoring and verification-Ten years on[END_REF]. It constitutes furthermore one of the rare examples of "eternal flames" directly emerging from shale formations, together with those from the Appalachians (Etiope et al., 2013). More specifically, the Fontaine Ardente main seep case represents an example where the leakage is focused to a 1D pathway, represented by the former FA2 borehole.

Here, the leakage does not occur from the casing itself, because of the presence of a concrete platform at surface, or from defective completion of the casing, which will be the most frequent occurrence of failure. Leakage occurs through the open borehole and this drillhole has strongly modified the gas emission on the site. From a spotty and rather diffuse leakage situation, attested by old photographs, it has been changed to a gas vent, with the borehole as main emitting structure, with minor contribution of diffuse degassing. It is thus possible that a hypothetical closure of the well will induce a highly different distribution of the seepage with strongly enhanced diffuse degassing, probably occur along fractures/faults following the main structural directions [START_REF] Roberts | Surface controls on the characteristics of natural CO 2 seeps: implications for engineered CO 2 stores[END_REF][START_REF] Bond | The physical characteristics of a CO 2 seeping Accepted Manuscript fault: The implications of fracture permeability for carbon capture and storage integrity[END_REF].

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This natural analogue highlights the potential difficulties of baseline assessment and site monitoring during exploration or exploitation of unconventional gas. The expected spatial extension of the leakage will condition the resolution of the monitoring grid (e.g. observation boreholes) and wrong initial hypotheses may lead to an incapacity to detect any anomaly related to leakage. The strongly focused fluxes at the Fontaine Ardente seepage area, further enhanced by the drilling, are somewhat similar to what might be expected in case of shale gas well failure, with a very restricted area of leakage around the borehole.

Other leakage pathways, e.g. out-of-formation hydraulic fracturation, may lead to a much more diffuse pattern at surface, due to the predominantly diffusive or pervasive migration, from the host formation, via overlying deep aquifers to shallow formations. Timing of such migration can be also very different from direct pathways via leaking wells.

Transposed to a real shale gas or CCS site, the difficulty to anticipate and detect gas seepage areas will be a major challenge for baseline assessment and site monitoring. The area studied here is very small (around 240 m²), vegetated and set in a narrow thalweg. Surveying methods used directly at ground level may not be operative (too large mesh size) or they should require so much manpower that their cost will dramatically increase in case the grid has a very low mesh size. There is an extensive literature that points out the narrowness of gas emissions from faults and fractures and very recent data still agree with this feature (e.g. [START_REF] Klinkmueller | Linking gas fluxes at Earth's surface with fracture zones in an active geothermal field[END_REF][START_REF] Bond | The physical characteristics of a CO 2 seeping Accepted Manuscript fault: The implications of fracture permeability for carbon capture and storage integrity[END_REF]. The disadvantage of ground monitoring techniques may be overcome by airborne/UAV operated methods or methods having wider aperture (such as Eddy-Covariance -EC, laser operated methods…). Such methods are capable of giving better geographical coverage provided that they are appropriately settled. In the case of the EC method, changes of gas flux are monitored in an upwind direction. Nevertheless such methods may suffer from other disadvantages such as the decrease in sensitivity as a Accepted Manuscript function of distance to the emission point or the existence of photosynthetic processes on land surface or in the canopy [START_REF] Klusman | Surface geochemical measurements applied to monitoring, verification, and accounting of leakage from sequestration projects[END_REF].

When working in a blind manner during baseline studies, in a setting similar to that of Fontaine Ardente area, it seems very difficult to detect gas anomalies related to minor or even major gas seeps. Topography and vegetation play an important role in the detectability of anomalies. Natural seeps from shale formations, as it is the case for the Fontaine Ardente, are frequently located on creeks, themselves following fault structures (Etiope et al., 2013a).

Their detectability by landborne methods may be simply limited by logistic obstacles and sampling resolution. With the available techniques, it seems virtually impossible to discover unknown seeps through a systematic research. Etiope et al., (2013a) used the Tunable Diode Laser Absorption Spectroscopy (TDLAS) detectors in accumulation flux chambers to locate methane anomalies in ambient air but only in the vicinity of known eternal flames. Using a statistically stringent measurement pattern (grid) for landborne surveys would be hindered by [START_REF] Krautwurst | Methane emissions from a Californian landfill, determined from airborne remote sensing and in situ measurements[END_REF] the accessibility of natural areas with rough terrain and (2) the possible resolution of such a grid compared to the size of anomalies. Some hope can be placed in vehicle-born surveys using CRDS (Cavity Ring-Down Spectroscopy) detectors but this approach fits best to urban areas with a high road density [START_REF] Gallagher | Natural Gas Pipeline Replacement Programs Reduce Methane Leaks and Improve Consumer Safety[END_REF] as well as to anthropogenic sources identified previously to the survey [START_REF] Zazzeri | Plume mapping and isotopic characterisation of anthropogenic methane sources[END_REF]. It is unlikely that natural point-source emissions from gas seeps of a similar size to the Fontaine Ardente could be detected by such a technique. Airborne techniques, again using CRDS, are capable of detecting major methane plumes from known sources (e.g. landfills; [START_REF] Krautwurst | Methane emissions from a Californian landfill, determined from airborne remote sensing and in situ measurements[END_REF] at kilometric scale but, again, their use in the present case seems to be highly limited by the vegetation cover and the topography.

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To approximatively assess the vertical extension of the plume emitted by the Fontaine Ardente we used infrared cameras with specific lenses for targeting CH 4 or CO 2 gas phases (FLIR GF320). Such devices can image the gas plume emitted by the borehole and its dispersal in the atmosphere. Some wisps of gas do reach the base of the canopy, some meters above ground level, but they rather rapidly vanish afterwards (Figure 10). If the leakage flux rate is lower than the one of the Fontaine Ardente main seep (FA2 borehole), the detection by plume imaging becomes virtually impossible in practice. Thorough scanning of the miniseepage area did allow detecting a possible leakage area only after minutes of careful monitoring. It is established that such infrared cameras are not conceived for the detection of diffuse minor leaks in a natural environment (contrast problems) and that other monitoring techniques have considerably lower detection limits. Nevertheless, such cameras allow visualizing in real time the shape of the gas emissions especially vertically. Given our preliminary results, any other monitoring at the upper level of the canopy will be very difficult even using techniques with low detection ranges, including low-altitude airborne techniques.

Figure 10 The question on the best baseline and monitoring strategy for leakage assessment remains thus largely open. Landborne monitoring will always miss anomalies because of restricted accessibilityin our case the ability to perform accurate measurement in forested areaor because of lack of sensitivityin our case quick dilution of the gas in the atmosphere.

Consequently, appropriate monitoring strategy should rely on dense monitoring of structures and/or areas that are suspected to be potential pathways for leakage (known Accepted Manuscript faults, creeks…; Etiope et al., 2013a). As dense monitoring cannot cover a large area, monitoring should be adapted to the size and location of the expected leak. This requires a realistic conceptual model of the leakage mechanisms and pathways, relying on a statistically sound database of gas emissions in the context of the storage to monitor. The temporal resolution of monitoring and its duration is another crucial issue, depending on [START_REF] Krautwurst | Methane emissions from a Californian landfill, determined from airborne remote sensing and in situ measurements[END_REF] the temporal variations of the baseline emissions and (2) the kinetics of potential gas migration via the different suspected pathways (e.g. borehole failures, out-of-formation fracturation…, existence natural fractured zones). In case of leakage allegation, even if the alleged zone has not been surveyed before, the knowledge of local behavior of soil gas emissions will allow to adequately identify an anomaly in the soil gas signalsan approach defined as process-based approach by [START_REF] Romanak | Process-based approach to CO 2 leakage detection by vadose zone gas monitoring at geologic CO 2 storage sites[END_REF]. The example on how the leakage allegation in Weyburn was withdrawn shows how powerful such an approach can be. Furthermore, it may not only apply to soil gas using the established CO 2 vs. O 2 relationship but also to soil gas fluxes using the derivative relation CO 2 flux vs. O 2 depletion is the soil efflux.

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Figure captions.

Figure 1 A: Location of methane-rich gas seeps in the W-subalpine chains (Blavoux et al., 1990[START_REF] Etiope | Origin and flux of a gas seep in the Northern Alps[END_REF], Gal et al., 2017a) Figure 9: contour maps of soil gas species (concentrations in % vol.) and soil gas flux (in g/m 2 /h); kriging is done using data reported in supplementary information. Locations of measurements are reported on each map: area investigated during soil gas measurements: 400 m 2 (20x20 m); area investigated during soil flux measurements: 240 m 2 (dashed polygon in bottom maps). The red dot indicates the location of the "Fontaine Ardente". The striped area in bottom right map shows the emplacement of the Vernant creek bed; the dark grey area shows the area (84 m 2 ) used for quantification of the gas emissions (no flux measurement possible in the light grey area: old concrete plugs from former FA2 borehole).

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Figure 10: infrared camera monitoring using specific lens sensitive to CO 2 (FLIR); the picture in the right shows the conditions of monitoring, with shooting of the CO 2 plume just above the flame (as can be seen in the background). The insets in the left show, from bottom to top: the CO 2 plume directly issued from the burning seep; the CO 2 plume reaching the first branches in the trees (3 to 4 m above ground); the dilution of the CO 2 plume at around 5 m height. Uncorrected and P-T-normalized CO 2 fluxes only differ by 0.65% as a mean (uncorrected fluxes are slightly higher), the whole range being between -0.24% and +1.41% (69 CO 2 flux measurements). The influence of temperature and pressure changes is thus very low in the present case and the dataset can be considered as non-temperature or pressure dependent. Not all the fluxes were calculated using the initial slope obtained from the 3 minutes duration of the build-up of gas concentrations inside the chamber. Contrarily to soil gas concentration measurements, some flux measurements were subject to short time scale variations of both the CO 2 and the CH 4 fluxes. In such cases, the fluxes have been calculated considering the linear fit that was prevailing during the major part of the acquisition. These short term variations concerned only a minor part of the data (respectively 17 of 69 CO 2 flux measurements and 12 of 56 CH 4 flux measurements).

Oxygen concentration decreases have been monitored at 10% of the monitoring pointsthis may be a consequence of a too short duration of the measurement sequence, typically set at 3 minutes in order to remain in the domain of linear increase (or decrease) of the gas concentrations. This may also be simply the consequence of low soil O 2 (progressive dilution by an oxygen-poor gas in the chamber). The point (label 17 in Figure 1) where O 2 concentration decrease was at maximum is also the unique point where the CH 4 sensor reached saturation within only 2 minutes thus the linear regression from which the flux is calculated was established only over this 2 minutes period.

As earlier reported [START_REF] Chiodini | Soil CO 2 flux measurements in volcanic and geothermal areas[END_REF][START_REF] Bloomberg | Soil gas flux exploration at the Rotokawa geothermal field and White Island[END_REF]Sanci et al., 2009), soil moisture may also have a strong influence on the evaluation of gas emissions. The most obvious influence is via soil biological activity. In the present case -CH 4 -rich seepagethis influence may eventually exist for the CO 2 gas phase but is less obvious for the CH 4 gas phase Accepted Manuscript as there are no potentially CH 4 -emitting wetlands in the vicinity of the investigated sites.

Field monitoring was performed during summertime thus the influence of humidity may be expected as reduced. Soil moisture directly influences soil gas concentrations through gas dilution by the water vapor content of the gas phase. Values of relative humidity of the air inside the chamber generally fall between 55% and 70% after the purge of the flux chamber depending on the hour of the day. Humidity variations measured during 3 minutes did not exceed +10% of the initial value and the observed maximum value was 75%. No apparent influence of moisture on CH 4 , CO 2 or O 2 measurements has been stated during the graphical examination of raw data. This is consistent with the statement reported by [START_REF] Parkin | Sampling Protocols -Chapter 3 -Chamber-Based Trace Gas Flux Measurements[END_REF]. They found that an increase in relative humidity within a chamber from 0% to 100% will dilute the gas by only 3.05%. In the present case, it may be postulated that the influence of gas dilution will be considerably lower than 1% and can therefore be reasonably neglected.

Last, permanent visualization of the flow rate inside the system did not highlight any biases linked to the system itself during the measurement session. Nonetheless, such issues were sometimes encountered during CH 4 and H 2 measurements. A purge time is allocated between two measurements in order to refresh the air inside the system. The recovery of near atmospheric values is established in between 1 and 3 minutes for CO 2 and O 2 . This may be somewhat longer for CH 4 as some memory effect on the sensor may delay the return to near zero values in case the atmospheric CH 4 background is elevated. This residual effect, sometimes reaching 200 to 300 ppm, may have induced some underestimates of CH 4 fluxes (no flux was measured instead of measuring a small flux). From Figure A.2.a it can be seen that data distribution is frequently bimodal. These polymodal data distributions may exist e.g. for O 2 concentrations with a cut-off at 6.8% vol. or for CH 4 concentrations with a cut-off at 4.36%. One distribution is linked to the existence of the gas seep and the second is linked to common biological processes in soils. Consequently, the use of arithmetic parameters (mean and standard deviation), defined for a normal distribution, for the calculation of threshold levels cannot be used. showing heterogeneous data distribution.

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A.2 Statistics

As shown by

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Here we refer to the quantile-quantile (Q-Q) plots to define threshold values (Figure A.2.b).

The log-transformed data help to highlight changes in data distribution associated with strong deviations in the slope of the Q-Q plots. Anomaly thresholds are defined as follows:

17.5% vol. (O 2 ), 3.8 and 7.5% vol. (CO 2 ), 65% vol. (CH 4 ), 190 ppm (He) and 360 ppm (C 2 H 6 ). As CO 2 and CH 4 flux data are linearly distributed, a clear definition of a threshold is not possible albeit deviations around the 0.25 quantile may have some significance (1.23 and 0.21 g/m 2 /h resp.). A reference to the thresholds obtained using the basic approach (mean + 2SD) gives levels of 5.14 and 5.66 g/m 2 /h respectively and such levels are not meaningful (only 1 point is above the CO 2 threshold and 2 points above the CH 4 flux threshold). Interestingly, the CO 2 and O 2 soil gas threshold levels (3.8% and 17.5%) are in good agreement with the pivot values defined in Figure 3

  Figure 2

Figure 3

 3 Figure3

Figure

  Figure 3C indicates the existence of two distinct behaviors, evidenced mainly by the 2015

Figure 4 4

 4 Figure 4

Figure 5 A

 5 Figure 5

1 .

 1 Figure 6

  Figure 7

  plotted on the geological map of France (1:1 000 000, BRGM). Geological formations in blue range from Lower Jurassic to Upper Cretaceous; B: close-up view of the Fontaine Ardente du Gua gas seep: different views in 2015 and 2016 showing the bare soil zone where CH 4 emissions occur and the pebbly area where the Vernant creek usually flows (arrow indicates the location of the Fontaine Ardente corresponding to the location of former wellhead of FA2 borehole); C: location of the sampling points in 2015 and in 2016 (satellite photograph from Google Earth).

Figure 2 :

 2 Figure 2 : whisker plots for the 2015 and the 2016 surveys. CO 2 , O 2 and CH 4 soil gas

Figure 3 :

 3 Figure 3: A: O 2 vs CO 2 binary plot; B: location of points as a function of the slope of the O 2 vs

Figure 4 :

 4 Figure 4: A: CH 4 vs. O 2 binary plot; B: CH 4 vs. CO 2 binary plot. Plots are drawn using filtered

Figure 5 :

 5 Figure 5: A: left: relation between CO 2 and CH 4 fluxes for all measurements; right: location of

Figure 6 :

 6 Figure 6: A: O 2 depletion as a function of CO 2 flux or CH 4 flux; B: geographical location of

Figure 7 :

 7 Figure 7: typology of CO 2 concentration increase and CH 4 concentration increase (shaded

Figure 8 :

 8 Figure 8: chrono-photography of the "Fontaine Ardente" gas seep in the manner of Étienne Jules Marey, French physiologist (end of XIX th ) -06 October 2016.

Figure 1

 1 Figure 1 Click here to download high resolution image Accepted Manuscript

  Figure A.2.a, the whole dataset is skewed for all gases. Consequently, the data are not distributed under normal law (Gaussian distribution) nor lognormal law (logtransformed data show Gaussian distribution). The situation is better when the dataset is reduced; He and C 2 H 6 concentration distributions are then normal (p-values of 0.100 and 0.188 resp. using Kolmogorov-Smirnov test) and CO 2 and CH 4 fluxes distributions rather lognormal (p-values of 0.929 and 0.729 resp.). As the data measured hundreds of meters from the Fontaine Ardente main seep are likely not influenced by the gas seep (background data), the use of a dataset restricted to the vicinity of the seep (square of 20 m of side) is a way to better define data distribution. The only change by using sub-sampled dataset is the appearance of a slight tendency for CO 2 concentration to follow a normal distribution (pvalue of 0.05). Another change is a higher p-value for CO 2 fluxes (0.965).

Figure

  Figure A.2.a: histograms of soil gas data (2015 and 2016 data) and soil flux data (2016)

  thus giving more confidence on the concentrations from which influence of the gas seep can be noticed. Prior to establishing contour maps, variograms were calculated for each parameter on a restricted dataset (data falling in the 20x20 m side square around the seep). Results are given in Figure A.2.c and show the parameters of variogram calculation used for krigged maps.

Figure

  Figure A.2.d presents classed post maps built using threshold levels given in Figure A.2.b.

Acknowledgments

This research was conducted in the frame of the G-baseline project, co-funded by the French Research Agency (ANR-14-CE05-0050 grant) and the Natural Sciences and Engineering Research Council of Canada (NSERC grant n° 463605).

The representative of FLIR Systems France (Stéphane Thiebaut) is thanked for the demonstration of infrared cameras.

Accepted Manuscript

Supplementary material A.1 Methods

A.1.1 Soil gas measurements

In 2016, only IRGA device was used to monitor soil gas concentrations. In order to take into account of deviations from the gas concentrations resulting from these IRGA measurements, IRGA data were calibrated against gas chromatography data. Six samples, representative of the whole CH 4 concentration range as monitored in the field (from 0 to 77% vol.), were taken in appropriate containers (Isotubes®) and analyzed in the lab (Thermo Trace GC). This allowed correcting the bias inherent to IRGA analyses at high concentrations and to homogenize the 2016 dataset in view of comparing 2015 and 2016 campaigns. The same procedure was applied to CO 2 and O 2 gas concentrations. R 2 correlation coefficients between IRGA field measurements and laboratory analyses are 0.9952, 0.9941 and 0.9992 for CO 2 , O 2 and CH 4 respectively (n=6). The mean error, evaluated using the slope of the linear fit between IRGA field measurements and lab measurements, is close to +4.8% for CO 2 (field data are slightly higher), -16.5% for CH 4 (field data are strongly underestimated) and +2.8%

for O 2 (field data are slightly higher).

A.1.1 Soil gas flux measurements

The flux is calculated by considering the linear build-up of the gas concentration in the chamber during 3 minutes and by determining the initial slope of the gas concentration vs. time relationship, as described by many authors (e.g. [START_REF] Chiodini | Soil CO 2 flux measurements in volcanic and geothermal areas[END_REF]Etiope et al., 2013a, and references therein). Fluxes were calculated according to the equation detailed by Lewicki et al. (2005) to correct the influence of pressure and temperature changes.


Anes Lazri
email: anes.lazri@centralesupelec.fr

Mohamed Maghenem
email: mohamed.maghenem@cnrs.fr

Elena Panteley
email: elena.panteley@cnrs.fr

Antonio Loria
email: antonio.loria@cnrs.fr

Antonio Loria Global

Global Uniform Ultimate Boundedness of Semi-Passive Systems Interconnected over Directed Graphs

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ẋi = fi(xi) + ui xi ∈ R, i ∈ {1, 2, • • • , n}, (1)
where fi : R → R is continuous for all i ∈ {1, 2, • • • , n} and the control inputs are set to

ui := -γ n i=1 aij(xi -xj) ∀i ∈ {1, 2, • • • , n}, (2)
where γ > 0 is a coupling gain and aij ≥ 0 are interconnection weights. We assume that the closed-loop interconnected systems form a network with an underlying connected directed graph that contains a directed spanning tree. For these systems, we establish global uniform ultimate boundedness of the solutions, under the assumption that each system (1) defines a semi-passive [START_REF] Pogromsky | On difffusion driven oscillations in coupled dynamical systems[END_REF] map ui → xi. As a corollary, we also establish global uniform global boundedness of the solutions.

Preliminaries

Notations. For x ∈ R n , x denotes its transpose, |x| denotes its Euclidean norm, blkdiag{x} ∈ R n×n denotes the diagonal matrix whose ith diagonal element is the ith element of x. For a set K ⊂ R n , |x| K := min{|x -y| : y ∈ K} denotes the distance of x to the set K. For a symmetric matrix Q ∈ R n×n , λ i (Q) denotes the ith smallest eigenvalue of Q. For an invertible matrix M ∈ R n×n , M -or M -1 denotes its inverse. Given N ∈ R n×n , Ker(N ) := {v : N v = 0} denotes the kernel of N . A class K ∞ function α : R ≥0 → R ≥0 is continuous, strictly increasing, unbounded, and α(0) = 0. Furthermore α -denotes the inverse function of α.

On Some Classes of Matrices

A matrix M := [m ij ], (i, j) ∈ {1, 2, ..., n} 2 , is a Z-matrix if m ij ≤ 0 whenever i = j.
It is an M -matrix if it is a Z-matrix and its eigenvalues have non-negative real parts. Equivalently, M := λI n -B, where B is a non-negative matrix and λ ≥ ρ(B), where ρ(B) := max {|λ i (B)| : i ∈ {1, 2, ..., n}} is the spectral radius of B. M is a non-singular M -matrix if it is a Z-matrix and its eigenvalues have positive real parts. Equivalently, M := λI n -B, where B is a non-negative matrix and λ > ρ(B) > 0; see [START_REF] Chen | Controllability of formations over directed time-varying graphs[END_REF][START_REF] Javed | Excitation conditions for uniform exponential stability of the cooperative gradient algorithm over weakly connected digraphs[END_REF] for more details. 1

Graph Notions

A directed graph or a digraph G(V, E) is characterized by the set of nodes V = {1, 2, ..., n}, and the set of directed edges E. The edge set E consists of ordered pairs, of the form (k, i), that indicate a directed link from node k to node i. Given a directed edge (k, i) ∈ E, then node k is called an in-neighbor of node i. We assign a positive weight a ik to each edge (k, i). That is, a ik = 0 if (k, i) is not an edge. The Laplacian matrix of a digraph is given by

L :=        d 1 -a 12 • • • -a 1n -a 21 d 2 • • • -a 2n . . . . . . . . . . . . -a n-11 • • • d n-1 -a n-1n -a n1 • • • -a nn-1 d n        =: D -A, (3)
where

d i := n j=1
a ij for all i ∈ {1, 2, ..., n}, D is the diagonal part of L and A is called the adjacency matrix.

A digraph is strongly connected if, for any two distinct nodes i and j, there is a path from i to j. The Laplacian matrix of a strongly connected graph admits λ 1 (L) = 0 as an eigenvalue with the corresponding right and left eigenvectors

1 n = 1 1 • • • 1 and v o := v 1 v 2 • • • v n ,
respectively, where v i > 0 for all i ≤ n.

Graph and Matrix Decomposition

Suppose that the digraph G is connected and contains a spanning tree. Then, it admits a decomposition into a leading strongly connected subgraph G = Ø and a subgraph G f := G\G l of followers; namely, the agents that do not belong to the leading component, and which we call the follower agents. In this case, up to a permutation, the Laplacian L admits the lower-block decomposition

L = L 0 -A f M f , (4)
where L := D -A ∈ R n ×n is the Laplacian matrix of the strongly connected component G , the lower-left block A f ∈ R n f ×n-n f , n f := n -n , is a non-negative matrix, and the lower-right block M f ∈ R n f ×n f is a non-singular M-matrix. The block M f can be seen as the sum of the Laplacian matrix L f corresponding to G f and a diagonal matrix D f gathering the weights of the interconnections between nodes in G and the nodes in G f . That is,

M f = L f + D f , where L f = D f -A f .

Lyapunov Analysis of a Directed Graph

Consider a network of n single integrators of the form ẋi = u i interconnected according to the classical consensus protocol

u i := - n i=1 a ij (x i -x j ) ∀i ∈ {1, 2, • • • , n}.
In closed loop, the network is governed by the linear system ẋ = -Lx, where L ∈ R n is the Laplacian matrix of a connected di-graph G that contains a directed spanning tree. According to Section 1, we can decompose the state x into x := x l x f , where x l ∈ R n l gathers the states of the leading component and is governed by

Σ : ẋ = -L x ,
and the non-leading component whose state is

x f ∈ R n f , are governed by Σ f : ẋf = -M f x f ,
on the manifold {x f = 0}. In the rest of this section, we overview some Lyapunov-function constructions allowing to prove uniform exponential stability of A for Σ , where

A := {x l ∈ R n l : x l1 = x l2 = • • • = x ln }, (5)
and exponential stability of the origin for Σ f .

1.4.1 Proof of uniform exponential stability of A for Σ let v o := v 1 v 2 • • • v n be a left eigenvector associated to λ 1 (L ) = 0 and V o := blkdiag{v o }. Based on Lemma 1 in the Appendix, Q o := L V o + V o L
is symmetric and positive semi-definite, and its kernel is spanned by 1 n . Then, the derivative of the Lyapunov function candidate W (x ) := x V o x , along the solutions to Σ , satisfies

Ẇ (x ) = -x (L V o + V o L )x ≤ -λ 2 (Q o )|x | 2 A . Now, we let Z(x ) := x -1 n v o x V o x -1 n v o x ,
which is positive definite. Its derivative along the solutions of ẋ = -L x satisfies

Ż(x ) = -x Q o x ≤ -λ 2 (Q o )|x | 2 A . (6)
To obtain the previous expression we used

v o L = 0, v 1 1 n = 1 and that 1 n is in the kernel of I n -1 n v o . Moreover, I n -1 n v o
is the Laplacian matrix of an all-to-all graph; hence, 1 ns spans the kernel of I n -1 n v o . Therefore, there exist z, z > 0 such that

z|x | 2 A ≤ Z(x ) ≤ z|x | 2 A ∀x ∈ R n . (7)
Uniform exponential stability of A from ( 6) and ( 7) and standard Lyapunov-stability theory.

Y (x f ) := x f R f x f , where R f := blkdiag M f -1 n f blkdiag M -1 f 1 n f -1
, which is positive definite. Furthermore, along the solutions to Σ f , we have

Ẏ (x f ) = -x f [M f R f + R f M f ]x f . Now, since (M f R f +R f M f ) is positive definite, exponential stability of the origin for Σ f follows.

Problem formulation

Consider the systems (1)-( 2), with γ > 0 and a ij ≥ 0. Then, defining

x := [x 1 • • • x n ] , and F (x) := f 1 (x 1 ), f 2 (x 2 ), • • • , f n (x n ) ,
we may write the closed-loop system in compact form as

ẋ = F (x) -γLx, ( 8
)
where L is defined as in (3). This is a networked system with an underlying topology that may be represented by a graph G.

Assumption 1

The graph G is connected and contains a directed spanning tree.

•

We are interested in verifying the following two boundedness properties for (8).

(P1) Global Uniform Boundedness (GUB). The solutions t → x(t) to (8) are globally bounded, uniformly in γ, if, for every r o > 0 and γ o > 0, there exists

R = R(r o , γ o ) ≥ r o such that, for all γ ≥ γ o , |x(t o )| ≤ r o ⇒ |x(t)| ≤ R ∀t ≥ 0.
(P2) Global Uniform Ultimate Boundedness (GUUB). The solutions t → x(t) to (8) are ultimately bounded, uniformly in γ, if given γ o > 0, there exists r = r(γ o ) > 0 such that, for all r o > 0, there exists

T = T (r o , γ o ) ≥ 0 such that, for all γ ≥ γ o , |x(t o )| ≤ r o ⇒ |x(t)| ≤ r ∀t ≥ T.
To verify the latter two properties, we make the following assumption on the individual nodes' dynamics in [START_REF] Pogromsky | Passivity-based design of synchronizing systems[END_REF].

Assumption 2 (State strict semi-passivity) For each i ∈ {1, 2, ..., n}, the input-output map u i → x i defined by the dynamics (1) is state strict semipassive [START_REF] Pogromsky | Passivity-based design of synchronizing systems[END_REF]. Furthermore, there exists a continuously differentiable storage function V i : R n → R + , a class K ∞ function α i , a constant ρ i > 0, a continuous function H i : R → R, and a continuous function

ψ i : R ≥0 → R >0 , such that α i (|x i |) ≤ V i (x i ), Vi (x i ) ≤ 2u i x i -H i (x i ), (9)
and

H i (x i ) ≥ ψ i (|x i |) for all |x i | ≥ ρ i . • Remark 1
The property described in Assumption 2 is called strict quasipassivity in [START_REF] Polushin | Strict quasipassivity and ultimate boundedness for nonlinear control systems[END_REF]. In [START_REF] Pogromsky | On difffusion driven oscillations in coupled dynamical systems[END_REF] the authors define a similar concept named strict semi-passivity, but radial unboundedness of the storage function is not imposed. See also [START_REF] Pogromsky | Passivity-based design of synchronizing systems[END_REF]. •

Main result

Theorem 1 (Uniform ultimate boundedness) The solutions of the networked system (1)-( 2) are globally uniformly ultimately bounded, i.e., Property (P2) holds, if Assumptions 1 and 2 are satisfied.

Proof: Under Assumption 1, the Laplacian matrix L admits a permutation, such that (4) holds. Therefore, the state x may be decomposed into x := [x x f ] and the system (8) takes the cascaded form

ẋ = f (x ) -γL x , f (x ) := f 1 (x 1 ) • • • f n (x n ) (10a) ẋf = f f (x f ) + γA f x -γM f x f , f f (x f ) := f n +1 (x f1 ) • • • f n +n f (x fn f ) . (10b)
Equation (10a) corresponds to the dynamics of a leading component, a networked system with an underlying strongly connected graph G , and a follower component, with dynamics (10b). The proof of the statement is constructed using a cascades argument and proving, firstly, global uniform ultimate boundedness for the solutions of (10a) and, consequently, the same property for (10b).

To that end, let r o > 0 be arbitrarily fixed and let |x(0

)| ≤ r o . Then, |x (0)| ≤ r o and |x f (0)| ≤ r o .
1) Uniform ultimate boundedness for the leading component: after Assumption 2, for each i ∈ {1, 2, ..., n }, there exists a storage function V i such that its total derivative along the trajectories of (1) satisfies

Vi (x i ) ≤ 2u i x i -H i (x i ), H i (x i ) ≥ ψ i (|x i |) ∀|x i | ≥ ρ i . (11) Next, let W (x ) := n i=1 v i V i (x i )
, where v i corresponds to the ith element of v o , which is the left eigenvector associated to the zero eigenvalue of L . Since the graph G is strongly connected, then v i > 0 for all i ≤ n , so W is positive definite and radially unbounded. Now, from (11), we obtain

Ẇ (x ) ≤ 2 n i=1 v i u i x i - N i=1 v i H i (x i ), ∀x ∈ R n . ( 12
)
The first term on the right-hand side of (12) satisfies

n i=1 v i u i x i = u V o x , (13)
where V o := blkdiag{v o } and, since u = -γL x , it follows that

Ẇ (x ) ≤ - n i=1 v i H i (x i ) -γx [L V o + V o L ]x ≤ - n i=1 v i H i (x i ) -γx Q o x , (14)
with

Q o := V o L + L V o ,
which is positive semi-definite-see Lemma 1 in the Appendix. Furthermore, we note that

-x Q o x = -x -1 n 1 n x /n Q o x -1 n 1 n x /n ≤ -λ 2 (Q o )|x | 2 A ,
where |x | A denotes the distance of x to the set A and λ 2 (Q o ) is the second smallest eigenvalue of Q o . Now, on one hand, we have that v i > 0 for all i ∈ {1, 2, . . . , n} and, on the other,

-H i (x i ) > 0 only if |x i | ≤ ρ i . Therefore, the constant H := - n i=1 max |xi|≤ρi v i H i x i > 0. Therefore, after (14), we get Ẇ (x ) ≤ H -γλ 2 (Q o )|x | 2 A ∀ x ∈ R n . ( 15
)
In turn, given γ o > 0 and > 0, for all γ ≥ γ o , we have

Ẇ (x ) ≤ H -γ o λ 2 (Q o )|x | 2 A ≤ - ∀x / ∈ C, (16)
where

C := x ∈ R n : |x | A ≤ √ n R e := + H γ o λ 2 (Q o )
.

Next, let ρ := arg max i∈{1,2,...,n } ρ i and

B β := {x ∈ R n : |x | ≤ β := √ n ρ + 2R e )}.
Note that for all x / ∈ B β , we have |x | > √ n (ρ + 2R e ) and, for all x ∈ C\B β ,

|x | > √ n (ρ + 2R e ) and |x | A ≤ √ n R e . ( 17
)
Furthermore, we use the fact that x = 1 n (1 n x )/n + x -1 n (1 n x )/n , and the fact that

|x | A = |x -1 n (1 n x )/n |, to conclude that |x | ≤ |x | A + |1 n x |/ √ n . (18)
Now, combining ( 17) and ( 18), we conclude that for all

x ∈ C\B β , √ n (ρ + 2R e ) < |x | ≤ |x | A + |1 n x |/ √ n ≤ √ n R e + |1 n x |/ √ n . (19)
So, for all x ∈ C\B β , |1 n x |/n > ρ + R e . Next, we use the fact that

x i = 1 n x /n + x i -1 n x /n ∀i ∈ {1, 2, ..., n } to conclude that |x i | > |1 n x |/n -| x i -1 n x /n |. Hence, |x i | > ρ + R e - √ n R e > ρ ∀i ∈ {1, 2, ..., n }
for all x ∈ C\B β . The latter, under Assumption 2, implies that

- n i=1 v i H i (x i ) ≤ - n i=1 v i ψ i (|x i |) ≤ 0 ∀x ∈ C\B β .
As a result, setting Ψ(x ) :

= n i=1 v i ψ i (|x i |)-note that Ψ is continuous and positive-we conclude that Ẇ (x ) ≤ -Ψ(x ) ∀x ∈ C\B β .
Combining the latter inequality to (16), we conclude that

Ẇ (x ) ≤ -min{Ψ(x ), } ∀x ∈ R n \B β .
The latter is enough to conclude global attractivity and forward invariance of the set

S σ := {x ∈ R n : W (x ) ≤ σ}, σ := max{W (y) : y ∈ B β }.
Furthermore, since W : R n → R ≥0 is continuous and B β is bounded, we conclude that σ is well defined. Consequently, the ultimate bound is

r := min i {α i } - (σ),
where, with an abuse of notation, min i {α i } corresponds to the function s → ψ(s) defined as ψ(s) := min i {α i (s)} for each s ≥ 0 and α i is defined in Assumption 2, so ψ : R ≥0 → R ≥0 is strictly increasing and radially unbounded, hence, globally invertible. Thus, W (x ) ≤ σ implies that |x | ≤ r .

Next, we compute an upperbound T (r o , γ o ) on the time that the solutions to (10a), with γ ≥ γ o and starting from

B ro := {x ∈ R n : |x | ≤ r o }, take to reach the compact set B β ⊂ S σ .
For this, we assume without loss of generality that r o ≥ β, and we define

ro := min{min{Ψ(x ), } : |x | ≥ β, x ∈ S σo },
where

S σo := {x ∈ R n : W (x ) ≤ σ o }, σ o := max{W (y) : y ∈ B ro }. (20)
Clearly, S σo is compact; hence, ro is positive.

Therefore, along every solution t → x (t) to (10a) starting from x (0) ∈ B ro \B β , we have Ẇ (x (t)) ≤ro , up to the earliest time when x reaches B β . For any earlier time, we have

W (x (t)) ≤ -ro t + W (x (0)), (21)
so we can take T (r o , γ o ) = σ o / ro . Clearly, T depends only on (r o , γ o ) and r depends only on γ o . Thus, the ultimate bounded guaranteed for the solutions of (10a) is uniform in γ.

2) Uniform ultimate boundedness for the follower dynamics: following up the previous computations and arguments, establish global uniform ultimate boundedness for the non-leading component, determined by (10b).

Using Lemma 2, we conclude that the matrices S := P M f + M f P and P := blkdiag M f -1 n blkdiag M - f 1 n are symmetric and positive definite. We also note that P is diagonal. Then, let p i , for i ∈ {1, 2, ..., n f }, be the ith diagonal element of P . In addition, let Z(x f ) :=

n f i=1 p i V i (x f i )
. Its total derivative along the trajectories of (10b) satisfies

Ż(x f ) ≤ - n f i=1 p i H i (x f i ) -γx f [P M f + M f P ]x f + 2γx f [P A f ]x . ( 22
)
On one hand, we already established the existence of r (γ o ) > 0 and T (γ o , r o ) such that

|x (t)| < r ∀t ≥ T .
On the other, for all |x | ≤ r l ,

Ż(x f ) ≤ H f -γλ 1 (S)|x f | 2 + 2γ pr |x f |, (23)
where p :

= |P A f | and H f := n f i=1 max |xi|≤ρi {v i H i x f i }. Now, from this and (22), we obtain Ż(x f ) ≤ H f -γx f Sx f + 2γx f [P A f ]x ≤ H f -γ x f Sx f /2 -2x A f P S -P A f x .
At the same time, integrating (15), we obtain that, for each t ∈ [0, T ],

W (x (t)) ≤ H T + W (x (0)) ≤ H T + σ o ,
where σ o comes from (20). Defining

R := min i {α i } - (H T + σ o ) , we have Ż(x f ) ≤ H f -γ λ 1 (S)|x f | 2 /2 -2|A f P S -P A f |R 2 , for all |x | ≤ R .
Note that, for all x f such that

|x f | 2 > d 2 f := 4|A f P S -P A f |R 2 λ 1 (S) + 42H f λ 1 (S)γ o , Ż(x f ) ≤ 0. This implies that, for all t ∈ [0, T ], Z(x f (t)) ≤ max {σ f o , σ f } , σ f := max{Z(x f ) : |x f | ≤ d f } σ f o := max{Z(x f ) : |x f | ≤ r o }. (24)
In turn, for each t ∈ [0, T ],

|x f (t)| ≤ ro := n f i=1 p i min i {α i } - (max {σ f o , σ f }) . (25)
Clearly, the previous upper bound is uniform in γ ≥ γ o .

Next, we focus on the solutions' behaviour after T (i.e., once |x | ≤ r ). Given > 0, we see that, for all γ ≥ γ o and for all x f and x such that

|x f | > β 1 := 1 + 2pr λ 1 (S) + + H f γ o λ 1 (S)
and |x | ≤ r , after (23), we conclude that Ż(x f ) ≤ -. Furthermore, |x (t)| ≤ r for all t ≥ T , then the set

S σ1 := {x f ∈ R n f : Z(x f ) ≤ σ 1 }, σ 1 := max{Z(y) : y ∈ B β1 }, B β1 := {x f ∈ R n f : |x f | ≤ β 1 },
is attractive and becomes forward invariant after time T .

Since Z : R n → R ≥0 is continuous and B β1 is bounded, we conclude that σ 1 is well defined. As a result, the ultimate bound for x f (t) is

r f = n f i=1 p i min i {α i } - (σ 1 ). Indeed, Z(x f ) ≤ σ 1 implies |x f | ≤ r f .
Finally, as for t → x (t) we give next an upperbound, denoted by T f (r o , γ o ), on the time that the solutions to (10b), with γ ≥ γ o and starting from

B ro := {x f ∈ R n : |x f | ≤ r o }, take to reach B β1 ⊂ S σ1 .
Let a solution t → x f (t) to (10b) starting from x f (0) ∈ B ro . Now, we use the fact |x f (T )| ≤ ro with ro coming from (25) and ro is uniform in γ. As a result, along the solution t → x f (t), we have Ż(x f (t)) ≤ -from T and up to when it reaches B β1 for the first time after T . Hence, before reaching B β1 , we have Z(x f (t)) ≤ -t + Z(x f (T )) (26) and, thus, using (24), we can take T f = T + max{σ f o , σ f }/ . Clearly, T f and r f depend only on (r o , γ o ). Thus, the ultimate bounded guaranteed for the solutions of (10b) is also uniform in γ.

Corollary 1 (Uniform boundedness) Under Assumptions 1 and 2 the solutions of the closedloop system in (8) are globally uniformly bounded, i.e., Property (P1) holds.

In turn, for each t ≥ 0, we have

|x f (t)| ≤ R f := n f i=1 p i min i {α i } - (max {σ f o , σ f }) . (28)

Proof:

  The statement of Theorem 1 holds, therefore, given r o > 0 and γ o > 0, for all γ ≥ γ o , we have|x (0)| ≤ r o =⇒ |x (t)| ≤ r (γ o ) ∀t ≥ T (r o , γ o ).Furthermore, we were able to show that on the interval [0, T (r o , γ o )], we haveW (x (t)) ≤ H T + W (x (0)).Hence, if we let σ := max{W (y) :|y| ≤ r o }, it follows that |x (t)| ≤ R := min i {α i } σ + H T + r ∀t ≥ 0.Next, for the solutions to (10b), for any γ > γ o and |x f (0)| ≤ r o , we know that|x f (t)| ≤ r f ∀t ≥ T f (γ o , r o ).At the same time, from the previous proof, we know thatŻ(x f ) ≤ H f -γ λ 1 (S)|x f | 2 /2 -2|A f P S -P A f |R 2 .As a result, when|x f | 2 > d 2 f := 4|A f P S -P A f |R 2 λ 1 (S) + 42H f λ 1 (S)γ o , then Ż(x f ) ≤ 0. Hence, for each t ≥ 0, Z(x f (t)) ≤ max {σ f o , σ f } , σ f := max{Z(x f ) : |x f | ≤ d f }, σ f o := max{Z(x f ) : |x f | ≤ r o }.

  AbstractWe analyse the solutions of networked heterogeneous nonlinear systems1

	Global Uniform Ultimate Boundedness of
	Semi-Passive Systems Interconnected over
	Directed Graphs
	Anes Lazri Mohamed Maghenem Elena Panteley Antonio Loría *
	September 21, 2023

Appendix

The following lemma is proposed in [START_REF] Javed | Excitation conditions for uniform exponential stability of the cooperative gradient algorithm over weakly connected digraphs[END_REF], see also [START_REF] Qu | Cooperative control of dynamical systems: applications to autonomous vehicles[END_REF].

Lemma 1 Let L ∈ R n×n be the Laplacian matrix of a directed and strongly connected graph.

the left eigenvector of L associated to the null eigenvalue of L.

Then, the vector v has strictly positive entries and, for

The next result can be deduced from [6, Section 4.