Laurent André

Mohamed Azaroual
email: m.azaroual@brgm.fr

Christian Bernstone
email: christian.bernstone@vattenfall.com

Andrea Wittek
email: andrea.wittek@vattenfall.de

Laurent Andre
email: l.andre@brgm.fr

MODELING THE GEOCHEMICAL IMPACT OF AN INJECTION OF CO 2 AND ASSOCIATED REACTIVE IMPURITIES (SO 2 ANDO 2 ) INTO A SALINE RESERVOIR

Keywords: gas storage, CO 2 -SO 2 -O 2 gas mixtures, saline reservoir, coupled modeling, geochemical reactivity

published or not. The documents may come

Modeling the geochemical impact of an injection of CO2 and associated reactive impurities (SO2 and O2) into a saline reservoir

INTRODUCTION

Carbon dioxide capture and storage (CCS) is anemerging technology for reducinggreenhouse gas emissions and addressing climate change. CO 2 geological storage in deep reservoirs isa promising greenhouse gas sequestrationmethod. However, in order to ensureeffective containment, investigations need to be carried out on reservoir behavior when subjected to physical and chemical perturbations causedby CO 2 . Numerical studies have shownthat massive and continuous injection of pure CO 2 leads toa disequilibrium of the physical (temperature, gas saturation, pressure, capillary pressure, etc.) [START_REF] Bielinski | monitoring of CO 2 plumes during storage in geological formations using temperature signals: numerical investigation[END_REF]Yamamoto et al., 2009;[START_REF] Hansen | Monitoring CO 2 injection into a fluvial brinefilled sandstone formation at the Snøhvit field[END_REF]and geochemical (dissolution of supercritical CO 2 in the brine, pH variations, dissolution/ precipitation reactions of the porous rock minerals) characteristics of the host reservoir [START_REF] Kharaka | Gas-water-rock interactions in Frio Formation following CO 2 injection: Implications for the storage of greenhousegases in sedimentary basins[END_REF][START_REF] Audigane | Two-dimensional reactive transportmodeling of CO2 injection in a saline Aquifer at the Sleipner site, North Sea[END_REF][START_REF] Gaus | Geochemical and solute transport modelling for CO 2 storage, what to expect from it ?[END_REF][START_REF] André | Numerical modeling of fluid-rock chemical interactions at the supercritical CO 2 -liquid interface during supercritical carbon dioxide injection into a carbonated reservoir, the Dogger aquifer (Paris Basin, France)[END_REF][START_REF] André | Numerical Simulations of the Thermal Impact of Supercritical CO 2 Injection on Chemical Reactivity in a Carbonate Saline Reservoir[END_REF][START_REF] Okuyama | Reactive transport simulation study of geochemical CO 2 trapping on the TokyoBay model -Withfocus on the behavior of dawsonite[END_REF].

If CO 2 injection can disrupt the physico-chemical equilibrium of the deep reservoir, the presence of impurities in the CO 2 gas mightalso impair the capture, transport, and injectionprocesses. Some associated reactive gases, i.e. chemical species other than CO 2 in the injected streamnamely SOx, NOx, H 2 S, or O 2 , might require adaptations of the processes used for theinjection and storageof a pure CO 2 stream.

The presence of associated gases in the CO 2 stream in relatively high proportions can affect the compressibility of the injected gas and reduce the CO 2 storage capacity of the reservoir.This is due both to the space taken up by these gases and the unfavorable volume balance of induced geochemical reactions. The effect of impurities on storage capacity has been evaluated using different gas mixture scenarios (IEAGHG, 2011) and results show that, for the unlikely case of high-impurity streams (e.g. 15% non-condensable gases like N 2 , Ar, O 2 ), the storage capacity can be reduced by 10 to 40%, depending on the storage depth (i.e. pressure conditions). An opposite effect is however observed for supercritical CO 2 mixed with a condensable gas such as SO 2 :2.9% SO 2 in the CO 2 stream can increase the storage capacity by 5% at 110 bars because of intrinsic SO 2 properties (such as the critical temperature of SO 2 , which is higher than that of CO 2 ). Furthermore, depending on the type of geological storage, the presence of ancillary gases mighthave other specific effects such as trapping performances. When CO 2 is storedin deep saline formations, the presence of gas impurities affects both the gas solubility in the aqueous phase andthe rate and amount of CO 2 stored through mineral dissolution and precipitation. The presenceofSO x andNO x , which causes a dramatic pH decrease of brine, decreases CO 2 solubility.

The behavior of these accessory gases in the context of CO 2 storagehas only recentlyinterested the scientific community.Previously, the most important studieswere carried outby the oilindustry and concerned the storage of sourgases in geological structures. [START_REF] Azaroual | Thermo-kinetic modelling of mass-exchange mechanisms between water-rock-gas induced by acid gas injection in carbonate reservoirs[END_REF] analyzed the impact of the injection of a gaseous H 2 S-CO 2 mixture (50:50) in a carbonate reservoir. This high proportion ofH 2 S was selected to highlight the chemical reactionslikely to occur within the reservoir, along with their impact on reservoir properties and integrity. [START_REF] Gunter | Aquifer disposal of acid gases: modelling of water-rock reactions for trapping of acid wastes[END_REF][START_REF] Gunter | Acid gas injection in the BrazeauNisku Q carbonate reservoir: Geochemical reactions as a result of the injection of an H 2 S-CO 2 waste stream[END_REF] and [START_REF] Bachu | Acid gas injection in the Alberta basin, Canada: a CO 2 storage experience[END_REF] used a coupled numerical and experimental approach to studythe injection of gas mixtures (CO 2 -SO 2 and CO 2 -H 2 S) in Canada, but with H 2 S and SO 2 proportions higher than those observed in CO 2 storage.

When CO 2 is stored in deep geological reservoirs, the gas compositionsare expected tovary depending on the treatment processesused and the industrial origin of the stream [START_REF] Anheden | CO 2 quality requirement for a system with CO 2 capture, transport and storage[END_REF]. Over the last decade, R&D efforts have been made to reduce the energy penalty, and CO 2 capture in power plants will result in CO 2 streams with high CO 2 contents. Today's technologies would give >95% CO 2 purity.However, despite continuous efforts to develop costeffective capture processes to clean the CO 2 stream before transport and injection, traces of accessory gases are notentirely eliminated. For example, at a certain level of cleaning of theoxycombustion process, traces of argon and oxygen are assumed to remain in the injected gas, as well as oxidized forms of sulfur,SO x ,and nitrogen, NOx (IEAGHG, 2011).

Oxygen sulfide (SO 2 ) is one of theancillary gasesthat is most often studied. Palandri and Kharaka (2005a) were among the first to use a numerical approachto study the impact of SO 2 on ferric iron-bearing sediments. They showed that the SO 2 content in the CO 2 -dominated gas phase couldinvolve different geochemical pathways. If there is enoughSO 2 in the gas phase, dissolved iron is totally reduced as Fe 2+ , which reacts with dissolved CO 2 to form siderite. However, if there is not enoughSO 2 , the iron is only partially reduced and less CO 2 is trapped by siderite precipitation. [START_REF] Knauss | Evaluation of the impact of CO 2 , co-contaminant gas, aqueous fluid and reservoir rock interactions on the geologic sequestration of CO 2[END_REF] also used a numerical approach toinvestigate the impact of SO 2 (and H 2 S)on the mineralogical assemblage modification of a geologic reservoir based on observations of the Frio pilot site (USA). The presence of SO 2 with CO 2 in the injected stream generates significant impacts, in particular near the injection well. The lowpH resulting from SO 2 and CO 2 dissolution leads to massive carbonate dissolution (non-stability of calcite) and, under oxidizing conditions, to the precipitationof the sulfate mineralanhydrite. [START_REF] Xu | Numerical modeling of injection and mineral trapping of CO 2 with H 2 S and SO 2 in a sandstone formation[END_REF] developed a more advanced conceptual model for the injectionof CO 2 with SO 2 (andH 2 S) in a sandstone formation. These authors compared numerical modeling results for pure CO 2 and CO 2 -SO 2 injections. They showed that theacidified zone is extensivewhen SO 2 is co-injected with CO 2 . They also reported the potential precipitation of alunite and traces of anhydrite and pyrite,whereasCO 2 trapping is due to the precipitation of small quantities of ankerite, dawsonite, and siderite.

Otherlaboratory experiments have been carried out undertemperature and pressure conditions similar to thoseof geological storage in order to gather more information on the impactof the coinjection of SO 2 and CO 2 on water and minerals. Palandri et al. (2005b) carried out laboratory experiments (at 150°C and 300 bars) to test their numerical approach (Palandri and Kharaka, 2005a). These confirmed that the presence of SO 2 in the gas phase can favor the precipitation of stable siderite and, depending ontheproportion of SO 2 , the potential precipitation of metastable pyrite and elemental sulfur (S°). [START_REF] Murphy | Ferrihydrite phase transformation in the presence of aqueous sulfide and supercritical CO 2[END_REF] and [START_REF] Murphy | Hematite reactivity with supercritical CO 2 and aqueous sulfide[END_REF] focused their experimental work on the reactivity of ferrihydrite and hematite nanoparticles in the presence of supercritical CO 2 and sulfide-bearing solutions. They showed that both iron-bearing minerals react to form iron carbonate (siderite), iron sulfide (pyrite) and in specific conditions (at 100 °C with supercritical CO 2 ) elemental sulfur (S°). They alsoemphasized that the kinetics of the reaction are relatively rapid at temperatures ranging between 70 and 100°C. KummerowandSpangenberg (2011) studiedthe injection of pure CO 2 and a CO 2 -SO 2 mixture (99:1 vol/vol)at laboratory scale on sandstone media from the Ketzin reservoir (Germany). They observedthat the injection of pure CO 2 does not drastically alter rock properties and identifiedno mineral reactions. However, with the CO 2 -SO 2 mixture, rockproperties are greatlymodified due, in particular, to the dissolution of intergranularcementation. According to XRD observations, the decrease inrock permeability is not due to precipitation of secondary minerals but to the remobilization of endogen fine particles in the samples. [START_REF] Erzinger | Experimental studies on the interaction of scCO 2 and scCO 2 -SO 2 with rock forming minerals at conditions of geologic carbon storages-First results[END_REF] also observedan increaseinthe chemical reactivity when a supercritical CO 2 -SO 2 mixture (99.5:0.5vol/vol) reacts withdifferentminerals (calcite, siderite, hematite, muscovite, biotite), with a particular increase in sulfate and cation (Fe, Si and Al) concentrations inaqueous solutions. [START_REF] Risse | Alteration of carbonates in saline aquifers due to CO 2 and accessory gases at geological storage conditions[END_REF] also observed an increase inchemical reactivity in batch experiments at 120°C and 200 bars. An IEAGHG report (2011) estimated,usinga modeling approach,that 1.5% vol total concentration ofSO x andNO x in the injection steam can favor reactivity in limestone, where calcite dissolution increases by about 50%.Atthe same time, clogging can occur, in particular in the dry-out zone, thus limiting injectivity. According to these authors (IEAGHG, 2011), sulfate precipitation can play a role in porosity variations butthese impactsmightbe less significant than those generated by S 0 (Claus reaction). The impact of SO 2 on the chemistry of the brine was also studied by [START_REF] Ellis | Limitations for brine acidification due to SO 2 co-injection in geologic carbon sequestration[END_REF]. Using thermodynamic data such as SO 2 solubility, the authors developed a model to calculate the spatial distribution of sulfur in the reservoir depending on storage conditions (T, P, salinity, etc.)., [START_REF] Ellis | Limitations for brine acidification due to SO 2 co-injection in geologic carbon sequestration[END_REF] used the same approach to analyzethe impact of SO 2 on pH, disregarding the potential mineral reactions that can buffer the pH of aqueous solutions. SO 2 may form sulfuric acid (H 2 SO 4 ), which is stronger than carbonic acid. [START_REF] Wilke | On the interaction of pure and impure supercritical CO 2 with rock forming minerals in saline aquifers: An experimental geochemical approach[END_REF] went further by testing the reactivity of such acidic solutions onreservoir rocks.

Among theother impurities that might be injected with CO 2 , traces of oxygen are expected in streams coming from the Oxyfuel capture technology. Previous studies have shown that the presence of high proportions ofO 2 (such as the O 2 concentration in air) can generate oxidative reactions (Yu et al., 2010). However, underCO 2 storage conditions, the role of O 2 is poorly understoodand laboratory experimental resultsare scarce [START_REF] Heeschen | Mineral alteration due to accessory gases in the geological storage of CO 2[END_REF][START_REF] Langlais | Geological and mineralogical effects of the presence of oxygen on the caprock and reservoir rocks for CO 2 storage application. 3 ème Colloquefranco-espagnol[END_REF]. [START_REF] Renard | Rôle des gaz annexes sur l'évolution géochimique d'un site de stockage de dioxyde de carbone[END_REF] and [START_REF] Renard | Geochemical study of the reactivity of a carbonate rock in a geological storage of CO 2 : implications of co-injected gases[END_REF] studiedthe impactof SO 2 and O 2 with CO 2 on carbonate rocksunder reservoir conditions. The gas composition they studied was similar to thatof gas comingfrom an oxycombustion process prior toany conditioningprocedure: 82% CO 2 , 4% SO 2 , 4% O 2 , 4% N 2 and 6% Ar(Note that conditioning will always be done, thus increasing the capture rate to >95%). These experiments showedthatthe chemical reactivity of the system increasedwhen SO 2 and O 2 wereadded to the gas mixture (with respect to pure CO 2 ). With pure CO 2 , minerals wereslightly altered, whereas massive dissolution of pyrite and calcite wereobserved in the presence of SO 2 and O 2 [START_REF] Renard | Geochemical study of the reactivity of a carbonate rock in a geological storage of CO 2 : implications of co-injected gases[END_REF]. Some precipitation of anhydrite and hematite was alsoobserved.

All of these studies show how important it is that we study the potential impact of these accessory gases on the reservoir. Numerical estimations are needed so that we canpredicthow the gas composition might impact the geochemical reactivity withinthe host reservoir, potentially modifying the porosity and permeability of the medium and influencing long-term well injectivity. This paper describesnumerical simulations using coupled hydrogeological, thermal, and geochemical codes such as TOUGHREACT version 1 [START_REF] Xu | TOUGHREACT-A simulation program for nonisothermalmultiphase reactive geochemical transport in variablysaturated geologic media: Applications to geothermalinjectivity and CO 2 geological sequestration[END_REF]. Thesetools aresuitablefor exploringdifferent evolution scenarios of gas injection in deep geological structures, and determining the behavior of the near-wellbore region of the targeted reservoir. These simulations will provide a methodological basis for future industrial-scale CO 2 storage projects nowbeing developed. The simulationresults (positive or negativeimpact on the caprock integrity or reservoir injectivity) will condition subsequent simulations and show whether such simulations need to be repeated/improved in order to obtain standard practices (methodologies and tools).

The final objective of this studywas to identify the potential reactivity ofany targetedsystem.Our calculationswere not, therefore, done for a specific site. We modeled a generic (hypothetical) site inspired by existing sitesa sandstone aquifer anda clay caprock formation with properties similar to those of the Nordland Shale formation.Below, we describe the reservoir, caprock and fluid properties in each formation, give the assumptions usedfor modelcalculations, anddescribe two simulations done to study the impact of CO 2 , SO 2 and O 2 mixtures on the main physicochemical characteristics of a deep saline aquifer.The results are compared to those of previous studies that did not consider the presence of impurities in the CO 2 injected gas.

RESERVOIR PROPERTIES

Reservoir characteristics

The saline aquifer defined in the modelis a sandstone reservoir approximately 200 m thick.

Before injection, the effective pressure is 150 barsand the temperature is 50°C.

Theporosityof both the reservoir rockand the caprock is around15%. This caprockvalueistaken from papers describing the Nordland Shale formation overlyingthe Sleipner reservoir. It is averaged from the porosity values estimated by [START_REF] Johnson | Reactive Transport Modeling of Geohazards Associated withCO 2 Injection for EOR and Geologic Sequestration[END_REF] and Gauss et al. (2005) (5 %) and those measuredbyHarrington et al. ( 2006) and [START_REF] Springer | Caprock properties of the Nordland Shale recovered from the 15/9-A11 well, the Sleipner area[END_REF] (about 40 %).The caprock layer has a mean permeability of 0.003 mD [START_REF] Johnson | Reactive Transport Modeling of Geohazards Associated withCO 2 Injection for EOR and Geologic Sequestration[END_REF], whereas the permeability of the saline reservoir is about 300 mD.

Due to the injection of a gas phase (non-wetting phase) intothe deep system, the relative permeability and the capillary pressure of the medium must be defined to accurately describe the relative flow of gas with respect to the aqueous solution. For the sandstone reservoir, we used literature data to define the representative curves and include them in the numerical code (e.g. [START_REF] Bachu | Effects of in-situ conditions on relative permeability characteristics of CO 2 -brine systems[END_REF]). Because of the chosen characteristics of the modeled reservoir (k mean # 300 mD;  # 15%; P reservoir = 150 bars; T = 50 °C), the Cardium sandstone was used as the reference sandstone (Fig. 1).The characteristic curves for the relative permeability and capillary pressure of the caprock are assumed to be the same as those of the reservoir. The caprock's impermeability is represented by its low intrinsic permeability (0.003 mD).

Figure1. Relative permeability (left) and capillary pressure (right) curves for Cardium sandstone (from [START_REF] Bennion | Relative permeability characteristics for CO 2 displacing water in a variety of potential sequestration zones in the WesternCanadaSedimentaryBasin[END_REF]. Fitted curves are calculated using TOUGHREACT for both the reservoir and caprock.

Experimental capillary pressure and liquid relative permeability values were fitted using the Van Genuchten model [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF]. The parameters used in the simulations for relative permeability and capillary pressure lawsare summarized in Table 1. The selection of the secondary minerals that are able to precipitate in the reservoir and caprock formationsis a key parameter since this can have an impact on the overall reactive pathways. As CO 2 is the major component of the gas phase, variouscarbonates werechosen for the reservoir (dolomite, dawsonite, siderite) and for the caprock (calcite, dolomite, dawsonite). Because of the high redox potential (Eh) of the injected solution due to the presence of O 2 in the gas phase, only oxidized iron-bearing minerals were selected (hematite). The high Eh alsohas an impact on the SO 2 , which is fully oxidized as sulfate. Consequently,although we did monitor pyrite behavior during these simulations, we focused primarily on a sulfate mineral. Because of the temperature of the reservoir (50°C), anhydrite was selected. Aluminosilicates (illite, kaolinite and chlorite)were also included in the calculations as potential secondary minerals (newly precipitating) in the reservoir formation.

Chemical composition of the initial water

Details concerning thephysico-chemical characteristics of the formation water are givenin Table 3. The water is representative of the saline solutions found in the Gassum reservoir (Kjoller et al., 2011).

The water composition for in situ conditions (150 bars, 50°C) wasdeterminedusing anappropriatehypothesis and thermodynamic calculations for highly saline aqueous systems using the SCALE2000 code [START_REF] Azaroual | SCALE2000 (V3.1): Logiciel de calculs thermodynamiques et cinétiques ; application aux saumures pétrolières, hydrothermales et industrielles[END_REF]. The pH and bicarbonate concentration of thebrine wererecalculated for the reservoir and caprockpressure and temperature conditions, assuming that the brine is in thermodynamic equilibrium with respect to primary minerals. This gave a bicarbonate concentration of 103 mg L -1 in the reservoir. Under these conditions, the brine remains slightly undersaturated with respect to the evaporitic (halite, sylvite, etc) andiron sulfide (pyrite, pyrrhotite)minerals, but is slightly supersaturated with respect to dolomite. In the caprock, the aqueous solution is supersaturated with respect to dolomite but undersaturated with respect to sulfide (pyrrhotite) and sulfate (anhydrite) minerals. These two compositions were specifically chosen so that we could determine the consequences of relatively large amounts of impurities. The likely compositions are constantly being updatedand the CO 2 content would currently exceed95%. TOUGHREACT simulates the chemical reactivity of systems using an extended thermodynamic database. We used the database providedwith the program, i.e., a modified version of the EQ3/6 database [START_REF] Wolery | EQ3/6: Software package for geochemical modeling of aqueous systems: Package overview and installation guide (version 7.0)[END_REF].

NUMERICAL APPROACH

Calculation code

Thermodynamic equilibrium between the gas phase and the aqueous phase is assumed for CO 2

dissolution. An extension of Henry's law, including salting-out effect, is used to estimate the solubility of CO 2 in the aqueous phase at high pressure and high salinity.

-13 -For this work, the advancement of dissolution and precipitation reactions of minerals wasmodeled under kinetic constraints.The general form of the rate law proposed by [START_REF] Lasaga | Chemical kinetics of water-rock interactions[END_REF] was applied for both the dissolution and precipitation of minerals:

      n n n n 1 A k r
Positive values for r n correspond to the dissolution of the mineral n (negative values for precipitation), k n isthe rate constant (mol m -2 s -1 ), A n is the specific reactive surface area per kg H2O ,  n isthe saturation ratio of the mineral n ( n = Q n /K n ), and and are empirical parameters determined from experiments, usually taken as 1.

The dependence of the rate constant k n ontemperature is calculated using the Arrhenius equation:

               15 . 298 1 1 exp 25 T R E k k a n
whereE a is the activation energy (J mol -1 ), k 25 is the rate constant at 25°C, R is the gas constant (J mol -1 K -1 ), and Tis the temperature in Kelvin.

For some minerals, specifically alumino-silicates and salts, dissolution and precipitation also depend on H + (acid mechanism) and OH -(alkaline mechanism) concentrations, in addition to the neutral mechanism. In this case, k n is calculated using the following equation:

where superscripts and subscripts nu, H, and OH indicate neutral, acid, and alkaline mechanisms, respectively, and a is the activity of the species.

For carbonate minerals, dissolution/precipitation mechanisms are catalyzed by HCO 3 -, and reaction rates depend on the activity of aqueous CO 2 . k n is calculated according to:

OH H n H OH a OH n H H a H nu a nu n a T R E k a T R E k T R E k k                                              15 .
a T R E k a T R E k T R E k k                                             
The kinetic parameters used for the last two equations are given in Table 5 and come from [START_REF] Palandri | A compilation of rate parameters of water-mineral interaction kinetics for application to geochemical modelling[END_REF]. The dissolution rates depend on the three mechanisms proposed in Table 5, whereas the precipitation rate depends only on the neutral mechanism. The kinetic parameters for siderite are assumed to be the ones of dolomite rates [START_REF] Gunter | Aquifer disposal of acid gases: modelling of water-rock reactions for trapping of acid wastes[END_REF] Table 5 -Kinetic parameters for mineral dissolution and precipitation (C stands for minerals from the caprock and R for minerals from the reservoir). k (calculated at 25°C and pH=0) = Rate constant in mol m - 2 s -1 , E = Arrhenius activation energy in kJ mol -1 , n H and n OH = reaction order with respect to H + and OH -, n CO2 = reaction order with respect to CO 2

Acid mechanism

Neutral Assumed to be at equilibrium

The reactive surface area of the dissolving minerals is a complex factor since the dissolution rate depends on the available surface area in contact with the aqueous solution. This contact surface is difficult to estimate. According toWhite and Peterson(1990), it can vary between 1 and 3 orders of magnitude with respect to the total surface area measured using the BET method. We assumed that all of the minerals except illite have the same specific surface area (about 10 cm².g - 1 ). A value of 100 cm².g -1 wasused for illite. These values are consistent with the data chosen by [START_REF] Xu | Numerical modeling of injection and mineral trapping of CO 2 with H 2 S and SO 2 in a sandstone formation[END_REF].

Mineral dissolution and precipitation reactions causetemporal changes in reservoir porosity.

Indeed, variations in the mineral volume fraction owing to chemical reactions make it possible tocalculatethe resulting porosity. Permeability variations were not calculated in this study because of the poor knowledge of the relationship between permeability evolution and porosity changes.

3.2Geometrical Model

The saline aquifer is represented by a 2D-radial model that includesthe reservoir and caprock units (Fig. 2). The injection rates of supercritical CO 2 and SO 2 -O 2 dissolved in brinewere 30 kg s -1 and 15 kg s - 1 , respectively. The mass ratio of 2 was arbitrary chosen for the simulations. Gas and brine were injected over total thickness of the reservoir during an assumed exploitation period of 30 years. The specific injection rates were 0.15 kg s -1 m -1 for CO 2 and 0.075 kg s -1 m -1 for brine containing dissolved SO 2 -O 2 because of the thickness of the reservoir formation. Since the reservoir had a high permeability (about 300 mD), the pressure buildup was limited to about 10 bars and enabled a long injection period.

Two gas stream mixtures are studied: 4)

 -Low
The chemical composition of the SO 2 -O 2 -acidified brine(co-injected with supercritical CO 2 )

wasdetermined in different steps:

 First, the injected brine wasequilibrated with the reservoirminerals at the reservoir temperature.

 SO 2 and O 2 gases werethen dissolved in the brine. Given the chosen mass ratio of 2 between the injected mass of supercritical CO 2 and the mass of SO 2 -O 2 brine, the SO 2 and O 2 concentrations in the solution weredefined to ensure consistency with the gas composition of an oxy-combustion capture process (given data from the industry).

These preliminary simulations were done with the batch option of the PHREEQC code (Parkhurst and Appelo, 1992). The hypothesis that SO 2 is totally dissolved in the SO 2 -O 2 brine is acceptable since its solubility is very high at such temperatures [START_REF] Ellis | Limitations for brine acidification due to SO 2 co-injection in geologic carbon sequestration[END_REF].

3.4Initial Conditions

The following assumptions were used fornumerical modeling:

 Hydrostatic equilibrium: no regional flow is considered, and a hydrostatic pressure is imposed in the outermost column of the mesh.

 Thermal equilibrium: before starting gas injection, reservoir and caprock are at the same temperature, i.e., 50°C.The numerical simulations are donein isothermal mode.

 Geochemical equilibrium: the aqueous solutions initially present in the reservoir and caprock are in equilibrium with their respective mineral assemblages at the temperature of the system.

The Davies modelwas used as the activity model in the batch simulations with PHREEQC.The TOUGHREACT code uses an extended Debye-Hückel model, although these models are poorlyadapted tohighly saline solutions such as those in the reservoir (Table 2). For this reason, and for more reliable results, the elemental concentrations of each species of the acidified water weredivided by 10 in order to decrease the ionic strength of the solution (Table 6). This artificial dilution does not drastically modify the simulation results and thereaction paths, but it can change thedissolution/precipitation magnitudes of minerals.Another change related to the dilution is the initial saturation state of water with respect to rock minerals. After dilution, the "low SO 2 -O 2 mixture" is undersaturated with respect to anhydrite, whereas the "high SO 2 -O 2 mixture" is supersaturated with respect to anhydrite. The consequences on numerical results will be analyzed in paragraph 4.2. The "SO 2 -O 2 mixtures" are highlyacidic, with oxidative and sulfateconcentrations higher than in the reservoir brines. These brines wereexpected to be highly reactive with reservoir minerals.

RESULTS AND INTERPRETATION

4.1 Case 1: Injection of a"Low SO 2 -O 2 mixture"

The injection ofgas and water in the reservoir changes the physical parameters (gas saturation, pressure, etc.)and geochemical properties at different scales, with a major impact on the nearwellbore zone.

The injection of the gas/solution mixture changesthe gas saturation around the injection well.

Because the supercritical CO 2 densityislower than that of the formation brine, CO 2 reachesthe top of the aquifer and spreads outalong it (Fig. 3). Although the caprock isless permeable than the reservoir, the supercritical CO 2 is able topenetrate itovera few centimeters.The pressurearound the injector, initially between 143 and 160 bars between the top and the bottom of the reservoir,increasesto 167 bars by the end of the injection period.

The pH of the formation water iscontrolled by water-rock interactions. Initially, the formation water isinequilibrium with the mineral assemblage of the reservoir: the pH isclose to 7.9 (Table 6). Co-injection of supercritical CO 2 and SO 2 -O 2 brinemodifiesthis equilibrium.The evolution of pH followsthe evolution of the gas saturation in the reservoir, with the impacted zone extending up to 3000 m from the injector after an injection period of 30 years (Fig. 3b). Consequently, the formation water around the injector acidifies,becomesundersaturated with respect to all minerals, and dissolvesthem (preferentially the carbonates).Calcite dissolution is illustrated by the increase in the calcium concentration in the low pH zone (Fig. 3c). As long as reactive carbonates arepresent, the aqueous solution isin equilibrium with them and the pH isbuffered. However, continued injection exhaustssome minerals.All of the carbonates areconsumed around the injector and the buffering stops. pHisno longercontrolled and decreasesto very low values. Near the well, the mineral assemblage isdrastically modified, and the pH decreasesto a minimum value of 1.4, similar to that of the injected fluid (Table 6). Further within the impacted zone of the reservoir, the pH isbuffered to a value ranging between 4 and 6, since not all of the carbonatesareconsumed.The pH does not change in the non-impacted zone.

Because of this acidification of the medium, calcite isthe mineral most affected by the injection of the acid solution. It dissolvesnear the injection well (50 m around it), but isnot significantlyimpacted elsewhere in the reservoir (Fig. 3e).Anhydrite is also affected by gas and water injection.The behavior of anhydrite differsin time and space.Atfirst, anhydrite precipitates near the injection well because the injected fluid contains sulfur,which is oxidized tosulfateby oxygen.Calcite is dissolved by the acidified injected water, releasing Ca 2+ in solution (Fig. 3c).

With SO 4 2-present in the injected water (Fig. 3d), these ions combine to form anhydrite according to the following successive geochemical reactions:

CO 2(g) + H 2 O → H + + HCO 3 - CaCO 3 (calcite) + H + → Ca 2+ + HCO 3 - SO 2(g) + H 2 O → SO 3 2-+ 2 H + SO 3 2-+ 0.5 O 2(aq) → SO 4 2- Ca 2+ + SO 4 2- → CaSO 4 (anhydrite)
However, when all of the calcite has beenconsumed, the Ca 2+ source disappears and anhydrite precipitation stops. Because the injected solution is undersaturated with respect to anhydrite, the -20 -secondary precipitated anhydrite and the original anhydrite (initially present in reservoir)

dissolve. This is why some zones showa deficit inanhydrite, whereas others showa positive balance of anhydrite compared to its initial amount (Fig. 3f). However, because of the amount of sulfate injected, the dissolution and precipitation of anhydrite have a limited impact on sulfate concentration.

As a consequence of calcite and anhydrite reactivity, porosity increases near the injector (Fig. 3g). After 30 years of injection, it is estimated that the porosity 10 m from the injection well is around 19 % (due to calcite and anhydrite dissolution). In any case, these values must be considered to beindicative (qualitative) and not quantitative, because of some limiting working hypotheses (dilution of the initial brine, limitations of kinetic parameters, restrictions of multiphase reactive transport, etc.).

-21 - All other minerals in the reservoir react, but less than calcite and anhydrite (by 2 to 5 orders of magnitude) and their impact on porosity isnegligible.

Because of the assumptions made regarding caprock permeability, supercritical CO 2 is able to penetrate it slightly, which causes some reactivity, in particular at the interface between reservoir -22 -and caprock formations. The amount of CO 2 infiltrating the caprockistoo low and the impact on chemical reactivity and porosity variations are too limited to be seen in the graphs.Some reactivity of Fe minerals is however observed. In relation to the extent of the CO 2 gas bubble (spreading about 2,000 m from the injection well) and the decrease in pH, siderite and chlorite are the most impacted minerals, with dissolution of these two minerals over a few centimeters inside the caprock. There is little dissolution of minerals but it causes an increase in the iron (III) concentration and a potential precipitation of hematite at the interface.

One key concern associated withCO 2 storage in geological reservoirs is the potential durable trapping of gas as minerals. For this specific case,CO 2 could be stored as carbonate but because of the acidity of the injected fluid, no mineral precipitation is observed. The primary carbonates are totally dissolved (close to the injection well),which leads to a release of carbon in the reservoir instead of trapping. The only gas storage observed was that of SO 2 , which is stored as anhydrite (after oxidation as sulfate). However, because of the low amount of calcium in solution (coming from carbonate dissolution), the trapped fraction of SO 2 is relatively low and has a low impact on the sulfate concentration (Fig. 3d).

4.2Case 2: Injection of a "High SO 2 -O 2 mixture"

At the temperature of the reservoir (i.e. 50°C), the "High SO 2 -O 2 mixture" is slightly oversaturated with respect to anhydrite. Consequently, after 12 years ofinjection, anhydrite precipitation has fully cloggedthe porosity nearthe injection well,putting an end to injection if no corrective measures have beentaken.

In order to studyanhydrite precipitation conditions, we rananother simulation,injectinga differentsolution. This time, the reservoir brine, once again diluted 10-fold,was not initially equilibrated with the mineralogical assemblage of the reservoir before theSO 2 and O 2 were added, and the injected solution was undersaturatedwith respect to anhydrite. The impact of the co-injection of supercritical CO 2 and acidified brine is a decrease in pH around the injection well and a co-dissolution of both carbonates and anhydrite (since the solution is undersaturatedwith respect to anhydrite) (Fig. 4). Since the acidity is higher than that ofthe "low SO 2 -O 2 mixture" (Table 3), the impacted zone is larger than in the first case (about 50 m in case 1 and 200 m in case 2).

The other minerals of the assemblage arealso impacted by this strong acidification. They areless impacted than carbonates or anhydrite, but the dissolution of some (K-feldspar) and the precipitation of others (quartz, kaolinite) must be carefully monitored.

The higher acidity of the injected fluid in this caseis neutralized to a lesser extent by reactions with minerals. Smaller variationsin porosity are observed in the near-well region, whereas the impacted zone is enlargedby the mineral dissolution and precipitation reactions. assemblage of iron-bearing minerals such as pyrite [START_REF] Renard | Geochemical study of the reactivity of a carbonate rock in a geological storage of CO 2 : implications of co-injected gases[END_REF] or daphnite [START_REF] Thibeau | Geochemical assessment of the injection of CO 2 into Rousse depleted gas reservoir[END_REF], which could release iron in solution.

Althoughsimulation results seem to corroborateexperimental results, they must be interpreted with caution because of the many assumptions made to do the calculations.First of all, the gases are not all injected as a gas phase. Some of them (O 2 and SO 2 ) are dissolved in brine. This has an impact on the variations in gas saturation inside the reservoir and on the chemical reactions that might occur in the aqueous phase. Secondly, the activity model used in TOUGHREACT does not handle highly saline solutions. As demonstrated by [START_REF] André | Numerical modeling of fluid-rock chemical interactions at the supercritical CO 2 -liquid interface during supercritical carbon dioxide injection into a carbonated reservoir, the Dogger aquifer (Paris Basin, France)[END_REF] by calculations with different numerical codes and activity models, the use of a specific approach (such as the one proposed in the Pitzer formalism) might increase the accuracy of geochemical predictions, in particular the quantitative results. This is crucial because the predicted amounts of minerals able to precipitate or dissolve depend entirely on these calculations in the aqueous phase.

Furthermore, there is noEquation of State (EoS) for the fugacity correction for the complex gas mixture (CO 2 -SO 2 -O 2 -(H 2 O)).The calculation of porosity variations might, therefore,be highly uncertain, even though the overalltrend seems to make sense. Thirdly,there are few data concerning the reactivity and the kinetic behavior of alumino-silicate minerals when these are exposed to acidified brines. Many studies emphasize the high reactivity of carbonates and intergranular cementation in sandstone. Studies (numerical and experimental) must be done to analyze the possible long-term effect of these acid gases on alumino-silicates after the buffering effect of carbonates has disappeared.

5CONCLUSIONS

The objectives of this paper were to present the results ofnumerical simulations of the coinjection of acid gases in a deep saline aquifer. Since the TOUGHREACT code cannot represent the co-injection of these components in a gaseous (non-wetting phase) mixture, some limiting hypotheses were used: CO 2 injection was simulated in a supercritical form, whereas SO 2 and O 2 injections were simulated as dissolved species in an aqueous solution. After an injection period of 30 years, simulation results indicate:

 The injected supercritical CO 2 dissolves in solution, increasing its ability to dissolve carbonates. Since calcite is one of the components of the mineralogical assemblage, it dissolves around the injection well and therebyincreases porosity. However,because of the negligibleinitial amount of calcite in the mineralogical assemblage, the impact on pore volume is limited.

 SO 2 reacts rapidly around the injection well, forming sulfatesdue tothe traces of oxygen in the injected gas. Bythe recombining of Ca 2+ (from calcite) and SO 4 2-(from SO 2 and O 2 ), anhydrite precipitates. The SO 2 concentration determinesthe extent of the anhydrite deposition occurring around the injection well.

 The higher the SO 2 concentration in the injected stream, the larger the radius of anhydrite deposition around the well. However, the SO 2 concentration seems to have no effecton the geochemical mechanisms in terms of, for instance,reaction paths.

 Calcite and anhydrite are the most reactive minerals. All of the other initial minerals are also affected by the injection of the acid solution, but to a lesser extentand with a minor impact on porosity.

 When calcite and anhydrite haveopposite behaviors, numerical simulations predictthat calcite dissolution will have a greater impact on porosity than anhydrite precipitation.

Consequently, increasing porosity is expected around the injection well, assumingan increase in well injectivity and a potential long-term injection period.

However, these results are onlyqualitative and highlight only the global trends of the system under investigation. Indeed, the real (quantitative)impact is still very hardto predict. For example, how the dissolution/precipitation of a particular mineral will affect rock porosity depends on the characteristics of the rock (e.g., micro/macro porosity, connected porosity). Our results must, therefore, be interpreted with caution.

Figure 2 .

 2 Figure 2. The geometrical 2-D radial model (vertical cross-section)

Figure 3 .

 3 Figure 3.State of the reservoir and caprock after a 30-yearinjection period. a)Gas saturation (Sg), b) pH pattern, c) Calcium concentration (mol/kg H2O )with x-axis up to 4000 m, d) Sulfate concentration (mol/kg H2O ), e) Volumic fraction of calcite, f) Volumic fraction of anhydrite, g) Porosity with x-axis up to 70 m from injection well

Figure 4 .

 4 Figure 4.State of the reservoir and caprock after an injection period of 30 years. a) pH pattern with x-axis up to 4000 m, b) Volumic fraction of anhydrite, c) Volumic fraction of calcite, d) Porosity with x-axis up to 400 m from inejction well.

Table 1 :

 1 Van Genuchten parameters used for fitting the characteristic curves of relative permeability and capillary pressure for brine and gas phases For the numerical simulations, the mineral assemblages ofthe caprock and the reservoir weredefinedusingliterature data. The caprock minerals were taken from[START_REF] Gaus | Reactive transport modelling of the impact of CO 2 injection on the clayey caprock at Sleipner (North Sea)[END_REF] andthe reservoir minerals fromKjoller et al. (2011), who described the mineralogy of the sandstone Gassum reservoir.

		1		kr gas				1.E+08
		0.9		kr water
		0.8		Modeled kr water			1.E+07
	Relative permeability	0.3 0.4 0.5 0.6 0.7		Modeled kr gas		Pressure (Pascal)	1.E+05 1.E+06
		0.2
		0.1						1.E+04
		0
		0	0.2	0.4	0.6	0.8	1	1.E+03
				Water saturation, Sw			0	0.2	0.4 Water saturation 0.6	0.8	1
			Relative permeability parameters for brine
			m = 1 -1/n				0.466
			Residual liquid saturation			0.197
			Liquid saturation				1.000
			Residual gas saturation			0.050
			Capillary pressure parameters

The caprock is made up for the most part of illite and quartz (74wt%) while the reservoir rock contains mostly quartz and K-feldspar (91 wt%). In the reservoir and caprock formations, the end-membersalbite (NaAlSi 3 O 8 ) and anorthite (CaAl 2 Si 2 O 6 )aresubstituted for theplagioclases series (solid solution)as stated by

Gauset al. (2005)

. The carbonate end member is represented by calcite in the reservoir and by siderite in the caprock.

Chlorite (Mg 2.5 Fe 2.5 Al 2 Si 3 O 10 (OH) 8 ),siderite and pyrite are iron-bearing minerals in the caprock formation, whereas the reservoir formation does not contain any Fe minerals. Rutile and anatase are common detrital minerals. Anatase, which comes from Ti-bearing minerals, waschosen for this study (Table

2

).

Table 2 .

 2 Mineralogical composition of the caprock and the reservoir rocks

		Minerals	Wt%
		Illite	60
		Quartz	14
		Albite	6
	Caprock	Anorthite	3
		Chlorite	5
		K-Feldspar	4

Table 3 .

 3 Chemical composition (mg L -1 )of the waters One has a low SO 2 content (0.08 % vol) and is calledhere "Low SO 2 -O 2 mixture", whereas the otherhas a higher SO 2 content (1.53 % vol) and is called"High SO 2 -O 2 mixture". For O 2 , the same concentration is considered for the two injected gases. The other impurities are not discussedhere.

	Element	Aquifer	Caprock
	Na	125,764	78,085
	K	21	196.2
	Ca	18,872	18,896
	Mg	3,059	28,042
	Fe	176	540.5
	Cl	236,252	238,063
	Al	0.002	0.0002
	Si (as SiO 2 )	10.2	7.4
	S (as SO 4 2-)	723.7	785
	2.4 Injected gas mixture

We used two gas steam mixtures provided by Vattenfallin 2008 (Table

4

). They contain mostlyCO 2 (90.28 to91.61 % vol) and a range of 8.38 to 9.71 % vol of impurities (i.e. other chemical species). These two mixtures differ mainly in theirSO 2 concentrations (due to different capture options).

Table 4 :

 4 Gas stream mixture compositions (in vol%) to be injected. Data provided by Vattenfall.

		Low SO 2 -O 2 mixture	High SO 2 -O 2 mixture
	CO 2	91.61	90.28
	Ar	5.70	5.62
	O 2	1.60	1.58
	N 2	0.61	0.60

Reservoir Impervious cap rock 3.3 Injecting supercritical CO 2 with SO 2 and O 2 impurities

	TOUGHREACT cannot handle CO 2 -SO 2 -O 2 -H 2 O gas mixtures -only CO 2 -H 2 O mixtures
	(specificity of the ECO2n module).A mixture of supercritical CO 2 and reservoir brine containing
	dissolved SO 2 and O 2 (hereafter called "SO 2 -O 2 mixture") were thereforeco-injected into the
	reservoir.

Table 6 .

 6 Chemical characteristics of the fluids used for numerical calculations

			Reservoir brine	Low SO 2 -O 2 mixture High SO 2 -O 2 mixture
	pH		7.9	1.5	0.2
	pe		-3.9	17.7	19.0
	TDS (g/kg H2O )	38.6	~38.6	~38.6
	[SO 4	-2 ](mol/L)	1.7e-02	5.0e-02	7.2e-01

AcknowledgmentThis work is part of the OXYGAS (phase II) project funded by VATTENFALL. The authors thankthe VATTENFALL Directorate and BRGM Research Directorate for allowing publication of these results.

Discussion

In order to determine the effect of impurities (SO 2 and O 2 ) co-injected with CO 2 in deep saline reservoirs, we comparedour results with those of [START_REF] Audigane | Two-dimensional reactive transportmodeling of CO2 injection in a saline Aquifer at the Sleipner site, North Sea[END_REF], which were obtained with the injection of pure CO 2 . The initial mineralogical assemblage and the secondary minerals in our study were the same as those of [START_REF] Audigane | Two-dimensional reactive transportmodeling of CO2 injection in a saline Aquifer at the Sleipner site, North Sea[END_REF]. These authors investigated both the injection period (25 years) and the post injection period (10,000 years). During the injection period, theypredicted a minor alteration of the alumino-silicates (chlorite, albite, muscovite).

Calcite is the most reactive mineral with dissolutionmainly in the reservoir formation and some precipitation of secondary calcite near the shale layers at the interface between the CO 2 -saturated brine and the initial brine. Our study gives the same results with predominantly carbonate dissolution (no traces of secondary precipitations) and only weak reactivity of alumino-silicates.

The main difference is the reactivity of sulfur-bearing minerals. When SO 2 is co-injected with CO 2 , the geochemistry of the system is controlled primarily by the sulfur, in particular its reduced form S(-II). Redox reactions are activated by the presence of sulfide in solution.

Because of acidification, iron-bearing minerals like ferrihydrite (Fe(OH) 3 ) and hematite (Fe 2 O 3 ) are dissolved, releasing Fe(III) in solution. Because of reduced conditions, iron III is reduced to iron II, which can react with dissolved CO 2 to form siderite (mineral trapping of CO 2 ) or H 2 S (coming from the solubilizationof SO 2 ) to form pyrite.

In our calculations, enough oxygen is injected to fully oxidize the injected SO 2 .

Consequently,the geochemical system is dominated by the reactions involving sulfates and carbonates with complex links with oxygen (through pH and redox conditions). Sulfate minerals play a major role, in relation tothe carbonates -the precipitation of the former being correlated with the dissolution of the latter. Simulations predict anhydrite precipitation, in agreementwith [START_REF] Pironon | Dehydration of Gypsum under Dry CO 2 Injection[END_REF]. We can then assumethat, in a deep CO 2 storage reservoir, anhydrite will be the mainsulfate mineral that is able to precipitate.

Brines present in the reservoir are modified by acid gases. CO 2 dissolution generates carbonic acid and SO 2 + O 2 producessulfates, both of which lead to adecreasein pH. Because of mineralreactions (carbonatedissolution), the pH does not dropbelow 4, which is in agreementwith the observations of [START_REF] Ellis | Limitations for brine acidification due to SO 2 co-injection in geologic carbon sequestration[END_REF]. This pH-buffering capacity is, however, controlled by carbonates only as long as these minerals are present in the mineralogical assemblage [START_REF] Gunter | Aquifer disposal of acid gases: modelling of water-rock reactions for trapping of acid wastes[END_REF].

The high solubility of SO 2 causesa massive dissolution of this gas, which is oxidized by the O 2 co-injected with the CO 2 . This oxidation produces sulfates, as described by [START_REF] Rumpf | Solubility of sulfur dioxide in aqueous solutions of sodium-and ammonium sulfate at temperatures from 313.15 K to 393.15 K and pressures up to 3.5MPa[END_REF], whereas no precipitation of hematite is predicted, despite the presence of iron in the initial solution (Table 3). This is probably due to the absence in the initial mineral simulation of CO 2 geologic storage and its impact on regional groundwater flow: A hypothetical case study at Tokyo Bay, Japan. 9 th International Conference on Greenhouse Gas Control Technologies(GHGT9), Washington DC, USA,in Energy Procedia, 1, 1871-1878[START_REF] Thibeau | Geochemical assessment of the injection of CO 2 into Rousse depleted gas reservoir[END_REF].


Livio De Luca
email: livio.deluca@map.archi.fr

Philippe Véron
email: philippe.veron@aix.ensam.fr

Michel Florenzano
email: michel.florenzano@map.archi.fr

A generic formalism for the semantic modeling and representation of architectural elements

Keywords: Architectural heritage, Architectural knowledge, Surveying, Feature-based modeling, Semantic shape

This article presents a methodological approach to the semantic description of architectural elements based both on theoretical reflections and research experiences. To develop this approach, a first process of extraction and formalization of architectural knowledge on the basis of the analysis of architectural treaties is proposed. Then, the identified features are used to produce a template shape library dedicated to buildings surveying. Finally, the problem of the overall model structuring and organization using semantic information is addressed for user handling purposes.

Introduction

Studies on architectural heritage can be supported today by three-dimensional reconstructions of real buildings. The 3D digital model can bring an effective support to documenting the current state of historical buildings and also to create resources for the researchers who lead analyses on their historical evolution. With the purpose of confirming this assumption, several problems must be approached. As a result of using instruments such as the laser scanner for the building documentation, making the 3D models of architectural elements focuses on automatic procedures of reconstruction which consist primarily in a geometrical interpolation of metric data [START_REF] Remondino | From point cloud to surface: the modeling and visualizaion problem[END_REF]. However, the objective of an architectural representation is to lead a semantic reading of the recorded data through the geo-metrical interpretation of the shape it describes. So, in order to attain these objectives, the process of threedimensional restitution of an observed reality must necessarily refer to a universe of knowledge. It is thus necessary to reaffirm the historical distinction between the acquisition phase and the data interpretation phase [START_REF] Saint-Aubin | Le relevé et la représentation de l'Architecture[END_REF] while taking into account the relations of dependence that exist between these two stages. To give an example, many patrimonial buildings testify an intensive use by their builders of the concept of moulding. The importance of this concept is, for the architectural theory, largely established: it is described there initially as an essential tool in the composition of the shapes, but also as the means by which the shape gathers sense, by which the shape adds to its geometry properties of significant expression -softening, underlining, etc. In this direction, the coherence of the procedures of digital processing compared to the codes of the architectural representation constitutes an important issue we will address. In this article, we present an approach and the tools we have developed for the geometrical and semantic description of classical architecture elements. This approach guides the development of two applications: 1 a tool for the three-dimensional reconstruction of buildings, starting from a joint acquisition by laser scanning and photography, and a system for consulting on the Internet of the surveying and of the representations. This article is organized as follows: Section 2 presents the basic concepts for the understanding of geometrical nature of elements of classical language. Section 3 describes a generic formalism for geometrical and semantic description of profiled elements while Sect. 4 tackles the problem of the semantic structuring of the model with the aim of exploiting the description model for multiple objectives of analysis.

2 Basic concepts to understand the geometrical nature of the elements of classical architecture

Architectural surveying as a reverse-engineering process

The analysis of an architectural shape may be led by the identification of the process allowing its geometrical construction. However, the codes of architectural representation can always help in this stage which remains the most difficult phase of building reconstruction. To give an example one can quote the geometrical representation of a column (see Fig. 1). This element, far from being a simple cylinder, is characterized by a pace (truncated, bent bottom, bent third, or reinflated) and transitions as the cimbia (moulding softening the meeting of the shaft with the base) or the astragale (body of mouldings separating the capital from the shaft) [START_REF] De Montclos | Architecture vocabulaire. Principes d'analyse scientifique[END_REF]. Moreover, certain proportions regulate its dimensions [START_REF] Forssman | Palladio e le colonne[END_REF]. This type of observation can be made for almost every part of a building. The study of the form has a double finality: the first one is the representation, the second one is the surveying of the object. If one analyzes these two moments of the analysis, one easily realizes that they are in a strictly interdependent relation which is neither hierarchical, nor sequential. To draw an element its shape should be known, to know its shape it should be measured, but to measure it, it is necessary to decipher its geometrical nature [START_REF] Migliari | Geometria e Architettura[END_REF]. In this sense, one of the most effective ways to define the architectural surveying is to regard it as the rebuilding of the project. The surveying is indeed a reverse process in which, starting from an existing object, one rebuilds the process of its realization, and one interprets the idea of design which is upstream of its realization.

Five key concepts to understand the geometrical nature of classical buildings

The application of the geometry to the description and analysis of the architectural shape obliges to reduce the multiplicity to intelligibility. Examples of this approach may be found in various fields and applied to the geometrical understanding of shapes or varied phenomena [START_REF] Thompson | On Growth and Form[END_REF]. In the same way, starting from a geometrical analysis of the various parts of a building and by having as an objective its geometrical and semantic description, we have proposed a method for the geometrical reconstruction starting from profiles [START_REF] De Luca | Reverse engineering of architectural buildings based on a hybrid modeling approach[END_REF]. This method is founded on the analysis of invariant and morphological specificities one can extract from a semantic cutting of the building. We saw that throughout the history of architecture, the morphological complexity of the shapes was always influenced by the methods of geometrical control that made their conception possible. Examples of these methods are the descriptive geometry [START_REF] Monge | Géométrie descriptive[END_REF], or stereotomy [START_REF] Desargues | Bruillon projet d'exemple d'une manière universelle du S.G.D. touchant la pratique du trait à preuves pour la coupe des pierres en Architecture[END_REF][START_REF] Vallée | Spécimen de coupe de pierres, contenant les principes généraux du trait et leur application aux murs, aux plate-bande, aux berceaux, aux voûtes spheriques, aux voûtes de révolution[END_REF]. Based on a study of the principles subjacent to these control methods of the architectural shape, one can then identify on one side relevant information to extract from a survey (profiles in a point cloud for example) and on the other side the process of construction better adapted to ensure the geometrical restitution of the elements.

Here, starting from an analysis of the architectural treaties relating the classical language [START_REF] Alberti | De re aedificatoria libri decem[END_REF][START_REF] Palladio | The Four Books of Architecture (re-edn. 1965[END_REF][START_REF] Serlio | Tutte l'opere d'architettura et prospettiva[END_REF], we identify five key concepts for understanding the geometrical nature of classical buildings and for the restitution of their shape.

• Dominant surface. Each building presents a dominant surface which characterizes its space extension and its principal internal divisions (see Fig. 2). • Transition. In the classical language, architectural shapes are based on geometrical transitions. Profiles Fig. 2. Reconstruction of a dominant surface shared by two elements are then to be distinguish in the general composition. In the specific case of the 3D reconstruction, the problem consists in identifying the transitions between the elements throughout a dominant direction (see Fig. 3). • Plans of construction. For Quintrand et al. [START_REF] Quintrand | La CAO en architecture[END_REF], although the objects handled by the architects during the centuries are primarily three-dimensional, figurative practice in design always has as support the plan. The profiles extracted from relevant plans (see Fig. 4) thus constitute the descriptors of surfaces that can be generated by traditional functions of modeling (sweeping, revolution, interpolation, etc.). • Repetition. The composition of a great number of architectural buildings is based on the distribution of repeated elements. These elements are often organized following geometrical layouts: symmetry, rhythm and other rules of composition can for example organize elements following a dominant direction. Moreover, these repetitions can interest various scales: the hierarchical relations that the architectural composition expresses organize the typified elements around the concepts of order, module, stage, or frontage. With regard to the classical language this organization results from the fundamental principles of the architectural composition [START_REF] Wittkower | Architectural principles in the age of humanism[END_REF]. • Mouldings. Understanding the role the mouldings play in the definition of the shape of classical architecture elements is essential [START_REF] Ungers | Ordo, pondo et mensura: criteri architettonici del Rinascimento[END_REF]. Firstly, that makes it possible to rebuild profiles by comparing them with a description language that belongs to the architectural representation. Secondly, that makes it possible to describe the building as a collection of objects identified by a precise To treat in a relevant way the restitution of the profiled elements, it is now necessary to look further into the aspects concerning the role of architectural knowledge in the interpretation of the shape.

Mouldings: atomic units of the classical architecture

The problem of the formalization of the geometrical nature of architectonic objects consists above all in the identification of the atomic entities that compose them. However, a classification of these basic elements cannot be led by an analysis of the morphology that refers to the descriptive systems used for their design. It is thus necessary to call upon the treaties of architecture, according to the period the studied object belongs to. Rattner [START_REF] Rattner | Parallel of the classical orders of architecture[END_REF] developed a classification of the mouldings showing in a systematic way the role they play in the design of clas-Fig. 5. Classification of mouldings of the classical language suggested by Rattner [START_REF] Rattner | Parallel of the classical orders of architecture[END_REF] sical buildings. The author defines the mouldings as the smallest physical units -the atoms -of the classical architecture and provides a way to understand the shape of the architectural elements according to their combination. This classification is based on a variety of 14 mouldings (see Fig. 5) and uses several criteria of regrouping. At the most elementary level, one can gather this variety of atoms according to the analysis of the shape. One distinguishes initially the straight mouldings, then the concave ones, the convex ones and finally the complex (see Fig. 5 from the left column to the right). One can also gather the same variety of mouldings, according to their architectural function in the composition of the building. In Fig. 6a a basic moulding is first used as a visual "foot" with the weight of the wall that rises above. Then it modulates the transition between the vertical and horizontal planes of the wall and floor, and finally it punctuates the bottom of the wall.

In Fig. 6b, the entablature, as all the elements extended on a important dimension (vertical or horizontal) of the frontage, aim to create structurally significant divisions of surface. In Fig. 6c, while it is illogical to have two different scales in a particular sequence of mouldings, it is common to have profiled elements pertaining to various scales. In the classical buildings one generally distinguishes two main scales that mark the relation between human and architectural dimension. From a geometrical point of view, the aspect that interests us most is certainly the coordination -the relative localization of a moulding, like its proportions, implies the coordination of various parts to compose an ordered unit. This is usually reached by means of alignments of plans. In the example of a base (see Fig. 7), when a torus is placed above a plinth, its external point is typically aligned to the prolonged vertical plan of the surmounted element. A horizontal axis on the basis of the barycenter of the torus in intersection with this plan then identifies the point of tangency of the half-round pearl that characterizes its profile.

Formalization of architectural primitives

General remarks

In this section, we present the method developed to formalize architectural primitives starting from the analysis of historical sources. The starting point of the formalization lies in the observation of the geometrical nature of the majority of classical elements. Indeed, in several cases, their shape derives from the revolution around an axis or the extrusion of a profile with a strongly ordered and connected geometry. The definition of our approach of formalization takes into account three distinct dimensions:

• The interpretation of knowledge relating to the shape • The definition of the methods allowing its geometrical modeling • The identification of the relations between the constitutive parts of the shape Several authors contributed to the classification of profiled elements [START_REF] Ginouvès | Dictionnaire methodique d'architecture Grecque et romaine. Tome II. Athènes[END_REF][START_REF] De Montclos | Architecture vocabulaire. Principes d'analyse scientifique[END_REF][START_REF] Rattner | Parallel of the classical orders of architecture[END_REF]. These classifications concentrate, however, on the analysis of the composition language over a specific period. Moreover, as the authors indicate it, a certain number of mouldings results from the combination of basic mouldings. With the geometry like central point of view, a comparative analysis of the traditional mouldings enables us to extract a series of reports:

• Any moulding results from a combination of linear segments and/or arcs. • Any transition in a profile (or inside a moulding) is based on the orthogonality between two construction plans. • Any curved part in a moulding (concave or convex) results from the deformation of its bounding box or from a polycentric construction of tangent arcs.

In regard to the surfaces of a profiled element, one can observe that:

• Any surface results from a generating profile and a directing path. • The generating profile and the directing path are always in condition of orthogonality (even when the second one is a curve).

• Any directing path results from the combination of the same geometrical entities that compose the mouldings, on a different scale.

It is on these reports that we base our approach of the analysis of the formalization of knowledge related to the architectural shape, with an aim of extracting the fundamental principles of its geometrical construction. However, the shape division resulting from the requirements of geometrical modeling is inevitably not coherent with the one that allows its semantic description. An architectural term can for example identify a part of the object whose surfaces are generated starting from two distinct modeling procedures. To solve this problem we organize the formalization of the element according to the relations between two parallel levels of description: geometrical and semantic. The first makes it possible to rebuild the shape in three dimensions; the second makes it possible to organize its parts according to the vocabulary of the architect. As we have the requirement to handle the entities we formalize in three dimensions, we base our approach on the development of a network of nodes developed in MEL (Maya Embedded Language). A node is a structure that can organize, receive and provide information through its attributes. A polygonal surface or NURBS (non-uniform rational B-splines) can be recorded in a node in the form of an attribute. By connecting simple nodes a computation channel is created in the DG (dependency graph) that will produce the final result. The DG is based on a push-pull model: when one provides data to a node, it causes an update of all the nodes which receive and/or process these data [START_REF] Gould | Complete Maya Programming: An Extensive Guide to MEL and C++API[END_REF].

Geometrical description

With an aim of extending the framework of formalization to a higher number of cases (possibly relative to Fig. 8. Geometrical atoms for the construction of mouldings of classical architecture other historical periods) we introduce a level of description lower than that of the mouldings: geometrical atoms.

They constitute a set of geometrical primitives that, by combining, allow the reproduction of any kind of moulding of the classical architecture language (see Fig. 8). These atoms are the only entities of the formalization for which we provide geometrical information (control points of a line or of a curve): indeed, the generation of the whole surface of the element is based exclusively on constraint relations, grouping operations and modeling functions. Geometrical description is based on the relations between three types of nodes which describe the construction of the element from the definition of its geometrical atoms until the complete generation of its surfaces.

• Atoms. This node is characterized by a structure of information concerning the geometrical construction, spatial transformation and constraints of an atomic entity.

• Profiles. This level allows grouping the mouldings according to the plans of construction.

• Surfaces. This level uses specific nodes for the generation of surfaces starting from the profiles.

First level: geometrical atoms

A node atom contains essential information for its representation in space (position, rotation, scale) and controls four under-nodes which share attributes (see Fig. 9):

• The bounding box constitutes a deformable support on which an atom is built. It is based on a 9-point grid that is also used as dimensional unit of reference. • The origin defines the first control point of the geometrical atom. • The anchoring defines its last control point and is exploited to constrain the position of the atom that is connected to it. • The geometrical representation contains the control point of a B-spline curve defining the geometry of the atom.

Second level: profiles and paths

The second geometrical level of description results from a combination of atoms. This combination is based on Fig. 11. Construction of a moulding by combination of a set of geometrical atoms an upward aggregation according to the vertical axis of a construction plan and it is carried out by a heritage of the position attribute (translation vector): the origin node of an atom is constrained with the anchoring node of the atom which precedes it in the chain (see Fig. 10). To generate profiles, on each level of the progressive combination, we modify the attributes of width and height of the corresponding atom. These attributes deform the bounding box of the atom to which the geometrical representation is connected. Figure 11 shows the modeling of one of the most complex mouldings: the beak. The re-grouping of the atoms in profiles is established according to the common modeling functions. For most elements Fig. 12. A vertical profile and two horizontal build by combination of geometrical atoms two profiles are sufficient. It is necessary to identify the generator and the director of surface. As in the case of the generating profile node, the directing path node can also be defined by combination of geometrical atoms (see Fig. 12).

Third level: surfaces

This level of the geometrical description is based on the concept of construction history: a property of our development environment that we exploit to save information concerning the entries of a modeling process. Surface is indeed generated by a modeling function node (linear, curved or combined extrusion), which receives geometrical information from a generating profile node and a directing path node. Once the surface is generated the two entry nodes are consequently connected to the node surface. That allows updating the surface according to each transformation of an atom. We base the procedures of modeling on three extrusion functions (see Fig. 13). In the three cases the geometry of the generating profile is swept along the directing path. The alternatives relate to the composition of the directing path and the degree of the resulting surface. We break up the directing path according to this last attribute. The generation of surface advances per piece, and, when a change of degree (from 1 • to 3 • or vice versa) is detected, a new surface is created in agreement with the degree of the directing path. This new surface uses as generating profile the last isoparametric curve of the preceding surface. That ensures continuity between the various pieces of surface. The same technique is used to create planar filling on the ends of the surfaces.

An example of formalization: a doric capital from the Palladio's treaty

Figure 14 shows the geometrical construction of a doric capital described in the Palladio's treaty [START_REF] Palladio | The Four Books of Architecture (re-edn. 1965[END_REF] and obtained by a simple combination of geometrical atoms gathered in two generating profiles and extruded along two directing paths. For the formalization of the element, we use the description provided by the author and on the relative drawing (see Fig. 15): we connect node atoms according to the described mouldings. Once the mouldings are suitably proportioned -this is possible at the same time by using the dimensions present on the drawing -we gather them in two generating profile nodes according to the respective modeling procedures.

Semantic description

Semantic description is defined by a structure of concepts organized around geometrical description. The nodes of the semantic description are connected on one side to an architectural term, and on the other they gather In Fig. 16 one can notice the differences between logics of cutting between the geometrical and the semantic description of the formalized element. First of all it is noticed that an architectural atom (moulding) often results from a series of geometrical atoms. Then one can observe that geometrical description must take into account all discontinuities along the profile (on the two axes) and to separate the transitions, concave and convex, from the linear segments with which they are in continu- ity. On the other hand a semantic description results from the assignment of an architectural term to the parts of the object according only to a vertical division. The architectural term identifying a moulding indeed indicates a transition in surface from the element, but it does not specify with precision its space perimeter, nor all the geometrical atoms that belong to him. We deduce from it that a semantic description relates to a mental representation of the shape that it is not always sufficient for an exhaustive description of its morphology. That justifies a part of our approach: we add a semantic layer to a geometrical representation. But that suggests at the same time the difficulty that an opposite approach can pose (to generate a geometrical representation starting from a semantic description). We used formalized primitives for two distinct applications:

• To instantiate architectural primitives in a point cloud with the double aim of reconstructing the shape of the analyzed element and automatically obtaining relevant measurements • To constitute a library of architectural elements by the simple declaration of a sequence of mouldings

In the first case, one builds a semantic layer on the geometrical representation of a scanned object. In the second case, one builds a geometrical representation starting from a semantic description.

Instantiation of architectural primitives in a point cloud

The concept of construction history, on which our formalization approach is based, guarantees the relationship be- element without being concerned with effective coincidences between the primitive and the point cloud (see Fig. 17). • The second stage consists in deforming under-constraint the geometrical atoms of the primitive. This stage also exploits the intersection planes. The logic of deformation follows an upward sequence. Thanks to the chain of formalized constraints, the deformations applied to the first atom determine the update of the positions on the remainder of the chain (see Fig. 18). The degrees of freedom of each atom are defined in the moment Fig. 19a,b. The primitive instantied on the point cloud. On the left b, the vertical profile with dimensional information. On the right a, the primitive projected on a photography oriented on the point cloud of formalization. However, in the case of the application of a rotation, the deformation of surfaces can cause a degradation of the final form of the primitive. For this reason it would also be necessary to take into account the formalization of constraints of orientation (limits of displacement, rotation and deformation of the geometrical atoms). • The third stage (optional) allows controlling the lowest level of geometrical description: atoms. In this case, the transformations can be applied to the control points of B-splines. This stage is important to finely adapt the profiles on the point cloud: indeed, the deformations applied to the geometrical atoms in the second stage of instantiation process relate to the bounding boxes that contains them. That constitutes an important limitation if one wants to adapt the profiles of the primitive to the co-ordinates resulting from the laser scanning. Indeed, each part of the surveyed shape presents various imperfections due to the decay of materials or more simply to the errors of realization.

Thanks to a hybrid registration of different sources [START_REF] De Luca | Reverse engineering of architectural buildings based on a hybrid modeling approach[END_REF], the whole procedure of instantiation can also exploit an oriented photograph as visual support where the point cloud remains a limited support of interpretation (see Fig. 19a). This procedure is currently manual, but its organization in progressive stages of adjustment offers the possibility of integrating automatisms. Finally, the relations established between the two levels of description (geometrical and semantic) enable us to extract from the geometrical representation a set of relevant dimensional information (see Fig. 19b). For example, after the instantiation process, we can know the height of the abaco of the capital instead of the simple distance between two points.

Modeling of architectural primitives per declaration of mouldings

Another type of exploitation of our approach relates to the constitution of a library of elements by simple declarations of a sequence of mouldings. We focused on the problem of adapting the declarative modeling approach [START_REF] Lucas | Les modeleurs déclaratifs[END_REF] for the semi-automatic generation of profiled elements of classical architecture. Selected logic is based on the sequential aggregation of mouldings forming a profile in an upward order and according to the principal parts of the object. For better explaining our problem we show the example of the balusters. The baluster is a small vertical element normally divided into three parts (base, body and capital) and whose form is generally based on the revolution of a profile around a vertical axis (see Fig. 20). This revolution can have various levels of smoothing starting from a horizontal square, hexagonal, octagonal or circular profile. According to the cases, different horizontal profiles can represent the shape of the various parts of the element. From the identification of this simple structure of description, a big variety of balusters can be described (see Fig. 21). An in-depth study on this element is provided by Wittkover [START_REF] Wittkover | Il balustro rinascimentale e il Palladio[END_REF]. The procedure developed allows to automatically generate balusters starting from a sequential description of mouldings associated with a path. For the dimensioning of the atomic units three strategies are possible: Another approach that we experimented uses an aggregation of entities of a higher nature, the mouldings. In this case, the geometrical primitives are geometrical sets of atoms already connected in groups. This king of modeling per declaration appears very effective for the geometrical construction of entities by a declaration of a sequence of architectural terms (mouldings) and without a direct handling of the geometry. For example, the base is an element characterized by a sequence of mouldings in revolution around the vertical axis: its modeling can result from the simple list of the mouldings composing its generating profile. Figure 22 illustrates three bases modeled starting from the declaration of a sequence of mouldings.

Library of elements

One of the most important advantages of our formalization approach resides in the possibility of separating the description of the elements from their geometrical representation. That makes possible the exploitation of the same description by various procedures of geometrical interpretation and also preserves the data independently of a specific modeling language. Indeed, several methods of geometrical translation could be studied with an aim of exploiting the collected information. We developed a solution for sharing a library of elements based on the interaction between a relational database and a 3D modeler. This procedure interprets the descriptions stored in the database by translating them into a MEL script. The recording in the data base organizes the geometrical and parametric description of each entity according to a pre-established structure with five fields (atoms, constraints, profiles, surfaces and vocabulary terms). We intend to entirely reproduce this structure in XML format. A MEL function allows introducing a web browser inside an interface lay- out. This browser can receive MEL scripts starting from an URL. For the automatic generation of a primitive a PHP procedure extracts the values stored in the database and uses them to generate a modeling script in MEL. Once generated, the primitive keeps the relation with their construction history allowing under-constraints deformations (see Fig. 23).

Model structuring and multi-representation

The building as a system of knowledge

To associate semantics to the architectural shape forces above all to regard the building as a system of knowledge, then to extract a model from it description, and finally to define its representation according to the objectives of the analysis. A system of architectural knowledge can be described as a collection of structured objects identified by a precise vocabulary [START_REF] Mitchell | The Logic of Architecture. Design, Computation, and Cognition[END_REF][START_REF] Quintrand | La CAO en architecture[END_REF]. Several researches concen-trated on the development of classifications of architectural elements in theoretical frameworks [START_REF] Tzonis | Classical Architecture -The Poetics of Order[END_REF] or in applications of the geometrical modeling [START_REF] Gaiani | Translating the architecture of the world into virtual reality and vice-versa: 7 years of experimentation with "conservation and representation" at OFF[END_REF][START_REF] Mitchell | The Logic of Architecture. Design, Computation, and Cognition[END_REF]. Beyond the identification of the architectural primitives, the description of the structuring of these elements in an ordered unit requires observations of a more complex nature. Goulette [START_REF] Goulette | Sémantique formelle de l'espace: Application au raisonnement spatial qualitatif en architecture[END_REF] identified three types of relations between the elements one can find into the vocabulary of the architect:

• Topological relations. They are the relations that express the relative positioning of an element, or a part of element, inside a unit device. • Part-unit relations. The description of a building can be led by regrouping simple elements in a unitary whole or ordered sets. These relations express the hierarchical organization which one can read in the composition of a building: the elements started from a homogeneous whole, itself being able to have started from a unit of a higher nature. • Composition relations. They are diagrams of composition resting on a set of rules controlling the relative positioning and the dimensioning of a set of elements in a general geometrical layout.

We turn our attention on part-unit relations because the other types of relations relate rather to the problems of architectural composition than to the analysis of existing buildings where these relations are already explicit. We distinguish three types of part-unit relations:

• Piece-all. The piece-all relations express the way in which the under-parts of an object are organized to form a single element. The treaties of architecture normally offer a term for each under-part of the element. This cutting corresponds sometimes to the true physical divisions of the object, and in other cases, it refers to the geometrical transitions that the shape expresses. • Member-collection. These types of relations are at the base of typological classifications. Compared to the identification of a type, these relations can gather the elements of categories according to comparison criteria. In architecture, these classifications are based normally on the study of the stylistic currents related to historical periods or geographical areas. • Component-assembly. They are relations that characterize the logic of presentation of treaties of classical architecture. They express the architectural sets according to a logic based on the analysis of the composition of the building. The composition of "harmonious" sets is led by principles of scheduling like proportions, rhythm and symmetry.

Semantic description of existing buildings

In the research we just quoted, it is obvious that the logic of the semantic cutting of a building is always constrained to the description of ideal models. These approaches are indeed founded on the structuring of symbolic shapes. However our objective consists in adding a semantic layer to the description of the morphology of existing buildings. Blaise [START_REF] Blaise | Le formalisme objet appliqué à l'étude de l'edifice patrimoniale: problèmes de modélisation et d'échanges de données sur le réseau Internet[END_REF] showed that for the architectural analysis, approaches of classification by intension (association of an ideal model to a real element) are sometimes incompatible with the requirements of architectural heritage studies, whereas approaches by extension (extraction and grouping of descriptors common to several elements) seem more relevant. In fact, the architectural element, as a symbolic shape, belongs indeed mainly to the field of the mental representation [START_REF] Goulette | Sémantique formelle de l'espace: Application au raisonnement spatial qualitatif en architecture[END_REF]: it is an abstracted object used to think and conceive an architectural project. Moreover, it should be noticed that knowledge that intervenes in a semantic cutting of a building always relates to an objective of analysis. As an example (see Fig. 24) the logic of cutting according to the vocabulary of the architect is in contrast with divisions between the physical units that make the object: divisions between the base, the shaft and the capital of the column do not coincide with the physi-Fig. 24. Ambiguities between the vocabulary of the architect and physical divisions of the elements cal borders of the elements. The astragale and the cimbia, which in the vocabulary of the architect constitute the ends of the shaft of the column, form, in the carried out building, part of the capital and respectively the base (due probably to problems of realization). This type of ambiguity explains the difficulties that a semantic description may meet within a framework of structuring of three-dimensional entities of buildings. With the aim of designing a methodological framework for the representation of buildings based on their semantic description, we introduce the "point of view" as a concept determining the structuring of the model. The concept of point of view includes three aspects:

• The use of a vocabulary to isolate elements from the building • The identification of a logic for the regrouping of these elements in a coherent unit • The choice of a type of representation able to extract from the shape the information necessary to the analysis

In agreement with the principles introduced by Falcidieno et al. [START_REF] Falcidieno | A shape-abstraction paradigm for modelling geometry and semantics[END_REF], we built around the morphology of the building a description model defined by three distinct levels: semantics, structure, representation. Consequently, we distinguish three phases passing from the restitution of the morphology of the object to its possible representations.

• Semantic level. A first phase consists of isolating concepts (terms of description) and associating to them parts of the shape. That underlines the need to lead a morphological cutting according to a precise vocabulary used by a specific analysis (see Fig. 25). • Structural level. The second phase makes it possible to establish a graph of relations between these concepts and to organize the elements of the scene compared to a requirement of description. To support this phase we use a symbolic notation, in 3D space, of the part-unit relations established between the isolated elements. We use as a starting point the logic of structuring introduced by Heine [START_REF] Heine | High precision building documentation: Element definition and data structuring[END_REF]. A 3D graph (tree structure in space), whose configuration depends on the handling (hierarchical relations) of a set of terms in a list, is established using specifically defined nodes (see Fig. 26):

• A morphological entity, which results from a semantic cutting, constitutes a concept identified by the user for the description of the building which can ing from three geometrical bases (points, curves, polyhedrons), we classify various types of representations (basic and colored point clouds, depth maps, parametrical surfaces, enriched polyhedrons, etc.) focusing on the type of description they allow, the techniques necessary to their generation and the information they convey (see Fig. 27). As we will see, this division confers an important degree of freedom in the exploitation of the data resulting from the surveying of the building because, starting from the same geometrical model, one can work out various semantic descriptions that exploit a multirepresentation system. Moreover, the model of description of the building is not only used for the structuring of the three-dimensional model, but may become the common denominator for the establishment of bilateral relations between morphology of the object and heterogeneous information.

Points of view on the building

Various researches concentrate today on the multi-representations of entities in various research fields. Among this works, some are applied to cartography [START_REF] Spaccapietra | GIS Databases: From Multiscale to Multirepresentation[END_REF]. In this case, the multi-representations of two-dimensional entities are exploited to guarantee a level of graphic in-formation adequate to a given scale. Others exploit the multi-representations to structure various levels of polyhedric complexity according to the various exploitations of the same 3D model [START_REF] Léon | Structure d'un environnement de Conception Multi-vues et Multi-représentation[END_REF]. All these works organize several representations of the same object while being always based on the same geometrical base (2D vectors or 3D polygons). The difference among the multiple representations thus relates to the quantity of geometrical information they contain. However, the analysis of the architectural heritage can use techniques of representation that exploit various geometrical bases. It is therefore important to introduce a distinction between the concepts of resolution and representation: a representation results from a development technique that allows describing the shape of an object according to a geometrical base and to a resolution level. A second problem relates to the possibility of making the collected data and information accessible to an important variety of users (researchers, administrators, general public, etc.). To do this, a representation system must necessarily reflect the various points of view of disciplines that are interested in the study of the building, its conservation and its valorization. Division between three parallel levels of description (semantic, structure and representation) can thus be exploited to construct a representation of the building according to a specific

Conclusions

The problems tackled in this article relates to an approach to understanding, reconstructing, structuring and representing architectural elements. This approach suggests an-swers to a certain number of problems in the field of the architectural representation, but it presents some gaps that must be analyzed. Indeed, even if the restitution of the shape based on a priori models can appear as an interesting solution for the dimensional comparison of repetitive elements, its implementation presents problems not yet solved. The positioning of the architectural primitive in the point-cloud is currently based on arbitrary estimations of the user. With an aim of making this procedure more robust it will be necessary to implement best-fit techniques according to various aspects: surfaces [START_REF] Eck | Automatic reconstruction of B-Spline surfaces of arbitrary topological type[END_REF], characteristic profiles [START_REF] Plass | Curve-fitting with piecewise parametric cubics[END_REF] or construction history [START_REF] Ramamoorthi | Creating generative models from range images[END_REF]. In the Fig. 29. Ideal models (in wireframe) instantiated on the archaeological rests (reconstructed) of the Roman theater in Arles same spirit, the methods using graph grammar for the automatic rebuilding of shapes [START_REF] Fuchs | Efficient Subgraph Isomorphism with 'A Priori, Knowledge. Application to 3D Reconstruction of Buildings for Cartography[END_REF] could be taken into account.

The principle introduced for the description of the building was tested only on buildings of classical language whose architectural composition is characterized by a clear organization of the composing elements. Other stylistic currents are likely to be studied by a formalization of this type. In the same way, it is necessary to hold into account the composition relations, which rest on a set of rules controlling the relative positioning and the dimensioning of elements in a general geometrical layout. This aspect may, in the same time, appear relevant to understanding the architectural composition that characterizes an existing building and to the development of restitution assumptions of a partially or totally destroyed building. The problem consists of formalizing the proportion systems that interest the buildings of a great part of the history of architecture [START_REF] Scholfield | The Theory of Proportions in Architecture[END_REF]. This type of reasoning may for example be applied to the representation of restitution assumptions resulting from a comparison between ideal models and reconstructed fragments [START_REF] Blaise | Architectural Modelling and Information Interfacing: Learning from Three Case Studies[END_REF] (see Fig. 29). In this direction, the concept of multi-representation could play an important role: to organize around the description of the building a set of representations corresponding to the various levels of its hypothetical restitution.

Fig. 1 .

 1 Fig. 1. Representation of the Tuscan order in the John Shute's treaty "The First and Chief Grounds of Architecture". London, 1563

Fig. 3 .Fig. 4 .

 34 Fig. 3. Reconstruction of a transition

Fig. 6 .

 6 Fig. 6. Diagram illustrating the role of mouldings in the composition of a classical building. Illustration elaborated on a drawing of the Scamozzi's treaty. Venice, 1615

Fig. 7 .

 7 Fig. 7. Construction of the Composite order in the Blum's treaty. Zurich, 1550

Fig. 9 .Fig. 10 .

 910 Fig. 9. Diagram of relations of the "geometrical atom" node's attributes

Fig. 13 .

 13 Fig. 13. Modeling functions for the surface generation

Fig. 14 .

 14 Fig. 14. Geometrical description of a Doric capital starting from the Palladio's treaty

Fig. 15 .

 15 Fig. 15. Description of the Doric capital in the Palladio's treaty[START_REF] Palladio | The Four Books of Architecture (re-edn. 1965[END_REF]

Fig. 16 .

 16 Fig. 16. Semantic description of a Doric capital starting from the Palladio's treaty

Fig. 17 .Fig. 18 .

 1718 Fig. 17. First stage of instantiation of the primitive. From left to right: the horizontal intersection plans, the primitive in the point cloud, the vertical intersection plan

Fig. 20 .•

 20 Fig. 20. The modeling of a baluster by declaration of a sequence of geometrical atoms

Fig. 21 .Fig. 22 .

 2122 Fig. 21. Balusters modeled by declaration of a sequence of geometrical atoms

Fig. 23 .

 23 Fig. 23. Diagram of relations between the data base and the 3D modeler

Fig. 25 .

 25 Fig. 25. Morphological cuttings of the analyzed object according to two different analyses: the vocabulary of the architect and the construction techniques

Fig. 26 .

 26 Fig. 26. 3D graph for the semantic description of the building

Fig. 27 .

 27 Fig. 27. Multiple representations organized around a morphological entity

Fig. 28 .

 28 Fig. 28. Four "points of view" on the studied object elaborated starting from the same morphological cutting

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