Christian Bruneau

Rafael Gramage-Doria
email: rafael.gramage-doria@univ-rennes1.fr

Ruthenium-catalyzed ortho-C-H bond arylation reactions

Keywords: ruthenium, C-H bond functionalization, arylation, ortho-selectivity, reaction mechanisms

The highly selective ruthenium-catalyzed ortho-C-H bond arylations have met a level of maturity that makes them an attractive tool to achieve unique organic transformations.

By means of the manifold mechanisms operating in the presence of different arylating agents and reaction conditions, these methodologies have succeeded in reducing the wastes/steps towards the desired molecules as well as enlarging the chemical space by affording molecules impossible to obtain otherwise.

Introduction.

Biaryl and terphenyl cores are widespread structures in many biologically active substances and have useful physical properties in material sciences. The seminal results on the synthesis of biphenyl by connecting two aryl iodide molecules via reductive coupling in the presence of copper were reported in 1901 [1] and this concept was further developed [2,3]. Since this discovery, continuous efforts have been devoted to the improvement of these aryl-aryl couplings, mainly based on palladium-catalyzed cross-coupling reactions of aryl (pseudo)halides with organometallic reagents leading to the formation of a new C(sp 2 )-C(sp 2 ) bonds (Figure 1a) [4][5][6][7][8].

Figure 1 Various strategies for the construction of biaryls by metal-catalyzed approaches.

These catalytic methods based on cross-coupling of an aryl halide (or triflate) and an organometallic substrate (B, Sn, Zn, Si, Mg derivative) have brought drastic improvements for the preparation of aryl-aryl bonds but they suffer from some disadvantages. Indeed, the organometallic substrate has to be prepared, which is sometimes not straightforward and the reaction generates metal-containing wastes produced in stoichiometric amounts, which is not compatible with the criteria of sustainable green chemistry in terms of atom economy and environmental aspects (Figure 1a). During the last twenty years, alternative methods involving direct C-H bond activation have emerged. Intermolecular oxidative coupling of two arenes or cross dehydrogenative coupling (CDC) represents a very appealing process since it proceeds via activation of two C-H bonds without prefunctionalization, but requires the presence of an oxidant and encounters regio-selectivity issues, which have however been recently tackled by the use of directing groups (Figure 1b) [9][10][11][12][13]. Cross-coupling involving the activation of only one aryl C-H bond has been realized using as cross-coupling partner either an electrophilic aryl (pseudo)halide or an organoboron substrate (Figure 1c). These direct arylation methods have been developed with several transition metal catalysts [14][15][16][17][18][19][20][21][22] and, typically, involve inexpensive substrates. As such, they are carried out in less steps by reducing waste production compared to the traditional cross-coupling methods. The critical point in these methods is the site-selectivity for the activation of a targeted aromatic C-H bond. This has been solved by the use of directing groups, which temporarily coordinate the metal centre, and orientate the activation of a specific C-H bond. Besides the use of catalysts based on Pd, Rh and Ir, the less expensive and more affordable Ru-based ones appeared similarly powerful due to similar oxidation states and geometries [23]. In this chapter, we discuss the ortho-selective C-H bond arylation of phenyl groups (aryl-aryl bond formation) catalyzed by ruthenium catalysts with the assistance of directing groups (Scheme 1).

Scheme 1 Ruthenium-catalyzed ortho-C-H bond arylation.

There are many reviews dedicated to directed ruthenium-catalyzed C-C bond formation involving aromatic C-H bond activation/functionalization, most of them include arylation but also alkenylation and alkylation as well as C-heteroatom bond (C-halide, C-N, C-O) formation [24][25][26][27][28][29][30][31][32][33][34][35]. Arylations proceed via various reaction mechanisms and catalytic cycles depending on the nature and especially the oxidation state of the ruthenium catalyst (Ru(0) or Ru(II)).

After presenting the different mechanistic aspects of the ruthenium-catalyzed arylation elaborated from experimental and theoretical studies, examples illustrating each type of mechanism are given. We discuss the Ru(0)-catalyzed ortho-arylation directed by ketones, and various aspects of the Ru(II)-catalyzed ortho-arylation including the nature of the directing groups, arylating agents, site selectivity (mono-and bis-arylation) and show some applications in industry. The directed arylation of five-membered heterocycles such as pyrroles, indoles, furans, thiophenes, and benzofurans are not included in this chapter [36][37][38][39][40][41][42][43].

2. Ru-catalyzed ortho-C-H bond arylations using low-valent ruthenium pre-catalysts.

2.1. Ru-catalyzed ortho-C-H bond arylation using low-valent ruthenium pre-catalyst: mechanistic considerations.

Following the seminal results obtained in the group of Murai on aromatic ortho-C-H bond activation/alkylation by a terminal olefin directed by a ketone with RuH2(CO)(PPh3)3 or Ru(CO)2(PPh3)3 as catalyst precursors [44], this type of ruthenium precursors have found catalytic applications in various directed C-H bond functionalization, in particular in arylation with aryl boronates. The accepted mechanism is based on the initial coordination of the carbonyl group of the aromatic ketone to the ruthenium(0) center of the active species, either starting from a ruthenium(0) catalyst or generated upon reduction of a ruthenium(II) dihydride complex, bringing the ortho C-H bond close to the ruthenium center. By direct interaction of the ruthenium atom with the C-H bond or by 1,4-addition of the ruthenium moiety followed by It can be noted that several cyclometalated RuH(o-C6H4C(O)Me)(CO)(PR3)2 complexes (R= Cy (I) [49], Ph (II) [50]) have been isolated and characterized. Among them, those presenting the two phosphine ligands in trans-position (I and II) exhibited poor catalytic activity whereas the cis-isomer III was identified as a key intermediate in the ortho-alkylation of aromatic ketones with olefins (Figure 2) [51]. In addition, it was found that the four-membered ruthenacycle IV generated from RuH2(CO)(PPh3)3 and resulting from ortho-C-H bond activation in a triphenylphosphine ligand was an excellent catalyst for the same reaction. In ortho-directed arylation of ketones and imines, this C-H bond activation with ruthenium(0) pre-catalysts has been essentially used with aryl boronate as arylating reagents. In their seminal report, Kakiuchi and co-workers proposed a catalytic cycle where two equivalents of ketone with respect to the arylating agent were necessary to achieve high yield in arylated ketone (Scheme 3) [52]. Indeed, one equivalent of ketone was involved in the formation of an alkoxy ligand by insertion of the carbonyl bond into the ruthenium hydride bond initially generated.

Further transmetalation installed the aryl group on the metal center and the final arylated product was obtained by reductive elimination, which concomitantly regenerated a catalytically active ruthenium(0) species. The drawback of this protocol was the loss of one equivalent of valuable aromatic ketone in the overall transformation. In order to circumvent this problem, the use of a reactive sacrificial aliphatic ketone able to produce the alkoxy ligand and act as a scavenger of both the B(OR)2 moiety and the ortho-hydrogen of the ketone substrate made possible the use of equimolar amounts of substrates. The best arylating reagents were aryl boronic acid neopentylglycol esters (Scheme 3), and the most efficient sacrificial ketone was tert-butylmethyl ketone (pinacolone) [53]. It must be noted that the catalytic cycle described in Scheme 3 involves Ru(0)/Ru(II) species and that no base/acid is required. Scheme 3 General catalytic cycle for the ortho-C-H bond arylation of ketones with aryl boronates starting from a Ru(0) pre-catalyst.

Examples of Ru-catalyzed ortho-C-H bond arylations using low-valent ruthenium pre-catalysts.

Ten years after the discovery of the ortho-directed C-H bond activation/alkylation of aromatic ketones with olefins in the presence of RuH2(CO)(PPh3)3 as precursor of catalytically active Ru(0) species [44], the corresponding arylation was shown with arylboronate as source of nucleophilic aryl group [53]. The catalytic cycle for this arylation has been described in Scheme 3 with 5,5-dimethyl-2-phenyl-1,3,2-dioxaborinane, which is the most efficient and used arylating boronate so far. Because, a catalytic cycle consumed 2 equivalents of ketone, the first results were obtained in refluxing toluene with a twofold excess of ketone in order to reach satisfactory yields (conditions A, Scheme 4). The utilization of pinacolone under refluxing conditions (106 °C) both as solvent and sacrificial ketone allowed to work under stoichiometric ratio of substrates with similar efficiencies (conditions B, Scheme 4) [53,54]. With both systems, bis-arylation was observed only with the less sterically hindered ketones, namely acetophenone and phenyl isopropyl ketone, but only monoarylated products were formed from pivalophenone. Electron-donating or withdrawing groups present on the ketone aromatic ring or on the arylating group were well tolerated. It is noteworthy that the directed ortho-arylation proceeded also with bicyclic substrates such as 1-acylnaphthalene, -tetralone and benzosuberone. The fact that the reaction also worked with 1-acetyl-1,2,3,4tetrahydronaphthalene as substrate (framed, Scheme 4) suggests that large six-membered ruthenacycles are at play in the key C-H bond activation step. With acetophenone substrates, the selectivity towards the bis-arylation that was observed even without an excess of arylboronate could be reversed towards mono-arylation by using the RuH2(CO)(PPh3)3/pinacolone system in the presence of a styrene as additive. For instance the reaction of acetophenone (3 equiv.) with 5,5-dimethyl-2-phenyl-1,3,2-dioxaborinane (1 equiv.) provided a mono-/bis-arylation ratio of 15:64 in the absence of styrene, ratio which was reversed to 56:17 in the presence of 1 equivalent of styrene. Mono-arylation products were thus produced in good yields with high selectivity with electron-rich arylboronates [55]. The conversion and selectivity towards mono-arylation of acetophenone was even improved to 70:8 when 2.5 mol% of RuHCl(CO)(P i Pr3)2 containing a trialkylphosphine ligand was used in combination with 5 mol% of CsF and 1 equiv. of styrene [56].

Scheme 4 Ruthenium-catalyzed ortho-C-H bond arylation of aromatic ketones with arylboronates with a low-valent ruthenium pre-catalyst under two different reaction conditions.

Starting from 1-acylnaphthalene and 2-(4-bromophenyl)-5,5-dimethyl-1,3,2-dioxaborinane, both experimental conditions from Scheme 4 led to very low conversion. In contrast, the presence of a catalytic amount of base (Cs2CO3, RbCO3, CsF, t BuOK, KOH, K3PO4) promoted the ortho-C-H bond arylation up to 83% yield with a very low level (< 5%) of dehalogenation of the bromoaryl group (Scheme 5) [57]. The reactions were carried out at 150 °C with an excess of the ketone without additional solvent. This reactivity of the halogenated arylboronate indicated that, in the presence of the base, the oxidative addition of the aryl bromide to the ruthenium center did not take place significantly and confirmed that the arylboronate was the source of the nucleophilic aryl group. This catalytic system could be applied to various bromoand iodo-arylboronates but was limited by substitution at the ortho-position of the arylboronate.

Because arylboronic acids are available for industrial applications, the arylation of 1acylnaphthalene was successfully carried out with the in situ generated halogenated arylboronate at 150 °C in toluene, yielding 78% of the expected brominated ortho-arylated product, directly from 4-bromophenylboronic acid and 2,2-dimethylpropane-1,3-diol in the presence of RuHCl(CO)(PPh3)3/Cs2CO3.

Scheme 5 Ruthenium-catalyzed ortho-C-H bond arylation of 1-acylnaphthalene with halogenated arylboronate partners.

In most cases, PPh3 is used as the ligand for the ruthenium pre-catalyst. However, RuHCl(CO)(PAr3)3 and RuH2(CO)(PAr3)3 containing bulky triarylphosphines (Ar= 3-MeC6H4, 4-MeC6H4, 4-EtC6H4, 4-FC6H4, 3-MeOC6H4, 4-MeOC6H4, 3,5-Me2C6H3, 4-MeO-3,5-Me2C6H2) also generated efficient catalysts for the arylation with arylboronates. RuHCl(CO)(PAr3)3 catalysts operated better in acetone/mesitylene mixture at 140 °C with CsF as additive and were especially useful for the selective mono-arylation of sterically demanding ketones such as benzophenone. The arylation of the sterically congested 3,5-dimethylphenyl tert-butyl ketone was achieved in high yields with RuH2(CO)(P(3-MeC6H4)3)3 [58]. Selective tetra-orthoarylation of anthraquinone has provided an elegant entry to the preparation of polyaromatic structures. The general methodology for the access to hexaarylanthracenes was based on tetraarylation with an arylboronate followed by arylation of the ketonic groups with an aryl lithium reagent and then reduction of the corresponding alcohol. On the other hand, the synthesis of coronenes involved first ortho-directed C-H bond arylation followed by dimethylenation and finally oxidative cylization promoted by FeCl3 (Scheme 6) [59,60]. Under similar conditions, the use of symmetrical diboronates made possible the ring extension in a linear manner starting from acetophenone substrates to give para-diarylbenzene products (Scheme 7) [61]. Dibenzo[a,h]anthracene and picene derivatives, potential organic field-effect transistor materials, were then produced in two steps after ethylation of the acetyl group and platinum-catalyzed cycloaromatization. Catalytic systems based on ruthenium(0) active species have also been used with other directing groups than ketones. In particular the ortho-C-H bond arylation of aldimines has been performed in toluene at 125 °C with Ru3(CO)12 as Ru(0) pre-catalyst in the presence of benzylideneacetone (BA) as hydride acceptor [62]. The final imines derived from aniline were further hydrolyzed by HCl to recover the ortho-arylated aldehydes (Scheme 8). The arylation was tolerant to many functional groups of the arylboronate reagent such as alkyl, alkenyl, vinyl, alkoxy, alkoxycarbonyl, acetyl, halide, trifluoromethyl placed in meta and para positions.

When one ortho-position of the aldimine was protected by a methyl, phenyl, trifluoromethyl, methoxy or fluoro group, the second ortho-position was efficiently arylated to give the expected o,o'-disubstituted aldehyde. Substitution of the aldimine at the meta-position of the aryl group by an electron-attracting group (F, CF3) or a methyl group led to mono-arylation at the less sterically hindered ortho-position, whereas bis-arylation took place in the presence of an electron donating group such as MeO or TsO. Similarly, the aldimine arising from benzaldehyde led to bis-arylation in 90% yield. It is also interesting to note that the ketamine obtained from unsubstituted acetophenone selectively led to the mono-arylation product in 91% yield contrasting with the direct arylation from acetophenone, which gave bis-arylation [53,54].

Rather than moving towards the aldehyde by hydrolysis, profit has been taken from this efficient ortho-arylation to include it in a one-pot tandem transformation aiming at the production of sterically hindered aromatic amines. Ruthenium(0)-catalyzed C-H bond arylation was followed by catalytic reduction of the imine in the presence of the catalytically active ruthenium residue either via hydrosilylation with Et3SiH or via hydrogen transfer process with isopropanol [63]. It is noteworthy that in all cases, the aldimine could be generated in situ from the aldehyde and the aniline derivative in the presence of MgSO4 as H2O scavenger in the same pot before starting the catalytic sequence. Scheme 8 Ruthenium-catalyzed ortho-C-H bond arylation of aldimines using a Ru(0) precatalyst.

Amides, esters, and nitrile have also been investigated as directing groups for this ruthenium(0)catalyzed ortho-arylation but the results were less conclusive. Amides are not applicable directing groups since in the presence of 4 mol% of RuH2(CO)(PPh3)3 at 125-135 °C in toluene for 85 h, only 7% of the ortho-phenylated product was obtained from 4-MeOC6H4CONEt2 [64].

Aromatic esters have been arylated by 5,5-dimethyl-2-aryl-(1,3,2)dioxaborinanes in the presence of a catalytic amount of RuH2(CO)(PPh3)3 (5 mol%) in refluxing pinacolone to form ortho-arylated benzoate derivatives. From isopropyl benzoate substrates, both mono-and diarylated esters were formed from 2 equivalents of arylboronate with moderate selectivity towards the diarylated products. From ortho-substituted benzoates, moderate yields were obtained and it was noted that the introduction of an electron-withdrawing group such as CF3 promoted the arylation. Easy hydrolysis under acidic conditions gave access to ortho-aryl benzoic acids [65]. The nitrile group appeared as a trickier directing group. The best conditions for arylation of aromatic nitriles included the use of RuH2(CO)(P(4-MeC6H4)3)3 in high catalyst loading (10-20 mol%) in pinacolone at 120 °C but also in the presence of KHCO3 as a base [66]. Depending on the substitution pattern of the nitrile, mono and di-ortho-arylation were observed but also the unexpected arylation at the para-position of the nitrile. Finally, it can be mentioned that the same catalytic systems based on ruthenium(0) catalysis are also very efficient under related experimental conditions for ortho-arylation of aromatic ketones esters and amides based on C-O and C-N bond cleavage of ortho-alkoxy and ortho-dialkylamino aromatic susbtrates [64,67]. In some cases, C-O and C-H bond activation have found complementary applications [60,67c].

3. Ru(II)-catalyzed ortho-C-H bond arylations with aryl (pseudo)halides. RuCl2(p-cymene)(phosphine) complexes reacted at room temperature with K2CO3 in polar solvent to generate [Ru( 2 -OCO2)(p-cymene)(PR3)] featuring carbonate as a coordinated base [72]. Inorganic bases such as K2CO3, Na2CO3, and Cs2CO3 have been extensively used in C-H bond functionalization either as sole base [13,73] or associated with a coordinated or free carboxylate [74] When the substrate possessed only one available ortho-position, the main focus of the studies have been devoted to investigation of functional group tolerance and improvement of the efficiency of catalytic systems to reach high conversion [86][87][88][89][90][91]. When two ortho-positions were available for direct functionalization, it has been shown that working with an excess of arylating reagent in most cases favored the bis-arylation process. Another recurrent observation is that when the nucleophile bears a meta-substituent on the aromatic ring, an ortho-C-H bond mono-arylation takes place at the para-position of this substituent, due to steric constraint at the other ortho-position. However, some parameters inherent to the nature of the directing group and the arylating reagent as well as the catalytic system were found to be crucial to favor mono-arylation processes.

In the presence of a strongly coordinating directing group, typically an aromatic N-heteroaryl group (2-pyridyl, 2-oxazolyl, imidazolyl, pyrazolyl, tri-and tetrazolyl) or an imine, the formation of diarylated products is highly favored and can be driven to completion in the presence of an excess of arylating reagent [71a,76b,92-97]. However, this general trend can be reversed when the directing group presents some steric hindrance in the neighborhood of the As soon as it was discovered, the pioneers of direct C-C bond formation involving rutheniumcatalyzed C-H bond activation pointed out the role of phosphine in the site-selectivity of arylation of aromatic substrates featuring a strong coordinating directing group. In Table 1 are gathered examples related to mono-arylation of 2-phenylpyridine. The general trend is that the presence of the phosphine coordinated to the ruthenium center slows down the rate of the second arylation leading to monoarylated products when equimolar amounts of substrates are used. In many cases, the presence of a simple triarylphosphine [99,102,103] provide this selectivity and more bulky phosphine have also been used for this purpose [100,101,104]. [Ru1]= [RuCl2(arene)]2 (arene = benzene or p-cymene).

[Ru2]= RuCl2(p-cymene)(PR3). Additionally, it is worth mentioning that whatever the other criteria favorable to mono-or bisarylation, the nature of the leaving group of the arylating reagent may have some influence not only on reactivity but also on site-selectivity towards mono-or bis-arylation. This is illustrated in the two examples described in Scheme 17. The ortho-arylation of 2-phenylpyridine with a ruthenium(II) precursor combined with a sterically hindered diaminophosphine oxide provided the monoarylated product with an aryl tosylate as electrophile source whereas the aryl chloride gave the bis-arylated product selectively even at lower temperature with shorter reaction time (Scheme 17) [88]. Similarly, with the same catalytic system, the aryl bromide as electrophile led to the sole formation of the bis-arylated product. On the other hand, the aryl chloride gave the mono-arylated product (Scheme 18) [89]. The mono-vs bis-arylation was also explored by utilizing biologically-relevant 6-phenylpurine and 6-arylpurine nucleosides as substrates and aryl iodides as arylating agents [112]. Besides aryl (pseudo)halides, phenols [113] and diaryliodonium salts [114] were found as suitable arylating agents, the formers provided they were in situ transformed in tosylate derivatives in water media [113]. A rare case of hydroxyl-directed ruthenium-catalyzed ortho-C-H bond arylation has been reported by using 1-naphthols as substrates and aryl halides under phosphane-free conditions [115]. It gives access to non-trivial, mono-arylated 8-aryl-1-naphthols via peri-C-H activation (Scheme 19). Whereas electron-poor aryl iodides reacted smoothly at 100 o C, a higher reaction temperature of 130 o C was required for electron-rich or sterically-demanding aryl iodides as well as aryl bromides that need higher energy for oxidative addition when compared to aryl iodides. Importantly, this methodology broadens the functional group tolerance reported under palladium catalysis [116]. manifold involving bis-cycloruthenated species prior to oxidative addition. Ackermann and co-workers later demonstrated that a similar reaction mechanism operates using [RuCl2(p-cymene)]2 as the pre-catalyst at higher temperatures of 120 o C [84]. A combination of experimental studies and DFT calculations suggested that p-cymene-free biscycloruthenated species are key intermediates upon oxidative addition with aryl halides (Scheme 23). Noteworthy, the C-H bond cleavage event is in line with a base-assisted internal electrophilic substitution (BIES) instead of a concerted metalation/deprotonation (CMD) pathway. The formation of key octahedral ruthenium(IV) species appear to be the turnoverlimiting step and the nature of the aryl ligand controls whether ortho-C-H bond arylation or oxidative homo-coupling occur. In the former case the aryl moiety acts as an electronwithdrawing spectator ligand (Scheme 23). This behavior is strongly highlighted using aryl halides with electron-withdrawing substituents in the 2-position, such as 2-CN, and with iodides rather than bromides. The groups of Greaney and Ackermann, independently, showed that bis-cycloruthenated species prior to oxidative addition also form under catalytic conditions starting from [RuCl2(pcymene)]2 pre-catalyst complex under photo-irradiation (blue LED) [123,85]. Importantly, the ruthenium-catalyzed ortho-C-H bond arylations were conducted at mild reaction temperature (r.t. to 35 o C) thanks to the dissociation of the p-cymene ligand from the ruthenium pre-catalyst for further engaging in a catalytic cycle similar to that reported by Larrosa (vide supra) [121].

H N R 1 R 1 R 2 R 2 X [Ru(MeCN) 4 (dmba)](PF 6 ) ( 10
The reaction conditions required high catalytic loadings of ruthenium (20 mol%), which is twice the loading reported by Larrosa's work [121]. Greaney's conditions also needed the use of H2O All the above described findings related to bis-cycloruthenated species in ortho-C-H bond arylations have been demonstrated for substrates bearing strong sp 2 -nitrogen-containing directing groups such as pyridine, imidazopyridine, imine, dihydrodibenzoazepin-2-ones, purines, oxazoles, pyrazoles, and triazoles. However, a recent example based on carboxylic directing group has also been reported [124]. In this case, the bis-cycloruthenated species forms thanks to a remote non-covalent ion pair interaction between an ammonium additive and a carboxylate group in the substrate as suggested by computations.

3.4. ortho-C-H bond arylations with aryl (pseudo)halides employing arene-free ruthenium pre-catalysts.

Other powerful ruthenium-based pre-catalysts involve ruthenium complexes that contain no arene ligands coordinated to ruthenium. From the many ruthenium complexes readily available, RuCl3(H2O)n occupies a place of choice, as it is directly extracted from the ores after chemical manipulation and it is the common precursor used for the synthesis of well-defined, mononuclear or dinuclear ruthenium complexes [125]. RuCl3(H2O)n is air and moisture stable, features that are interesting for storing purposes. As such, it appears useful and practical for synthetic applications, although its action mode (reaction mechanism) is not fully understood in many cases (in situ reduction to Ru(II) species is sound). In this context, Ackermann and coworkers reported the first examples of its successful use as pre-catalyst in ortho-C-H bond arylations [126]. This was applied to arylations in NMP with pyridines, oxazolines and pyrazoles as directing groups (Scheme 27). The catalysis was compatible with (hetero)aryl bromides and tosylates, but not that much with aryl chlorides [127]. To circumvent this issue, Yu and co-workers demonstrated that the catalysis was efficient using (hetero)aryl chlorides in the presence of PPh3 as ligand in NMP as solvent (Scheme 27) [128] and later Gimeno and coworkers employed the RuCl3(H2O)n-NaOAc-Zn catalytic system in water [129]. A general methodology was applied to aryl tosylates in the presence of catalytic amounts of MesCO2H (Scheme 27) [130]. The efficiency of RuCl3(H2O)n pre-catalyst was enhanced in ionic liquids media [103,131] and its recyclability has been achieved by means of heterogeneization as Ru/CeO2 species [132] and in combination with PEG [133], respectively.

Scheme 27 ortho-C-H bond arylation with aryl (pseudo)halides using RuCl3(H2O)n as the precatalyst.

Alternative para-cymene-free ruthenium pre-catalysts include the ruthenium(II) complex [RuH(cod)2]BF4 [134] and the ruthenium(IV) alkylidene complex [Cl2(PCy3)2Ru=CHPh] for reactions similar to those presented in Scheme 27 [135], the [RuCl2(PPh3)3] complex with triazoles as directing groups [136], and the [Ru( t BuCN)6](BF4)2 complex with carboxylic acids as directing groups and aryl iodides as arylating agents (Scheme 28) [137]. It is worthy to mention that ruthenium-catalyzed ortho-C-H bond arylations with aryl (pseudo)halides have successfully been coupled with other ruthenium-catalyzed transformations such as meta-C-H bond alkylations [138], oxidations [139] and hydrogenations [119b,140], respectively, via one-pot procedures, which is appealing from a sustainable point of view.

H R 1 R 1 R 2 R 2 I [Ru( t BuCN) 6 ](BF 4 ) 2 (3-6 mol%) K 2 CO 3 (2-3 equiv.) KOC(CF 3 ) 3 (
4. C-H bond activation followed by oxidation via Ru(II/0) catalysis enabling orthoarylation.

A way to achieve ruthenium-catalyzed ortho-C-H bond arylations is by using arylboronic acids The above-described reaction design has been well applied for substrates bearing so-called weak directing groups such as amides, ketones or thioamides. For instance, Jeganmohan and co-workers reported the ruthenium-catalyzed ortho-C-H bond arylation of N-alkylbenzamides with aryl boronic acids (Scheme 30) [142]. The group of Szostak showed that the same type of reactivity was observed for N,N-dialkylbenzamides with very similar reaction conditions (Scheme 31) [143]. Analogously, the groups of Jeganmohan and Ackermann, independently, demonstrated the ability of acetanilides as suitable substrates for this type of transformation (Scheme 32) [144,145]. In addition, the Ackermann group also showed that diarylborinic acids and potassium aryltrifluoroborates behaved similarly to arylboronic acids. All these methodologies proved tolerant to a wide range of functional groups (alkyl, aryl, ether, ester, alcohol, halide, nitrile, nitro, aldehyde, ketone, thiophene, furan) at different positions of both the substrate and the arylating agent. However, the reactions were not compatible with benzamide (primary amide) as the substrate. Sulfoximines (-SONHMe) as directing groups were equally efficient as secondary amides [146]. Boronic acids were found to react with N-arylisoindolinones (cyclic tertiary amides) under ruthenium catalysis to afford selective ortho-C-H bond arylation in the anilide ring (Scheme 33) [147]. This constitutes an example of the control of the site-selectivity of the catalysis as two aromatic rings are likely to undergo arylation a priori [148]. In this case, a six-membered ruthenacycle is favored over a five-membered one. The methodology was applied in the latestage functionalization of indoprofen, a blockbuster drug displaying anti-inflammatory properties.

Scheme 33 Ruthenium-catalyzed ortho-C-H bond arylation of tertiary amides with aryl boronic acids.

Acyclic tertiary amides such as N,N-dialkylbenzamides underwent ruthenium-catalyzed regioselective ortho-C-H bond arylation only upon using an arylsilane such as ArSi(OMe)3 [149]. The key for the success of this transformation was the use of over-stoichiometric amounts of CuF2 as activator and re-oxidant (Scheme 34). As in the previous example, even when using an excess of aryl coupling partner, selective monofunctionalization was observed. The same reaction conditions were applied to substrates with nitrogen-containing directing groups (i.e.

pyridine, pyrazole, quinoline) affording the corresponding ortho-C-H bond arylated products [150]. In this case, an excess of arylsilane led to the formation of bis-arylated products.

Scheme 34 Ruthenium-catalyzed ortho-C-H bond arylation of tertiary amides with aryl silanes.

Szostak and co-workers reported the ruthenium-catalyzed ortho-C-H bond arylation of N,Ndialkyl thiobenzamides with boronic acids (Scheme 35) [151]. Control experiments revealed the role of sulfur for ruthenium-coordination as a key step in the catalysis. Interestingly, the reaction was performed using 2-MeTHF as eco-friendly and sustainable solvent. The presence of over-stoichiometric amounts of K2CO3 was critical for an efficient catalytic system, although the exact role of this reagent is not clear. Maheswaran and co-workers reported the same type of reactivity for 2-phenylpyridine derivatives using also [RuCl2(p-cymene)]2 as the precatalyst but using iodosobenzene bis(trifluoroacetate) as an organic chemical oxidant (Scheme 37) [152]. The reaction was applied successfully to 1-phenyl-1H-pyrazole as well as 9-(pyridin-2-yl)-9H-carbazole derivatives.

Scheme 37 Ruthenium-catalyzed ortho-C-H bond arylation of 2-phenylpyridine derivatives with aryl boronic acids.

Substrates appended with pyridine or pyrimidine strong directing groups such as N-pyridyl-2pyridones, N-pyridyl-1-isoquinolinones or 9-(pyrimidin-2-yl)-9H-carbazole were found suitable substrates towards ruthenium-catalyzed ortho-C-H bond arylation using copper(II) or silver(I) salts as chemical oxidants, respectively (Scheme 38 and Scheme 39) [153,154]. These chemical motifs are found as central cores in many biologically relevant molecules, so these methodologies might be of interest for late-stage functionalization and drug discovery. The group of Chang showed that ruthenium-catalyzed ortho-C-H bond arylations using aryl glycol boronic esters are possible using quinoline N-oxide derivatives as substrates (Scheme 40) [155]. Besides a broad substrate scope -including polycyclic quinoline derivatives-, in depth mechanistic studies demonstrated that the oxygen atom linked to the nitrogen atom of the quinoline core coordinated to ruthenium prior to the C-H bond activation step. Transmetallation with aryl boronic esters and oxidation afforded a ruthenium(III) species that underwent oxidatively induced reductive elimination (ORE) to afford selective ortho-C-H bond arylation [156]. Then, the remaining ruthenium(I) species was oxidized toward the active ruthenium(II) species, whereas a second ruthenium(II) species catalyzed the deoxygenation towards the final product with trifluoroethanol solvent behaving as a reducing agent (Scheme 41).

Scheme 40 Chang's ruthenium-catalyzed ortho-C-H bond arylation of quinoline N-oxide derivatives with aryl glycol boronic esters. The utilization of direct C-H bond functionalization avoiding synthetic steps for preparation of reactive substrates as well as purification steps presents advantages for industry in terms of green chemistry and sustainability [158]. This is why industry has been rapidly interested in application of direct C-H bond functionalizations. At the Merck company, the synthesis of Anacetrapib, a potent and selective inhibitor of cholesteryl ester transfer protein, has been envisioned at a preparative scale by reaction of a functional bromoaryl substrate with 2-(3fluorophenyl)oxazoline [159]. This procedure took profit of the oxazoline moiety, which served first as directing group for the ortho-C-H bond arylation and was then used for the final steps of the synthesis. The catalytic system based on the [RuCl2(benzene)]2 dimer associated with triphenylphosphine as ligand, potassium acetate as source of acetate ligands and potassium phosphate as a base operated in NMP at 120 °C and gave 96% yield of the arylated oxazoline at a 4 kg scale (Scheme 44 -route a) [159]. Later on, with the objective of proposing a more sustainable procedure avoiding the use of a phosphine additive and the toxic, flammable and costly NMP solvent, the water-tolerant, ionic ruthenium catalyst [Ru2Cl2(HCOO)3(pcymene)]Na was used in water with sodium formate as basic additive to perform the same ortho-C-H bond arylation reaction in water under reflux [160]. With this greener catalytic system developed by Chicago Discovery Solutions LLC, starting from equimolar amounts of the two substrates, a conversion of 85% was obtained after 8 hours and the product was isolated in 73% yield (Scheme 44 -route b). thanks to a Piv-Val-OH-based ruthenium(II) catalyst (Scheme 47). Importantly, the reaction was also efficient using the less reactive, although more affordable, 4-chloro-aryl amino-acid coupling partner (Scheme 47). From a mechanistic point of view, it was suggested that the carboxylate site from the amino-acid ligand Piv-Val-OH participated in the C-H bond activation step prior to the cycloruthenation. It is relevant to note that ruthenium-catalyzed ortho-C-H bond arylations have been implemented in a continuous-flow system [164], although its applicability is scarce.

[LRu II (OTs) 2 ] L = p-cymene H N O R 1 Ru II N O R 1 L OTs B R 2 O O Ru II N O R 1 L R 2 Ru III N O L R 2 [LRu I (OTs)] N O R 1 R 2 [LRu II (OTs) 2 ] HO CF 3 O H CF 3 N R 1 R 2 + H 2 O Ag + Ag 0
Scheme 46 Ruthenium-catalyzed ortho-C-H bond arylation of 1-N-benzyl 2-phenyltetrazole for accessing N-benzyl Valsartan.

Conclusions.

It is now well accepted that ortho-directed aromatic C-H bond arylations proceed via an intermediate (five-or six-membered) ruthenacycle arising from coordination of a directing group followed by C-H bond activation and metal-carbon bond formation. This rational, which is based on decades of experimental and computational studies, has enabled a large set of given reaction conditions that enables to use many type of arylating agents as it is described in this chapter. The manifold mechanisms highlighted here demonstrates the compatibility of ruthenium-catalyzed ortho-C-H bond arylations with many functional groups, which resulted in important applications for active pharmaceutical ingredients as well as laying the foundation for late-stage arylation. Currently, for the ruthenium(II)-catalyzed methodologies strong nitrogen-containing directing groups have been mostly studied. As such, the exploration of weak directing groups (i.e. ketones, aldehydes, esters, amines) that are prominent in organic synthesis, as well as controlling the site-or positional-selectivity in cases where multiple arylations can occur, remain as two of the major challenges in ruthenium-catalyzed ortho-C-H bond arylation chemistry.

a 1 , 2 -Scheme 2

 122 Scheme 2 Reaction pathway for the ortho-C-H bond activation involving oxidative addition at the ruthenium center.

Figure 2

 2 Figure 2 ortho-Ruthenated complexes relevant for ortho-C-H bond arylations.

4 -Scheme 6

 46 Scheme 6 Ruthenium-catalyzed tetra-ortho-C-H bond arylation of anthraquinone en route towards polyaromatics.

Scheme 7

 7 Scheme 7 Ruthenium-catalyzed ortho-C-H bond arylation of acetophenones with arenediboronates.

3. 1 .

 1 Ru(II)-catalyzed ortho-C-H bond arylations with aryl (pseudo)halides: mechanistic considerations. Most of the catalytic processes involving ruthenium(II) catalysts are also based on the initial formation of a five-membered ruthenacycle via directed base-assisted C-H bond activation. The most effective directing groups contain a coordinating nitrogen atom that is incorporated in a heteroaromatic ring (pyridine, oxazoline, diazine, azole) or an aliphatic imine. A few examples include a bidentate dinitrogen directing group based on an amino N-heterocycle coordinating structure. Some other functional groups such as carboxylic acid, N-oxide, acetanilide, thiocarbonyl, featruring an O-, S-or P-coordinating atom have also been evaluated as directing groups. Concerning the key deprotonation event, the presence of a basic ligand such as a carboxylate coordinated to the metal center can induce an inner-sphere concerted metalationdeprotonation (CMD) [68]. The deprotonation can also occur in the presence of an external base via an outer-sphere mechanism (external-CMD) [69]. Scheme 9 illustrates these two simplified possibilities from Ru(OAc)2(p-cymene) as catalyst and 2-phenylpyridine as model substrate. It is often difficult to assess the real nature of the interaction since it strongly depends on the experimental conditions [69]. The external deprotonation of 2-phenylpyridine, 2-phenylpyrazole and 2-phenyl-2-oxazoline has been shown experimentally with Ru(OAc)2(p-cymene) as catalyst precursor and in acetonitrile an autocatalytic process where the liberated acetic acid facilitated the coordinated acetate release from the ruthenium centre was observed [70]. Computational studies have also shown that the deprotonation by an external base was a realistic possibility [71]. It has been demonstrated that mononuclear

  Scheme 9 Proposed transition states for the ruthenium-mediated ortho-C-H activation based on internal and external carboxylate deprotonation.

Figure 3

 3 Figure 3 Selected examples of isolated and characterized ortho-C-H bond cyclometalated ruthenium(II) complexes with their associated references in brackets.

Scheme 10

 10 Scheme 10 Reactivity of relevant intermediates in ruthenium-mediated ortho-C-H bond arylation starting from [RuCl2(p-cymene)]2 precursor.

  Scheme 12 Ruthenium-catalyzed ortho-C-H bond mono-arylation using aryl bromides and thiazole as the directing group.

Scheme 15

 15 Scheme 15 Ruthenium-catalyzed ortho-C-H bond mono-arylation of azoarenes using aryl bromides.

  Scheme 16 Ruthenium-catalyzed ortho-C-H bond mono-arylation of benzoic acid derivatives using aryl bromides.

Figure 4

 4 Figure 4 Selected ruthenium(II) catalyst enabling ortho-C-H bond mono-arylation of 2-phenylpyridine derivatives with aryl (pseudo)halides, with their associated references in brackets.

Scheme 17

 17 Scheme 17 Influence of the leaving group (Cl vs OTf) on the arylating reagent for the monovs bis-arylation in ruthenium-catalyzed ortho-C-H bond arylation.

Scheme 19

 19 Scheme 19 Ruthenium-catalyzed ortho-C-H bond arylation of 1-naphthol with (hetero)aryl iodides and bromides.

Scheme 22

 22 Scheme 22 General and mild ruthenium-catalyzed ortho-C-H bond arylation with aryl (pseudo)halides towards late-stage functionalization of pharmaceuticals.

Scheme 23 A

 23 Scheme 23 A common ruthenium(IV) intermediate leads to direct ortho-C-H bond arylation (top) or oxidative homo-coupling (bottom) depending on the stereoelectronic nature of the aryl ligand.

in 2 -

 2 Scheme 24 Ruthenium-catalyzed ortho-C-H bond arylation with aryl bromides under visible light irradiation at room temperature.

[

  Scheme 29 Mechanism of the ortho-C-H bond arylation with aryl boronic acids via Ru(II/0) manifold.

Scheme 30

 30 Scheme 30 Ruthenium-catalyzed ortho-C-H bond arylation of N-alkyl benzamides with aryl boronic acids.

Scheme 35

 35 Scheme 35 Ruthenium-catalyzed ortho-C-H bond arylation of thiobenzamides with aryl boronic acids.

Scheme 38

 38 Scheme 38 Ruthenium-catalyzed ortho-C-H bond arylation of N-pyridyl-2-pyridone and Npyridyl-1-isoquinolinones derivatives with aryl boronic acids.

Scheme 41

 41 Scheme 41 Proposed reaction mechanism for the ruthenium-catalyzed ortho-C-H bond arylation of quinoline N-oxide derivatives with aryl glycol boronic esters.

Scheme 44

 44 Scheme 44 Ruthenium-catalyzed ortho-C-H bond arylation as the key step in the synthesis of Anacetrapib.

Table 1 .

 1 Influence of phosphane ligands in the ruthenium-catalyzed ortho-C-H bond arylation of 2-phenylpyridine with aryl (pseudo)halides.

	R	X	x	[Ru] cat.	PR3	Solvent	Conv. (%)	mono:bis ratio	Ref.
	H	Br	1	[Ru1]	PPh3	NMP	82	71:11	[99]
	H	OTf 1	[Ru1]	PPh3	NMP	68	61:7	[99]
	H	Cl	1	[Ru1]		NMP	96	96:4	[100]
	H	Cl	1	[Ru2]		NMP	R= H, 85	76:9	[101]
							R= Ph, 78	70:8
	H	Cl	0.8	[Ru2]	PPh3	H2O	92	95:5	[102]
	OMe Br	0.8	[Ru1]	P(p-Tol)3	Bu4NOAc	76	64:12	[103]
	OMe Cl	2.2	[Ru1]		NMP	94	81:13	[104]


Krzysztof Kurdyka
email: kurdyka@univ-savoie.fr

Adam Parusi Ński
email: adam.parusinski@univ-angers.fr

ON THE NON-ANALYTICITY LOCUS OF AN ARC-ANALYTIC FUNCTION

Keywords: Mathematics Subject Classification. 14Pxx, 32S45, 32B20. This research

A function is called arc-analytic if it is real analytic on each real analytic arc. In real analytic geometry there are many examples of arc-analytic functions that are not real analytic. Arc analytic functions appear while studying the arc-symmetric sets and the blow-analytic equivalence. In this paper we show that the non-analyticity locus of an arc-analytic function is arc-symmetric. We discuss also the behavior of the nonanalyticity locus under blowings-up. By a result of Bierstone and Milman a big class of arc-analytic function, namely those that satisfy a polynomial equation with real analytic coefficients, can be made analytic by a sequence of global blowings-up with smooth centers. We show that these centers can be chosen, at each stage of the resolution, inside the non-analyticity locus.

Introduction.

Let X be a real analytic manifold. A function f : X → R is called arc-analytic, cf. [START_REF] Kurdyka | Ensembles semi-algébriques symétriques par arcs[END_REF], if for every real analytic γ : (-1, 1) → X the composition f • γ is analytic. The arc-analytic functions are closely related to blow-analytic functions of Kuo, cf. [START_REF] Kuo | On classification of real singularities[END_REF]. In particular, we have the following result, conjectured for the functions with semi-algebraic graphs in [START_REF] Kurdyka | Ensembles semi-algébriques symétriques par arcs[END_REF], and shown in [START_REF] Bierstone | Arc-analytic functions[END_REF].

Theorem 1.1. Let X be a nonsingular real analytic manifold and let f : X → R be an arc-analytic function on X. Suppose that G(x, f (x)) = 0, where G(x, y) = p i=0 g i (x)y p-i is a nonzero polynomial in y with coefficients g i (x) which are analytic functions on X. Then there is a mapping π : X ′ → X which is a composite of a locally finite sequence of blowings-up with nonsingular closed centers, such that f • π is analytic.

Let f : X → R be an arc-analytic subanalytic function. In this paper we study the set S(f ) of non-analyticity of f . By definition, S(f ) is the complement of the set R(f ) of points p ∈ X, such that f as a germ is real analytic at p. It is known (cf. [START_REF] Tamm | Subanalytic sets in the calculus of variation[END_REF], [START_REF] Kurdyka | Points réguliers d'un ensemble sous-analytique[END_REF], [START_REF] Bierstone | Semianalytic and Subanalytic sets[END_REF]) that S(f ) is closed and subanalytic. It follows from [START_REF] Bierstone | Arc-analytic functions[END_REF] or [START_REF] Parusiński | Subanalytic functions[END_REF], that dim S(f ) ≤ dim X -2. As we show in Theorem 3.1 below, S(f ) is arc-symmetric in the sense of [START_REF] Kurdyka | Ensembles semi-algébriques symétriques par arcs[END_REF]. Theorem 3.1 is shown in section 3.

We also study how the set of non-analyticity behaves under blowings-up with smooth centers. This depends on whether the center is entirely contained in S(f ) or not. If it is not then the non-analyticity lifts to the entire fiber, see Proposition 3.10. Note that Theorem 1.1 can be also derived from [START_REF] Parusiński | Subanalytic functions[END_REF]. Using the method of [START_REF] Parusiński | Subanalytic functions[END_REF] and Proposition 3.10 we show the following refinement of Theorem 1.1.

Theorem 1.2. In Theorem 1.1 we may require that the mapping π : X ′ → X, that is a locally finite composite π = • • • • π k • • • • • π 0 of blowings-up with smooth centers, satisfies additionally:

for every k the center of π k+1 is contained in the locus of non-analyticity of f

•π 0 •• • ••π k .
1.1. Algebraic case. Theorem 1.1 can be stated in the real algebraic version, see [START_REF] Bierstone | Arc-analytic functions[END_REF]. In this case if we assume that X is a nonsingular real algebraic variety and that the coefficients g i are regular then we may require that π is a finite composite of blowings-up with nonsingular algebraic centers.

In the algebraic case we cannot require that the centers of blowings-up are entirely contained in the non-analyticity loci as Example 1.5 shows.

An analytic function on X is called Nash if its graph is semialgebraic. It is called blow-Nash if it can be made Nash after composing with a finite sequence of blowing-ups with smooth nowhere dense regular centers. Thus the algebraic version of Theorem 1.1, cf. [START_REF] Bierstone | Arc-analytic functions[END_REF], says that the function with semi-algebraic graph is arc-analytic if and only if it is blow-Nash. Nash morphisms and manifolds form a natural category that contains the algebraic one, cf. [START_REF] Bochnak | Géométrie algébrique réelle[END_REF]. We note that our refinement of the statement of Theorem 1.1 holds in the Nash category.

Theorem 1.3. Let X be a Nash manifold and let f : X → R be an arc-analytic function on X. Suppose that G(x, f (x)) = 0, where

G(x, y) = p i=0 g i (x)y p-i
is a nonzero polynomial in y with coefficients g i (x) which are Nash functions on X. Then there is a finite composite

π = • • • • π k • • • • • π 0 of
blowings-up of nonsingular Nash submanifolds, such that for every k the center of π k+1 is contained in the locus of non-

analyticity of f • π 0 • • • • • π k , and f • π is Nash. 1.2. Subanalytic case.
Less is known for an arc-analytic function with subanalytic graph if it does not satisfy an equation (1.1). It is known that an arc-analytic subanalytic function has to be continuous and can be made real analytic by composing with finitely many local blowings-up with smooth centers, see [START_REF] Bierstone | Arc-analytic functions[END_REF] or [START_REF] Parusiński | Subanalytic functions[END_REF] (we refer the reader to these papers for a precise statement). It is not known whether these blowings-up can be made global that is whether the arc-analytic subanalytic functions coincide with the family of blow-analytic functions of T.-C.Kuo, see e.g. [START_REF] Kuo | On classification of real singularities[END_REF], [START_REF] Fukui | Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities[END_REF], [START_REF] Fukui | Arc Spaces and Additive Invariaants in Real Algebraic and Analytic Geometry[END_REF]. It is also not known, whether the centers of such blowings-up can be chosen in the locus of non-analyticity of the function. We present below in Example 1.6 a subanalytic arc-analytic function that cannot be made analytic, even locally, by a blowing-up of a coherent ideal. In particular, it cannot satisfy an equation of type (1.1).

Examples.

Examples 1.4. The function f : R 2 → R , f (x, y) = x 3

x 2 +y 2 for (x, y) = (0, 0) and f (0, 0) = 0, is arc-analytic but not differentiable at the origin.

The function g(x, y) = x 4 + y 4 is arc-analytic but not C 2 . This example is due to E. Bierstone and P.D. Milman.

The function h : R 2 → R , h(x, y) = xy 5 x 4 +y 6 for (x, y) = (0, 0) and h(0, 0) = 0 is arc-analytic but not lipschitz. This example is due to L. Paunescu. We generalize the first example as follows. Fix a real analytic Riemannian metric on X and let Y be a nonsingular real analytic subset of X. Then d 2 Y : X → R, the square of the distance to Y , is a real analytic function on X. Suppose that Y is of codimension ≥ 2 in X and let f : X → R be an analytic function vanishing on Y and not divisible by d 2 . Then, f 3 d 2 vanishes on Y , is arc-analytic and not analytic at the points of Y . Note that f 3 d 2

composed with the blowing-up of Y is analytic.

Example 1.5. Let g(x, y) = y 2 + x(x -1)(x -2)(x -3). Then g -1 (0) ⊂ R 2 is irreducible and has two connected compact components, denoted by X 1 and X 2 . These connected components that can be separated by h(x, y) = x -1.5, that is h < 0 on X 1 and h > 0 on X 2 . For ε > 0 sufficiently small, h 2 + εg is strictly positive on R 2 . Define

g 1 (x, y) = h 2 + εg + h.
Then g 1 is analytic, 0 is a regular value of g 1 and g -1

1 (0) = X 1 . Moreover, g 1 is Nash. Then f : R 3 → R defined by f (x, y, z) = z 3 z 2 + g 2 1 (
x, y) for (x, y, z) = 0 and f (0) = 0, is arc-analytic and S(f ) = X 1 × {0}. The function f becomes analytic after blowing-up of S(f ).

Example 1.6. Let π 0 : R 3 → R 3 be the blowing-up of the origin and let E be the exceptional divisor of π 0 . Let C ⊂ E be a transcendental (the smallest algebraic subset of E that contains C is E itself) non-singular analytic curve and let π C : M → R 3 be the blowing-up of C. Let f be an arc-analytic function on R 3 such that the set of non-analyticity of f • π 0 is C and f • π 0 • π C is analytic. Such a function can be constructed as follows. Using the last remark of Examples 1.4 we may construct an arc-analytic function g : R 3 → R such that S(g) = C. Then we may set f (x, y, z) = (x 2 + y 2 + z 2 ) g(π -1 0 (x, y, z)). Such f , as a germ at 0, cannot be made analytic by a single blowing-up of an ideal. Indeed, suppose contrary to our claim that there exists an ideal I of R{x 1 , x 2 , x 3 } such that f • π I is analytic, where π I denotes the blowing-up of I. Multiplying I by the maximal ideal at 0 we may assume that π I factors through π 0 , i.e. π I = π J • π 0 , where J is a sheaf of coherent ideals centered on an algebraic subset Y of E. We may assume that dim Y ≤ 1. Thus the blowing-up of J , π J : M J → R 3 is an isomorphism over the complement of Y that contradicts the construction of f .

Arc-meromorphic mappings.

In this section subanalytic mean subanalytic at infinity. Let us recall, [START_REF] Tamm | Subanalytic sets in the calculus of variation[END_REF], [START_REF] Kurdyka | Points réguliers d'un ensemble sous-analytique[END_REF], that a subset A of R n is called subanalytic at infinity if A is subanalytic in some algebraic compactification of R n . (Then in fact it is subanalytic in every algebraic compactification of R n .) All functions and mappings are supposed to be subanalytic, that is their graphs are subanalytic at infinity. Definition 2.1. Let U be an open subanalytic subset of R n . An everywhere defined subanalytic mapping f : U → R m is called arc-meromorphic if for any analytic arc γ : (-1, 1) → U there exists a discrete set D ⊂ (-1, 1) and ϕ an meromorphic function on (-1, 1) with poles contained in D and such that f • γ = ϕ on (-1, 1) \ D. Note that it may happen that f • γ does not coincide with ϕ at some points of D and may be at these points discontinuous.

Example 2.2. The function f : R 2 → R defined by f (x, y) = xy x 2 +y 2 for (x, y) = (0, 0) can be extended to an arc-meromorphic function on R 2 by assigning any value at the origin. Then it becomes discontinuous at (0, 0) even if for every analytic arc γ : (-1, 1) → R 2 , γ(0) = (0, 0), f • γ extends to an analytic function.

(1) |f (x)| ≤ C dist (x, Γ) -N , x ∈ U \ Γ.
In particular we can take as U the complement of the non-analyticity locus of f .

Proof. It is well-known (cf. e.g. [START_REF] Hironaka | Subanalytic sets, in Number Theory, Algebraic Geometry and Commuta-tive Algebra[END_REF], [START_REF] Lojasiewicz | Sur la géométrie semi-et sous-analytique[END_REF]) that there exists a stratification of R n which is compatible with U and such that f is analytic on each stratum contained in U . We take as Γ the union of all strata contained in U of dimension less than n. Let us consider the function defined as follows:

g(x) = |f (x)| if |f (x)| ≤ 1, and g(x) = |f (x)| -1 if |f (x)| ≥ 1. Then h(x) := dist(x, Γ)g(x)
is a subanalytic and continuous function on U which is compact. Moreover, if dist (x, Γ) = 0 then h(x) = 0. Therefore, by the classical Lojasiewicz's inequality (cf. e.g. [START_REF] Hironaka | Subanalytic sets, in Number Theory, Algebraic Geometry and Commuta-tive Algebra[END_REF], [START_REF] Bierstone | Semianalytic and Subanalytic sets[END_REF]) for subanalytic functions, there exist N ∈ N and c > 0 such that

(2) h(x) ≥ c dist (x, Γ) N +1 , x ∈ U.
Thus inequality (1) follows with C = max{1/c, M }, where M = sup x∈U dist(x, Γ) N .

We state now an auxiliary lemma on arc-meromorphic functions in two variables.

Lemma 2.6. Let U be an open subanalytic subset in R 2 and let f : U → R m be an arcmeromorphic mapping. Then for any a ∈ U there exists a neighborhood V of a and an analytic function ϕ : V → R, ϕ ≡ 0, such that ϕf is arc-analytic.

Proof. Let Γ be the subanalytic set associated to f by Lemma 2.5. Clearly we may assume that a ∈ Γ, otherwise f is analytic at a and the statement is trivial. Since dim Γ = 1, by a result of Lojasiewicz's [START_REF] Lojasiewicz | Ensembles semi-analytiques[END_REF] (see also [START_REF] Kurdyka | Zurro Stratifications distinguées comme un util en géométrie semianalytique[END_REF]), the set Γ is actually semianalytic. Then there exists a neighborhood V of a and an analytic function ψ : V ′ → R, ψ ≡ 0, which vanishes on V ′ ∩ Γ. Hence for some compact neighborhood V ⊂ V ′ of a there exists c > 0 such that

|ψ(x)| ≤ c dist (x, Γ), x ∈ V.
(This is a consequence of the main value theorem). Put ϕ = ψ N +1 , then by Lemma 2.5 the function ϕf is continuous on V . Clearly ϕf is arc-meromorphic, so by Remark 2.3 this function is arc-analytic.

Proposition 2.7. Let f : U → R be an arc-meromorphic function, where U is an open subset in R n . Assume that f is analytic with respect to the variable x 1 . Then the function

∂f ∂x 1 : U → R is again arc-meromorphic.
Proof. First observe that by [START_REF] Kurdyka | Points réguliers d'un ensemble sous-analytique[END_REF] the function ∂f ∂x 1 is (globally) subanalytic. To prove that ∂f ∂x 1 is arc-meromorphic let us fix an analytic arc γ : (-1, 1) → U . We define an arcmeromorphic function g :

V → R by g(s, t) = f (γ(t) + se 1 )
, where e 1 = (1, 0, . . . 0) and

V is an open neighborhood of {0} × (-1, 1) in R 2 . Clearly ∂f ∂x 1 (γ(t)) = ∂g ∂s (0, t).
By Lemma 2.6 there exist a neighborhood V of (0, 0) and an analytic function ϕ : V → R such that h := ϕg is arc-analytic on V . Since dim S(h) ≤ 0, for any t = 0 sufficiently small h is analytic at (0, t), but of course also ϕ is analytic at (0, t). Since g(s) is analytic with respect to s it follows that g = h/ϕ is actually analytic at (0, t) for any t = 0 sufficiently small. By [START_REF] Bierstone | Arc-analytic functions[END_REF] there exists a map π : M → R 2 , which is a finite composition of blowing-ups of points, such that h • π is analytic. Consider the arc η(t) := (0, t) and let η(t) ∈ M be the unique analytic arc such that π • η = η. The chain rule gives

(3) d η(t) h • π = (d η(t) h) • (d η(t) π).
Note that d η(t) π is invertible for t = 0, moreover the map t → (d η(t) π) -1 is meromorphic. It follows that t → d η(t) h is meromorphic. In particular t → ∂h ∂s (0, t) is meromorphic. We have ∂h ∂s (0, t) = ϕ ∂g ∂s (0, t) + g ∂ϕ ∂s (0, t).

Since ϕ(0, t) = 0 for t = 0, the map t → ∂g ∂s (0, t) is meromorphic and Proposition 2.7 follows.

Remark 2.8. A repeated application of Proposition 2.7 shows that for every k ∈ N,

∂ k f ∂x k 1 : U → R
is arc-meromorphic. Moreover, there exists a subanalytic stratification S of U such that for every stratum S ∈ S and every x ∈ S there is ε > 0 and a neighborhood V of x in S such that f (x + se 1 ) is an analytic function of (x, s) ∈ V × (-ε, ε). In particular, for every k ∈ N, ∂ k f /∂x k 1 : U → R is analytic on the strata of S.

3. The non-analyticity locus of an arc-analytic function is arc-symmetric.

Let U ⊂ R n be open and let f : U → R be arc-analytic with subanalytic graph. We denote by S(f ) the non-analyticity set of f and by R(f ) its complement in U . Then S(f ) is closed in U and by [START_REF] Tamm | Subanalytic sets in the calculus of variation[END_REF] (see also [START_REF] Kurdyka | Points réguliers d'un ensemble sous-analytique[END_REF], [START_REF] Bierstone | Arc-analytic functions[END_REF]) it is a subanalytic set. It follows from [START_REF] Bierstone | Arc-analytic functions[END_REF] or [START_REF] Parusiński | Subanalytic functions[END_REF] that dim S(f ) ≤ n -2.

Theorem 3.1. Let γ : (-ε, ε) → U be an analytic arc such that γ(t) ∈ R(f ) for t < 0. Then γ(t) ∈ R(f ) for t > 0 and small. In other words, S(f ) is arc-symmetric subanalytic in the sense of [START_REF] Kurdyka | Ensembles semi-algébriques symétriques par arcs[END_REF].

For the proof we need some basic properties of Gateaux differentials. For each k ∈ N we consider (4)

h k (x, v) = 1 k! ∂ k v f (x) = 1 k! d k dt k f (x + tv) |t=0 .
Proposition 3.2. Let f : U → R be an arc-analytic function. Then for any k ∈ N the function

h k (x, v) : U × R n → R is arc-meromorphic.
Proof. Let (x(t), v(t)) be an analytic arc in U × R n . Define an arc-analytic function g(s, t) = f (x(t) + sv(t)). Then

h k (x(t), v(t) = 1 k! ∂ k ∂s k g(t, s) |s=0
that is meromorphic by Proposition 2.7.

For x ∈ U , k ∈ N we denote h x,k (v) = h k (x, v) = 1 k! ∂ k v f (x)
Note that h x,k is k-homogeneous function. If f is analytic at x, then h x,k is polynomial. We have also the inverse which is Bochnak-Siciak Theorem, see [START_REF] Bochnak | Analytic functions in topological vector spaces[END_REF], which states that if h x,k is polynomial for each k ∈ N, then f is analytic at x. Traditionally if h x,k is polynomial then it is called the Gateaux differential of f at x of order k. We call h x,k generically polynomial if it is equal to a polynomial except on a nowhere dense subanalytic (and homogenous) subset of R n . Note that, by Remark 2.4, h x,k is generically polynomial if it coincides with a polynomial on an open nonempty set. Proposition 3.3. Let f : U → R be an arc-analytic function, where U is an open subset in R n . Let γ : (-ε, ε) → U be an analytic arc and k ∈ N. If h γ(t),k is generically polynomial for t ∈ (-ε, 0), then there exists a finite set F k ⊂ (0, ε) such that h γ(t),k is generically polynomial for each t ∈ (0, ε) \ F k .

Proof. Let R k [x 1 , . . . , x n ] denote the space of homogenous polynomials of degree k and let d k = n+k-1 n denote its dimension. We need the classical multivariate interpolation.

Lemma 3.4. There exists an algebraic nowhere dense subset

∆ ⊂ (R n ) d k such that for V = (v 1 , . . . , v d k ) ∈ (R n ) d k \ ∆ the map Ψ V : R k [x 1 , . . . , x n ] → R d(k) given by Ψ V (P ) = (P (v 1 ), . . . , P (v d k )).
is a linear isomorphism.

Fix V = (v 1 , . . . , v d k ) ∈ (R n ) d k \∆ generic and denote Φ V = Ψ -1 V : R d(k) → R k [x 1 , . . . , x n ]. We define an arc-meromorphic map P k : (-ε, ε) → R k [x 1 , . . . , x n ] by P k (t) := Φ V (h k (γ(t), v 1 ), . . . , h k (γ(t), v d(k) )). The map p k ; (-ε, ε) × R n → R, where p k (t, v) = P k (t)(v) is arc-meromorphic. If V is
sufficiently generic then, for t ∈ (-ε, 0) \ {finite set}, p k (t) coincides with h γ(t),k . Since they both are arc-meromorphic, by Remark 2.4 they coincide on (-ε, ε) × R n \ Z k , where Z k is a closed subanalytic set with dim Z k ≤ n. Hence there exists a finite set

F k ⊂ (0, ε) such that for t ∈ (0, ε) \ F k the intersection Z k ∩ ({t} × R n ) is of dimension less than n.
Thus, for each t ∈ (0, ε) \ F k the function h γ(t),k is generically polynomial, as claimed.

The following proposition is a version of the mentioned above Bochnak-Siciak Theorem, [START_REF] Bochnak | Analytic functions in topological vector spaces[END_REF].

Proposition 3.5. If for every k there is a nonempty open subset V k ⊂ R n and a homogeneous polynomial P k of degree k such that h x,k ≡ P k on V k , then f is analytic at x.

Proof. We first show that

k P k (v) is convergent in a neighborhood of 0 ∈ R n .
We may assume that x is the origin. Let π 0 be the blowing up of the origin, π 0 (y, s) = (sy, s), s ∈ R, y ∈ R n-1 , in a chart. The function f (y, s) := f (π(y, s)), defined in a neighborhood U ′ of the exceptional divisor E : s = 0, is arc-analytic. The set of nonanalyticity of f , denoted by S, is closed subanalytic and of codimension at least 2. For y / ∈ S, f is analytic in a neighborhood of (0, y) and, moreover, by analytic continuation, (5)

h x,k (v) = P k (v) for v = t(y, 1), t ∈ R, y / ∈ S.
Fix A ′ an open non-empty subset of E such that the closure of A ′ does not intersect S. Let A ⊂ R n be the cone over A ′ . Then, by [START_REF] Bochnak | Analytic functions in topological vector spaces[END_REF], k P k (v) is convergent in any compact subset of A. The convergence in a neighborhood of 0 in R n follows from the following lemma.

Lemma 3.6. Let V ⊂ R n be starlike with respect to the origin, a ∈ V , and suppose that

|P k (v)| ≤ L on V ′ = a + V. Then |P k (v)| ≤ L on 1 2e V.
Proof. Since P k is homogeneous of degree k

(6) P k (v) = 1 k! s=k s=0 (-1) k-s k s P k (a + sv).
Indeed, ( 6) can be shown recursively on k using Euler's formula as follows. First note [START_REF] Fukui | Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities[END_REF] holds for a = 0 and the derivative of the RHS of ( 6) with respect to a equals

(7) 0 = 1 k! s=k s=0 (-1) k-s k s Q(a + sv),
where

Q(x) = n i-1 a i ∂P k ∂x i (x) is a homogeneous polynomial of degree k -1. By the inductive assumption s=k s=0 (-1) k-s k s Q(a + sv) = s=k-1 s=0 (-1) k-1-s k -1 s Q(a + sv) + + s=k s=1 (-1) k-s k -1 s -1 Q(a + sv) = -Q(v) + Q(v) = 0
This shows [START_REF] Fukui | Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities[END_REF].

Thus, if v ∈ 1 k V , |P k (v)| ≤ 1 k! L k s=0 k s = L 2 k k! , that means that for v ∈ 1 2e V |P k (v)| ≤ L (2k) k k! 1 (2e) k ≤ L.
This ends the proof of lemma 3.6.

Then k P k (v) is an analytic function in a neighborhood of the origin that coincides with f on a set with non-empty interior. Hence f (v) = k P k (v) in a neighborhood of the origin. This shows proposition 3.5.

Proof of theorem 3.1. We may assume that γ is injective otherwise the image of t > 0 equals the image of t < 0 and the statement is obvious. Let F := F k , where F k are finite subsets of (0, ε) given by Proposition 3.3. Clearly the complement of F is dense in (0, ε), so by Proposition 3.5 our function f is analytic at γ(t) for t ∈ G, where G is an open dense subset of (0, ε). Hence theorem 3.1 follows.

Consider the subanalytic sets

Rk 0 (f ) = {x ∈ U ; ∀k ≤ k 0 , h x,k is generically polynomial }, R k 0 (f ) = {x ∈ U ; ∀k ≤ k 0 , h x,k is polynomial }.
Clearly Rk+1 (f ) ⊂ Rk (f ) and R k+1 (f ) ⊂ R k (f ). We recall from [START_REF] Kurdyka | Points réguliers d'un ensemble sous-analytique[END_REF] the following result

compact K ⊂ U there is k ∈ N such that R(f ) ∩ K = R k (f ) ∩ K. Proposition 3.8. For any compact K ⊂ U there is k ∈ N such that R(f )∩K = Rk (f )∩K.
Proof. By Remark 2.8 there exists a stratification S of U × S n-1 such that for every k, h k is analytic on the strata. Refining the stratification, if necessary, we may suppose that for every stratum S ⊂ U × S n-1 its projection to U has all fibers of the same dimension. In the proof we use only these strata for which all the fibers of projection to U are of maximal dimension n -1. We denote the collection of them by S n and their union as Z. Now it is easy to adapt the proof of Lemma 6.1 of [START_REF] Kurdyka | Points réguliers d'un ensemble sous-analytique[END_REF] (based on multivariate interpolation) and show the following lemma. Lemma 3.9. There are analytic subanalytic functions

w i : U × S n-1 → R, i ∈ N,
analytic on each stratum of S such that h x,i is generically polynomial if and only if w i ≡ 0 generically on {x} × S n-1 . Now Proposition 3.8 follows from Lemma 2.5 of [START_REF] Kurdyka | Points réguliers d'un ensemble sous-analytique[END_REF] that shows that for every stratum there exist k such that

∞ i=1 {w i = 0} = k i=1 {w i = 0}.
We complete this section with two results, one that controls the change of non-analyticity locus by blowings-up. This result will be crucial in the next section. The last result of this section, Proposition 3.11, though not used in this paper, indicates a possible analogy between our approach and the theory of complex analytic functions.

Proposition 3.10. Let T = {x k = x k+1 = • • • = x n = 0}
and let π T be the blowing-up of T . Suppose that the origin is in the closure of R(f ) ∩ T and that f • π T is analytic at least at one point of π -1 T (0) (hence on a neighborhood of this point). Then f is analytic at 0. Proof. Let Π : R n ×R×R n → R n be given by Π(x, t, v) = x+tv and let Π T : T ×R×R n → R n be the restriction of Π. First we show that if f • Π T is analytic at some points of Π -1 T (0) ∩ {t = 0} and 0 is in the closure of R(f ) ∩ T then f is analytic at 0. Indeed, suppose that A ′ ⊂ R n has non-empty interior and suppose that f

• Π T is analytic in a neighborhood {0} × {0} × A ′ . Let h k (x, v), x ∈ T, v ∈ R n ,
be defined by [START_REF] Bochnak | Géométrie algébrique réelle[END_REF]. Then h k is arc-meromorphic and analytic on A = U ′ × A ′ , where U ′ is a small neighborhood of 0 in T . For each k, we define by Lemma 3.4, (8)

P k (x, v) = Ψ -1 V (h k (x, v 1 ), . . . , h k (x, v d(k) ))(v)
, where v 1 , . . . , v d k ∈ A ′ are generic. Each P k is analytic on A and equals h k for x ∈ R(f )∩T . Therefore h k (0, v) = P k (0, v) for v ∈ A ′ and the claim follows from proposition 3.5.

Thus it remains to show that f • Π T is analytic at some points of Π -1 T (0) ∩ {t = 0}. For this we factor Π T restricted to {v n = 0} through π T and use the assumption on π T . Write π T in an affine chart π T (x, y, s) = (x, sy, s), where x = (x 1 , . . . , xk-1 ), y = (y k , . . . , y n-1 ) and s ∈ R. Then on these charts Π T = π T • ϕ, where

(x, y, s) = ϕ(x, t, v) = (x + tv ′ , 1 v n v ′′ , tv n ), where v ′ = (v 1 , . . . , v k-1 ), v ′′ = (v k , . . . , v n-1 ). Restricted to t = 0, ϕ is a surjective projection (x, v) → (x, 1 vn v ′′ ) onto s = 0. Hence R(f • Π T ) ∩ Π -1 T (0) ∩ {t = 0} ⊃ ϕ -1 (R(f • π T ) ∩ π -1 (0)) is non-empty. Proposition 3.11. Let x = (x 1 , x ′ ) ∈ R × R n-1
and suppose that for every x 1 > 0 and small, f (x 1 , x ′ ) is analytic at (x 1 , 0) as a function of x ′ . Moreover, suppose that for x 1 > 0 and small we have a uniform bound

|h k ((x 1 , 0), v ′ )| ≤ c k , for v ′ ≤ ε, k ∈ N,
where v ′ = (v 2 , . . . , v n ). Then f is analytic at the origin.

Proof. The function h k ((x 1 , 0), v ′ ) is arc-meromorphic as a function of x 1 , v ′ . Moreover, since continuous arc-meromorphic functions of one variable are analytic, using polynomial interpolation lemma, Lemma 3.4, we may show that each h k ((x 1 , 0), v ′ ) extends to an analytic function Ψ(x 1 , v ′ ) defined in a neighborhood of (0, 0), such that for each

x, v ′ → Ψ(x 1 , v ′ ) is a homogeneous polynomial in v ′ . Moreover, for x 1 > 0 and x ′ < ε/c f (x 1 , x ′ ) = k h k ((x 1 , 0), x ′ )
and the series on the right-hand side is convergent. Fix any k ∈ N and v ′ < ε/c. Then for

v = (1, v ′ ), 0 < t < 1, f (tv) = ∞ j=0 h j ((t, 0), tv ′ ) = ∞ j=0 t j h j ((t, 0), v ′ ) = k j=0 t j h j ((t, 0), v ′ ) + ϕ(t, v ′ ),
where ϕ is subanalytic and O(t k+1 ). Therefore for such v

(9) H k (0, v) := 1 k! d k dt k f (tv) |t=0 = 1 k! d k dt k k j=0 h j ((t, 0), tv ′ ) |t=0 .
Note that the right-hand side, and hence H k (0, v) as well, is a polynomial in v. Indeed, this follows from the fact that x → k j=0 h j ((x 1 , 0), x ′ ) is an analytic function of x and H k (0, v) coincides with its Gateaux differential. Thus proposition 3.11 follows from proposition 3.5.

Proof of Theorem 1.2.

We may suppose that U is connected. We suppose also that the coefficients g 0 and g p of G and the discriminant ∆(x) of G are not identically equal to zero. By the resolution of singularities [START_REF] Hironaka | Resolution of Singularities of an algebraic variety over a field of characteristic zero[END_REF], [START_REF] Bierstone | Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant[END_REF], [START_REF] Lodarczyk | Simple Hironaka resolution in characteristic zero[END_REF], there is a locally finite sequence of blowings-up π : U ′ → U with nonsingular centers such that (g 0 g p ∆) • π is normal crossings. Thus Theorem 1.1 follows from the following. Proposition 4.1. Let an arc-analytic function f (x) satisfy the equation (1.1) with analytic coefficients g i . If g 0 , g p and ∆(x) are simultaneously normal crossings (and hence not identically equal to zero) then f is real analytic. Proposition 4.1 was proven in [START_REF] Parusiński | Subanalytic functions[END_REF] under an additional assumption g 0 ≡ 1, see the proof of Theorem 3.1 of [START_REF] Parusiński | Subanalytic functions[END_REF]. It is easy to reduce the proof to this case by replacing f by g p f . Then, an argument of [START_REF] Parusiński | Subanalytic functions[END_REF] shows that locally f can be expand as a fractional power series. Finally, an arc-analytic fractional power series is analytic, see the proof of Theorem 3.1 of [START_REF] Parusiński | Subanalytic functions[END_REF]. If the discriminant of G vanishes identically then we replace it by the first non-vanishing higher order discriminant.

To show Theorem 1.2 we follow, for the product h(x) = g 0 (x)g p (x)∆(x), the monomialisation procedure of W lodarczyk or Bierstone-Milman. In this procedure the centre of blowing-up is defined as a the locus of points where a local invariant is maximal. Thus suppose that we have the following data described in a local system of coordinates x 1 , • • • , x n at the origin. The function h • π, where π = π k • • • • • π 0 , is of the form h • π = x A h k , where h k is the controlled transform by the preceding blowings-up. Let m = ord x h k . We may assume that H = {x n = 0} is a hypersurface of maximal contact. Then, using the notation x = (x ′ , x n ),

h k (x) = x m k + m-2 j=0 c j (x ′ )x j k ,
and mult 0 c i ≥ mi.

Let C be the next centre given by the procedure and denote by π C the blowing-up of C. We show that it cannot happen that 0 ∈ S(h • π) and 0 ∈ C \ S(f • π). Suppose, contrary to our claim, that this is the case. Then, by Proposition 3.10, the fibre over the origin of the blowing-up π C = π k+1 of C is contained in S(f • π • π C ). Since C is contained in the equimultiplicity locus of h k , at the generic point π -1 C (0) the strict transform of h k is nonzero, and hence h • π • π C is normal crossing. This contradicts Proposition 4.1.

Let C ′ denote the connected component of C containing 0. Then either C ′ ⊂ S(h • π) or C ′ ∩ S(h • π) = ∅. Thus Theorem 1.2 proven.

Remark 2 . 3 .

 23 If f is an arc-meromorphic and continuous function on an open connected set U ⊂ R n , then f is arc-analytic. Remark 2.4. Let f and g be arc-meromorphic functions on an open connected set of U . Assume that f = g on an open non-empty subset U ⊂ R n , then f = g except on a nowhere dense subanalytic subset of U . Lemma 2.5. Let U be an open bounded subanalytic subset in R n and f : U → R m be an arc-meromorphic mapping. Then there exists Γ ⊂ R n a closed nowhere dense subanalytic set, N ∈ N and C > 0 such that

Proposition 3 . 7 .

 37 [ [11], Proposition 4.4] Let f : U → R be a subanalytic (not necessarily arc-analytic) function on an open bounded U ⊂ R n . Then for any


Saideep Pavuluri

Christophe Tournassat

Francis Claret

Laurent André
email: l.andre@brgm.fr

Cyprien

Cyprien Soulaine

porousMedia4Foam: An open-source multiphase reactive transport platform based on OpenFOAM for geothermal applications

Keywords: Reactive transport modelling, flow and transport in geothermal well, porousMedia4Foam

INTRODUCTION

Renewable sources of energy are seen to play a vital role in the foreseeable future to address concerns regarding the emissions of contaminant gases and to further reduce extensive dependence on fossil fuels. World Energy Resources (2013) has reported that there is a rising trend in tapping geothermal energy over the past few decades. However, geothermal energy has not been exploited to its full potential yet [START_REF] Payam | A review on geothermal wells: Well integrity issues[END_REF]. A major concern for geothermal wells is their longevity. On the one hand, maintaining the wells integrity at high temperatures and pressures is a challenge [START_REF] Kiran | Identification and evaluation of well integrity and causes of failure of well integrity barriers (A review)[END_REF]. On the other hand, scaling within the geothermal wells [START_REF] Jamero | Mineral scaling in two-phase geothermal pipelines: two case studies[END_REF] has been a cause for the premature closure of many wells before exploiting their full potential.

A priori knowledge of the potential minerals that precipitate within the wells is key to explore solutions to avoid clogging the wells during the operating cycles. In that regard, numerical simulators with hydrogeochemical modelling capabilities come in handy. Though there exist many studies related to scaling in geothermal reservoirs [START_REF] Wagner | Numerical simulation of pore space clogging in geothermal reservoirs by precipitation of anhydrite[END_REF] and many studies analysing fluid flow in geothermal wells without considering effects of scaling [START_REF] Bjornsson | A multi-feedzone geothermal wellbore simulator[END_REF], there is limited literature where scaling is studied exclusively in a geothermal well.

In this work, porousMedia4Foam is used to investigate the minerals that may precipitate within geothermal wells inspired from different studies. We compute the saturation indices of several minerals according to the solution injected through the feedzone. Two sets of simulations are considered. First, the saturation index is computed in a one-dimensional (1D) well using a crosssection-averaged formulation (Darcy-like). Second, we use high-resolution modelling using Navier-Stokesbased Direct Numerical Simulations (DNS) to investigate the impact of turbulent flow mechanisms on the scaling formation at the junction of two well segments of different diameters.

In the following (Section 2), we describe briefly the hydro-geochemical coupling used to model the scaling formation in geothermal wells. In Section 3, we discuss the large and fine-scales results. Finally, we end with conclusions and future outlook.

HYDRO-GEOCHEMICAL COUPLING STRATEGY

Simulations of scaling formation are run with porousMedia4Foama multi-scale open-source hydro-geochemical simulation platform developed by the authors to model reactive transport phenomena at multiple scales i.e. at the pore-, hybrid-and Darcyscales [START_REF] Soulaine | porousMedia4Foam: Multi-scale open-source platform for hydro-geochemical simulations with OpenFOAM®[END_REF]. porousMedia4Foam is built on the skeletal framework of OpenFOAM (https://www.openfoam.org) for solving for flow and transport and has been successfully coupled with PHREEQC to account for geochemistry (https://www.usgs.gov/software/phreeqc-version-3). The code is available at https://github.com/csoulain/porousMedia4Foam.

Flow and transport mathematical model

The package comprises of several flow solvers such as the 'constantVelocityFoam'assigns a constant velocity field within the flow domain, 'dbsFoam'solves the Darcy-Brinkman-Stokes equation and 'darcyFoam'solves the Darcy's equations. porousMedia4Foam also includes an extensive set of models that describes the evolution of porous media and fluid properties.

In this study, we investigate flow and scaling formation, first at the scale of the well using a 1D approach, and then, using high-resolution 3D simulations to interrogate emerging processes at the junction between two segment of different diameters.

In the first case, governing equations are cross-sectionaveraged. The resulting equations form a Darcy-like system that is solved using darcyFoam solver. For an incompressible fluid, the mass balance equation in a well comprising of varying cross-sections (Ac) is given by,

𝜕 𝜕𝑧 (𝐴 𝑐 𝑣 𝑓 ) = 0 [1]
where vf refers to the fluid velocity defined by crosssection averaged momentum that looks like Darcy's law,

𝑣 𝑓 = -𝑘 µ ( 𝜕𝑝 𝜕𝑧 -𝜌𝑔 𝑧 ) [2]
k is a friction factor that depends on the well diameter and flow rates, µ is the fluid viscosity and 𝜕𝑝 𝜕𝑧 ⁄ is the pressure gradient. The friction factor is computed by a Colebrook-White model. Though we are interested in 1D analysis, the cross-sectional area Ac is integrated into the mass balance equation (Eq. 1) to account for change in cross-sectional areas. Substituting Eq. 1 in Eq. 2 results in a Laplacian equation to be solved for the pressure. Once the pressure is computed, the velocity is calculated following Eq. 2, and transport equations for species are computed to update the concentration profile.

In the second case, we use Navier-Stokes-based simulations to investigate the saturation index at the vicinity of the corners of the expanding cross-section of the well. The simulation is performed in two consecutive steps. First, a steady-state Reynolds Average Navier-Stokes (RANS) simulation based on k-Epsilon model is run with simpleFoam. Then, resulting velocity profile is used in porousMedia4Foam (using constantVelocityFoam) to transport chemical species and compute geochemistry.

Coupling of OpenFOAM with PHREEQC

In this work, we use the phreeqcRM geochemical package that acts as an interface between OpenFOAM and PHREEQC. The coupling between the packages relies on an operator splitting strategy. We use the Strangs algorithm in which advection-dispersion of chemical species is solved for half a time-step initially, the geochemistry for a full time-step, and the transport equation is computed for the remaining half time-step.

𝜕𝐶 𝑖 𝜕𝑡 + 𝛻. (𝑣 𝑓 𝐶 𝑖 ) -𝛻. (𝐷 𝑖 . 𝛻𝐶 𝑖 ) = 0 [3]
In the transport equation, 𝐶 𝑖 is the species concentration, 𝐷 𝑖 is the effective dispersion tensor that accounts for molecular diffusion and hydrodynamic dispersions. The geochemical time step is run by PHREEQC. The sequential order of steps followed in porousMedia4Foam is illustrated in the form of a flowchart shown in Fig. 1.

The hydro-geochemical coupling strategy implemented in porousMedia4Foam has been extensively benchmarked against state-of-the-art reactive transport packages for well established cases [START_REF] Pavuluri | Reactive transport modelling with a coupled OpenFOAM®-PHREEQC platform[END_REF].

HYDRO-GEOCHEMICAL INVESTIGATION OF GEOTHERMAL WELLS

In this section, we discuss specifics of two geothermal wells, well #1 that is straight and well #2 with three subsections of different diameters. Both the hydraulic conditions and the chemical reactivity of the pumped solutions are investigated.

Geothermal well #1

Well #1 is straight with a constant diameter of 0.244 m and a length of 2,600 m. The reservoir temperature is about 120°C whereas pressure is 250 bar. Wellhead pressure is 10 bar and temperature is 104°C. According to temperature and pressure conditions at the boundaries of the well, the fluid is supposed to remain under liquid phase (no degassing). We consider a linear temperature gradient in the well between the feedzone and the wellhead.

The geometry of the well allows testing the code on a simple application case. First, the fluid velocity is estimated to about 0.40 m.s -1 . It is constant all along the well and the impact of fluid density variations is negligible. Then, we perform reactive transport calculations by considering the chemical composition of the geothermal fluid. The fluid pumped in the reservoir is undersaturated with respect to main minerals such as calcite and quartz but at equilibrium with respect to barite (Fig. 3). A second simulation is then proceeded by activating precipitation of minerals (Fig. 3 -right). Since barite is oversaturated, this mineral precipitates along the well and the deposited amounts of mineral can be estimated.

Geothermal well #2

The geothermal well is 540 meters deep and it is made of three segments of different cross-sections whose dimension is shown in Table 1 [START_REF] Baba | Hydrochemical and isotopic composition of Tuzla geothermal field (Canakkale-Turkey) and its environmental impacts[END_REF].

Table 1: The aqueous solution that enters through the feedzone comprises of carbon (C), calcium (Ca), magnesium (Mg), barium (Ba), sulfate (SO4) and silica (Si). The pH of the solution is 5.34. At this stage, a simplified composition of fluid is used. For the specified solution composition, there exists potential to precipitate calcite (CaCO3), dolomite (CaMg(CO3)2) and chalcedony (SiO2) within the well. The chemical reactivity of barite is also followed during the simulations.

1D simulation in geothermal well #2

The well is discritized by 540 cells along the length of the channel (y = 1 m). The three different crosssectional areas of the well have been assigned to the setup according to the data provided in Table 1. The friction coefficient is computed through the crosssection diameter and the flow rate using the Colebrook-White formula.

According to the available field data, the feedzone is set a Dirichlet boundary condition for pressure pfeedzone = 50 atm and pwellhead = 3 atm. The velocity at the boundaries is a zero gradient Neumann boundary. The temperature at the feedzone is Tfeedzone = 180°C and Twellhead = 130°C.

Fig. 4 shows the pressure and temperature profiles in the geothermal well #2. The pressure drop is more pronounced in the thinner cross-section of the well due to larger viscous dissipation. On the other hand, the thicker cross-section of the well shows a lower pressure drop due to lesser viscous dissipations. The impact of degassing in the well can be observed on the temperature profile: temperature suddenly decreases at 100 m depth because of phase change. The saturation index of calcite and chalcedony is shown in Fig. 5.

High-resolution simulations observations

The impact of hydrodynamics on the scaling formation at the change of cross-section has received little attention so far. We use Navier-Stokes-based simulations to interrogate emerging processes in this area of the well. . The corner zones are therefore a favourable location for the creation of seeds that move inside the well, eventually. These disseminated seeds can then become ideal precursors for larger precipitations.

CONCLUSIONS

A novel hydro-geochemical simulation platform porousMedia4Foam has been used to address questions related to minerals that may precipitate within the geothermal well. porousMedia4Foam benefits from the coupling of OpenFOAM with PHREEQC where the flow physics is solved using OpenFOAM whereas the geochemistry is solved using PHREEQC.

The code was applied to different geometries of geothermal wells, with various pressure and temperature conditions. It allows calculating the pressure and temperature profiles in the well, fluid velocities and predicting the degassing. The coupling between transport and chemistry allows estimating the reactivity of geothermal fluids in the well and the potential precipitation of minerals due to temperature changes or phase changes.

High-resolution simulations (Navier-Stokes-based) have been run to account for bridging of scales. This high-resolution simulation of hydrochemical processes is the first attempt to model the scale formation in geothermal well. We noticed mixing zones as consequence of vortices developed at expanding crosssections of the well that are favorable for the development of precipitable minerals. The coupled hydrogeochemical OpenFOAM-PHREEQC platform introduced in this paper has all the capabilities to investigate these specific locations in detail in future studies.

Fig

  Fig. 1: Sequential steps followed during solving a hydro-geochemical time step in porousMedia4Foam.

Fig. 2 :

 2 Fig. 2: The temperature (in red) and pressure (in blue) profiles in the geothermal well #1.

Fig. 3 :

 3 Fig. 3: Saturation indices of quartz, calcite and barite in the well without (left) and with chemical reactivity (right)

  and density are µ = 0.001 kg/m.s and ρ=1000 kg/m 3 , respectively.

Fig. 4 :

 4 Fig. 4: The pressure (in red) and temperature (in blue) profiles in the geothermal well #2.

Fig. 5 :

 5 Fig. 5: The saturation index profile for mineralscalcite (in blue) and chalcedony (in red) in the geothermal well #2. The dashed black line indicates equilibrium (saturation index of 0) for reference.

  The geometry consists in a 2m long subset of the geothermal well #2 centered on the change of diameter from 0.5 to 0.35 m. It is described by a 2D axisymmetric domain meshed with 9800 hexahedral cells with refinement near the walls. A constant velocity of 0.44 m.s -1 is applied at the bottom boundary. The top boundary has a fixed pressure value (0 Pa). Noslip conditions is used on the walls. We use the k-Epsilon model to compute turbulent flow in the well. The resulting steady-state velocity profile is then used in our coupled hydrogeochemical package to compute the geochemistry.

Fig. 6 :

 6 Fig. 6: Left: flow profile in the well where the crosssection expands with vortices developed at the corners of the expanding cross-section of the well. Right: Barium concentration profile and barite saturation index

ACKNOWLEDGEMENTS

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement nº 850626.