Abir Chihaoui
email: abir.echihaoui@ujf-grenoble.fr

Etienne Jaillard
email: etienne.jaillard@ujf-grenoble.fr

Jean-Louis Latil

Ihsen Zghal
email: ihsen_zghal@yahoo.fr

Anne-Sophie Susperregui
email: anne-sophie.susperregui@insa-rennes.fr

Jamel Touir
email: jamel.touir@fss.rnu.tn

Jamel Ouali
email: ouali_j@yahoo.fr

STRATIGRAPHY OF THE HAMEIMA AND LOWER FAHDENE FORMATIONS IN THE TADJEROUINE AREA (NORTHERN TUNISIA)

Keywords: Tunisia, Early Albian, ammonites, stratigraphy, biostratigraphy, foraminifera

In the Tadjerouine area (NW of Central Tunisia) the incipient Albian transgression is materialised by alternating marls, limestones and sandstones (Hameima Fm) that overly massive platform carbonates (Serdj Fm). The ongoing transgression then gives way to a thick series of marls and shales with subordinate carbonate beds (Fahdene Fm). A detailed study of the Hameima Fm, previously ascribed to the Late Aptian, shows that it can be subdivided into three members, all of earliest Albian age (Leymeriella tardefurcata Zone), and its top is slightly diachronous. A similar study of the lower Shales of the Fahdene Fm demonstrates that they belong to the Leymeriella tardefurcata Zone and to the lower part of the Douvilleiceras mammillatum superzone. The overlying Allam limestones, formerly considered Middle Albian in age, are mainly of late Early Albian age (upper part of Douvilleiceras mammillatum Superzone and Lyelliceras pseudolyelli Zone), although an extension into the lower Middle Albian cannot be ruled out. The Allam limestones are abruptly overlain by a diachronous series of Late Albian shales (Middle Shales of Fahdene Fm), thus evidencing a stratigraphic hiatus of most of the Middle Albian and part of the early Late Albian. Our new data will lead to revise the range of some Aptian/Albian orbitolinids and Albian planktic foraminifera.

Introduction

Facies changes between platform and basin facies have long caused problems to stratigraphers because of lithologic lateral changes and of difficult biostratigraphic correlations between shelfal and basinal faunas. As an example, the transition between the Central Tunisian platform and Northern Tunisian basin is marked by both complete argillaceous successions and lateral facies changes, which make difficult the use of the current stratigraphic nomenclature. Although ammonite successions are known for a long time from the Aptian-Albian interval in northwestern Tunisia [START_REF] Pervinquière | Étude de paléontologie tunisienne. 1 : Céphalopodes des terrains secondaires[END_REF][START_REF] Dubourdieu | Etude géologique de la région de l'Ouenza (Confins Algéro-Tunisiens)[END_REF][START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF][START_REF] Memmi | L'Aptien, et l'Albien de Tunisie: Biostratigraphie à partir de ammonites[END_REF], the ammonite biostratigraphy, the transition between platform and basin facies, and the chronology of the Albian transgression are still poorly understood.

Moreover, the Aptian-Albian boundary is still a matter of debate, since the ammonite faunas are commonly endemic, the sedimentary environments are marked by abrupt changes related to widespread anoxic events or to sharp regressions followed by major eustatic transgressions (e.g. [START_REF] Erba | The Aptian stage[END_REF][START_REF] Hart | The Albian stage and substage boundaries[END_REF][START_REF] Kennedy | Integrated stratigraphy across the Aptian-Albian boundary in the Marnes Bleues, at the Col de Pré-Guittard, Arnayon (Drôme), and at Tartonne (Alpes de Haute-Provence), France: a candidate Global Boundary Stratotype Section and Boundary point for the base of the Albian stage[END_REF][START_REF] Hancock | A proposal for a new position for the Aptian/Albian boundary[END_REF][START_REF] Owen | The base of the Albian stage; comments on recent proposals[END_REF][START_REF] Mutterlose | The Vörhum section (northwest Germany) and the Aptian/Albian boundary[END_REF], and significant synsedimentary tectonic events obscure the sedimentary or eustatic signals, provoking hiatuses, erosions and unconformities (e.g. [START_REF] Cotillon | Les temps médio-crétacés : une singularité au cours de l'histoire du Phanérozoïque ? Geobios[END_REF][START_REF] Guiraud | Early Cretaceous rifts of Western and Central Africa: an overview[END_REF][START_REF] Chaabani | Aptian sedimentation : an example of interaction between tectonics and eustatics in Central Tunisia[END_REF].

The aim of this paper is to present a detailed stratigraphic study of the Hameima Formation and of the lower part of the Fahdene Formation, and a reassessment of the age and nature of the stratigraphic successions deposited during the Lower to Middle Albian. This study made us able: (1) to refine the stratigraphic nomenclature in the transition zone between platform and basin in Central and Northern Tunisia, (2) to refine the age of the lithologic units, and thus, (3) to provide a more reliable chronology of the tectonic and eustatic events that affect Tunisia during the latest Aptian -lower Upper Albian time-span. For this purpose, we studied several selected sections located around the city of Tadjerouine (36°6' N, 8°30' E), by means of field observations and extensive collections of ammonites.

Geological setting

During the Cretaceous, Tunisia is classically divided into three main paleogeographic zones, which are from south to North [START_REF] Fournié | Esquisses sédimentologiques et paléogéographiques sur le Crétacé inférieur de Tunisie, du Berriasien au Barrêmien[END_REF]Zghal and Arnaud, 2005; Fig. 1) : -the Saharian epicratonic platform, covered by thin, mainly clastic, continental to shallow marine deposits;

-the central Tunisian platform, marked by shallow marine, first siliciclastic, then calcareous deposits, which is bounded to the North by the present-day NE-SW trending thrust and folds belt that characterizes Northern Tunisia ; Central Tunisia incudes the southern Chott depression that separates the Saharian and Central Tunisian platforms;

-the north Tunisian basin, which received thick marly deposits during most of the Cretaceous (Fig. 1).

During Albian times, the regional tectonic regime is thought to be dominated by a NE-SW to E-W extension, dominated by normal faults enhancing tilting of blocks and evaporite diapirs, both controlled by inherited structures (e.g. [START_REF] Ouali | Caractères de la tectonique crétacée en distension au jebel Kebar (Tunisie centrale)[END_REF][START_REF] Burollet | L'évolution des bassins sédimentaires de la Tunisie centrale et orientale[END_REF][START_REF] Martinez | Le contexte géodynamique de la distension albo-aptienne en Tunisie septentrionale et centrale : structuration éocrétacée de l'Atlas tunisien[END_REF][START_REF] Ladeb | Structures synsédimentaires mésoscopiques associées à la distension aptienne en Tunisie centrale (Dj. Semmama)[END_REF][START_REF] Zouari | Tectonique transtensive de la paléomarge tunisienne au cours de l'Aptien-Campanien[END_REF], although a short termed, intra-Albian transpressional regime has been locally reported [START_REF] Bouaziz | Tectonic evolution of the northern African margin in Tunisia from paleostress data and sedimentary record[END_REF].

Tertiary deformations increase toward the North; they are marked in central Tunisia by open folds affecting previously faulted series, and in northern Tunisia by tighter folds associated with salt diapirs. The latter area is overthrusted by the south verging Tellian nappes, which crop out along the coast. Our study area is located in northwestern Tunisia, at the transition between the central and northern Tunisian paleogeographic domains [START_REF] Souquet | Séquences et cycles d'ordre 2 en régime extensif et transtensif: exemple du Crétacé inférieur de l'Atlas tunisien[END_REF]. The little town of Tadjerouine is located south of Le Kef, close to the Algerian border. It is surrounded by scattered, faulted and folded massifs (Jebels), which morphologically dominate less deformed Late Cretaceous to Quaternary sedimentary cover (Fig. 2). They are usually made of massive Aptian carbonates (Serdj Formation), overlain by limestones, shales and marls of the Hameima and Fahdene formations. Six sections have been studied around these jebels, which are from Northwest to Southeast (Fig. 2) : (1) Jebel Harraba on the algerian border; (2) Henchir El Goussa area (North of the El Gara section of Burollet 1956, p. 71), (3) North of Jebel Hameima (Dubourdieu, 1956, pp. 160-161 and 197-201;Burollet, 1956, p. 66 and pp. 71-75), (4) Northwest of Jebel Slata (Burollet, 1956, pp. 75-77; Charens section of [START_REF] Vila | Découverte d'Albien inférieur (marnes et calcaires à mélobésiées) et moyen (marno-calcaires à débris de Trias) au NW du Djebel Slata: vers l'abandon du modèle en champignon anté-vraconien[END_REF], (5) south of Jebel Djerissa (Burollet, 1956, p. 77), and (6) across Jebel Bou el Haneche (Burollet, 1956, p. 71) located North of Thala (Fig. 2).

Previous works

Following multidisciplinary scientific explorations carried out in the late XIXth century, the Cretaceous stratigraphy of Northern and Central Tunisia was first studied by [START_REF] Pervinquière | Étude de paléontologie tunisienne. 1 : Céphalopodes des terrains secondaires[END_REF], who attempted to recognise the succession of the stratigraphic stages, through lithological and fossil collections [START_REF] Pervinquière | Étude de paléontologie tunisienne. 1 : Céphalopodes des terrains secondaires[END_REF]. But it is not before the 1950, that the biostratigraphy and lithostratigraphy of the Cretaceous series of Tunisia were studied and described in detail. [START_REF] Castany | Etude géologique de l'Atlas tunisien oriental[END_REF] published a geological synthesis and a geological map of the area, while [START_REF] Sainfeld | Le gîtes plombo-zincifères de Tunisie[END_REF] studied Lead-Zinc mineralizations, mainly hosted in Aptian-Albian strata. [START_REF] Dubourdieu | Etude géologique de la région de l'Ouenza (Confins Algéro-Tunisiens)[END_REF] published a detailed description of numerous sections in western Algeria and Eastern Tunisia, associated with numerous fossil determinations, while [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF], by means of numerous detailed field sections, established the stratigraphic nomenclature of the Cretaceous succession of Tunisia, which is still in use today (see M 'Rabet, 1981;[START_REF] Burollet | Le Crétacé inférieur de Tunisie. Aperçu stratigraphique et sédimentologique[END_REF][START_REF] M'rabet | Catalog of type sections in Tunisia: Cretaceous[END_REF]Zghal and Arnaud, 2005).

Since then, [START_REF] Bismuth | Réflexions stratigraphiques sur l'Albo-Aptien dans la région des djebels Douleb et Semmama et son environnement (Tunisie du Centre-Nord)[END_REF] made a helpful synthesis of the Aptian-Albian stratigraphy, refining the location of faunas formerly quoted in the Hameima section, and correlating the latter with southern areas. Short synthesis on the stratigraphy and ammonite paleontology of the Lower Cretaceous of Tunisia were published by [START_REF] Burollet | Le Crétacé inférieur de Tunisie. Aperçu stratigraphique et sédimentologique[END_REF] and [START_REF] Memmi | L'Aptien, et l'Albien de Tunisie: Biostratigraphie à partir de ammonites[END_REF]), respectively. Robaszinsky et al. (1993a, 1993b[START_REF] Robaszynski | The Albian (Vraconnian)-Cenomanian boundary at the western Tethyan margin (central Tunisia and southeastern France)[END_REF] published detailed stratigraphic and sedimentological studies focused on the Albian-Cenomanian boundary, and on the Late Cretaceous, along sections located Southwest of Tadjerouine. [START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF][START_REF] Zghal | Foraminifères et Ostracodes de l'Aptien-Albien du Jebel Hameima (region de Tadjerouine, Tunisie) : Biostratigraphie, paleoécologie[END_REF] extensively studied the microfossils along the Hameima section (Tadjerouine area), confirming previous age assignments [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF][START_REF] Bismuth | Réflexions stratigraphiques sur l'Albo-Aptien dans la région des djebels Douleb et Semmama et son environnement (Tunisie du Centre-Nord)[END_REF]. Finally, [START_REF] Vila | Découverte en Tunisie, au SW du Kef, de matériel triasique interstrati-fié dans l'Albien : extension du domaine à " glaciers de sel " sous-marins des confins algéro-tunisiens[END_REF][START_REF] Vila | Découverte d'Albien inférieur (marnes et calcaires à mélobésiées) et moyen (marno-calcaires à débris de Trias) au NW du Djebel Slata: vers l'abandon du modèle en champignon anté-vraconien[END_REF][START_REF] Vila | Découverte d'Albien inférieur à floridées et d'Albien inférieur et moyen marneux, dans le dôme de sel-pli de l'ensemble Djebel el Ajered, Henchir Adjerat Saouda, Djebel el Hamra (ouest de la Tunisie) : comparaisons, conséquences, essai de rétrotectonique salifère[END_REF][START_REF] Ghanmi | Vers l'abandon du modèle de diapir en champignon anté-vraconien au Djebel Slata: découverte sur son flanc nord-ouest d'Albien inférieur (marnes et calcaires à mélobésiées) et d'Albien moyen (marno-calcaires à débris de Trias), à leur place dans une série à l'endroit[END_REF] provided numerous, although scattered, biostratigraphic data on the Aptian-Albian interval, and evidenced the presence of Lower Albian strata in the Hameima Formation, as did El Euchi (1993) farther South. More recently, [START_REF] Zghal | The Hameima Formation. Aptian-Turonian events in Central Tunisia, Field trip guidebook[END_REF] and [START_REF] Jaillard | Albian sedimentation in the Tadjerouine area. Livretguide excursion, Aptian-Turonian events in Central Tunisia[END_REF] presented preliminary results on the Aptian-Albian succession in the Tadjerouine area, and the paleontological study of our ammonite collections is under publication [START_REF] Latil | Lower Albian ammonites from Central Tunisia and adjacent aereas of Algeria[END_REF].

In the Tadjerouine area, [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF] divided the Aptian -Albian interval into: (1) The Serdj reefal massive limestones of probable Aptian age, the top of which may reach locally the Albian (Burollet 1956, p. 66), (2) the Hameima Formation (≈ 300 m), made of shales, dolomites, limestones and sandstones of approximately Late Aptian age, and (3) the Fahdene Formation, mainly made of dark marls and shales, which comprises several informal units:

-Lower shales (300 m): dark shales and marls, which include a noticeable "ammonite-rich horizon" ascribed to the "Clansayes zone", considered by Burollet (1956, p. 72) as earliest Albian, but now refered to the latest Aptian [START_REF] Flandrin | Rapport sur l'étage Aptien[END_REF][START_REF] Burollet | Le Crétacé inférieur de Tunisie. Aperçu stratigraphique et sédimentologique[END_REF] The studied sections allowed us to examine the Hameima Formation, and the Lower Shales and Allam limestone units of the Fahdene Formation.

Lithostratigraphy and fauna

1. Serdj Formation [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF] defined the " Serdj Limestones " as massive limestones and dolomites, rich in reefal to peri-reefal faunas such as rudistids, corals and benthic foraminifera. The Serdj Formation [START_REF] M'rabet | Stratigraphie, sédimentation et diagenèse carbonatée des séries du Crétacé inférieur de Tunisie Centrale[END_REF], mainly of Aptian p.p. age is assumed to be diachronous, and its top may reach the Albian [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF]. This assumption has been supported by a detailed study of the formation in its type location (Jebel Serdj, Northeastern Tunisia) by [START_REF] Tlatli | Etude des calcaires de l'Albo-Aptien des Djebels Serdj et Bellouta[END_REF].

In Jebel Harraba, 30 meters below the top of the Serdj Formation, we found several specimens of Acanthohoplitinae, and two Epicheloniceras sp. of Late Aptian age (jacobi Zone). However, the latter seem to be reworked, since they are eroded, bored and phosphatised, and this latter lithology does not correspond to that of the surrounding shaly succession. [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF] defined the "shales with sandy interbeds of Hameima" (now known as Hameima Formation, e.g. [START_REF] M'rabet | Catalog of type sections in Tunisia: Cretaceous[END_REF] as a "black to dark green shale succession, with numerous thick interbeds of sandstone, dolomite and biogenic limestone".

Hameima Formation

In the Tadjerouine area, the 150 to 230 m thick Hameima Formation directly overlies the Serdj Formation (Fig. 3). It is not exposed in El Goussa (Fig. 2), where the section begins at the top of the Hameima Formation. The basal contact is at the top of the uppermost thick, massive, rudistids and chondrodonts bearing limestone (or dolomite) bed of the Serdj Formation. The upper contact is at the top of the uppermost thick bed of sandstone (El Goussa and Hameima areas) or limestone (other sections), and at the base of the lowermost thick argillaceous marly interval of the Fahdene Formation. The Hameima Formation is mainly made of massive limestone levels, locally rich in orbitolinids, separated by thick marl intervals, and usually comprises sandstone intercalations. A laterally constant sandstone interval ("Clansayes Sandstones" of [START_REF] Vila | Découverte d'Albien inférieur (marnes et calcaires à mélobésiées) et moyen (marno-calcaires à débris de Trias) au NW du Djebel Slata: vers l'abandon du modèle en champignon anté-vraconien[END_REF] allows us to subdivide the Hameima Formation into three lithologic members (Fig. 3).

The Lower

Member is 70 to 120 m thick, and rests on the massive carbonate bed of the top of the Serdj Formation. The top of this unit is marked in all sections by a thick limestone bed, usually caved by deep karsts, locally infilled by iron-rich mineralizations. This member mainly consists of marls and limestones, rich in orbitolinids at the base, with few beds of calcareous sandstones in the lower, and upper parts. Marly beds are rich in echinoids (sea urchins), bivalves, some brachiopods and gastropods, and scarce ammonites, while limestone beds display orbitolinids, pectinids and oysters. The Hameima section exhibits a dominantly marly succession, while limestone beds are more numerous in the Slata and Djerissa sections (Fig. 3). In the same way, the Djerissa and Bou el Haneche sections commonly exhibit crinoid-rich calcarenitic beds.

In the nearby section of Jebel Ouenza (Eastern Algeria), Dubourdieu (1956, p. 56) collected from this member Knemiceras compressum and Douvilleiceras gr. albense-monile, and concluded surprinsingly with a latest Aptian age (Dubourdieu, 1956, p. 90-92). During our work, the Lower Member yielded few ammonites, identified as Acanthohoplitinae, Parengonoceras sp. and Douvilleiceras sp. nov.? from the Djerissa section. In the Slata section, the top of this Member yielded "Hypacanthoplites" paucicostatus, and Parengonoceras sp..

The 35 to 70 m thick

Middle Member of the Hameima Formation is marked by the occurrence of abundant sandstone beds ("Clansayes Sandstones" of [START_REF] Vila | Découverte d'Albien inférieur (marnes et calcaires à mélobésiées) et moyen (marno-calcaires à débris de Trias) au NW du Djebel Slata: vers l'abandon du modèle en champignon anté-vraconien[END_REF]. The lower part of the unit is usually made of marls and sandstones, with limestone beds in the Djerissa and Bou el Haneche sections (Fig. 3). The middle part is dominated by sandy marls, which grades upward into cross laminated sandstones. The top of the unit is made of thick, orbitolinid-rich limestones, which become thicker toward the Southeast (Djerissa and Bou el Haneche sections). In marls and sandstones, macrofossils mainly consist of oysters and pectinids, associated with surbordinate echinoids, annelids, trigonids and other scarce bivalves, corals, gastropods and plant remains. In the Slata, Djerissa and Bou el Haneche sections, the uppermost orbitolinid limestones exhibit scarce annelids, oysters and corals. In the El Goussa and Hameima sections, the Hameima Formation ends up with thick sandstone beds, whereas in the Slata, Djerissa and Bou el Haneche sections, the top is marked by massive orbitolinid rich limestones (Fig. 3). The Middle Member yielded scarce, poorly preserved ammonites (Djerissa section), identified as Acanthohoplitinae. In the Hameima section, the top of the Middle Member yielded Orbitolina (Mesorbitolina) parva and O. (M.) minuta [START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF][START_REF] Zghal | Foraminifères et Ostracodes de l'Aptien-Albien du Jebel Hameima (region de Tadjerouine, Tunisie) : Biostratigraphie, paleoécologie[END_REF]. However, [START_REF] Zghal | The Hameima Formation. Aptian-Turonian events in Central Tunisia, Field trip guidebook[END_REF] quoted Mesorbitolina parva and Praeorbitolina wienandsi, the association of which is classically ascribed to the lower part of the Upper Aptian.

The Upper

Member is 0 to 55 m thick, and exhibits variable successions. It mainly comprises marls and limestones, with subordinate sandy limestone beds (Djerissa section, Fig. 3), and includes glauconitic/phosphatic, or clast rich layers. The fauna consists of oysters, pectinids, pinnids, brachiopods (Bou el Haneche) and scarce ammonites. However, some massive limestone beds are still rich in orbitolinids, and locally in red algae and annelids. This unit seems to be thinner and richer in marls toward the Northwest (Slata section). Shallow karsts are visible on top of some beds of the Southeastern sections (Djerissa and Bou el Haneche, Fig. 3).

In a faulted outcrop located Southwest of Jebel Bou el Haneche, the Upper Member of the Hameima Formation yielded a Douvilleiceras sp. (determination Arnould-Saget, in [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF]p. 71), indicating a Lower Albian age. Our ammonite fauna is made of Acanthohoplitinae. In Jebel Djerissa, we collected "Hypacanthoplites" sp. nov. A in the middle part of this member (Fig. 3). [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF] defined the "Fahdene marls and shales" (usually nammed Fahdene Formation, M 'Rabet, 1981;[START_REF] Burollet | Le Crétacé inférieur de Tunisie. Aperçu stratigraphique et sédimentologique[END_REF][START_REF] M'rabet | Catalog of type sections in Tunisia: Cretaceous[END_REF] as "a very thick series of marine, grey to black marls and shales, with a few beds of limestones or marly limestones", which "rest on the Serdj limestones, where present". The type locality of the Fahdene Formation is located in the centre of the Oued Bahloul anticline, Southeast of Makthar [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF]. In this paper, we shall examine the Lower part of the Fahdene Formation, i.e. the Lower Shales, the Allam Limestones, and the base of the Middle Shales (Fig. 4).

Fahdene Formation

The Lower Shale

Member is 180 m (Bou el Haneche) to 400 m thick (El Goussa). Although it consists mainly of monotonous marl and shale accumulations, the unit exhibits calcareous to marly intercalations, thus defining a basal, median and upper shaly unit. This member has not been studied in the Djerissa section. The base of the Lower Shale Member is defined as the first thick shaly level overlying the uppermost massive sandstone or limestone bed of the Hameima Formation.

Figure 4 about here -The basal shales unit is mainly known in the El Goussa and Hameima section, where it is 55-60 m thick (Fig. 4). It consists mainly of dark green shales, with few thin calcareous interbeds, and locally phosphatic nodules. In the Hameima section, the lower part of this unit is marked by a thickening upward series of bioturbated, sandy limestones, exhibiting storm laminations near the top. There, the fauna associated with the sandy limestones consists in ammonites, oysters, pectinids, trigonids, sea urchins, and scarce plant remains. This sandy interval seems to be lacking in the El Goussa section, where large scale slump structure can be observed. The upper shaly part of the unit yielded only ammonites, with scarce oysters and gastropods. In the Slata section, this unit may be represented by a 15 m thick, undated shaly interval.

Ammonites collected from the basal shale units were determined as "Hypacanthoplites" paucicostatus, ?Parengonoceras sp. and Acanthohoplitinae indet. in the lower part, and "Hypacanthoplites" ouenzaensis, Douvilleiceras sp nov.? , ?Parengonoceras sp. and "Hypacanthoplites" sp. in the upper part (Fig. 4). The occurrence of both "H." paucicostatus and "H." ouenzaensis indicates that the basal shales unit is at least in part coeval with the Upper Member of the Hameima Formation of the southeastern sections, thus indicating a lateral facies change from the Northwest to the Southeast.

-The first carbonated interval is 45 to 60 m thick, and corresponds to the "Ammonite-rich horizon" of [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF]. In the Djerissa and Bou el Haneche sections, the Fahdene Formation seems to begin directly with the "Ammonite-rich horizon". In contrast to the underlying shales, the "Ammonite-rich horizon" consists of alternating limestones and dark bioturbated, argillaceous marls, locally rich in pyrite. In the Slata section, this unit is represented by a 1 to 20 m thick series of coarse grained, bioclastic limestones, locally sandy. In the Hameima, Slata and Bou el Haneche sections, it contains ammonites, associated with belemnites, sea urchin fragments, brachiopods, pectinids, oysters and other bivalves (Fig. 4), whereas in the El Goussa section, it only yielded ammonites and scarce belemnites.

From these layers, Burollet (1956, p. 74) and Dubourdieu (1956, p. 152, 160, 163) mentioned ammonite faunas made of Douvilleiceras and Acanthohoplitinae, ascribed to the "Clansayes zone". In the "Ammonite-rich horizon", we collected numerous specimens of "Hypacanthoplites" nov. sp., associated with Douvilleiceras sp. and desmoceratids (Fig. 4). This horizon has been recognized in all the studied sections. -The intermediate shale unit is 70 to 130 m thick (Fig. 4). It consists of dark shales, with scarce thin beds of marly limestones, containing abundant pyrite concretions, and locally dark green phosphatic nodules. No fauna but ammonites have been observed in the Hameima and El Goussa sections, while echinoids, oysters, pectinids and other bivalves occur in the Bou el Haneche section. Bioturbations are rare, and horizontal laminations are commonly observed. In the Slata section, the intermediate shaly unit is represented by a 35 m thick series of dark shales, yielding few poorly preserved pyritous ammonites.

The base of this unit is marked by the first local occurrence of Prolyelliceras gevreyi. The other collected ammonites consist of desmoceratids, Prolyelliceras gevreyi, Mirapelia advena and scarce reworked Acanthohoplitinae (Fig. 4).

-The second carbonated interval is 20 to 70 m thick (Fig. 4). In the El Goussa and Hameima sections, it consists of alternating shales with pyritous concretions, and bioturbated marly limestones. The base is usually marked by a calcareous bed containing phosphatized clasts, which forms a conspicuous correlation bed [START_REF] Dubourdieu | Etude géologique de la région de l'Ouenza (Confins Algéro-Tunisiens)[END_REF][START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF][START_REF] Zghal | Foraminifères et Ostracodes de l'Aptien-Albien du Jebel Hameima (region de Tadjerouine, Tunisie) : Biostratigraphie, paleoécologie[END_REF]. The fauna content is dominated by belemnites, irregular echinoids and ammonites to the Northwest, and by pectinids, ammonites and subordinate brachiopods to the Southeast. Oysters and gastropods are also present. This unit seems to be reduced in the Bou el Haneche section (20 m). In the Slata section, the second carbonated interval is made of a 25 m thick succession of phosphatic, coarse grained bioclastic limestones, with few shaly intervals.

In this unit, we collected the following ammonites: Prolyelliceras gevreyi (only in the lower part), Mirapelia cf. alticarinata, Beudanticeras sp. and Desmoceras (Desmoceras) latidorsatum.

-The upper shale unit is 20 to 70 m thick, and is chiefly made of dark shales, rich in pyrite concretions, locally bioturbated, with few thin beds of marl (Fig. 4). Macrofossils are dominated by pyritous ammonites localised in few layers, and belemnites associated with subordinate brachiopods, pectinids and other bivalves. In the Slata section, it is 55 m thick and contains a few sea urchins. This unit thickens toward the Northwest.

The pyritous ammonite levels have been recognized by [START_REF] Pervinquière | Étude de paléontologie tunisienne. 1 : Céphalopodes des terrains secondaires[END_REF], [START_REF] Dubourdieu | Etude géologique de la région de l'Ouenza (Confins Algéro-Tunisiens)[END_REF] and [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF]. In this unit, we collected the following ammonites : Prolyelliceras radenaci, Douvilleiceras mammillatum, Beudanticeras revoili, B. dupinianum africanum, Phylloceras (Hypophylloceras) velledae, Ptychoceras hamaimensis, Pictetia astieriana, ?Silesitoides thos, Desmoceras sp., Protanisoceras (Protanisoceras) cf. acteon, Anisoceras sp. and Puzosiinae (Fig. 4). In the Bou el Haneche section, the top of the unit yielded Tegoceras mosense (Fig. 5).

The Allam Limestone

Member is 110 m (Hameima, Djerissa) to 180 m thick (El Goussa, Bou el Haneche) (Fig. 4 and5). [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF] described the "Allam Limestones" as hard, finegrained black limestones, which alternate with dark shales and marls, containing belemnites and poorly preserved ammonites (Puzosia sp.). He ascribed the Allam Limestones to "approximately the Middle Albian". [START_REF] Dubourdieu | Etude géologique de la région de l'Ouenza (Confins Algéro-Tunisiens)[END_REF] also ascribed this unfossiliferous unit to the Middle Albian.

According to our observations, the Allam Member is made of fine-grained black laminated limestones, interbedded within dark grey shales rich in pyrite, and containing locally small scale oil seeps. In the Djerissa section, the base of the unit is locally marked by a limestone bed containing phosphatized clasts, whereas in the Slata section, it is marked by bioclastic and finely sandy limestones, rich in phosphate and glauconite (Fig. 5). The uppermost part of the unit is commonly silicified (El Goussa, Hameima, Djerissa, Bou el Haneche), and often contains abundant, large limestone nodules. The fauna mainly consists of belemnites, associated with scarce ammonites, costulated or inoceramid-like small bivalves, and fich remains.

Figure 5 about here

From the Allam Limestone Member, we collected at the base : Beudanticeras sp., Ulighella? sp., ?Douvilleiceras sp., Oxytropidoceras sp., Pusozia sp. and Desmoceratids (Fig. 4 and5). In the Djerissa and Bou el Haneche sections, the lower part (60 to 70 m) of the Allam Limestone Member yielded Tegoceras camatteanum, while its middle part yielded one specimen of Lyelliceras pseudolyelli in Bou el Haneche (Fig. 5). The upper part is barren of ammonites.

The Middle Shale

Member of the Fahdene Formation rests conformably of the top, often silicified, of the Allam Limestone Member. Only the base of this member has been studied in detail, since good outcrops are scarce.

The Middle Shale Member is not silicified, and consists of dark shales with few thin beds of bioturbated marl. The macrofauna consists of numerous crushed ammonites, associated with less common belemnites, and scarce echinoids and bivalves. The very base of the unit is locally rich in glauconitic grains (El Goussa) or fine-grained sandstone (Hameima, Slata).

In the Hameima section, the very base (5 m) of the Middle Shale Member yielded ?Hamites sp., Kosmatella sp. and Mirapelia sp., together with Venezoliceras sp. and (Oxytropidoceras) sp. (Fig. 4). Higher up in the Hameima section, and in other sections, the ammonite association consists of Mortoniceras (Deiradoceras) sp., Mortoniceras (Mortoniceras) sp., Oxytropidoceras sp., Venezoliceras sp., Hysteroceras sp., Neokentroceras sp., Puzosia sp., indicative of a Late Albian age (Mortoniceras pricei Zone).

Biostratigraphy 1. Top of the Serdj Formation

The occurrence in Jebel Harraba, 40 m below the top of the Serdj Formation, of reworked Epicheloniceras sp. (that normally indicates the Late Aptian) indicates that the age of the top of the Serdj Formation must be considered as latest Aptian or younger, i.e. close to the Aptian-Albian boundary.

In various places of Eastern Algeria and Central Tunisia, and especially in Jebel Hameima, the top of the Serdj Formation (or its equivalent) yielded the following association of benthic foraminifera : Archaealveolina reicheli, Paracoskinolina tunesiana, Mesorbitolina texana, M. minuta, as well as calpionellids: Colomiella mexicana and C. recta [START_REF] Bismuth | Réflexions stratigraphiques sur l'Albo-Aptien dans la région des djebels Douleb et Semmama et son environnement (Tunisie du Centre-Nord)[END_REF][START_REF] Fourcade | Crétacé du Kef Hahouner et position stratigraphique de " Ovalveolina " reicheli P. de Castro (série septentrionale du môle néritique du Constantinois, Algérie)[END_REF][START_REF] Tlatli | Etude des calcaires de l'Albo-Aptien des Djebels Serdj et Bellouta[END_REF][START_REF] Cherchi | Dictyoconus algerianus n. sp., grand foraminifère de l'Aptien supérieur de la plaque africaine (marge septentrionale)[END_REF][START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF][START_REF] Zghal | Foraminifères et Ostracodes de l'Aptien-Albien du Jebel Hameima (region de Tadjerouine, Tunisie) : Biostratigraphie, paleoécologie[END_REF]. Although A. reicheli is locally mentionned in earliest Albian deposits (Dinarides; [START_REF] Husinec | Mid Cretaceous orbitolinid (Foraminiferida) record from the islands of Cres and Losinj (Croatia) and its regional stratigraphic correlation[END_REF][START_REF] Velic | Stratigraphy and palaeobiogeography of Mesozoic benthic forraminifera of the Karst Dinarides (SE Europe)[END_REF], A. reicheli and P. tunesiana are usually regarded as early Late Aptian in age [START_REF] Bismuth | Réflexions stratigraphiques sur l'Albo-Aptien dans la région des djebels Douleb et Semmama et son environnement (Tunisie du Centre-Nord)[END_REF][START_REF] Peybernès | Les orbitolinidés crétacés d'Afrique : Essai de synthèse[END_REF][START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF][START_REF] Arnaud-Vanneau | Larger benthic foraminifera[END_REF]. On the other hand, the association of M. texana and M. minuta would indicate a Late Aptian age in Northern and Western Africa, although it reaches the Early Albian in Eastern Africa [START_REF] Peybernès | Les orbitolinidés crétacés d'Afrique : Essai de synthèse[END_REF]. However, since C. recta first occurs soon after the base of the Albian, the association of C. mexicana and C. recta indicates an earliest Albian age [START_REF] Reháková | Evolution and distribution of calpionellids -the most characteristic constituents of Lower Cretaceous Tethyan microplankton[END_REF]Días-Brito, 1999 in Días-Brito and[START_REF] Días-Brito | Albian roveacrinids (stemless crinoids) in the Albian carbonates of the offshore Santos Basin, southeastern Brazil: stratigraphic, palaeobiogeographic and palaeocanographic significance[END_REF]. This, combined with our ammonite findings, suggests that the age of the top of the Serdj Formation is close to the Aptian-Albian boundary. As a consequence, A. reicheli and P. tunesiana, as well as the M. texana -M. minuta association should be considered as reaching the latest Aptian, or even the earliest Albian (see discussion below).

Hameima Formation

The Lower Member of the Hameima Formation yielded Acanthohoplitinae, Parengonoceras sp., Douvilleiceras sp. nov.? (Plate 1, Fig. 1 and2) and "Hypacanthoplites" paucicostatus (at the top). The Middle Member of the Hameima Formation did not yield diagnostic fauna, while the Upper Member of the Hameima Formation contains Douvilleiceras sp. [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF] and Acanthohoplitinae. Douvilleiceras of the group of D. leightonense are known from the middle part of the Leymeriella tardefurcata Zone (sensu [START_REF] Owen | The base of the Albian stage; comments on recent proposals[END_REF] to the lower part of the Douvilleiceras mammillatum Zone [START_REF] Owen | The ammonite zonal sequence and ammonite taxonomy in the Douvilleiceras mammillatum Superzone (Lower Albian) in Europe[END_REF][START_REF] Kennedy | Integrated stratigraphy across the Aptian-Albian boundary in the Marnes Bleues, at the Col de Pré-Guittard, Arnayon (Drôme), and at Tartonne (Alpes de Haute-Provence), France: a candidate Global Boundary Stratotype Section and Boundary point for the base of the Albian stage[END_REF], while parahoplitids (among which Acanthohoplitinae) became extinct before the end of the Leymeriella tardefurcata Zone [START_REF] Kennedy | Integrated stratigraphy across the Aptian-Albian boundary in the Marnes Bleues, at the Col de Pré-Guittard, Arnayon (Drôme), and at Tartonne (Alpes de Haute-Provence), France: a candidate Global Boundary Stratotype Section and Boundary point for the base of the Albian stage[END_REF]. Therefore, the Hameima Formation is ascribed to the Leymeriella tardefurcata Zone (first ammonite zone of the Early Albian, Fig. 6), except the very base that may be of latest Aptian age, since it yielded only Acanthohoplitinae. Note that the Douvilleiceras specimens found in the Lower Member of the Hameima Formation [START_REF] Dubourdieu | Etude géologique de la région de l'Ouenza (Confins Algéro-Tunisiens)[END_REF]this work;Pl. 1) are the oldest known representatives of the genus Douvilleiceras in North Africa, providing, however, an unequivocal Lower Albian age for these beds. The ontogeny of the tunisian specimen with persistent primitive morphology on early whorls, clearly differentiates this specimen from all other known Douvilleiceras [START_REF] Latil | Lower Albian ammonites from Central Tunisia and adjacent aereas of Algeria[END_REF]. It probably deserves a species status, but our material is not sufficiently well preserved for that. [START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF][START_REF] Zghal | Foraminifères et Ostracodes de l'Aptien-Albien du Jebel Hameima (region de Tadjerouine, Tunisie) : Biostratigraphie, paleoécologie[END_REF] reported Mesorbitolina parva and M. minuta from the upper part of the Lower Member of the Hameima Formation, and ascribed this unit to the Late Aptian. More recently, [START_REF] Zghal | The Hameima Formation. Aptian-Turonian events in Central Tunisia, Field trip guidebook[END_REF] mentionned Mesorbitolina parva and Praeorbitolina wienandsi from the same beds, suggesting an age close to the Early to Late Aptian boundary. The currently admitted age of benthic foraminifers such as A. reicheli, P. tunesiana, M. texana, M. minuta, and M. parva has been mainly determined using stratigraphic sections bearing both foraminifers and ammonites in Algeria and Tunisia (e.g. [START_REF] Fourcade | Crétacé du Kef Hahouner et position stratigraphique de " Ovalveolina " reicheli P. de Castro (série septentrionale du môle néritique du Constantinois, Algérie)[END_REF][START_REF] Bismuth | Réflexions stratigraphiques sur l'Albo-Aptien dans la région des djebels Douleb et Semmama et son environnement (Tunisie du Centre-Nord)[END_REF][START_REF] Cherchi | Dictyoconus algerianus n. sp., grand foraminifère de l'Aptien supérieur de la plaque africaine (marge septentrionale)[END_REF]. The conflicting ages indicated by ammonites and benthic foraminifers in the studied sections seem to arise mainly from two facts.

Biostratigraphic discussion

Plate 1. Albian ammonites from the Hameima Formation and from the Lower Shales of the Fahdene Formation. 1: Douvilleiceras sp. nov. ?, FSL.596316, Level 4b of SW Ouenza section, Algeria (Dubourdieu, 1956, p. 56) First, the ammonite fauna reported from these sections is mainly made up of species of Acanthohoplitinae, which are highly endemic forms, only rarely associated with cosmopolitan species. In the Ouenza massif (Algeria, about 30 km west of the Tajerouine area), [START_REF] Dubourdieu | Etude géologique de la région de l'Ouenza (Confins Algéro-Tunisiens)[END_REF] found, associated with numerous Acanthohoplitinae, a species determined as Douvilleiceras gr. albensemonile (Pl.1, Fig. 1) near the base of what is now called the Hameima Formation (40 to 80 m above the massive limestones equivalent to the Serdj Fm; [START_REF] Dubourdieu | Etude géologique de la région de l'Ouenza (Confins Algéro-Tunisiens)[END_REF]p. 56). Although he acknowledged that this ammonite usually indicates a late Early Albian to earliest Middle Albian age, he ascribed this unit to the Late (not latest) Aptian because of the admitted range of the associated Acanthohoplitinae ammonites (Dubourdieu, 1956, p. 92). Conversely, near the base of the Fahdene Formation of the Hameima section, [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF] quoted in his "ammonite-rich horizon", several Acanthohoplitinae and a Douvilleiceras bigoureti. He ascribed, therefore, this level to the Early Albian "Clansayes zone included" (e.g. [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF]p. 72). However, following the conclusions of the colloquium of Lyon, 1963, the "Clansayesian stage" has been included into the Late Aptian [START_REF] Flandrin | Rapport sur l'étage Aptien[END_REF], thus adding confusion to the age assignments of these Acanthohoplitinae faunas.

Second, in the sections described in the Constantine area, ammonites and benthic foraminifers have not been found in the same beds, and therefore, age assignments of the foraminifers remained speculative. In Kef Hahouner (NE of Constantine, [START_REF] Fourcade | Crétacé du Kef Hahouner et position stratigraphique de " Ovalveolina " reicheli P. de Castro (série septentrionale du môle néritique du Constantinois, Algérie)[END_REF], the uppert part of a 200 m-thick series of massive limestones ("terme E"), contains A. reicheli, P. tunesiana, and M. texana, and is overlain by a 40 m-thick series of thinly bedded limestones, and then by nodular, ammonite-bearing limestones ("terme F" of [START_REF] Fourcade | Crétacé du Kef Hahouner et position stratigraphique de " Ovalveolina " reicheli P. de Castro (série septentrionale du môle néritique du Constantinois, Algérie)[END_REF]. The ammonite fauna from the "terme F" was examined by Breistroffer (in [START_REF] Fourcade | Crétacé du Kef Hahouner et position stratigraphique de " Ovalveolina " reicheli P. de Castro (série septentrionale du môle néritique du Constantinois, Algérie)[END_REF]p. 233), who mentionned that "all these ammonites are too big and poorly preserved and do not allow reliable determinations", and concluded nevertheless, that the fauna is most probably of Late Aptian (Gargasian) age. After quoting that some ammonite identifications suggest an older age (p. 233), [START_REF] Fourcade | Crétacé du Kef Hahouner et position stratigraphique de " Ovalveolina " reicheli P. de Castro (série septentrionale du môle néritique du Constantinois, Algérie)[END_REF] concluded that A. reicheli and P. tunesiana are of Late Aptian (Gargasian) age.

Revising the same section, [START_REF] Cherchi | Dictyoconus algerianus n. sp., grand foraminifère de l'Aptien supérieur de la plaque africaine (marge septentrionale)[END_REF] questionned this age, and ascribed the "terme F" of Kef Hahouner to the latest Aptian ("Clansayesian"). On the other hand, [START_REF] Cherchi | Dictyoconus algerianus n. sp., grand foraminifère de l'Aptien supérieur de la plaque africaine (marge septentrionale)[END_REF] mentionned that in Jebel Dahar the A. reicheli-P. tunesiana association has been ascribed to the Albian. Finally, [START_REF] Cherchi | Dictyoconus algerianus n. sp., grand foraminifère de l'Aptien supérieur de la plaque africaine (marge septentrionale)[END_REF] mentionned that in Jebel Sidi Rehris (Constantine area), the A. reicheli-P. tunesiana association is found in the 50 m-thick upper part of a massive limestone series. These beds are located 20 m above a marly bed with Hypacanthoplites species of latest Aptian age (Late Clansayesian), and 10 m below a marl series that yielded Desmoceras species, of Early to Middle Albian age. [START_REF] Cherchi | Dictyoconus algerianus n. sp., grand foraminifère de l'Aptien supérieur de la plaque africaine (marge septentrionale)[END_REF] concluded that the A. reicheli-P. tunesiana association is of latest Aptian age, although the presented data only indicate a latest Aptian to Early Albian age for the A. reicheli-P. tunesiana association.

In the same work, [START_REF] Cherchi | Dictyoconus algerianus n. sp., grand foraminifère de l'Aptien supérieur de la plaque africaine (marge septentrionale)[END_REF] ascribed to the early Late Aptian (Gargasian) the microfauna found at the top of the Serdj Formation (A. reicheli-P. tunesiana and C. mexicana-C. recta) in the Hameima section, based on the fact that they are located 300 m below the "ammonite-rich horizon" of [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF], which they ascribed to the latest Aptian ("Clansayesian"). The same reasonning has been applied by [START_REF] Bismuth | Réflexions stratigraphiques sur l'Albo-Aptien dans la région des djebels Douleb et Semmama et son environnement (Tunisie du Centre-Nord)[END_REF], who ascribed to the early Late Aptian (Gargasian), the A. reicheli found at the top of the Serdj Formation, because they are located "much below a fossiliferous horizon bearing ammonites of the Clansayes zone" [START_REF] Bismuth | Réflexions stratigraphiques sur l'Albo-Aptien dans la région des djebels Douleb et Semmama et son environnement (Tunisie du Centre-Nord)[END_REF]p. 184). However, we already mentionned that the "ammonite-rich horizon" yielded a specimen of the Douvilleiceras genus [START_REF] Burollet | Contribution à l'étude stratigraphique de la Tunisie Centrale[END_REF], that ensures an Early Abian age for this level. This succession of misinterpretations from french-speaking workers explains probably why the M. parva -M. texana -M. minuta association has been assigned to the early Late Aptian in the frenchspeaking african countries (Tunisia, Algeria, Marocco, Sénégal), whereas it is considered of Late Aptian and earliest Albian age in english-speaking african countries (Tanzania, Somalia) [START_REF] Peybernès | Les orbitolinidés crétacés d'Afrique : Essai de synthèse[END_REF]. Whatever the case, our ammonite collections [START_REF] Latil | Lower Albian ammonites from Central Tunisia and adjacent aereas of Algeria[END_REF] indicate that the A. reicheli -P. tunesiana association, found at the top of the Serdj Formation, reaches the latest Aptian, and that M. parva and M. minuta, determined from the Hameima Formation, reach the earliest Albian (L. tardefurcata Zone). The mention of P. wienandsi in the Hameima Formation probably results from a wrong identification.

Fahdene Formation: Lower Shale Member

In the basal shales unit and for most of the first carbonate interval, the association of Douvilleiceras sp. nov.? (Plate 1, Fig. 3) and Acanthohoplitinae indicates an age included in the Leymeriella tardefurcata Zone. The upper part of the first carbonate interval records the last local occurrence of Acanthohoplitinae (bed 52), followed by the first local occurrence of P. gevreyi (Fig. 4). Because P. gevreyi is known elsewhere in the Leymeriella tardefurcata Zone [START_REF] Kennedy | Integrated stratigraphy across the Aptian-Albian boundary in the Marnes Bleues, at the Col de Pré-Guittard, Arnayon (Drôme), and at Tartonne (Alpes de Haute-Provence), France: a candidate Global Boundary Stratotype Section and Boundary point for the base of the Albian stage[END_REF], the first carbonate interval is ascribed to this time-span (Fig. 6).

The intermediate shale unit of the Lower Shale Member yielded P. gevreyi and M. advena. Because of the known range of these two species [START_REF] Kennedy | Integrated stratigraphy across the Aptian-Albian boundary in the Marnes Bleues, at the Col de Pré-Guittard, Arnayon (Drôme), and at Tartonne (Alpes de Haute-Provence), France: a candidate Global Boundary Stratotype Section and Boundary point for the base of the Albian stage[END_REF], the intermediate shales can be ascribed to the late part of the Leymeriella tardefurcata Zone (Fig. 6).

The second carbonate interval records the first local occurrence of M. cf. alticarinata (bed 67) and the last local occurrence of P. gevreyi (bed 68, Fig. 4). Since M. alticarinata is only known from the Douvilleiceras mammillatum Zone [START_REF] Owen | The ammonite zonal sequence and ammonite taxonomy in the Douvilleiceras mammillatum Superzone (Lower Albian) in Europe[END_REF], P. gevreyi seems to be not restricted to the Leymeriella tardefurcata Zone, as previously assumed [START_REF] Kennedy | Integrated stratigraphy across the Aptian-Albian boundary in the Marnes Bleues, at the Col de Pré-Guittard, Arnayon (Drôme), and at Tartonne (Alpes de Haute-Provence), France: a candidate Global Boundary Stratotype Section and Boundary point for the base of the Albian stage[END_REF]. Whatever the case, the co-occurrence of these two species suggests an age comprised within the Douvilleiceras mammillatum Zone (middle and upper part of the Lower Albian; Fig. 6). The occurrence of Tegoceras mosense at the top of the Lower Shale Member suggests an age close to the Lyelliceras pseudolyelli Subzone (last ammonite subzone of Lower Albian age) for this level. [START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF] and [START_REF] Zghal | Foraminifères et Ostracodes de l'Aptien-Albien du Jebel Hameima (region de Tadjerouine, Tunisie) : Biostratigraphie, paleoécologie[END_REF] quoted the first local occurrence of Hedbergella rischi and Ticinella primula a few meters above the base of the second carbonate interval. This has been taken as indicative of a Middle Albian age [START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF][START_REF] Zghal | Foraminifères et Ostracodes de l'Aptien-Albien du Jebel Hameima (region de Tadjerouine, Tunisie) : Biostratigraphie, paleoécologie[END_REF], as assumed by several workers (e.g. Premoli-Silva and Slitter, 1995;[START_REF] Bellier | Lower Cretaceous planktonic foraminiferal biostratigraphy of the Western North Atlantic (ODP LEG 171B), and taxonomic clarification of key index species[END_REF]. However, this association has been reported from Early Albian [START_REF] Luciani | Albian high-resolution biostratigraphy and isotope stratigraphy : The Coppa della Nuvola succession of the Gargano Promontory (Southern Italy)[END_REF] or even Late Aptian strata [START_REF] Cobianchi | Early Cretaceous nannofossils and planktonic foraminifera from northern Gargano (Apulia, southern Italy)[END_REF]. Our data suggest that T. primula and H. rischi co-occur in Tunisia within the Douvilleiceras mammillatum superzone and before the beginning of the Lyelliceras pseudolyelli Zone, of late Early Albian age.

Fahdene Formation: Allam Limestone Member

Beside undiagnostic ammonites collected near the base of the unit, the Allam Limestones yielded several T. camatteanum in the lower third part of the succession, while L. pseudolyelli was found in the middle part of the Allam Limestone Member (Fig. 5). Therefore, the lower part of the member can be ascribed to the upper part of the Douvilleiceras mammillatum superzone, while its middle part is of Upper Lower Albian age (Lyelliceras pseudolyelli zone; [START_REF] Latil | Les Lyelliceratinae Spath, 1921 (Ammonitina, Ammonoidea) de l'Albien inférieur et moyen dans le bassin de Paris et sur les bordures du bassin vocontien : Stratigraphie, paléobiogéographie et taxonomie[END_REF]. However, the undated upper part of the section (40 to 60 m in the Jerissa and Bou el Haneche sections) may reach the base of the Middle Albian (Fig. 6). Therefore, in the Tadjerouine area, at least part of the Middle Albian substage seems to be lacking, since Upper Albian strata directly overly the Allam Limestone Member (see below).

Fahdene Formation: Middle Shales Member

At the base of the Middle Shales Member of the Hameima section, evoluate forms of Venezoliceras sp. and Oxytropidoceras (Oxytropidoceras) sp. suggest an age within the Dipoloceras cristatum Zone (first ammonite zone of the Upper Albian; Fig. 6). This ammonite association has not been found in the other sections of the Tadjerouine area. A few meters above, the occurrence of Mortoniceras (M.) gr. pricei and Mortoniceras (Deiradoceras) sp. indicates the Mortoniceras pricei Zone (second ammonite zone of the Upper Albian). At the base of the Middle Shales Member of the Hameima section, the first occurence of planktic foraminifera such as Ticinella roberti, T. gr. raynaudi and Biticinella breggiensis support a Late Albian age [START_REF] Zghal | Etude microbiostratigraphique du Crétacé inférieur de la Tunisie du Centre Ouest[END_REF][START_REF] Zghal | Foraminifères et Ostracodes de l'Aptien-Albien du Jebel Hameima (region de Tadjerouine, Tunisie) : Biostratigraphie, paleoécologie[END_REF].

Therefore, the stratigraphic hiatus of Middle Albian age seems to be followed by a diachronous transgressive series, since the base of the Upper Albian Middle Shale Member of the Fahdene Formation would belong to the Dipoloceras cristatum Zone or to the Mortoniceras pricei Zone, according to the sections.

Conclusions

In the Tajerouine area, the latest Aptian -early Late Albian interval presents a rather constant lithological succession (except in Jebel Slata), the age of which, however, may vary according to the locations.

In all sections, the Hameima Formation is of Early Albian age (middle part of Leymeriella tardefurcata Zone). However, the upper Member of the Hameima Formation, made of marls and orbitolinid limestones in the southeastern sections (Fig. 3), is coeval with ammonite bearing shales, marls and subordinate sandstones deposited in the Northwestern sections (base of the Lower Shale Member of Fahdene Fm, Fig. 4). The boundary between the Hameima and Fahdene formations is, therefore, diachronous (older in the Northwest than in the Southeast; Fig. 6).

Although its base is heterochronous, the Lower Shale Member of the Fahdene Formation is of Early Albian age (Leymeriella tardefurcata Zone, and Douvilleiceras mammillatum Zone). This lithological unit thickens toward the Northwest, illustrating the paleogeographic transition between the Central Tunisian platform and the North Tunisian Basin.

Finally, the Allam Limestone Member is mainly of late Early Albian age (late Douvilleiceras mammillatum Zone and Lyelliceras pseudolyelli Zone), although its upper part may belong in part to the early Middle Albian (Fig. 5 and6). It is directly overlain by early Late Albian shales (Middle Shale Member of Fahdene Fm), the base of which seems to be slightly diachronous. This, together with the probable hiatus of most of the Middle Albian substage (Fig. 6), suggests a major tectonic and/or sedimentary event during the Middle Albian.

These new biostratigraphic data would led to revise the stratigraphic range of some foraminifers. The benthic foraminifers Archaealveolina reicheli and Paracoskinolina tunesiana seem to reach the latest Aptian, while Mesorbitolina texana and M. minuta reach the earliest Albian (Leymeriella tardefurcata Zone). In the same way, the planktic foraminifers Hedbergella rischi and Ticinella primula appear in the late Early Albian in Tunisia.

Figure 1 :

 1 Figure 1 : Paleogeographic sketch of Tunisia during Aptian -Albian times, and location of the studied area.

Figure 2 :

 2 Figure 2 : Location map of the main studied sections (location on Fig. 1). From Northwest to Southeast: Jebel Harraba, Sidi Embarka section; Henchir el Goussa, El Goussa section; Jebel Hameima section; Jebel Slata, Charens section; Jebel Djerissa, Djerissa section; Jebel Bou el Haneche, Bou el Haneche section.

Figure 3 :

 3 Figure 3 : Distribution of ammonites in the Hameima Formation, along the Djerissa and Slata sections.Grey dots: location of ammonites from the Harraba section; White dots: reworked ammonites.

Figure 4 :

 4 Figure 4 : Distribution of ammonites in the Fahdene Formation, along the Hameima section. Black dots: ammonites from the Hameima section; White dots: reworked ammonites. Shaded areas: carbonate intervals. Same caption as Fig. 3.

Figure 5 :

 5 Figure 5 : Distribution of ammonites in the Allam Limestones (Fahdene Formation), along the Djerissa and Bou el Haneche sections. Same caption as Fig. 3.

Figure 6 .

 6 Figure 6. Synthetic stratigraphic column of the Albian series in the Tadjerouine area, and comparisons between former, and proposed age attributions. Same caption as Fig. 3.

Acknowledgements

The results presented here were obtained thanks to scientific agreements between the universities of Grenoble (France) and Sfax (Tunisia) (CMCU project 2002-05, and Partenariat H. Curien Utique 2006-09). We are undebted to T. Saucède (Dijon) and M. Aly (Cairo, Egypt) for their participation to field work, to E. Robert (Grenoble) and L. Bulot (Marseille) for their contributions to ammonites identification, and to A. Arnaud-Vanneau and H. Arnaud (Grenoble) for highlighting discussions. We also thank two anonymous reviewers for their useful and very constructive remarks, which made more complete this work.


Suzanne Lapillonne
email: suzanne.lapillonne@inrae.fr

Georgios Fourtakas

Guillaume Piton

Guillaume Chambon

Vincent Richefeu

Lapillonne Suzanne

Fourtakas Georgios

Piton Guillaume

Chambon Guillaume

Richefeu Vincent

Validation of DualSPHysics for viscous Newtonian flow fronts

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Viscous flow fronts, i.e., sudden transition from dry conditions to near uniform flow depth, are encountered in many applications in geophysics such as debris-, mud-and lavaflows. The precise understanding and modelling of the behaviour of such flow fronts is of interest for instance in the study of impact forces and the design of hazard mitigation measures. These highly non-uniform free-surface flows are difficult to model at front scale by classical mesh-based methods, notably due to their complex free-surface requiring remeshing which results in high computational cost. Initiatives to numerically model mud-flow fronts have been either based on depth-averaged schemes [START_REF] Laigle | Numerical modeling of mudflows[END_REF] which perform satisfyingly in replicating the bulk behaviour of the flow (e.g. area covered by flow spreading), but do not provide information on the internal dynamics, or, using Lagrangian methods [START_REF] Laigle | SPH-based numerical study of the impact of mudflows on obstacles[END_REF], [START_REF] Leonardi | Coupled DEM-LBM method for the free-surface simulation of heterogeneous suspensions[END_REF] which are a viable alternative on modelling flow fronts dynamics. Initial works using SPH for very viscous fluids, that are eventually non-Newtonian, are promising but lack a detailed validation against experimental campaigns. Indeed, validating SPH models against high quality experimental data for mudlike behaviour opens up the road to applications for better capturing the internal dynamics of these complex flows that are very difficult to measure experimentally and cannot be studied internally at prototype scale due to their destructive nature. Ultimately, it will allow to tailor more efficient and resistant hazard mitigation measures. An SPH solver such as DualSPHysics [START_REF] Domínguez | Dual-SPHysics: from fluid dynamics to multiphysics problems[END_REF] has many advantages for geophysical applications due to explicit free-surface capturing, coupling with granular and discrete models, as well as being a highly optimised solver using GPU parallelisation techniques.

This work studies the predictions of DualSPHysics for such types of mud-flows. Validation is done against accurate measurements of the internal dynamics of flow front using data from Freydier et al. [START_REF] Freydier | Experimental characterization of velocity fields within the front of viscoplastic surges down an incline[END_REF]. Newtonian flows are studied using the classical DualSPHysics code and the modified dynamic boundary condition (mDBC) to ensure no-slip condition.

II. SPH MODELLING METHOD

A. Governing Equations

Herein we use the continuity and Navier-Stokes equation for a weakly-compressible fluid which, in their Lagrangian form, can be written as:

       Dρ Dt = -ρ∇ • u Du Dt = 1 ρ ∇P + 1 ρ ∇ • τ + g (1a) ( 1b
)
where ρ is the density, u is the velocity, P is the pressure, τ is the viscous stress tensor, g represents body forces, and D Dt stands for the material derivative. To close this system of equation, an equation of state is used to compute the pressure P :

P = c 2 0 ρ 0 γ ρ ρ 0 γ -1 (2)
where c 0 is the numerical speed of sound, γ = 7 and ρ 0 is the reference density. In the SPH scheme, the discrete governing equations read:

                             ∂ρ a ∂t =ρ a b (u a -u b ) • ∇W ab V b + hc 0 D a ∂u a ∂t = - b m b P a + P b ρ b ρ a ∇ a W ab + b m b 4ν 0 r ab • ∇ a W ab (ρ a + ρ b )(||r ab || 2 + η 2 ) u a + g ∂r a ∂t =u a (3a) (3b) (3c)
where subscripts a and b refer to the interpolating and neighbouring particles respectively , r i is the positional vector of particle i and r ij = r j -r i , V i the volume occupied by the particle, m i is the mass assigned to the numerical particle, W ij is the smoothing kernel, η = 0.01h with h the smoothing length, D i is a density diffusion term (see Section II-E) and ν 0 the dynamic viscosity of the fluid.

The smoothing kernel used is the quintic Wendland kernel [START_REF] Wendland | Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree[END_REF]:

W (r, h) = α D 1 - 1 2 r h 4 2 r h + 1 for 0 < r h < 2 (4)
where α D depends on the dimension of the model: α 2D = The time scheme used is the symplectic position Verlet scheme which reads [START_REF] Domínguez | Dual-SPHysics: from fluid dynamics to multiphysics problems[END_REF]:

                       r n+1/2 a = r n a + ∆t 2 v n a v n+1/2 a = v n a + ∆t dv a dt n a v n+1 a = v n a + ∆t dv a dt n+1/2 r n+1 a = r n+1/2 a + ∆t v n+1 a + v n a 2 (5a) (5b) (5c) (5d)
and the density evolution follows:

         ρ n+1/2 a =ρ n a + ∆t 2 dρ a dt n ρ n+1 a =ρ n a 2 -ϵ n+1/2 a 2 + ϵ n+1/2 a (6a) (6b) with ϵ n+1/2 a = -dρa dt n+1/2 /ρ
n+1/2 a ∆t The actual time-step is variable to allow for optimal description of the fluid while ensuring sensible computational time. The time-step is thus taken as follows:

                         dt = C CFL min(dt 1 , dt 2 , dt 3 ) dt 1 = min a h || dv dt || a,max dt 2 = h c s + max b hv ab •r ab r 2 ab +η 2 dt 3 = h 2 ν 0 (7a) (7b) (7c) (7d)
in which, dt 1 accounts for the CFL condition, dt 2 accounts for both CFL condition and the numerical viscosity and dt 3 accounts for the physical viscosity.

C. Wall Boundary condition

Due to the nature of the flow studied in this work, a noslip condition must be imposed at the boundaries. To ensure it, modified Dynamic Boundary Condition (mDBC) [START_REF] English | Modified dynamic boundary conditions (mdbc) for general-purpose smoothed particle hydrodynamics (sph): application to tank sloshing, dam break and fish pass problems[END_REF] is used. No-slip condition is implemented as a correction on the velocity of fluid particles close to the boundary.

D. Shifting

Due to the nature of SPH, particles in the flow cannot maintain a relatively homogeneous distribution. Voids tend to form leading to high risks of numerical instability. In order to counteract these effects inherent to the SPH method, a shifting algorithm is used. This is particularly important for highly viscous dominated flows.

At each time step, the particles are shifted in order to maintain a more uniform particle distribution. The distance by which the particles are shifted is dependent on the concentration of particles C i inside the kernel influence for each particle i, as well as, the velocity of the particle, as follows [START_REF] Lind | Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves[END_REF], [START_REF] Skillen | Incompressible smoothed particle hydrodynamics (sph) with reduced temporal noise and generalised fickian smoothing applied to body-water slam and efficient wave-body interaction[END_REF]:

δr a = -Ah||u|| a dt∇C a ( 8
)
where A is a shifting coefficient tuned by user (the value A = -2 is recommended in Dominguez et al. [START_REF] Domínguez | Dual-SPHysics: from fluid dynamics to multiphysics problems[END_REF]).

E. Density diffusion term

In Eq. (3a), the term hc 0 D a corresponds to the density diffusion term. This reduces density fluctuations, and thus oscillations in the pressure field. Indeed Eq. ( 2) is very stiff and with the natural disordering of SPH particles, small amplitude -high frequency oscillations are created in the pressure and density fields. In this work the density diffusion term introduced by Fourtakas et al. [START_REF] Fourtakas | Local uniform stencil (lust) boundary condition for arbitrary 3-d boundaries in parallel smoothed particle hydrodynamics (sph) models[END_REF] is used:

       D a =δ ϕ b ψ ab • ∇ a W ab m b ρ b ψ ab =2(ρ T ab -ρ H ab ) x ab ||x ab || 2 (9a) (9b)
with ρ T ab the total density, ρ H ab = ρ 0 γ P H ab +1 c 2 0 ρ0/γ -1 the hydrostatic component of the density and δ ϕ a diffusion coefficient controlling the magnitude of this diffusion term.

III. EXPERIMENTAL CONTEXT

A. Physical properties of the flow

The flows studied in this work are highly viscous, laminar flow fronts. Their highly non-uniform free-surface make their study very difficult using classical FEM methods, and thus SPH is a natural candidate to perform accurately for such flows.

The focus of this work is on a viscous laminar flow front with Reynolds number close to the creeping flow limits (Re ≈ 0.1). These flow front are assumed to be composed of a nonuniform front followed by a uniform zone, where the free surface elevation z max becomes steady longitudinally ∂ x z max = 0.

At the lowest order, the flow pressure can be assumed as :

p = ρg cos θ(z max -z) (10)
which gives the following force balance :

ρg sin θ -∂ x p + ∂ z τ xz = 0 (11)
τ xz = µ ∂u x ∂z = ρg sin θ(1 -cot θ∂ x z max )(z max -z) (12)
giving the following velocity profile and average velocity in the uniform zone, with α = ρg sin θ µ (1 -cot θ∂ x z max ):

     u x (z) = α(z max - 1 2 z)z ūx = α 3 z 2 max, unif (13a) (13b)

B. Selected experimental setup

The setup chosen for this validation is the experimental work by Freydier [START_REF] Freydier | Dynamique interne au front d'écoulements à surface libre[END_REF]. The characteristics of the experiments used as referenced are shown in Table I. The experimental setup is composed a conveyor belt tilted to a chosen angle, transparent side-walls, and a wall upstream the conveyor belt forcing a flow front to form steadily. This setup is thoroughly described in [START_REF] Chambon | Gravity-driven surges of a viscoplastic fluid: An experimental study[END_REF], [START_REF] Freydier | Experimental characterization of velocity fields within the front of viscoplastic surges down an incline[END_REF]. Fluids used are transparent mixtures of glucose and water allowing to measure accurate velocity fields and free surface profiles. Freydier [START_REF] Freydier | Dynamique interne au front d'écoulements à surface libre[END_REF] showed that these flows have a viscosity independent from strain and strain-rate, varying with concentration of glucose in the mixture. They are also highly influenced by drying [START_REF] Freydier | Dynamique interne au front d'écoulements à surface libre[END_REF], leading to uncertainties on the measured viscosity in a rheometer.

Velocimetry using STV technique (see Freydier [START_REF] Freydier | Dynamique interne au front d'écoulements à surface libre[END_REF]), and high definition, high velocity cameras at different location of the flow allows for a full description of both the free surface shape and the velocity fields within the flow.

C. Numerical setup

The numerical setup reproduces the experimental setup as a 2D flow front (Fig. 1). The 2D assumption is possible thanks to the tests operated by Freydier [START_REF] Freydier | Dynamique interne au front d'écoulements à surface libre[END_REF]. Volume of the experiment has been respected and the flow height is initialized at the theoretical height of the uniform zone (from Eq. 13b).

The conveyor belt starts a backwards motion in a periodic domain after a short resting time, using a ramp to reach the velocity of the experiment in order to avoid any instabilities caused by the sudden infinite acceleration of an instantaneous motion. The initial particle spacing allows for the flow front to have at least 15 particles in depth.

A full convergence study is made, including convergence in both d p the particle spacing distance and C h the smoothing coefficient applied in the simulation according to Table II.

The validation of the behaviour of the flow will be regarded as the accuracy to represent : the bulk flow behaviour as per (i) the velocity of the flow front, (ii) the free-surface elevation in the uniform zone and (iii) the free-surface shape, but also the internal kinetic of the flow through (iv) the velocity profiles at 3 different locations in the flow front (Fig. 1) and as provided by Freydier et al. [START_REF] Freydier | Experimental characterization of velocity fields within the front of viscoplastic surges down an incline[END_REF].

IV. RESULTS

A. Data processing

The numerical model is run for a sufficient period of time, ensuring steady state to be reached. For each error calculation, the results are averaged over a few seconds to obtain more representative data.

For all results, heights and lengths are normalized by h n , the theoretical flow height in the uniform zone so that h * = h/h n and x * = x/h n . This theoretical flow height is derived from Eq. 13b and is taken as h n = 18.6 mm. The velocities are normalized by the average velocities, i.e., by the velocity of the conveyor belt u b so that u * = u/u b . It is important to note that due to the numerical setup, the flow front is immobile in the eye of the observer. For more clarity, the reference frame is put back in the Eulerian framework so that u numerical = u sim + u b where u sim is the raw results from the simulation.

B. Macro-dynamics of the flow

The primary interest of the validation is the macro behaviour of the flow. Looking at the front velocity error gives us the performance of DualSPHysics on modelling such flows at the surge scale. The error (i) on the velocity of the flow front is characterized as:

ζ f = |u * f,numerical -u * b | u * b (14)
where u * f,numerical is the normalized velocity provided by the numerical model averaged over one second, taken as the derivative of the location of the toe of the front.

This error is expected to be null as the velocity u * b is set to the theoretical value of the flow front velocity.

In Fig. 2, ζ f is plotted for three different d p values and four different C h values. For the finest resolution, the error on the velocity of the front and thus location of this front becomes less than 1% for three smoothing coefficients which is acceptable for mud-flows behavior, considering the uncertainty on field and experimental measurements with which our future simulations will be compared. Overall, convergence In the uniform zone (upstream from the front), the height of the flow should reach the theoretical value, for which ∂ x z = 0. In this zone (taken as x * > 9), the error (ii) on the free surface elevation is:

ζ h = |h * u,numerical -h * n | h * n (15)
where h * u,numerical is the normalized height in the uniform zone. In Fig. 3, ζ h is plotted for numerical convergence. It is shown that the numerical solution converges and the error is diminished to only 5% of the theoretical value, which is largely acceptable for the applications this project is intended for.

Free surface profile is also of interest to be accurately reproduced for geophysical applications. Access to experimental accurate measurements of the free surface shape from the toe of the front to the uniform zone, as provided by Freydier in [START_REF] Freydier | Dynamique interne au front d'écoulements à surface libre[END_REF] is rare. To characterize globally the performance of DualSPHysics in reproducing the shape of the flow front, a root mean squared error on the free surface elevation (iii) is computed as follows:

ζ f s = x z max, Freydier (x) -z max, num (x) 2 N x ( 16
)
where N x is the number of points on the front shape measurements provided by Freydier [START_REF] Freydier | Dynamique interne au front d'écoulements à surface libre[END_REF].

In smoothing coefficient on the error value: this is a reminder that C h drives not only the numerical computation but the freesurface detection too. In applications where the free surface is very variable, and is the focus of the study, the value of the smoothing coefficient must be thoroughly investigated. In Fig. 5, the velocity profiles for C h = 1.8 are plotted for each location and resolution. The overall behaviour of the flow is well reproduced for all resolutions. Profiles are well within the error bars of the experimental data. Discrepancies on the height of the flow for x * f = 0.76 are due to the experimental measurements : close to the free-surface, access to the velocity of the flow, especially near the free surface, becomes difficult, as stated by Freydier in [START_REF] Freydier | Dynamique interne au front d'écoulements à surface libre[END_REF] making measurements in this section of the flow less accurate.

Even near the front, the shape of the velocity profile is accurately reproducing the dynamics, and allows to access to precise velocity mapping, which is rarely available experimentally, and not available using classical depth-averaged models.

A global criterion to characterize the convergence, for each of the location j is needed. A global root mean squared error of the velocity field (ii) is thus defined:

ζ u,j = z (u * j,Freydier (z) -u * j,num (z)) 2 N z,j (17)
where N z,j is the number of points in the vertical direction. In Fig. 6, ζ u,j for each (d p , C h ) is plotted. Convergence is shown for each smoothing coefficient. The convergence is similar regardless of the choice of a smoothing coefficient C h , and overall, the error becomes smaller than 5% when using the finest resolution which is acceptable considering the precision of the experimental measurements.

V. DISCUSSION

The accuracy of the results at different points in the flow show promising results in terms of possible applications, especially close to the toe of the front. Precise knowledge about the flow front characteristics is of major interest for applications on mud-flow behaviour as well as debris flow modeling.

A. Acceptability of the residual errors

The residual errors presented above could be further investigated using finer resolutions. Although this is of interest for theoretical convergence studies, this work is intended as an application project. The cost over benefit ratio of studying finer schemes becomes significantly less interesting in resolutions that are finer than ones presented in Table I. The knowledge on experimental data and data uncertainty for such mud-flows and debris flows in the field leads us to consider acceptable this error and consider it as an uncertainty of the numerical method in further applications. One can note that the particle distribution with this simulation forms layers near the front, distributing the particles along lines, eventually leading to instabilities in the pressure profile. In addition, there is a layer of particles at the bottom of the flow where a vertical ordering of the particles can be seen. This leads to the hypothesis that the formulation of shifting by Skillen at al. [START_REF] Skillen | Incompressible smoothed particle hydrodynamics (sph) with reduced temporal noise and generalised fickian smoothing applied to body-water slam and efficient wave-body interaction[END_REF] already implemented in the release version of DualSPHysics is not the optimal formulation for this application. Indeed, the non-uniform nature of this flow, coupled with the strong no-slip condition at the bottom of the channel leads to large differences in velocity magnitude As an alternative, the shifting formulation by Lind et al. [START_REF] Lind | Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves[END_REF] is used:

δr a = - A l 2 h 2 dt∇C a (18)
In this method, particles within C h • h of the detected free surface are not shifted. On Fig. 7(b), the particle distribution at the front for the shifting method of Lind et al [START_REF] Lind | Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves[END_REF] is shown. It is much smoother and less ordered near the free surface as expected. On Fig. 9 and 10, comparison of the different shifting methods for C h = 1.8 is plotted against the flow front velocity and the uniform flow depth, respectively. The shifting method by Lind et al. [START_REF] Lind | Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves[END_REF] performs better, impacting positively the bulk behavior of the flow. On the other hand, Fig 8 shows that the shifting method marginally affects the performance of the velocity profile error ζ u , while still ensuring satisfactory results.

In essence, this study demonstrates that this alternative method shows better results comparatively and these preliminary results show that a reflection on the shifting method is of major importance for such applications with very viscous, though surging flows, where velocities are null locally, and velocity gradient highly varies across the flow.

VI. CONCLUSION

In this work, we validated the results of DualSPHysics against accurate flow measurement of very viscous flows having a surging behaviour, i.e., steep free surface profiles. Overall, the global behavior of the front, as well as the dynamics of the flow, are well reproduced by the Dual-SPHysics solver. We additionally show that another shifting method than the classical one implemented in the release version of DualSPHysics is better suited, probably due to the peculiarities of these flows, notably their locally very low velocities. Applications of the DualSPHysics solver in mud-flows opens up a lot of possibilities for prototype-scale modeling to study e.g., impact forces and interaction with structures. The advantages of the DualSPHysics software, including its highly parallelized implementation, its growing community and the many coupling currently in place make it a good candidate for more application in the field of debris flow and mud-flow investigations.

VII. ACKNOWLEDGMENT

This project has received funding by the Labex TEC21 (Investissements d'Avenir program, grant agreement ANR-11-LABX-0030).

7 4 πh 2

 72 and α 3D = 21 16 πh 3 B. Time-stepping

Fig. 1 :

 1 Fig. 1: Flow front modelled in DualSPHysics, three different positions are studied precisely to be compared to the experimental data

Fig. 2 :

 2 Fig. 2: Error on the flow front velocity as defined in Eq. 14, d p in m

Fig. 4 ,Fig. 4 :

 44 Fig. 3: Error on the free surface elevation in the uniform zone as defined in Eq. 15, d p in m

Fig. 5 :

 5 Fig. 5: Velocity profile convergence for C h = 1.8 at three different locations in the front : x * f = 0.76 and x * f = 1.5 being close to the front tip, where ∂ x z is significant, and x * f = 9.7 in the uniform zone, d p in m

Fig. 6 :

 6 Fig. 6: Root mean squared error as defined in Eq. 17 on the velocity profiles at three different locations in the front: x * f = 0.76 and x * f = 1.5 being close to the front tip (i.e., where ∂ x z ̸ = 0), and x * f = 9.7 in the uniform zone (i.e., where ∂ x z ≈ 0), d p in m

Fig. 7 : 8 Fig. 8 :

 788 Fig. 7: Comparison of the particle spatial distribution for the two shifting methods for d p = 0.25 • 10 -3 m and C h = 1.8

Fig. 9 :Fig. 10 :

 910 Fig. 9: Comparison of ζ f for the shifting methods, d p in m

TABLE I :

 I Characteristics

	exp. ID	ρ	µ	θ	u b	Re
	150506	1383 5.6 15.3	75	0.34

of the experiment, density ρ [kg • m -3 ], viscosity µ [Pa • s], slope angle θ [°], conveyor belt velocity u b [mm • s -1 ], Reynolds number Re [-]

TABLE II :

 II Numerical parameters of the convergence study

	dp (m)	0.001 0.0005 0.00025
	C h (-)	1	1.3	1.5	1.8 2.2