Helene Miche
email: michemiche@cerege.fr

Ginette Saracco
email: saracco@cerege.fr

Adriano Mayer
email: adriano.mayer@univ-avignon.fr

Khaoula Qarqori
email: khqarqori@gmail.com

Mohamed Rouai
email: mohamed.rouai@laposte.net

Abdelilah Dekayir
email: dekayir@yahoo.fr

Konstantinos Chalikakis
email: konstantinos.chalikakis@univ-avignon.fr

Christophe Emblanch
email: christophe.emblanch@univ-avignon.fr

Hélène Miche

Hydrochemical constraints between the karst Tabular Middle Atlas Causses and the Sais basin (Morocco): implications of groundwater circulation

Keywords: Morocco, Hydrochemistry, Isotopic analyses, Conceptual models, Groundwater flow

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

Introduction

The study area is located at the junction of the Tabular Middle Atlas (TMA) mountains and the plain of Saïs. The plain of Saïs is bordered in the south by the Tabular Middle Atlas (TMA), in the north by the pre-Rif ridges, in the west by the Rharb plain, and in the east by the Fez-Taza corridor (Fig. 1). The plain of Saïs is very important for agriculture in Morocco and has a population of 1.6 million inhabitants, with two important cities, Fez to the east and Meknes to the west.

Population growth in the region, the intensive farming and industrial development, and the recurrence of drought [START_REF] Amraoui | Behaviour of a karstic spring subjected to a long drought period: Bittit spring (Morocco)[END_REF];B z i o u i2004; Amraoui et al. 2005;[START_REF] Luterbacher | Mediterranean climate variability over the last centuries: a review[END_REF][START_REF] Esper | Long-term drought severity variations in Morocco[END_REF][START_REF] Etebaai | Climatic and human impacts on the hydrogeochemical functioning of Lake Ifrah (Middle Atlas, Morocco)[END_REF] impact and affect groundwater management and the capacity to ensure a sufficient supply, in terms of both quantity and quality, for surrounding towns and villages over time. In this context, knowledge of water circulation pathways and of the seasonal variations of groundwater resources becomes a key issue for resource preservation and the mitigation of possible desertification or soil degradation.

The TMA is mainly constituted by Liassic limestonedolomite formations (Fig. 2), which have an important hydrological potential [START_REF] Kabbaj | Le domaine atlasique: présentation du domaine atlasique [The Atlas area: introduction to the Atlas area[END_REF][START_REF] Bentayeb | Le Causse moyen-atlasique [The Middle-Atlas Causse[END_REF]. These Liassic carbonates dip northward at the junction of TMA with the Saïs basin, beneath a thick Miocene marls deposit, and constitute a confined aquifer (Fig. 3). At the TMA/Saïs basin junction, several outlets as karst springs drain a part of the groundwater flow. "Ribaa-Bittit" complex is the most important one.

In the Saïs basin, the Liassic and Miocene formations overlay the Triassic clays, basalts and evaporites, and are overlain by Pliocene-Quaternary limestone and travertine [START_REF] Chamayou | Ressources en eau du Maroc, Plaines et bassins du Maroc atlantique, tome 2. Notes et Mémoires du Service Géologique, n°231 [Morocco water resources, plains and basins of Atlantic Morocco[END_REF];F i g s .2 and 3). These carbonates are affected by several fractures causing a dislocation of different Liassic blocs and sometimes Pliocene-Quaternary limestone-travertine [START_REF] Bentayeb | Le Causse moyen-atlasique [The Middle-Atlas Causse[END_REF]Martin1981). The groundwater flows in this region are conditioned by two major fracture networks, mainly NE-SW and NW-SE directions [START_REF] Bentayeb | Le Causse moyen-atlasique [The Middle-Atlas Causse[END_REF][START_REF] Essahlaoui | Preliminary survey of the structure and hydrogeology of the western Saïs basin, Morocco, using electrical resistivity[END_REF]Amraoui 2005), and probably with a NW-SE preferential flow direction [START_REF] Kh | Geoelectrical phy investigating and of fractures network around Bittit spring (Middle Atlas, Morocco)[END_REF]. A karstic complex water circulation is especially developed locally at the border of the two hydrogeological units [START_REF] Qarqori | Contribution à l'étude du réservoir discontinu et karstique des causses Moyen-Atlasiques et de sa jonction avec le bassin de Saïs par télédétection spatiale et imagerie géophysique[END_REF].

The deep and shallow aquifers of Saïs basin, supplied by rainfall and TMA water, and the springs located on the TMA/ Saïs basin contact boundary represent the most important water supply of the region for drinking and agricultural uses [START_REF] Bahaj | Aspects of groundwater geochemistry from Middle Atlas and Saiss basin (Northern Morocco)[END_REF][START_REF] Belkhiri | Gestion integrée des ressources en eau, protection de la ressource: bassin du Sebou [Integrated water resources management, resource protection: Sebou Basin][END_REF][START_REF] Benaabidate | Controls on ground water chemistry in the central couloir Sud Rifain, Morocco[END_REF]. However, the modalities of water transfer between the Middle Atlas Causses and Saïs basin and the interactions of water with the different geological facies are not well known.

The purpose of the study was to understand the groundwater flows and their renewal in this densely populated region suffering water resources overexploitation, and this paper explains the methods and outcomes. Constraints on the groundwater flow paths have been obtained by coupling hydrochemical and isotopic characterization of the water samples from springs and wells.

Geochemical and isotopic results (major and minor elements, δ 18 O, δDa n d 222 Rn) are presented for the main ten springs and three wells (from different geological origins) studied between 2009 and 2011 around the Ribaa-Bittit springs complex and along the transition zone (study area of 70 × 40 km 2 ) between the Saïs basin and the TMA reservoir (Figs. 1 and2). The ten springs studied included Bittit; Ribaa; Aguemguam; Sbaa; El Mir; Boucharmou; Bou Youssef; Fig. 2 Geological map of the study area, with springs and wells studied at the foot of the L'Hajeb Causse (East) and Guigo Causse (West; adapted from [START_REF] Qarqori | Contribution à l'étude du réservoir discontinu et karstique des causses Moyen-Atlasiques et de sa jonction avec le bassin de Saïs par télédétection spatiale et imagerie géophysique[END_REF]. Lias limestones behave as aquifers. Triassic clays behave as aquitards. Springs: 1, Bittit; 2, Aguemguam; 3, Ribaa; 4, Sbaa; 5, El Mir; 6, Boucharmou; 7, Bou Youssef; 8, Khadem; 9, Maarouf 1; 10, Maarouf 2. Wells: A, Haj Kaddour (deep well in Saïs Plain); B, Palaeozoic schist well; C, Basaltic schist well Fig. 3 Geological section (XY, Fig. 2) of the south rifain groove according to the geological map of El Hajeb 1:100,000 [START_REF] Chamayou | Ressources en eau du Maroc, Plaines et bassins du Maroc atlantique, tome 2. Notes et Mémoires du Service Géologique, n°231 [Morocco water resources, plains and basins of Atlantic Morocco[END_REF] Khadem; Maarouf 1; Maarouf 2, while the three water wells studied were Haj Kaddour, a deep well (700 m) drawing from a Lias carbonate aquifer in Saïs Plain; a shallow well drawing from Palaeozoic schist (Palaeozoic schist well); and a shallow well in the Basalt-schist environment (Basaltic schist well). After a recall of climatic and hydrogeological context, the results of hydrochemistry analyses (major elements and isotopes) are presented. The interpretation is followed by the building of a schematic model of flow paths, and then by conclusions and perspectives.

Climatic and hydrogeological context

Climatic context

The junction zone of TMA and the study area is characterized by a temperate climate. The TMA is characterized by low temperature compared to the surrounding regions and has a rainfall average of about 730 mm/year [START_REF] Bentayeb | Le Causse moyen-atlasique [The Middle-Atlas Causse[END_REF]. In winter, the rainfall is enhanced by a frequent west wind with high moisture, whereas at the east of the junction, the climate is dominated by hot and dry winds, the so-called "Sirocco" wind, which can produce more dry summers in TMA.

The Saïs basin is generally characterized by mild winters with average rainfall of about 500-600 mm/year [START_REF] Chamayou | Ressources en eau du Maroc, Plaines et bassins du Maroc atlantique, tome 2. Notes et Mémoires du Service Géologique, n°231 [Morocco water resources, plains and basins of Atlantic Morocco[END_REF]; Agence du bassin hydraulique de Sebou-Fes, 2008[START_REF] Savoy | Radon and CO 2 as natural tracers to investigate the recharge dynamics of karst aquifers[END_REF]. This climate regime is also characterized by relatively constant temperature across the entire Saïs basin, which becomes slightly higher in the east.

During the past 50 years, the frequency of drought periods has increased [START_REF] Esper | Long-term drought severity variations in Morocco[END_REF][START_REF] Damnati | Recent environmental changes and human impact since mid-20th century in Mediterranean lakes: Ifrah, Iffer and Afourgagh, Middle Atlas Morocco[END_REF]. In particular, the climate type has become humid to sub-humid in the Ifrane catchment in TMA-altitude 1,635 m above sea level (asl). Using calculations from about 30 years of meteorological data, the authors of this study observed a drop of the annual rainfall by 17% and an increase of the temperature of the same order in comparison with the period 1933-1963 described in [START_REF] Bentayeb | Le Causse moyen-atlasique [The Middle-Atlas Causse[END_REF].

Hydrogeological context

The Liassic carbonates outcropping in the TMA are sometimes intercalated with clay layers, thereby forming several local aquifers within the same hydrosystem. At the junction zone, in the Saïs basin, these carbonates are confined by thick Miocene marls, forming an aquifer which is intensively exploited for drinking water and for irrigation. The average water flux of all TMA aquifers is estimated at about 32-35 m 3 /s, with 10 m 3 /s transiting to the subterranean Saïs basin [START_REF] Bentayeb | Le Causse moyen-atlasique [The Middle-Atlas Causse[END_REF]. The flow rate of Bittit spring is from 1.3 to 1.5 m 3 /s. The flow of Ribaa spring is from 0.02 to 0.32 m 3 /s and that of Aguemguam spring is typical of a karst system, from 0 after a dry period to 0.53 m 3 /s (Agence du bassin hydraulique de Sebou-Fes 2008[START_REF] Savoy | Radon and CO 2 as natural tracers to investigate the recharge dynamics of karst aquifers[END_REF]. During this study (2009)(2010)(2011) there was an excess of rainwater compared to the normal value of 550 mm in Saïs Plain; the excess was 5% at the end of the study and 47% at the beginning in 2009. Despitethis,thewatertableatFez-Meknesexperiencedapiezometric level drop of about 5 m/year, showing a high stress of the system.

The granular porosity of Liassic limestone is low; therefore, the permeability of the matrix is mostly controlled by fractures and karstification. In karst areas it is assumed that about 30% of rainwater recharges groundwater [START_REF] Drogue | Contribution à la connaissance hydrogéologique de quelques karsts du midi-méditerranéen Français [Contribution to the hydrogeological knowledge of some karsts in the French Mediterranean[END_REF][START_REF] Jong | A transdisciplinary analysis of water problems in the mountainous karst areas of Morocco[END_REF]).

In the junction zone, several important springs of high water quality occur in the northernmost outcrops of Liassic limestone, which is overlain in some areas by Quaternary travertines. Two of these springs in particular, Bittit and Ribaa, almost exclusively provide the drinking water supply for Meknes and the surrounding villages (1.6 million inhabitants), as well as for agriculture. These springs experienced a significant drop in flow rate in the last decades. Although this water originates certainly in the Liassic karst formation, the drop in water table raises several questions regarding the mechanisms of water transport (influence of fractured and karst systems in particular) and the participation of other unknown groundwater reservoirs in the system, which could potentially cause deterioration of groundwater quality. Since 1994Since -1995, , the karst water has suffered from occasional turbidity problems [START_REF] Amraoui | Turbidity dynamics in karstic systems: example of Ribaa and Bittit springs in the middle atlas (Morocco)[END_REF].

Hydrochemistry: methods, results and discussion

Materials and methods

Water samples were taken from springs along the junction between Saïs plain and TMA along the NE-SW section, for both hydrochemical and isotopic analyses. Well-water samples characterize geological structures in the TMA and Saïs plain. Temperature, pH and electrical conductivity were measured in the field. Water samples were collected from springs by rinsing the bottle at least three times and filling it completely before capping, while water samples from wells were collected with a pump after a minimum of several hours pumping. The geochemical characterization of major (Ca 2+ ,M g 2+ ,N a + ,K + , Si 4+ ,Cl -,SO 4 2-,NO 3 -,HCO 3 -)andminor(Sr 2+ ,Ba 2+ )solutes was done at the Chemical Services of CEREGE by inductively coupled plasma emission spectrometry (Horiba Jobin-Yvon ICP-OES) for major and minor cations, and by capillary electrophoresis (Waters) for anions, from samples collected at the end of the dry and wet season (Table 1;Figs.4,5 and 6).

Saturation Indexes for phases in solution were determined by PHREEQC version 2.7 [START_REF] Parkhurst | User's guide to PHREEQC (version 2): a computer program for speciation, batch-reaction, onedimensional transport, and inverse geochemical calculations[END_REF]with database wateq.dat-Table S1 in the electronic supplementary 2;Fig. 7).

Radon activities in groundwater were measured via 218 Po counting using a RAD7 alpha spectrometer. Water samples had been collected using an immersion pump providing a constant water flow rate of 3.5 L/min into 250-ml glass bottles. In order to minimize radon degassing, the volumes of the bottles were displaced twice during continuous flow (Fig. 8).

To specify the types of waters and the mixing zones, multivariate statistical analysi so ft h ed a t aw a sp e r f o r m e d (Tables 3 and4;F i g .9) with an Excel application [START_REF] Georgin | Analyse interactive des données (ACP, AFC) avec Excel[END_REF]. Principal component analysis (PCA) was carried out on 25 individual samples (24 samples from springs and 1 sample from well A) for 11 variables (geochemical parameters and radon) listed in Tables 1 and2. Wells B and C, which were polluted by human activity, were not taken into account.

Major element geochemistry

The analyses allow characterization of the water-rock interactions in the aquifers (Figs. 4, 5 and 6;T a b l e1). Figures 4,5, 6 and 8 were developed using "Diagrammes" software V. 5.9 [START_REF] Simler | Software "Diagrammes" V. 5.9[END_REF]. All collected spring waters are calcium and magnesium bicarbonate waters. Speciation of these waters, calculated using PHREEQC version 2.7, indicates saturation conditions (see Table S1 of the ESM) with respect to calcite and dolomite for all springs and wells. A clear distinction among three groundwater types can be based on Ca 2+ ,Mg 2+ ,Na + ,K + ,Cl -and SO 4 2 -. Type I comprises calcium-magnesium bicarbonate waters collected in the southern sectors: Bittit (site 1), Ribaa (site 2), Aguemguam (site 3), Boucharmou (site 6), Bou Youssef (site 7), and Khadem (site 8). These waters are characterized by variable Mg/Ca ratios, indicating dissolution of limestone and dolomite in the aquifer to various extents (Fig. 6). Some of them, notably Boucharmou, have a Mg/Ca ratio of nearly one, indicative of pure dolomite dissolution. Type II waters-Sbaa (site 4), El Mir (site 5), Maarouf 1 (site 9), Maarouf 2 (site 10), and the deep well (A) Haj Kaddour 2360/15 in Saïs Plain-includes waters characterized by a clear enrichment in Na + and Cl -,with a slightly lower Mg/Ca, compared to type I.

Dolomite-log K s = -16.54 at 25 °C (K s :so lubi lityproduct)-is less soluble than calcite (log K s = -8.48 at 25 °C) and can hardly reprecipitate [START_REF] Michalowski | Thermodynamic modelling of dolomite behavior in aqueous media[END_REF]; thus, the Mg/Ca values between 1 and 1.1 in the western sector in type I springs seem to be a consequence of a more Springs and wells are referenced as in Fig. 2. Note: This figure was derived using "Diagrammes" software V. 5.9 [START_REF] Simler | Software "Diagrammes" V. 5.9[END_REF] in the Tertiary basalts (Basaltic schist well). Both of them are characterized by a high level of sulfate and low Mg/Ca ratio compared to type II. One can distinguish well B from well C by Na and Cl values that are high for B and lower for C (Table 1;Fig. 6). These two water samples are however clearly polluted by nitrate (probably recent anthropogenic pollution); hence, their chemistry cannot be taken as representative of an equilibrium condition for the Palaeozoic schist or the Tertiary basalts. Saturation conditions for anhydrite, gypsum and barite are not reached in any of these waters for the data set. a seasonal effect for springs is observed between the end of the dry season in autumn (solid circles in Fig. 7) and the end of the wet season in March (open circles in Fig. 7).

The deuterium excess is defined by d excess = δD -8×δ 18 O [START_REF] Dansgaard | Stable isotopes in precipitation[END_REF][START_REF] Craig | Deuterium and oxygen-18 variations in the ocean and the marine atmosphere[END_REF]. A higher deuterium excess (see Table 2) characterizes all spring waters with respect to the local meteoric water line (d excess < 8). This is particularly evident for samples collected after the rainy season-March/April 2010; Fig. 7 (open circles), see values in Table 2. This shift indicates a larger kinetic effect in the vapor source region for the winter recharge. Spring waters of type I have slightly lighter isotopic compositions than waters of type II. This difference most likely reflects a higher altitude of the recharge area for spring waters of type I, compared to spring waters of type II.

Moreover, it was observed that springs of water type I in the northeastern part (complex Ribaa-Bittit) have some isotopic compositions lower than springs of the same type in the southwestern part, which corresponds to higher recharge altitudes for the northeast. These altitudes [START_REF] Sefrioui | Utilisation des outils isotopiques pour la détermination de l'altitude de recharge des principales sources du bassin de Sebou-Maroc [Use of isotopic tools to determine the altitude of recharge of the main springs in the Sebou-Morocco basin[END_REF] correspond to values of vertical gradient for δ 18 Oof-0.27 ‰ per 100 m of rise and they are estimated for the studied area (in m asl) following the formula Z recharge = -(δ 18 O + 3.7)/0.27 [START_REF] El Ouali | Contribution of isotopes of the water molecule to determine recharge altitude of the Main Springs welling up in the Middle Atlas limestone (Morocco)[END_REF]; Table 2).

The Liassic aquifer has recharge areas located at ~1,400 m asl in the eastern sector (L'Hajeb Causse) and at ~1,000 m asl in the western sector (Guigo Causse). Waters in contact with the Triassic aquitard (El Mir spring) in the eastern sector are recharged around 1,000 m asl and the two mixed zones are highlighted at the ends of the basin: one corresponds to Sbaa spring with an intermediate recharge area (~1,200 m asl); the second to Maarouf spring with a more local recharge area (900 m asl; see Table 2), showing a net distinction between the eastern and western sectors. There is evidence of a drain (or preferential flow) of waters perpendicular to the Causses in the Bittit-Ribaa area, which could be confirmed with geophysical investigations.

Transit times from radon activity and tritium constraints

Radon activity and transit time

Radon measurements (Table 2;F i g .8)h a v eb e e nc a l i b r a t e d from B, the Palaeozoic schist well, in contact with the Triassic aquitard, to obtain the high limit of electrical conductivity. Type I waters in the Liassic aquifer yield a mean radon activity of about 3,500 Bq/m 3 , which is probably resulting from secular equilibrium with the aquifer carbonates. This level of activity is typical for fresh groundwater in Mesozoic carbonate formations [START_REF] Savoy | Radon and CO 2 as natural tracers to investigate the recharge dynamics of karst aquifers[END_REF]. The steady-state radon is achieved when the rate of radon disintegration matches the rate of radon generation supported by the decay of the parent isotope, 226Ra, present in the aquifer material [START_REF] Hoehn | Radon in groundwater: a tool to assess infiltration from surface waters to aquifers[END_REF]. Radon sources in karstic waters are the radium in limestone (due to trace concentration of uranium in calcite) and deposits of "terra rossa", which contain clays minerals, Al-Fe-Mn hydroxides and organic matter that may accumulate radium [START_REF] Tadolini | Relation between "terra rossa" from the Apulia aquifer of Italy and the radon content of groundwater: experimental results and their applicability to radon occurrence in the aquifer[END_REF]. Once the radon steady state is reached (around three half-lives, i.e. 12 days) no information about transit time is obtainable from radon. The transit time of these waters from the recharge zone to the spring is thus more than 12 days. The differences in the altitude of springs do not lead to any significant radon variations, so it is not possible to highlight shorter transit times for the higher altitude springs based on radon activity. However, two kinds of aquifers for this group are noted, as previously seen in isotopic analyses (δ 18 O, δD): one in the eastern sector correlated to the seasonal variations (around 3,200 ± 450 Bq/ m 3 ), the other in the western sector that seems constant around 5,000 Bq/m 3 . Type II waters are characterized by significantly higher 222 Rn activity, reaching a mean value of 15,000 with a seasonality of ±1,000 Bq/m 3 . To explain this higher radon activity compared to type I, a possibility is that these groundwaters are in equilibrium with an aquifer containing higher radium concentrations. The radium concentration in the Liassic aquifer [START_REF] Savoy | Radon and CO 2 as natural tracers to investigate the recharge dynamics of karst aquifers[END_REF]) is constantly independent of the two groups of water. Besides, there is no reason to think that the higher activity results from lesser degassing compared to type I. Therefore, it is assumed that the higher radon activity of waters of type II results from interactions with the Triassic evaporites at the base of the Lias. Moreover, this interpretation is supported by the enrichment in sulfate and chloride, which indicates an interaction with gypsum and halite, typically found in Triassic evaporites of the area. The enrichment in radon activity is specific for groundwater having a higher content in sulfate, because radium typically co-precipitates in the aquifer together with barite and other insoluble sulfates [START_REF] Rosenberg | Radium removal in a large scale evaporitic system[END_REF], which constitute a permanent radon source in the aquifer. Since Triassic formations in the area behave as aquitards (they essentially consist of lowpermeability evaporites and schist with few carbonates), one cannot consider a true Triassic evaporite aquifer, but rather an interaction-zone between carbonates and evaporites, located near the base of Liassic dolomitic limestone.

The two samples of the El Mir spring showing the larger enrichments in Na and Cl indicate that these waters interact the most with the evaporites of the Permo-Triassic sequence; furthermore these springs have elevated radon activity and a similar chloride to the water directly extracted from the Palaeozoic schist well (Figs. 6a-c and Fig. 8; Table 2). One may assume that the radon secular equilibrium activity in the interaction zone with the Permo-Triassic basement is equal or higher than 16,000 Bq/m 3 .

Three springs (Figs. 4, 5, 6, 7 and 8) represented by green (No. 4), orange (No. 9) and brown (No. 10) symbols have geochemical compositions (stable isotopes, major elements) intermediate between type I and type II waters. These spring waters result from a mixing zone between the two groups. Their radon activity is however higher than the expected activity from simple binary mixing (see Fig. 8), indicating that the mixing occurs while waters interact with the Triassic evaporites. The opposite possibility (i.e. mixing is in the Liassic carbonates) can be dismissed since in that case the radon activity would be lower than the values shown by the mixing zone. The excess of radon, with respect to the expected activity of the mixing, depends on the in-growth time of radon in the interaction zone. However, this radon excess declines during the successive transfers to the surface due to the radioactive decay. These waters would ultimately reach the radon secular equilibrium activity of the Liassic limestone (3,500 Bq/m 3 )onatimescaleof12days.

In other words, type II waters have the highest measured radon activity and major-element geochemistry most similar to the groundwater extracted from the Triassic schist (well B). If it is assumed that 16,000 Bq/m 3 is the secular-equilibrium activity in the mixing zone (Fig. one may assume, for the three intermediate water samples (green No. 4, orange No. 9 and brown No. 10 symbols), the upwelling time since these waters have left the contact with the Triassic evaporites to enter the Liassic aquifer and reach the surface is thus less than 12 days, on probably on the order of a few days. Remote sensing images of this area point out that these mixed springs or zones lie on important tectonic alignments [START_REF] Qarqori | Contribution à l'étude du réservoir discontinu et karstique des causses Moyen-Atlasiques et de sa jonction avec le bassin de Saïs par télédétection spatiale et imagerie géophysique[END_REF][START_REF] Dauteuil | Structural pattern of the Saïss basin and Tabular Middle Atlas in northern Morocco: hydrological implications[END_REF], which suggests that a fracture system could play a role in the fast upwelling of groundwater.

Tritium constraints on the recharge areas

It was noted that two types of recent sources of tritium exist (half-life = 12.32 years, from Lucas and Unterweger 2000; 3 H ~5.0 ± 0.3 TU from IAEA 2010) with a complete recharge in less than 5 years. The groundwater residence time verifies a "piston model" [START_REF] Hubert | Modélisation par une loi gamma de la distribution des temps de séjour de l'eau dans des systèmes hydrogéologiques en régime permanent [Modeling by a gamma law of the distribution of water transit times in steady-state hydrogeological systems[END_REF]: effective rainfall runs through the karst at a relatively constant velocity, and because the residence time is less than 5 years, the waters have an average age of less than three years in the aquifer. The tritium activity in the eastern sector is slightly weaker than the activity in the western sector; this seems to indicate a quicker recharge for the eastern than for the western sector. These results are in good accordance with the different recharge altitudes obtained with isotopic analyses (δ 18 O, δD). In the eastern sector (L'Hajeb Causse), the recharge areas have higher altitudes than in the west (Guigo Causse).

Principal component analysis (PCA) of data and implication for the mixing zones

The correlation matrix (Table 3)s h o w sas t r o n gc o r r e l a t i o n (>0.9) between Na + ,C l -and electrical conductivity (EC), which confirms the dissolution of Triassic halite and a good correlation (0.7; 0.9) for nitrate and sulfate coming from shales of the Triassic aquitard. Moderate correlation (~0.6) between Mg 2+ and SO 4 2-and between Mg 2+ and NO 3 -shows an interaction between the Liassic aquifer mainly composed of dolomitic carbonates and the shale-rich Triassic aquitard. Calcium and silicon are poorly correlated (~0.5), which is consistent with low clay content in the Liassic carbonate aquifer (eastern sector around Bittit). Calcium and sulfate are inversely correlated, which indicates that groundwater chemistry is dominated by dissolution of Liassic carbonates in some areas and of gypsum (from the Triassic aquitard) in others. Si 4+ and Mg 2+ as well as Si 4+ and SO 4 2-are anti-correlated. This seems to indicate that the aquifer rich in magnesium contains very low silicon and probably very low clays. Moreover, the presence of silicon is the mark of a bicarbonate aquifer poor in sulfates. This confirms the existence of two types of aquifers more or less rich in Ca 2+ and/or Mg 2+ .

From graphical representations of the principal component analysis for individuals and variables (Fig. 9a-d), it is possible to study the factorial planes formed by axes 1, 2 and 3 which represent a global inertia of 83.4%. These planes are associated with eigenvalues greater than 1 (Table 4).

In the main plane (Fig. 9a), the individuals form five distinct clouds, which means that sub-populations of water have been highlighted. These subpopulations correspond to (1) Maarouf 1 and 2 (9.1-10.2) and Khadem (8); (2) Boucharmou and Bou Youssef (6.1-7.2); (3) Sbaa (4.1-4.3); (4) Bittit, Ribaa, Aguemguam (1.1-3.3); and (5) El Mir (5.1-5.3) and the well A.

The results (Figs. 9a-d) allow the types of water to be specified: for the L'Hajeb Causse in the east, waters of the main Liassic aquifer (Bittit, Ribaa, Aguemguam) are calcium and magnesium bicarbonate with possible dissolved clays. They characterize type I waters. The clay content in waters is low but it explains problems of turbidity after heavy rains. The water represented by El Mir shows high sodium and chloride at a similar concentration to each over and a high radon content characteristic of Triassic aquitards, which characterize type II. Sbaa is just at the origin of axes 1 and 2, i.e. presents no main component. It represents a typical mixing zone between type I and type II waters of the L'Hajeb Causse.

For the Guigo Causse in the west, waters in the Liassic aquifer (Boucharmou, Bou Youssef) are marked by higher magnesium content than those of the L'Hajeb Causse. This is in agreement with a longer transit time. However,thesetypeIwatersareweakly marked by nitrate and sulfate, indicating that the Triassic interacts with the Liassic aquifer. This is in accordance with a discontinuous and more faulted structure of the Guigo Causse. Type II waters (Maarouf 1 and Maarouf 2) are marked by their magnesium, nitrate and sulfate content, which confirm a longer transit time in the Triassic aquitard and dissolution of sulfates. On the basis of their chloride, sodium and radon content, Maarouf 1 and 2 represent a mixing zone between the Triassic aquitard and the Liassic aquifer of the Guigo Causse. This mixing zone has a different composition from that of the mixing zone formed at Sbaa in the L'Hajeb Causse, due to a longer transit time.

Khadem seems to be closer to a mixing zone than type I water: the main PCA variables (magnesium, sulfate and nitrate) and the radon content are similar to those of Maarouf while its content in sodium and chloride is lower. The deep well A (Saïs basin), in contact with the Triassic, has an intermediate character between El Mir and, to a lesser extent, Maarouf. This indicates that the groundwater flows mainly take place from the southeast to the northwest but also from the southwest to the northwest.

Interpretation and schematic model of flow paths

Water-rock interactions and mixing zones A distinction between two spring water types can be made based on chemical analyses and principal component analysis (Figs. 4 and9). The first water type arises quasi-exclusively from water-rock interactions within the Liassic limestonedolomite aquifer and Quaternary travertines. Waters of this group are located in the alignment of Bittit, Ribaa, Aguemguam, Boucharmou and Bou Youssef springs. In contrast, the 2nd water type arises from water-rock interactions with Triassic evaporites. The type II springs are: Maarouf 1 and Maarouf 2, Sbaa, and El Mir. Khadem is distinctive and seems to be intermediate between type I and II waters. Lower Triassic rocks are not exposed in the study area. Water-rock interactions in the Triassic strata take place at a depth that can be obtained from estimates of spring travel time using radon isotopes analyses [START_REF] Mayer | A multi-tracer study of groundwater origin and transit-time in the aquifers of the Venice region (Italy)[END_REF].

The higher ratio (≤1.1) of Mg/Ca (Fig. 6) observed in some springs of type I in the western sector is a probable consequence of a more advanced dissolution of dolomite followed by precipitation of calcite, in an aquifer with waters having longer transit times. Another possible mechanism explaining the higher Mg/Ca ratio is the hydrolysis of Mg-rich silicates in Quaternary and Triassic basalts. From the results of principal component analysis (Table 3;F i g .9) it is shown that magnesium is anti-correlated with silicon; thus, this second mechanism must be eliminated, which is in accordance with a longer transit time in the western sector.

In contrast, the second water type represented by Maarouf, Sbaa, and El Mir (Figs. 4, 5 and 6), shows a net enrichment in Na + ,K + and Cl -and suggests an interaction with Triassic evaporite formations. Since Triassic rocks do not outcrop in this area, the interaction with Triassic evaporites probably takes place at depth, at the base of the Liassic dolomitic limestone. Groundwater extracted from well A in the Saïs Plain also shows evidence of interactions with Triassic evaporites.

Bicarbonate, nitrate and sulfate concentrations (Table 1;F i g s .4, 5 and 6c) are linked to seasonal variations. All springs have significantly lower concentrations of bicarbonate in March after the rainy season, which is consistent with a higher proportion of groundwater in contact with gypsum and schist near the Triassic aquitard. The increase of nitrate and sulfate after rainy season is particularly true for El Mir spring. Therefore, one can define a Liassic aquifer characterized by waters of type I previously described, and a Triassic aquitard (waters of type II), with two main transition or mixing zones: Sbaa, to the north, and, Maarouf, to the south (Figs. 4 and5). Rn allows us to highlight areas of rapid exchanges between waters flowing into the Liassic aquifer and the Triassic aquitard. The groundwaters in the main mixing areas at extremities (Sbaa, Maarouf) and in the small mixing area of Khadem have higher radon activities (5,500-10,000 Bq/m 3 ) than if mixing were simply occurring between waters from the Liassic aquifer and the Triassic aquitard. The upwelling time from Triassic aquitard to surface is less than 2 weeks.

Recharge regimes, groundwater reservoir and seasonality

As previously seen, there are recharge areas at five different altitudes. First, the Liassic aquifer contains water (type I) that is regularly renewed from the highest recharge area (1,400 m asl). This recharged water corresponds to Bittit, Ribaa and Aguemguam springs in the L'Hajeb Causse. Isotopic analyses (Fig. 7; Table 2) of these spring sources show they follow the local meteoric water line and suggest a confined aquifer with a rapid and local infiltration (i.e. low evaporation), which explains the observed turbidity problem described after rain events [START_REF] Amraoui | Turbidity dynamics in karstic systems: example of Ribaa and Bittit springs in the middle atlas (Morocco)[END_REF]. Secondly, groundwater in contact with the Triassic aquitard (El Mir) presents a recharge zone at lower elevation (1,000 m asl) and a stronger evaporation than the type I waters in the eastern sector. Third, the two mixing zones at the edges of the basin, corresponding to Sbaa and Maarouf springs, have respectively intermediate recharge altitudes of 1,200 and 900 m asl. Fourth, a fifth recharge zone at 1,040 m asl corresponds to another causse (Guigo Causse) and supplies Boucharmou, Bou Youssef and Khadem; however, isotopic analyses show that these waters are subparallel to the local meteoric line, which indicates high evaporation.

Seasonality is well marked by the change of sulfate (Fig. 6c) and water isotopes (δ 18 O, δD; Fig. 7). The water renewal is still sufficient, as shown by tritium analyses, which characterizes modern waters. However, periods of rain are more intense on shorter time scales [START_REF] Amraoui | Turbidity dynamics in karstic systems: example of Ribaa and Bittit springs in the middle atlas (Morocco)[END_REF][START_REF] Etebaai | Climatic and human impacts on the hydrogeochemical functioning of Lake Ifrah (Middle Atlas, Morocco)[END_REF], while the droughts are longer; thus, the infiltration in the causses is then less effective. In addition, the development of human activity

Conceptual model of flow paths

The set of geochemical and isotopic data and the multivariate analyses of data allow for the establishment of a first model (Fig. 10) of groundwater flow for the L'Hajeb Causse (east) and Guigo Causse (west). The model takes into account five highlighted recharge areas, groundwater flow paths to the and the mixing zones. It important to note that the causses of this study area were formed in the same period and have a similar geological composition (dolomite, calcite). From a geochemical point of view, dolomite that can hardly reprecipitate [START_REF] Michalowski | Thermodynamic modelling of dolomite behavior in aqueous media[END_REF] shows a difference between the region east and west. Reprecipitation of calcite is favored by longer transit times in the western part of the study area, leading to a higher Mg/Ca ratio in the Liassic aquifer of the Guigo Causse, as confirmed by the geochemical analyses of this study and tritium values from the International Atomic Energy Agency (type I waters).

Radon analyses revealed the existence of rapid exchange areas (less than 12 days) in the mixing zones between the Triassic aquitard and the Liassic aquifer. El Mir spring in L'Hajeb Causse would be typical of the Triassic aquitard of both causses. Indeed, the water of well B in the Triassic aquitard and in contact with the Palaeozoic schists of Guigo Causse, has the same electrical conductivity as type II waters, i.e. the same content in dissolved salt. The Triassic aquitard, probably very discontinuous around El Mir, allows an easy water circulation, as indicated by the radon activities of the mixing zones (Sbaa, Maarouf 1 and 2, and probably Khadem). Because these activities are higher than if it was a simple mixing of waters circulating in Lias and Triassic, one can say that the mixing takes place in the Triassic aquitard.

In L'Hajeb Causse, Sbaa water is in contact with the Liassic confined aquifer and circulates in one mixing zone with the Triassic. In Guigo Causse, which is less high and more rugged, the springs Maarouf 1 and 2 are in contact with the Liassic unconfined aquifer, which is more prone to evaporation, and water probably circulates in multiple mixing zones with the Triassic, as shown by elevated values of electrical conductivity. Khadem spring in Guigo Causse seems to be associated with another mixing zone with a more Liassic character than Maarouf.

The water of both aquifers at the junction between the TMA and the Saïs plain would be in contact between Ribaa and Bou Youssef, probably close to Oued Tizguit, and would circulate slowly from northeast to southwest, and also to a lesser extent from northwest to southwest perpendicular to the causses.

Conclusions and perspective

The geochemical and isotopic study associating a multivariate statistical analysis of the data and a thermodynamic balance of the phases in solution, allowed for characterization of the s t u d i e ds y s t e mo f1 0s p r i n g sa n d3w e l l s( o v e ra na r e a 70 × 40 km) at the junction of Saïs Basin and the Middle Atlas Causses. The study showed the existence of several aquifers of Liassic origin and a few reservoirs in the Triassic aquitard, and pointed toward two different karstic hydrological systems associated with L'Hajeb and Guigo Causses. Two main mixing zones at the edges of the system are highlighted: Sbaa to the east, and Maarouf 1 and 2 to the west. Khadem to the west seems to be also a small mixing zone.

The study highlighted the existence of areas of rapid exchange between waters of Liassic aquifers and waters in contact with the Triassic aquitard. The upwelling time from the Triassic aquifer is less than 12 days (from 222 Rn data). These results are consistent with the time scale of seasonal rainfall events observed in the geochemistry of spring waters and explain turbidity problems [START_REF] Amraoui | Turbidity dynamics in karstic systems: example of Ribaa and Bittit springs in the middle atlas (Morocco)[END_REF] after strong rain episodes.

Deuterium excess characterizes a local recharge of the groundwater. Waters in the Liassic aquifer have a lighter isotopic composition than waters in contact with Triassic evaporites. This difference characterizes a higher altitude of recharge for karstic waters. Five altitudes of recharge have been identified; moreover, it is inferred that there is a drain (or preferential flow) of waters perpendicular to the causses in the Bittit-Ribaa area [START_REF] Kh | Geoelectrical phy investigating and of fractures network around Bittit spring (Middle Atlas, Morocco)[END_REF]Qarqori2015).

The influence of climate is difficult to assess given the lack of data on previous years for the whole basin and causses. Groundwater is primarily recharged by large rainfall events; nevertheless, the impact of human activity plays a more active role than the one of seasonal variations in the Ribaa-Bittit complex.

Finally, a first conceptual model of the flow paths from recharge areas to springs in this karst system is presented. To improve the model and to obtain a better understanding of groundwater recharge and flow, installation of monitoring instruments at wells and springs, with recording of stage, temperature, and rainfall at regular intervals (at least daily), is necessary. -3.36 -3.22 -3.28 -3.34 -3.18 -3.36 -3.22 -3.39 -3.10 -2.96 -3.38 -2.93 -3.03 Aragonite -0.12 -0.09 0.02 0.26 0.34 -0.11 -0.18 0.05 -0.11 -0.26 -0.28 -0.27 -0.05 -7.46 -6.97 -6.69 -6.08 -5.58 -7.39 -7.35 -6.62 -7.34 -7.52 -7.69 -7.59 .72 Gypsum -3.12 -2.97 -3.04 -3.10 -2.93 -3.11 -2.97 -3.14 -2.86 -2.72 -3.14 -2.69 -2.79 H 2 (g) -22. 36 -22.90 -23.00 -23.34 -23.76 -22.44 -22.66 -23.08 -22.30 -22.26 -22.18 -22.32 -22.80 H 2 O(g) -1.73 -1.74 -1.74 -1.73 -1.73 -1.74 -1.75 -1.76 -1.68 -1.67 -1.67 -1.68 -1.68 Halite -8. .05 -8.80 Magnesite -0.64 -0.64 -0.50 -0.25 -0.17 -0.63 -0.75 -0.47 -0.62 -0.76 -0.81 -0.81 -0.58 Mirabilite -10. .44 Quartz -0.03 0.02 0.02 -0.00 0.06 -0.02 0.14 0.02 -0.19 -0.30 0.05 0.06 0.05 .32 Silicagel -1.03 -0.97 -0.97 -1.00 -0.93 -1.01 -0.86 -0.98 -1.18 -1.29 -0.94 -0.93 -0.94 SiO 2 (a) -1. 36 -1.30 -1.30 -1.33 -1.26 -1.34 -1.19 -1.31 -1.50 -1.61 -1.26 -1.25 -6.36 -5.11 -6.72 -6.46 -6.81 -6.65 -6.77 -6.52 -6.53 -6.36 -5.71 -7.80 -7.25 Barite -2.14 * -1.96 * -1.69 * -0.88 * * -1.73 -0.17 -0.40 -1.73 -5.36 -4.36 -5.56 -5.27 -5.61 -5.39 -5.62 -5.31 -5.32 -5.19 -4.64 -5.88 -5 -5.44 -3.43 -5.80 -5.07 -5.77 -5.45 -5.84 -5.28 -6.19 -5.06 -2.86 -5.39 -4 .04 H 2 O(g) -1.67 -1.68 -1.70 -1.71 -1.64 -1.64 -1.64 -1.64 -1.70 -1.47 -1.64 -1.51 -1.69 Halite -9.25 -9.00 -9. .73 Magnesite -0.28 -0.03 -0.45 -0.49 -0.47 -0.52 -0.42 -0.48 -0.50 -0.37 -0.34 -1.13 -0.81 12 Quartz -0.33 -0.34 -0.37 -0.31 -0.22 -0.28 -0.24 -0.26 -0.34 -0.24 0.10 0.03 0.10

HYDROGEOLOGY JOURNAL -SUPPLEMENTARY MATERIAL

Hydrochemical constraints between the karst Tabular Middle Atlas

Causses and the Saïs basin (Morocco): implications of groundwater circulation

Artinite

Sepiolite

Brucite

Mirabilite

Sepiolite

Fig. 1

 1 Fig. 1 Geographical localisation of the study area (red rectangle) between the Saïs Basin and Middle Atlas Causse (Morocco; Qarqori 2015)

Fig. 4

 4 Fig. 4 Piper diagram showing water geochemistry of springs and wells. Springs and wells are referenced as in Fig. 2. Characterization of two types of waters from Liassic and Triassic origins, including two Liassic aquifers, in the north-eastern sector (L'Hajeb Causse) and south-western sector (Guigo Causse). The effect of seasonality is mainly shown by sulfates and chlorides. Solid symbols represent dry season, and open symbols represent wet season.Note: This figure was derived using "Diagrammes" software V. 5.9[START_REF] Simler | Software "Diagrammes" V. 5.9[END_REF]

Fig. 5

 5 Fig. 5 Seasonality expressed by Schoëller-Berkaloff diagrams, in a dry season (October-November) and b wet season (March-April). Sites are referenced as in Fig.2. Types of spring waters and mixing zones for the eastern and the western sectors are shown. Note: This figure was derived using "Diagrammes" software V. 5.9[START_REF] Simler | Software "Diagrammes" V. 5.9[END_REF]

Fig. 6

 6 Fig. 6 Mg/Ca ratio as an indicator of limestone and dolomite dissolution versus a sodium, b chloride and c histogram of sulfates, showing seasonality. Mg/Ca < 1: L'Hajeb Causse. Mg/Ca ≤ 1.1: Guigo Causse.Springs and wells are referenced as in Fig.2. Note: This figure was derived using "Diagrammes" software V. 5.9[START_REF] Simler | Software "Diagrammes" V. 5.9[END_REF]

  Fig. 8 Electrical conductivity (EC) of water versus 222 Rn activity and transit time of water from the Triassic aquitard to the Liassic aquifer. Springs and wells are referenced as in Fig. 2.Note:This figure was derived using "Diagrammes" software V. 5.9[START_REF] Simler | Software "Diagrammes" V. 5.9[END_REF]

Fig. 9

 9 Fig. 9 Principal component analysis (PCA) representations of individuals (25 samples of water) (a-c) and variables (chemical parameters) (b-d). Main plane (71.1% of inertia) defined by axes 1 and 2(a-b). Second plane (41.8% of inertia) defined by axes 2 and 3 (c-d). a Five sub-populations of groundwater are identified in the main planes 1 and 2. Axis 1 discriminates the set of (i) [Bittit, Ribaa, Aguemguam (1.1-3.3)] from (ii) [El Mir (5.1-5.3), Haj Kaddour (A)] and from (iii) [Maarouf 1 and 2 (9.1-0.2), Khadem (8)]. Axis 2 discriminates (iii)

  222

Fig. 10

 10 Fig. 10 Conceptual model of groundwater circulation, for a L'Hajeb Causse in the east, and for b Guigo Causse in the west, with mixing zones between the Middle Atlas Causses and Saïs Basin, from five recharge areas to springs-I to III: L'Hajeb Causse; IV, V: Guigo

Table 1

 1 Chemical data of springs and wells for the sampling campaigns(2009)(2010)

	Code: name Sample Sample date Water Geological Alt. Z Temp EC pH Ca 2+ Mg 2+ Na + K + Si 4+ Sr 2+ Ba 2+ Cl -SO 4 2-NO 3 -HCO 3 -	(spring or number type area (m asl) (°C) (μS/cm) (mg/L) (mg/L) (mg/L) (mg/L) (mg/L) (μg/L) (μg/L) (mg/L) (mg/L) (mg/L) (mg/L)	well)	1: Bittit spring 1.1 Oct. 2009 I Lhajeb C. 775 16.8 591 7.18 69.7 31.3 6.33 0.89 4.36 43.97 3.59 9.49 3.19 12.2 405	1.2 March 2010 16.5 555 7.45 68.3 27.9 5. 7 71 . 1 24 . 9 5N D N D 9 . 0 64 . 2 01 6 . 12 3 4	1.3 Oct.-Nov. 2010 16.4 580 7.50 68.6 31.1 5.42 0.85 4.93 50.20 4.25 10.1 3.72 17.5 268	2: Ribaa spring 2.1 Oct. 2009 I Lhajeb C. 937 16.6 588 7.67 70.0 31.8 6.25 0.82 4.65 43.95 3.60 9.90 3.25 13.9 317	2.2 March 2010 16.9 565 7.88 67.6 30.5 4. 3 20 . 9 75 . 5 2N D N D 6 . 9 14 . 7 92 6 . 22 3 7	2.3 Oct.-Nov. 2010 16 583 7.49 72.8 31.1 5.21 1.91 4.94 52.01 5.32 8.65 3.12 15.0 284	3: Aguemguam 3.1 Oct. 2009 I Lhajeb C. 919 16.3 590 7.22 71.2 32.1 5.87 0.86 4.47 43.83 3.00 9.53 3.11 12.4 375	spring	3.2 March 2010 15.9 559 7.33 70.5 28.4 5. 2 71 . 1 86 . 3 1N D N D 8 . 4 74 . 1 21 6 . 42 4 6	3.3 Oct.-Nov. 2010 15.8 548 7.54 68.8 31.0 5.22 0.87 4.80 48.35 4.15 7.58 2.91 13.9 271	4: Sbaa spring 4.1 Oct. 2009 II Lhajeb C. 855 18.7 678 7.15 81.3 35.2 10.2 0.89 3.24 39.64 3.98 16.7 5.19 13.5 370	4.2 March 2010 18.9 661 7.13 77.2 34.1 8.10 1.49 2.55 ND ND 17.0 7.30 23.8 275	4.3 Oct.-Nov. 2010 18.5 638 7.30 80.7 35.1 9.65 1.81 3.21 43.16 5.85 16.6 5.15 15.8 319	5: El Mir spring 5.1 Oct. 2009 II Lhajeb C. 840 19 1,232 7.09 79.1 32.5 118 1.99 5.71 97.88 3.66 186 3.07 11.1 306	5.2 March 2010 18.5 1,086 7.16 76.1 31.3 102 1.89 5.78 ND ND 154 8.67 20.3 257	5.3 Oct.-Nov. 2010 18.4 1,276 7.40 78.9 32.5 129 2.64 5.63 105.4 5.50 215 6.95 12.6 264	6: Boucharmou 6.1 Oct. 2009 I Guigo C. 1,041 19.1 529 7.46 53.5 34.7 3.18 0.91 2.40 42.45 2.34 6.49 9.89 23.6 384	spring	6.2 March 2010 18.7 517 7.97 55.9 34.3 3.50 1.06 2.32 ND ND 10.5 11.2 27.9 204	7: Bou Youssef 7.1 Oct. 2009 I Guigo C. 1,059 18.3 551 7.41 63.2 33.6 3.11 0.93 2.10 32.72 3.15 6.84 10.6 24.8 320	spring	7.2 March 2010 17.4 528 7.58 60.0 30.6 3. 6 41 . 6 22 . 3 8N D N D 6 . 6 01 1 . 51 1 . 52 1 2	8: Khadem spring 8 March 2010 I (?) Guigo C. 1,017 17.7 578 7.52 64.0 33.7 4.60 1.13 2.23 ND ND 9.07 16.4 52.3 207	(El Hajeb)	9: Maarouf 1 9.1 Oct. 2009 II Guigo C. 781 20.1 701 7.27 59.7 40.4 22.3 1.17 3.17 56.01 5.12 39.5 14.4 37.9 320	spring	9.2 March 2010 20.1 698 7.38 58.1 39.6 22.7 1.16 2.78 ND ND 43.7 16.4 41.3 218	10: Maarouf 2 10.1 Oct. 2009 II Guigo C. 853 20.1 751 7.27 56.9 40.1 35.2 2.23 3.05 51.26 4.91 59.1 9.26 28.2 364	spring	10.2 March 2010 20.2 759 7.42 55.5 39.0 35.4 4.52 2.90 ND ND 63.7 12.0 39.9 225	A: Haj Kaddour A Oct. 2009 II Saïs Plain 644 26.6 991 7.31 56.2 33.3 88.7 1.13 3.82 93.92 7.36 130 12.2 10.1 315	(deep well	in Saïs)	B: Palaeozoic B.1 Oct. 2009 III Guigo C. 977 20 1,210 7.81 112.5 36.0 64.0 6.78 6.69 599.8 29.25 140 115 106 170	schist well	B.2 Oct.-Nov. 2010 ND ND ND 222.7 43.8 41.8 4.85 6.70 479.0 26.40 192 122 289 142	C: Basaltic schist C Oct. 2009 III Guigo C. 912 18.3 530 7.52 88.3 12.5 5.60 1.36 6.23 42.20 7.36 11.4 7.65 36.0 279	well	Site codes for the sampling campaigns (2009-2010). Springs and wells are referenced as in Fig. 2. Springs: 1, Bittit; 2, Aguemguam; 3, Ribaa; 4, Sbaa; 5, El Mir; 6, Boucharmou; 7, Bou Youssef; 8, Khadem; 9, Maarouf 1; 10, Maarouf 2. Wells: A, Haj Kladdour (deep well	in Saïs Plain); B, Palaeozoic schist well; C, Basaltic schist well. Sample numbers are shown in principal component analysis (PCA)	C. Causse; ND not determined

material (ESM). Stable isotopes (δ 18 O, δD) were measured at the Geochemical Laboratory of the Faculty of Science at the University of Avignon (France) by mass spectrometry. Stable isotopes were analyzed in spring and well water samples collected during the four sampling campaigns (Table

Table 2

 2 Isotopic data of springs and wells, d: deuterium excess (d = δD -8×δ 18 O), for the four sampling campaigns(2009)(2010)(2011) with some values ofFeb. 2007[START_REF] El Ouali | Contribution of isotopes of the water molecule to determine recharge altitude of the Main Springs welling up in the Middle Atlas limestone (Morocco)[END_REF]

	Code: name	Water type	Sample date	δ 18 O	δ 2 H	d ‰	222 Rn	SD 222 Rn	Z rech
	(spring or well)			(V. VSMOW)‰	(V. VSMOW)‰		(Bq/m 3 )	(Bq/m 3 )	(m asl)
	1: Bittit spring	I	Feb. 2007	-7.65	-42.87	18.3	--	1,415
			Oct. 2009	-7.58	-48.82	11.8	3,330	383
			March 2010	-7.40	-44.23	15.0	4,320	436
			Oct.-Nov. 2010	-7.39	-44.73	14.4	--
			Nov. 2011	-7.58	-45.55	15.1	--
	2: Ribaa spring	I	Feb. 2007	-7.42	-42.68	16.7	--	1,391
			Oct. 2009	-7.54	-48.21	12.1	1,976	295
			March 2010	-7.17	-42.58	14.8	3,820	410
			Oct.-Nov. 2010	-7.58	-46.47	14.2	--
			Nov. 2011	-7.57	-45.39	15.2	--
	3: Aguemguam spring	I	Oct. 2009	-7.60	-48.80	12.0	2,137	307	1,435
			March 2010	-7.51	-43.87	16.2	3,458	390
			Oct.-Nov. 2010	-7.57	-45.86	14.7	--
			Nov. 2011	-7.61	-46.1	14.8	--
	4: Sbaa spring	II	Oct. 2009	-6.93	-44.92	10.5	10,753	688	1,185
			March 2010	-6.78	-39.94	14.3	9,640	651
			Oct.-Nov. 2010	-6.92	-42.71	12.7	--
			Nov. 2011	-6.97	-42.02	13.7	--
	5: El Mir spring	II	Oct. 2009	-6.42	-42.08	9.3	12,894	753	1,063
			March 2010	-6.60	-39.54	13.3	16,464	851
			Oct.-Nov. 2010	-6.50	-40.59	11.4	--
			Nov. 2011	-6.76	-41.64	12.4	--
	6: Boucharmou spring	I	Oct. 2009	-6.54	-43.73	8.6	4,656	453	1,042
			March 2010	-6.49	-41.01	10.9	4,267	433
	7: Bou Youssef spring	I	Oct. 2009	-6.40	-41.35	9.9	4,351	437	997
			March 2010	-6.40	-38.09	13.1	2,597	338
			Nov. 2011	-6.37	-38.32	12.6	--
	8: Khadem spring (El Hajeb)	I (?)	March 2010	-6.65	-41.03	12.2	5,623	497	1,093
	9: Maarouf 1 spring	II	Oct. 2009	-5.90	-40.67	6.5	7,960	592	846
			March 2010	-6.07	-38.82	9.7	5,511	492
	10: Maarouf 2 spring	II	Feb. 2007	-6.16	-37.51	11.8	--	927
			Oct. 2009	-6.18	-42.15	7.3	3,043	366
			March 2010	-6.16	-38.30	11.0	2,776	349
			Nov. 2011	-6.31	-40.26	10.2	--
	A: Haj Kaddour (deep	II	Oct. 2009	-6.20	-40.69	8.9	3,227	377	981
	well in Saïs)		Nov. 2011	-6.50	-40.69	11.3	--
	B: Palaeozoic schist well	III	Oct. 2009	-5.44	-35.14	8.3	263.5	108	609
			Nov. 2011	-5.25	-33.66	8.3	--
	C: Basaltic schist well	III	Oct. 2009	-5.88	-37.79	9.3	1,120	222	807
	The main elevation of recharge zone is obtained as follow: Z rech (m) = -(δ 18 O + 3.7)/0.27 (Sefrioui et al. 2010). Stable isotopes are expressed versus
	VSMOW (Vienna Standard Mean Ocean Water)
	SD 222 Rn standard deviation for radon

Table 3

 3 Correlation matrix between geochemical parameters (see Table1) and radon (Rn, see Table2) for the 25 individual components of springs considered (1.1-10.2) and well A

	Correlation	EC	Ca 2+	Mg 2+	Na +	K +	Si 4+	Cl -	SO 4 2-	NO 3 -	HCO 3 -	Rn
	EC	1	0.319	0.116	0.989	0.462	0.383	0.991	0.035	-0.180	0.020	0.656
	Ca 2+		1	-0.417	0.240	-0.039	0.536	0.234	-0.661	-0.500	0.150	0.594
	Mg 2+			1	0.058	0.353	-0.609	0.090	0.631	0.615	0.166	0.035
	Na +				1	0.442	0.399	0.997	0.056	-0.201	0.022	0.597
	K +					1	-0.005	0.471	0.170	0.202	0.242	0.066
	Si 4+						10 . 3 8 0	-0.687	-0.521	0.026	0.231
	Cl -							10 . 0 7 3	-0.171	0.037	0.600
	SO 4 2-								1	0.760	0.342	-0.037
	NO 3 -									10 . 3 8 5	-0.024
	HCO 3 -										1	-0.012
	Rn											1

Table S1

 S1 Saturation indexes determined with PHREEQC version 2.7 (database Wateq.dat), for phases in springs and wells of the sampling campaigns (2009-2010). Samples of springs and wells are referenced with the same codes as in Table 1 of the main article. Phases of interest in bold characters: aragonite, calcite, dolomite, quartz and cristobalite

	Hélène Miche	1! , Ginette Saracco	1 , Adriano Mayer	1,2	, Khaoula Qarqori	3 , Mohamed Rouai	3 ,
		Abdelilah Dekayir	3 , Konstantinos Chalikakis	2 , Christophe Emblanch	2
	1 CNRS-UMR 7330, CEREGE, Aix-Marseille Université, IRD, Europôle de l'Arbois, BP 80,
	13545 Aix en Provence cedex 4, France, miche@cerege.fr, saracco@cerege.fr
	2 Université d'Avignon et des Pays de Vaucluse, INRA, EMMAH, UMR 1114, LHA, 84000
	Avignon, France adriano.mayer@univ-avignon.fr, konstantinos.chalikakis@univ-avignon.fr,
	christophe.emblanch@univ-avignon.fr
	3 Université Moulay Ismail, Faculté des Sciences, Département des Sciences de la Terre, BP
	11201,	50000	Meknès,	Maroc,	khqarqori@gmail.com,	mohamed.rouai@laposte.net,
	dekayir@yahoo.fr
	Hélène Miche

! : contact author, miche@cerege.fr

Table S1 (continued) Saturation indexes Boucharmou spring

 S1

				Bou Youssef	Maarouf 1	Maarouf 2	Khadem	HajKaddour	Palaeozoic	Basaltic
				spring		spring		spring	spring	(deep w.)	schist w.	schist w.
	Samples	6.1	6.2	7.1	7.2	9.1	9.2	10.1 10.2 8	A	B.1 B.2	C
	Anhydrite	-2.98 -2.49 -2.87 -2.83 -2.79 -2.74 -3.01 -2.89 -3.53	-2.88	-1.72 -1.52 -2.86
	Aragonite	0.05	0.32	-0.02 -0.04 -0.17 -0.22 -0.13 -0.19 -0.99	-0.07	0.29 -0.34 0.19
	Artinite

Acknowledgements We want to particularly thank Yves Travi (EMMAH-Avignon University PAV, France) for his advice in our isotopic analyses, Mustapha Boujamaoui (FST-Errachidia, University MI, Maroc) for his efficient help during our measurement acquisitions, and Clément Layet and Ghislain Gassier (CEREGE) for fruitful discussions on the PCA method. We also acknowledge the reviewers for their useful remarks. This work was developed in the Program of "Fédération de Recherche" ECCOREV No. 3058 of CNRS -Centre National de la Recherche Scientifique-France, and BREMEX project of the "Conseil Régional PACA, France".


V G Bashtovoi

P P Kuzhir

A Y Zubarev

V S Moroz

On the mechanics of magnetic fluids with field-induced phase transition: application to Couette flow

Keywords: magnetic fluid, magnetic field, magnetophoresis, brownian diffusion, phase transition, Couette flow

Influence of processes of Brownian diffusion and magnetophoresis which are followed by phase transition on characteristics of a stationary plane Couette flow of a magnetic fluid in non uniform magnetic field is considered. Conditions of phase transition in magnetic fluid are accepted as a natural restriction of increasing of concentration of particles in non-uniform magnetic field. Profiles of concentration of particles are calculated and dependences of volume magnetic force, and also viscous force from them are established.

Introduction

In this paper magnetic fluid is a stable suspension of magnetic nanoparticles with size of order 10 nm and classical magnetic and mechanical properties [START_REF] Rosensweig | Ferrohydrodynamics[END_REF][START_REF] Blums | Magnetic Fluids[END_REF][START_REF] Bashtovoy | Introduction to Thermomechanics of Magnetic Fluids[END_REF][START_REF] Taketomi | Motion of ferrite particles under a high gradient magnetic field in a magnetic fluid seal[END_REF]. Stability of suspension is provided by the special surfactant.

The nanosize of particles provides a Brownian nature of their behavior in fluid, exchange of a momentum with molecules of fluid, transfer of a momentum resulted by their interaction with external magnetic field to all volume of fluid. Due to this such suspension can be considered as a continuous medium with magnetization providing the volume magnetic force in fluid in external magnetic field.

In non-uniform magnetic field the natural processes taking place in such system are magnetophoresis and Brownian diffusion of magnetic particles leading to a redistribution of concentration of particles. On the importance of these processes in magnetic fluids the attention was paid in the very first works on this subject and in the subsequent monographs [START_REF] Rosensweig | Ferrohydrodynamics[END_REF][START_REF] Blums | Magnetic Fluids[END_REF][START_REF] Bashtovoy | Introduction to Thermomechanics of Magnetic Fluids[END_REF].

These processes become apparent significantly in magnetic fluid seals due to large gradients of magnetic field intensity using in them [START_REF] Taketomi | Motion of ferrite particles under a high gradient magnetic field in a magnetic fluid seal[END_REF][START_REF] Anton | Behavior of magnetic liquids in a inhomogeneous magnetic field[END_REF][START_REF] Bashtovoi | The effect of diffusion processes on the statics of magnetic fluids[END_REF].

Recently it is specified also an important role which these processes can play in the general problems of statics and dynamics of magnetic fluids due to their influence on distribution of magnetization, volume magnetic force, and viscous tension in fluids. It is important, for example, at a levitation of magnetic and non magnetic bodies in them as applied to magnetic fluid suspensions, supports, vibroprotective systems [6 -8], and also at a flow of magnetic fluids in channels [START_REF] Bashtovoi | Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid[END_REF]. In these works the complete system of the differential equations and boundary conditions providing the adequate description of these processes is formulated.

The main disadvantage of the presented description of processes of Brownian diffusion and magnetophoresis in magnetic fluids based on the classical theory of these processes is that they give unlimited increasing of concentration of magnetic particles in areas with high intensity of non-uniform magnetic field, more than at densest packing. It results, in particular, abnormally high values of their viscosity and, respectively, viscous tension at their flow [START_REF] Bashtovoi | Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid[END_REF]. From the physical point of view it cannot be accepted.

However, it is known that even in a non magnetic suspension the increasing of concentration of particles leads to a phase transition [START_REF] Russell | Colloidal Dispersions[END_REF]. It means a sharp transition of suspension to the high-concentrated state of particles in which proposed description of diffusive processes is not valid but which has fluidity and a finite viscosity.

The situation is equivalent to that which arises upon usual phase transition of the gaseous medium to a liquid phase at its condensation.

It is obvious that the corresponding critical values of concentration of particles leading to such phase transition have to be a natural limit of validity of the theory of Brownian diffusion for suspension.

It is known also that in magnetic fluid as a suspension of magnetic particles, the special phase transition to the high-concentrated state of particles takes place [START_REF] Blums | Magnetic Fluids[END_REF][START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF][START_REF] Zubarev | Phase transitions in electro-and magnetorheological fluids[END_REF][START_REF] Iskakova | On the theory of phase transitions in magnetorheological suspensions[END_REF][START_REF] Magnet | Behavior of nanoparticle clouds around a magnetized micro-sphere under magnetic and flow fields[END_REF][START_REF] Pshenichnikov | Magnetophoresis of particles and aggregates in concentrated magnetic fluids[END_REF][START_REF] Pshenichnikov | Magnetophoresis, sedimentation, and diffusion of particles in concentrated magnetic fluids[END_REF][START_REF] Pshenichnikov | Sedimentation of particles in concentrated magnetic fluids: Numerical simulation[END_REF].

It is caused by the interaction of magnetic particles with each other, and with external magnetic field. In spite of the fact that properties of such high-concentrated magnetic fluid still are studied insufficiently, there is rather large number of the experimental data demonstrating it behavior as a liquid medium. First of all it belongs to data on deformation of drops of this phase in magnetic fields [START_REF] Blums | Magnetic Fluids[END_REF] and a possibility to characterize them by the corresponding surface tension and viscosity.

General theory.

We believe that magnetic particles in magnetic fluid at steady state can be in two thermodynamic states: 1) disordered weak-concentrated phase with behavior of magnetic particles in fluid, similar to gaseous state (a "gas" phase), and 2) a highconcentrated phase with ordered structured packing of magnetic particles and their behavior, similar to a condensed incompressible fluid (the "condensed" phase).

Further the parameters relating to magnetic fluid with "gaseous" and the "condensed" phases of magnetic particles will be noted by indexes g and c respectively.

The main properties of these phases are accepted by the following. 2.1. Mass transfer processes. a) "Gas" phase.

According to the approach developed in [START_REF] Bashtovoi | The effect of diffusion processes on the statics of magnetic fluids[END_REF][START_REF] Bashtovoi | Influence of Brownian diffusion on statics of magnetic fluid[END_REF][START_REF] Bashtovoi | The effect of magnetophoresis and Brownian diffusion on the levitation of bodies in a magnetic fluid[END_REF][START_REF] Bashtovoi | Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid[END_REF] on the base of phenomenological theory of mass transfer the distribution of volume concentration Φ g of magnetic particles having density ρ in a "gas" phase is governed by the diffusion equation

0 g divi t ρ ∂Φ + = ∂  , (1)
where mass flux density of particles i  in motionless fluid at the presence of nonuniform magnetic field with value of intensity H is defined as follows 0 ( )

g g i D b m L H ρ ρ m ξ = -∇Φ + Φ ∇  , ( 2
)
where: t is time, D -diffusion coefficient, which is connected with the mobility of particles b by the Einstein's relation kTb D =

; μ 0 ≈1,26•10 -6 H/m -magnetic permeability of vacuum; m -magnetic moment of individual particle, J/T; k≈1,38•10

-23 J/K -Boltzmann constant; T -temperature, K; ( ) ( ) 1/ L cth ξ ξ ξ = - -Langevin function, 0 / mH kT ξ m = .
In general case diffusion coefficient D is a function of concentration g Φ . On boundaries impermeable for particles the normal component of mass flux density i n must be equal to zero.

In accordance with (1) at stationarity mass flux density i  is equal to zero in whole volume of fluid bounded of impermeable boundaries. Taking into account (2) and Einstein's relation it means 0 ( / ) ( )

g g m kT L H m ξ ∇Φ = Φ ∇ . ( 3
)
The dimension analysis of the equations [START_REF] Bashtovoy | Introduction to Thermomechanics of Magnetic Fluids[END_REF] shows that a stationary problem described by them is defined by one dimensionless parameter 0 * / U mH kT m = representing the ratio of magnetic energy of a particle to its thermal energy [START_REF] Bashtovoi | The effect of diffusion processes on the statics of magnetic fluids[END_REF][START_REF] Bashtovoi | Influence of Brownian diffusion on statics of magnetic fluid[END_REF][START_REF] Bashtovoi | The effect of magnetophoresis and Brownian diffusion on the levitation of bodies in a magnetic fluid[END_REF][START_REF] Bashtovoi | Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid[END_REF]. It includes characteristic value of magnetic field intensity * H specified for each concrete situation.

Most often typical magnetic fluid contains as a magnetic phase particles of magnetite with magnetization of 4,46•10 5 A/m [START_REF] Rosensweig | Ferrohydrodynamics[END_REF]. At their characteristic diameters d about 10 -8 m and, respectively, a volume of 5,2•10 -25 m 3 , the magnetic moment of particle m is about 2,3•10 -19 J/T. Then, at room temperature T=300 K value U of an order of one takes place at characteristic magnetic field intensity * H of about 10 4 A/m. In real situations, for example, in gaps of magnetic fluid seal in which strong magnetic field to 10 6 A/m with large gradients the values of parameter U can reach several tens.

Total volume of magnetic particles V mg in the volume V g , occupied by this phase is equal mg

g Vg V dV = Φ ∫ . b) "Condensed" phase.
Analysis shows that in the condensed phase the volume fraction of nanoparticles varies relatively insignificantly, and remains quite close to the maximum packing fraction of the considered particle network. At such condition we can assume that the "condensed" phase has constant concentration of magnetic particles, Φ c = Const corresponding to conditions of thermodynamic coexistence of phases.

The total volume of magnetic particles V mc in volume V c occupied by this phase, is equal to

mc c c c Vc V dV V = Φ = Φ ∫ .
Magnetic interaction of spherical ferromagnetic magnetic particles has a dipoledipole character. It is defined by the magnetic moment of a particle m, and characterized by dimensionless parameter ( ) ( )

2 3 0 / 4 m d kT λ m π =
representing the ratio of the energy of dipole -dipole interaction of two closely situated particles to the thermal energy.

For magnetite particles with the characteristics given above and at the room temperature parameter λ is about 1,3.

c) An average concentration of particles Φ 0 in all considered volume V of magnetic fluid is a constant and is determined by the next way 0 [START_REF] Rosensweig | Ferrohydrodynamics[END_REF] 1

g c g c c Vg Vc Vg dV dV dV V V V     Φ + Φ = Φ +Φ = Φ         ∫ ∫ ∫ . (4)
2.2. Phase co-existence. Interface between both phases and concentration of particles in the "condensed" phase are defined by thermodynamic conditions of coexistence of phases. The approach developed in [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF][START_REF] Zubarev | Phase transitions in electro-and magnetorheological fluids[END_REF][START_REF] Iskakova | On the theory of phase transitions in magnetorheological suspensions[END_REF][START_REF] Magnet | Behavior of nanoparticle clouds around a magnetized micro-sphere under magnetic and flow fields[END_REF] is adopted in the present paper. It leads to convenient analytical relationships of phase coexistence in the magnetic fluid taking into account dipolar interaction between magnetic particles. This approach was effectively used for describing of experimental results in [START_REF] Magnet | Behavior of nanoparticle clouds around a magnetized micro-sphere under magnetic and flow fields[END_REF]. The phase equilibrium between two phases is given by the equilibrium of their chemical potentials and osmotic pressures.

The chemical potential ζ(Φ,ξ,λ) and the osmotic pressure p(Φ,ξ,λ), both contain the contributions coming from and hard-sphere repulsion ζ hs, p hs and magnetic interactions ζ m , p m .

The appropriate expressions for the first kind of interaction have been developed on the basis of osmotic compressibility calculations carried out by Carnahan and Starling [START_REF] Carnahan | Equation of state for nonattracting rigid spheres[END_REF] for semidilute and Hall [START_REF] Hall | Another Hard-Sphere Equation of State[END_REF] for concentrated hard-sphere suspensions.

Based on the perturbation analysis, Buyevich and Ivanov [START_REF] Yu | Equilibrium properties of ferrocolloids[END_REF] have derived the expressions for the magnetic contribution to the chemical potential and osmotic pressure of magnetic particles in a magnetic fluid taking into account pair magnetic interactions between particles and neglecting higher order interactions.

Assuming that the steric interactions between particles in magnetic fluid respect the Carnahan-Starling law in the "gas" phase and the Hall law in the "condensed" phase, by using the results [START_REF] Yu | Equilibrium properties of ferrocolloids[END_REF], we arrive to the following expressions for quantities ζ and p in both phases. 1) sh

g g g g hs g m g g g g kT ξ ζ ξ λ ζ ζ ξ  -Φ + Φ Φ = + = Φ + Φ + -  -Φ   2 2 1 8 ( ) 3 g L ξ λ λ    - + Φ       , ( 5
) 2 3 3 1 ( , , ) ( ) ( ) (1 )
g g g g g hs g m g g p g kT p p p V ξ λ  + Φ + Φ -Φ Φ = + = Φ -  -Φ   2 2 2 1 4 ( ) 3 g L ξ λ λ    - + Φ       , ( 6
) ( , , ) ( ) ( ) ln ln sh c c с hs с m с m m c m c A A kT C ξ ζ ξ λ ζ ζ ξ  Φ Φ = + = + + + -  Φ Φ -Φ Φ -Φ  2 2 1 8 ( ) 3 c L ξ λ λ    - + Φ       , (7) 2 2 2 1
( , , ) ( ) ( ) 4 ( ) 3 c с hs с m с c c p m c kT A p p p L V ξ λ ξ λ λ     Φ = + = Φ - + Φ     Φ -Φ     . ( 8
)
Here V p is volume of the particle, Φ m is the volume concentration of densest packed particles (for the face-centered cubic (FCC) lattice Φ m =0.74), A and C are some numerical parameters, estimated in [START_REF] Russell | Colloidal Dispersions[END_REF] as A=2.2, C=1.255.

In reality, at high enough field, a body-centered tetragonal (BCT) structure exists instead of the FCC lattice, however, the energy and magnetic properties of both lattices are quite close, so that the assumption of the FCC lattice should not generate significant errors in calculations of the concentration profiles. The assumption of the FCC lattice allows us to recover the order-disorder phase transition in the absence of the magnetic field.

Phase co-existence is defined by the equality of their chemical potentials and osmotic pressure on the phase interface:

ζ g (Φ g , ξ, λ)= ζ c (Φ c , ξ, λ), p g (Φ g , ξ, λ)=p c (Φ c , ξ, λ). (9)
Taking into account ( 5)-( 8) it leads to the next equations for determination of concentrations on the interface:

( )

2 2 2 3 8 9 3 1 ln 8 ( ) 3 1 g g g g g g L ξ λ λ -Φ + Φ   Φ + Φ - + Φ =     -Φ = 2 2 1 ln 8 ( ) 3 c c m m c m c A A C L ξ λ λ Φ   + + - + Φ   Φ Φ -Φ Φ -Φ   , (10)
( )

2 3 2 2 2 3 1 1 4 ( ) 3 1 g g g g g g L ξ λ λ + Φ + Φ -Φ   Φ - + Φ =     -Φ 2 2 1 4 ( ) 3 c c m c A L ξ λ λ   = Φ - + Φ   Φ -Φ   . (11)
.

The equations ( 10), [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF] have to be solved numerically with respect to the concentrations Φ g and Φ c for arbitrary values of magnetic field intensity ξ and parameter λ. As a result the diagrams of phase equilibrium Φ g (ξ,λ) and Φ c (ξ,λ), defining interface between the "gas" and "condensed" phases in magnetic fluid can be found and constructed.

So, for example, in the absence of magnetic field (ξ=0) and for nonmagnetic particles (λ=0) these equations have a solution: Φ g =0,495 and Φ c =0,545 [START_REF] Russell | Colloidal Dispersions[END_REF].

In the case of magnetic fluid magnetized to saturation, when ξ >>1 and 1 ) ( ≈ ξ L , diagrams of phase equilibrium, calculated on the base of equations ( 10)- [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF], are presented in Fig. 1. The both binodal curves divide the space of diagram into the two regions: 1) the disordered fluid -"gas" phase as a continuum under the left curve, and 2) the "condensed" phase as a continuum under the right curve. At given value of λ the points on the left curve define a value of critical concentration of particles in "gas" phase Φ g at which phase transition arises. And the corresponding points on the right curve determine a value of concentration of particles in "condensed" phase Φ c .

In stationary equilibrium "gas" and "condensed" phases as a continuum coexist with interface between them. There is a jump of concentration between phases on this interface which is determined by their corresponding values on binodal curves.

Fig. 1 demonstrates that significant discrepancy between binodal curves is appeared at λ ≥ 3. That is, taking into account the interaction between particles ( 0 ≠ λ ) in the equations ( 10), [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF] does jump of concentration of particles between the "gas" and "condensed" phases to much more considerable, than 0,495 and 0,545 which is characteristic of suspensions of non magnetic particles [START_REF] Russell | Colloidal Dispersions[END_REF]. More precise critical values of Φ g and Φ c for some values of parameter λ corresponding to the Fig. 1 are presented in the table below.

Table 1. Critical concentration of particles in "gas" phase Φ g and corresponding concentration Φ c of particles in "condensed" phase at different values of λ at phase transition in magnetic fluid at magnetic saturation.

λ 0 0,5 1 1,5 2 2,5 3
3,5 4 4,5 5 Φ g 0,494 0,493 0,491 0,489 0,487 0,479 0,469 0,440 3,69·10 -8 2,37·10 -10 8,69·10 -13 Φ c 0,545 0,546 0,549 0,552 0,557 0,563 0,573 0,587 0,649 0,666 0,735

The analysis of the equations ( 10), [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF] shows that at large values of parameter λ critical value of concentration of particles in a "gas" phase Φ g asymptotically tends to zero, and concentration of particles in the "condensed" phase Φ c asymptotically tends to the maximum value Φ m . Equating the right part of eq.( 11) to zero, in the case of strong magnetic field (L(ξ)≈1) one can get

( ) 2 ( ) / 4 / 3 m c c A λ λ Φ -Φ Φ = + .
Similarly the equation [START_REF] Russell | Colloidal Dispersions[END_REF] taking into account [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF] in this case gives asymptotic dependence We suppose also that: 2.3. Magnetization of magnetic fluid M at any phase state of magnetic particles depends on their concentration Φ, magnetization of individual magnetic particle M m and on intensity of magnetic field under Langevin function: M=M m ΦL(ξ).

As shown in [START_REF] Yu | Equilibrium properties of ferrocolloids[END_REF], the dipole-dipole interaction between the particles leads to modification of the Langevin's law of magnetization of the fluid and the form ( ) ( ) ( ) 1 8 ( )

m dL M M x L x d ξ ξ λ ξ   = Φ + Φ     can be used.
Magnetization of magnetic particles M m is supposed uniform in their volume and is equal to the ratio of their magnetic moment m to their volume V p : M m = m/V p .

2.4. Magnetic fluid at any phase state of magnetic particles is Newtonian and is governed by the well known equations of mechanics of magnetic fluids [START_REF] Rosensweig | Ferrohydrodynamics[END_REF][START_REF] Blums | Magnetic Fluids[END_REF][START_REF] Bashtovoy | Introduction to Thermomechanics of Magnetic Fluids[END_REF].

2.5. The magnetic force m F  acting on the volume V of magnetic fluid in nonuniform magnetic field is determined by expression:

0 0 ( ) m m V V F M HdV M L HdV m m ξ = ∇ = Φ ∇ ∫ ∫  . ( 12
)
2.6. A dynamic coefficient of magnetic fluid viscosity η at any phase of magnetic particles will be described by the approximation of Krieger-Dougherty [START_REF] Krieger | Concentration dependence of the viscosity of suspensions[END_REF][START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF] 2,5

( 0) 1 m m η η -Φ   Φ = Φ = -   Φ  
, where η(Φ=0) is a dynamic viscosity of a base fluid, and Φ m is a сoncentration at densest packing of particles. At FCC lattice Φ m =0,74 and 1,85

( 0)(1 1,35 ) η η - = Φ = - Φ . ( 13
)
3. Couette flow of magnetic fluid with phase transition.

We will consider influence of mass transfer processes, leading to the phase transition in magnetic fluid, on the stationary plane Couette flow in locally non uniform magnetic field as is presented in Fig. 2.

It will allow us to describe problem by exact analytical solutions of equation ( 3) and find out all of the main features of this influence. Also this problem can be a good model for description of a magnetic fluid behavior in a gap of magnetic fluid seal [START_REF] Bashtovoi | The effect of diffusion processes on the statics of magnetic fluids[END_REF][START_REF] Bashtovoi | Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid[END_REF]. Let, as shown in Fig. 2, the volume of magnetic fluid ( ,

0 l x ≤ ≤ h y ≤ ≤ 0
) fill a gap between two plane-parallel solid boundaries y=0 and y=h. Side boundaries x=0 and x=l of the fluid volume are supposed plane too. It means that the curvature of these boundaries caused by the wetting of solid boundaries is not essential for the being studied processes. Magnetic fluid is held in the gap by the non-uniform magnetic field H(x) created by the local pole of a magnet system (MP) when pressure drop along the xaxis Δp=p 1 -p 2 is applied to it. This position of magnetic fluid volume corresponds to the maximum pressure drop which is held by magnetic fluid seal. If magnetic field depends only on coordinate x, the distribution of concentration of particles in fluid will be function only of coordinate x too.

The part of the fluid located in a x ≤ ≤ 0 , i.e. near a pole in the area with maximum intensity of magnetic field, can represent the "condensed" phase of magnetic particles, and the area l x a ≤ ≤ -the "gas" phase. Location of the interface between these phases x=a is to be determined in the process of the solution of equations ( 10)- [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF].

We suppose that the upper boundary (y=h) moves relatively to the bottom one (y=0) with the velocity v 0 in the direction of z axis. The Couette flow with velocity profile v z =v 0 y/h takes place in the fluid.

Viscous tension τ on the upper boundary is equal:

( ) 0 / / z v y v h τ η η = ∂ ∂ =
and is determined by the dynamic coefficient of viscosity of the fluid η[Φ(x)] which is a function of particle concentration in accordance to [START_REF] Iskakova | On the theory of phase transitions in magnetorheological suspensions[END_REF].

Further consideration will be implemented by using the dimensionless parameters:

x′ =x/l, * / H H H ′ =
, where * H is a maximum value of magnetic field intensity at x=0: * H = H(0), and the dimensionless coordinate s of interface between phases is s = a/l.

All boundaries of the magnetic fluid volume will be assumed impermeable for particles and, as it was mentioned above, the mass flux density in whole volume of the fluid is zero. In accordance with the equation ( 3), it means that distribution of particles concentration in the "gas" phase will be governed by the next dimensionless equation ( )

g g d dH U L UH dx dx Φ ′ ′ = Φ ′ ′ , 0 * / U mH kT m = , (14)
which has the general solution

[ ( )] ( ) g sh UH x B H x ′ ′ Φ = ′ ′ . ( 15
)
At the absence of phase transition constant B is defined from the condition (4) as

1 0 0 g dx′ Φ = Φ ∫ and [ ] 1 1 0 0 ( ) ( ) sh UH x B dx H x - ′ ′   ′ = Φ   ′ ′   ∫ . ( 16
)
At the presence of the phase transition we will use the next approach.

Let the critical value of particle concentration in the "gas" phase on the interface with "condensed" phase x=s be Φ g (s)= Φ g* . Than the constant B in ( 15) is defined as

* ( ) [ ( )] g H s B sh UH s ′ = Φ ′ . ( 17
)
In this case the coordinate s of the interface between the phases is determined from the condition of constancy of average concentration of particles (4)

1 0 c g s s dx′ Φ + Φ = Φ ∫ as 1 * 0 ( ) ( ) [ ( )] c g s H s sh UH s dx sh UH s H ′ ′ ′ Φ + Φ = Φ ′ ′ ∫ . ( 18
)
This equation must be solved for s at predetermined configuration of magnetic field ( )

H x′ , values of parameters U and Φ 0 , with taking into account equations ( 10), [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF] for Φ c and Φ g* .

In the case when overwhelming majority of particles at large values of magnetic field intensity is concentrated in the "condensed" phase near the pole of magnet in the region 0 x s ′ ≤ ≤ , the first term in the left part of ( 18) becomes much more then second one and this equation gives a linear dependence of s on Φ 0 :

s= Φ 0 / Φ c . ( 19
)
Influence of redistribution of particles concentration on viscous and magnetic forces acting in the considered volume of magnetic fluid is of considerable interest for this problem.

If w is a length of the layer in the direction of axis z viscous force F τ acting on the moving upper boundary of the layer is

0 0 0 ( ) ( / ) [ ( )] l l

F w y h dx wv h x dx

τ τ η = = = Φ ∫ ∫ .
Value of this force F τ0 for the initial homogeneous fluid with constant concentration of particles Φ 0 and constant dynamic viscosity η 0 = η (Φ 0 ), will be equal to F τ0 =(wv 0 l η 0 /h).

It is convenient to describe the influence of concentration redistribution on this force by using the dimensionless coefficient

1 0 0 / [ ( )] k F F x dx τ τ τ η′ ′ ′ = = Φ ∫ , where 0 / η η η ′ = .
At the absence of phase transition we get

1 0 [ ( )] g k x dx τ η′ ′ ′ = Φ
∫ , whereas at the presence of the phase transition

1 1 0 ( ) [ ( )] ( ) [ ( )] s c g c g s s k dx x dx s x dx τ η η η η ′ ′ ′ ′ ′ ′ ′ ′ ′ = Φ + Φ = Φ + Φ ∫ ∫ ∫ . ( 20
)
Value of magnetic force F m acting on the magnetic fluid volume along x axis will be defined from [START_REF] Zubarev | Phase transitions in electro-and magnetorheological fluids[END_REF] as:

0 0 1 ( ) ( ) m m dH F M wh x L dx dx m ξ = Φ ∫ .
Value of dimensionless magnetic force f m =F m /(whμ 0 M m H * ) will be defined by the next way: a) at the absence of phase transition

[ ] 0 1 ( ) m g dH f L UH x dx dx ′ ′ ′ ′ = Φ ′ ∫
, or with taking into account (

g m g g d f dx U dx U Φ   = = Φ -Φ   ′ ∫ , 14), 0 1 1 1 (0) (1)
H s m g c H s dH f L UH x dx L UH dH dx ′ ′ ′ ′ ′ ′ ′ ′ = Φ + Φ ′ ∫ ∫ = [ ] [ ] ( ) (0) 1 ( ) ( ) ( )
c g g H s sh UH s U U H sh UH s ′ ′ Φ   = Φ -Φ +   ′ ′ . (0) 1 ( ) (1) ln (0) ( )
Value of this force f m0 for initial homogeneous fluid with constant concentration of particles Φ 0 will be equal to

[ ] [ ] (0) 0 0 0 (1) (1) (0) ( ) ln (0) (1)
H m H H sh UH f L UH dH U H sh UH ′ ′ ′ ′ Φ ′ ′ = Φ = ′ ′ ∫ . ( 23
)
It is convenient to describe the influence of concentration redistribution on this force by using the dimensionless coefficient

k m = f m / f m0 .
Some of the obtained solutions become more simple if magnetic fluid is in a magnetic saturation at large values of magnetic field intensity when U>>1. In this case the Langevin function ( ) 1 L UH ′ ≈ and the solutions of above problem take a form: exp( )

g B UH ′ Φ =
. And: a) at the absence of phase transition

1 1 0 0 exp( ) B UH dx -   ′ ′ = Φ     ∫ , 0 (0) (1) 1 (0) (1)
g g m k U H H Φ -Φ = ′ ′ Φ - . (24)
b) at the presence of phase transition

* exp[ ( )] g B UH s ′ = Φ - , 1 * 0 exp[ ( )] exp[ ( )] c g s s UH s UH x dx ′ ′ ′ ′ Φ + Φ - = Φ ∫ , (25) 0 0 ( ) (1)
(0) ( ) 1 (0) (1) (0) (1)
g g c m s H H s k H H U H H Φ -Φ ′ ′ Φ - = + ′ ′ ′ ′ Φ - Φ - .

Results and discussion.

For further concrete calculations the following model approximation of distribution of magnetic field is accepted:

2 1 / (1 3 ) H x ′ ′ = +
. This approximation is based on our measurements of magnetic field intensity in a gap of real magnetic fluid seal. Compliance between this approximation and experimental data is presented in Fig. 3. Results of calculations fulfilled on the base of formulas ( 15)-(23) for λ=1.5 are presented in Figs 45678. As it will be shown below the phase transition in magnetic fluid with reasonable average concentration of particles is observed at large values of magnetic parameter U, when magnetic fluid is in saturation. In this case (λ=1,5 and L(U H ′ )≈1) the equations of the phase co-existence ( 10)- [START_REF] Zubarev | Theory of structural transformations in ferrofluids: chains and gas-liquid phase transitions[END_REF] and Fig. 1 give * g Φ =0,489 and Φ c =0,552.

Typical distribution of particles concentration Φ in magnetic fluid depending on magnetic parameter U is presented in Fig. 4 at initial average concentration Φ 0 = 0,1. A large number of particles are concentrated in the region of maximum value of magnetic field intensity, and concentration of particles decreases far from it.

With increasing of U concentration of particles in the region of maximum magnetic field intensity increases until reaches value Φ g* = 0,489 at U equal about 9.

Here the phase transition which is followed by concentration jump from Φ g* = 0,489 to Φ с = 0,552 is observed. Value * g Φ = 0,489 defining position x s ′ = of the phase transition interface is displaced to the right when U increases. To the left of it the "condensed" phase with constant value of concentration of particles Φ с = 0,552 takes place and the dimension of this area increases with the increasing of U. This fact is demonstrated by the curves 5, 6, 7 in Fig. 4. At large values of parameter U, practically all number of magnetic particles is collected in area with the maximum intensity of magnetic field so the remained area becomes practically without them, that is not magnetized. In magnetic fluid seals this not magnetized volume of fluid ceases to be kept in them and can flow outside practically in the form of pure base fluid. At early stages of research of magnetic fluid seals this phenomenon was connected with not good quality of magnetic fluid. As it appears from the given results, this phenomenon can take place also in a good magnetic fluid as a result of processes of magnetophoresis, diffusion and phase transition which are of their essential attributes.

Coordinate of phase transition boundary s depends also on initial average concentration of particles Φ 0 . It is obvious that with its increasing phase transition has to come at smaller values of magnetic field intensity that is at smaller values of U. The corresponding dependence is presented in Fig. 5. Areas to the left of the curves represented in Fig. 5 correspond to the states excluding phase transition in magnetic fluid at given value of magnetic parameter U. Points of crossing of the curves with axis Φ 0 define that minimum value of average concentration of particles Φ 0min at which phase transition in fluid can take place at given U: 0 min ( ) 0 s Φ = . Fig. 5 demonstrates that with increasing of average concentration of particles Φ 0 at the same value of magnetic parameter U the value of s increases too, that is the area of the condensed phase becomes larger. The same takes place and at increasing of magnetic field intensity (magnetic parameter U) for fluid with constant value of average concentration Φ 0 . At large value of U curves on the Fig. 5 tend to the straight line described by a formula [START_REF] Hall | Another Hard-Sphere Equation of State[END_REF] 0 / 0,552 s = Φ

, and all curves have almost the same inclination at large s.

The dependence of this minimum value of average concentration of particles Φ 0min on magnetic parameter U is presented in Fig. 6. This dependence can be used also for the solution of the inverse problemdetermination of minimum value of magnetic parameter min U only at excess of which phase transition in magnetic fluid can take place at a given value of average concentration of particles 0 Φ . At U>> 1 this dependence is determined from a formula (25) as follows

1 0 min * 0 exp( ) exp[ ( )] g U UH x dx ′ ′ ′ Φ = Φ -∫ .
The influence of the considered processes on the magnetic force acting on the volume of magnetic fluid, and also on force of viscous friction on moving boundary described by formulas ( 20) -( 23), is presented in Figs. 7,8.

As it can see from Fig. 7 a strong nonlinear dependence of a dynamic coefficient of viscosity of magnetic fluid on concentration of particles described by formula of Krieger-Dougherty (13) leads to essential increase in relative value of viscous force (coefficient of viscous force k τ ) with increasing of magnetic parameter U when concentration of particles in the area with large intensity of magnetic field increases. Φ further remain fixed. Further smooth growth k τ is generally caused by expansion of area of the "condensed" phase of magnetic particles.

More complicated influence the redistribution of concentration of particles in magnetic fluid has on the magnetic force acting on it. It is connected with the fact that the value of this force depends not only on magnetization of fluid, but also on the value of gradient of magnetic field intensity. As a result of magnetophoresis the concentration of particles, and, therefore, magnetization of fluid, have the largest values in the point of maximum value of magnetic field intensity. However, as one can see from Fig. 3 in this area the gradient of magnetic field intensity decreases to zero. Therefore, as it is seen from Fig. 8, at moderate values of magnetic parameter U < 5 coefficient of magnetic force m k increases approximately by 10% due to the fact that concentration of magnetic particles in the area with large gradients of magnetic field increases. This fact was noted also earlier in [START_REF] Bashtovoi | The effect of diffusion processes on the statics of magnetic fluids[END_REF][START_REF] Bashtovoi | Influence of Mass Transfer Processes on Couette Flow of Magnetic Fluid[END_REF]. At large values of U an increasing number of particles concentrate in the area with the maximum value of magnetic field intensity, that is in the area with its minimum gradient. It leads to monotonous decreasing of coefficient of magnetic force m k with a further increasing of U. This decreasing is more visible for fluids with less average concentration of particles. In the considered situations this decreasing can reach considerable values to 40%.

Conclusion

Thus, it is established that processes of magnetophoresis and Brownian diffusion can have significant effect on characteristics of flows of magnetic fluid, namely on the values of volume magnetic force and viscous tension. In this connection forces of viscous friction tend to increase, and magnetic forces can be increased, as well as decreased depending on value and configuration of the magnetic field.

Inclusion in consideration of possible phase transitions in magnetic fluid allows us to perform physically consistent calculation of characteristics of magnetic fluid flow taking into account processes of magnetophoresis and Brownian diffusion at large values of magnetic field intensity.

It is shown that phase transition in magnetic fluid is observed when the magnetic parameter exceeds a certain value, and the area of the condensed phase increases with its increasing. At large values of magnetic parameter the coordinate of phase transition interface almost linearly depends on average concentration of particles in the problem under consideration.

Fig. 1 .

 1 Fig. 1. Diagram of phase equilibrium in magnetic fluid with magnetization of saturation.

Fig. 2 .

 2 Fig. 2. Geometry of the problem. c -region of condenced phase, g -region of the gas phase, MP -pole of the magnet system.

  b) at the presence of phase transition [ ]

Fig. 3 .

 3 Fig. 3. Distribution of magnetic field intensity in a gap of real magnetic fluid seal. The solid lineapproximation by expression

Fig. 4 .

 4 Fig. 4. Typical distribution of particle concentration Φ in magnetic fluid depending on magnetic parameter U.

Fig. 5 .

 5 Fig. 5. Dependence of coordinate of phase transition point s in magnetic fluid on average particle concentration Φ 0 at different values of magnetic parameter U.

Fig. 6 .

 6 Fig.6. Dependence of minimum value of average concentration of particles Φ 0min on magnetic parameter U at which a phase transition is taken place.

Fig. 7 .

 7 Fig. 7. Dependence of viscous force coefficient k τ on magnetic parameter U for different values of average concentration of particles Φ 0 .

Fig. 8 .

 8 Fig. 8. Dependence of magnetic force coefficient k m on magnetic parameter U for different values of initial average concentration of particles Φ 0 .

Acknowledgement

The authors express their thanks to Belarusian Republican Foundation of Fundamental Researches (project No 01T17MC-013) and Russian Fund for Basic Research, 16-58-12003, the Program of Russian Federation Ministry of Science and Education, project 3.1438.2017/4.6 for financial support of this work.