Patricia Henri

Dominique Rumeau
email: dominique.rumeau@cea.fr

Inter-kingdom complementation discloses functional convergence of lipocalins

Keywords: Apolipoprotein D, Lipocalin, Chloroplast, Oxidative stress, Arabidopsis 1

Background

The chloroplastic lipocalin (LCNP) is induced in response to various abiotic stresses including high light, dehydration and low temperature. It contributes to protection against oxidative damage promoted by harmful conditions by preventing accumulation of hydroxy fatty acids and lipid peroxidation. In contrast to animal lipocalins, LCNP is poorly characterized and the molecular mechanism by which it exerts protective effects during oxidative stress is largely unknown. LCNP is considered the ortholog of human apolipoprotein D (APOD), a protein whose lipid antioxidant function has been characterized. Here, we investigated whether APOD could functionally replace LCNP in Arabidopsis.

Results

We introduced APOD cDNA fused to a chloroplast transit peptide encoding sequence in an Arabidopsis LCNP KO (knock-out) mutant line and challenged the transgenic plants with different abiotic stresses. We demonstrated that expression of human APOD in Arabidopsis can partially compensate for the lack of the plastid lipocalin. However, if the results obtained with the drought and oxidative stresses point to the protective effect of constitutive expression of APOD in plants lacking LCNP, this effect is not as effective as that conferred by LCNP overexpression. Moreover, when investigating APOD function in thylakoids after high light stress at low temperature, it appeared that APOD could not contribute to qH, a slowly reversible form of non-photochemical chlorophyll fluorescence quenching, as described for LCNP.

Conclusions

The results are consistent with a conserved function of APOD and LCNP under stressful conditions. The work contributes towards understanding the molecular mechanism of action of LCNP in oxidative stress protection and provides the basis for new research assumptions.

Background.

Lipocalins constitute an evolutionary conserved family of small proteins widely distributed in nature whose common feature is their ability to bind small hydrophobic molecules [START_REF] Flower | The lipocalin protein family: structure and function[END_REF].

Although their primary amino acid sequences are not highly conserved, they all display a repeated structurally conserved region and an eight-stranded antiparallel b-sheet that together form a compact barrel structure with a calyx-shaped ligand pocket [START_REF] Akestrom | Lipocalins: unity in diversity[END_REF]. Shape, size and properties of this pocket vary among family members in line with the different ligands that can be accommodated. A large variety of ligands including lipids, steroids as well as secondary metabolites such as vitamins, cofactors and odorants have been reported [START_REF] Flower | The lipocalin protein family: structure and function[END_REF][START_REF] Schiefner | The menagerie of human lipocalins: a natural protein scaffold for molecular recognition of physiological compounds[END_REF]. Related to their ligands, lipocalins can fulfill a wide variety of functions such as transport of small molecules, regulation of developmental processes, signal transduction and response to stress. Because of their possible involvement in various diseases including lipid disorders, neurodegenerative diseases and cancer [START_REF] Bratt | Lipocalins and cancer[END_REF][START_REF] Xu | Lipocalins as biochemical markers of disease[END_REF], human and animal lipocalins are extensively studied.

In striking contrast, plant lipocalins are poorly characterized. The first lipocalins reported in plants were the violaxanthin-de-epoxidase (VDE) and the zeaxanthin epoxidase (ZEP), two enzymes involved in the xanthophyll cycle [START_REF] Bugos | Xanthophyll cycle enzymes are members of the lipocalin family, the first identified from plants[END_REF]. Given the peculiar molecular and structural features of both enzymes and the strict definition of lipocalins, VDE and ZEP are now classified as lipocalin-like proteins [START_REF] Charron | Identification, expression, and evolutionary analyses of plant lipocalins[END_REF]. Genomic data mining demonstrated that plants possess also true lipocalins, which were classified as temperature-induced lipocalin (TIL) and chloroplastic lipocalin (LCNP 1 ) [START_REF] Charron | Identification, expression, and evolutionary analyses of plant lipocalins[END_REF]. Investigation of gene regulation, proteomic analyses and reverse genetic studies have related TIL and LCNP with stress response and tolerance. TIL appeared essential for thermotolerance [START_REF] Chi | Temperature-induced lipocalin is required for basal and acquired thermotolerance in Arabidopsis[END_REF], its accumulation during cold acclimation was described in Arabidopsis leaves [START_REF] Kawamura | Mass spectrometric approach for identifying putative plasma membrane proteins of Arabidopsis leaves associated with cold acclimation[END_REF] and in Siberian spruce needles [START_REF] Kjellsen | Proteomics of extreme freezing tolerance in Siberian spruce [Picea obovata][END_REF]. TIL gene expression has been demonstrated to increase with heat and cold treatment in wheat and Arabidopsis [START_REF] Charron | Molecular and structural analysis of a novel temperature stress-induced lipocalin from wheat and Arabidopsis[END_REF] and peach fruits [START_REF] Zhang | Crucial roles of membrane stability and its related proteins in the tolerance of peach fruit to chilling injury[END_REF]. In Oryza sativa, a TIL encoding gene was reported to be highly up regulated under heat stress in panicles of tolerant rice cultivar [START_REF] Prasanth | Trait and marker associations in Oryza nivara and O. rufipogon derived rice lines under two different heat stress conditions[END_REF]. TIL expression was also significantly induced by a cold treatment in Medicago sativa subsp. falcata, a forage legume cold tolerant [START_REF] He | A temperature induced lipocalin gene from Medicago falcata [MfTIL1] confers tolerance to cold and oxidative stress[END_REF] and in a grapefruit following a conditioning treatment that enhances chilling tolerance [START_REF] Maul | Temperature conditioning alters transcript abundance of genes related to chilling stress in "Marsh" grapefruit flavedo[END_REF]. In grape berries of Vitis labruscana, salicylic acid treatment up-regulated TIL preventing postharvest loss during cold storage [START_REF] Cai | Biochemical and proteomic analysis of grape berries [Vitis labruscana] during cold storage upon postharvest salicylic acid treatment[END_REF]. In mango fruits a TIL homolog and its corresponding gene appeared to be up-regulated during a brassinolide-mediated response to cold stress [START_REF] Li | Brassinolide enhances cold stress tolerance of fruit by regulating plasma membrane proteins and lipids[END_REF].

Moreover, TIL plays a role in salt tolerance in Populus euphratica, a salt-tolerant poplar species [START_REF] Brinker | Linking the salt transcriptome with physiological responses of a salt-resistant Populus species as a strategy to identify genes important for stress acclimation[END_REF][START_REF] Abo-Ogiala | Temperature-induced lipocalin [TIL] is translocated under salt stress and protects chloroplasts from ion toxicity[END_REF].

Considering LCNP, it has been demonstrated that high light, dehydration, oxidative stress and abscisic acid (ABA) induced transcript and protein accumulation in Arabidopsis [START_REF] Levesque-Tremblay | The chloroplastic lipocalin AtCHL prevents lipid peroxidation and protects Arabidopsis against oxidative stress[END_REF]. In wheat, cold acclimation induced LCNP mainly in freezing-tolerant cultivar while heat exposure downregulated gene expression [START_REF] Charron | Identification, expression, and evolutionary analyses of plant lipocalins[END_REF]. In Festuca arundinacea, salt stress induced a LCNP protein level increase, which was higher in a salt-tolerant genotype [START_REF] Pawlowicz | Remodeling of chloroplast proteome under salinity affects salt tolerance of Festuca arundinacea[END_REF]. Furthermore, LCNP abundance similarly decreased in response to drought treatment in two cultivars of F arundinacea, however, after rewatering lipocalin level returned to initial level only in the high-droughttolerant genotype [START_REF] Kosmala | Changes in the chloroplast proteome following water deficit and subsequent watering in a high-and a low-droughttolerant genotype of Festuca arundinacea[END_REF].

In spite of the accumulating knowledge regarding the contribution of TIL and LCNP to the resistance of plants to environmental constraints little is known about their role at the cellular and molecular levels. It has been suggested that plant lipocalins prevent accumulation of hydroxy fatty acids and thus stop the lipid peroxidation chain reactions initiated by reactive oxygen species (ROS) generated during stresses [START_REF] Chi | Temperature-induced lipocalin is required for basal and acquired thermotolerance in Arabidopsis[END_REF][START_REF] Charron | The plant Apolipoprotein D ortholog protects Arabidopsis against oxidative stress[END_REF]. In 2009, Levesque-Tremblay et al. [START_REF] Levesque-Tremblay | The chloroplastic lipocalin AtCHL prevents lipid peroxidation and protects Arabidopsis against oxidative stress[END_REF] demonstrated that when exposed to excess light, LCNP KO plants display a rapid accumulation of hydroxy fatty acids relative to the wild-type, whereas the lipid peroxidation level remains very low in LCNP overexpressing plants. Boca et al. [START_REF] Bocca | Arabidopsis lipocalins AtCHL and AtTIL have distinct but overlapping functions essential for lipid protection and seed longevity[END_REF] demonstrated that LCNP deficiency dramatically enhances the photosensitivity of mutants affected in lipid protection mechanisms strengthening the hypothesis that lipocalins prevent lipid peroxidation. Interestingly, recent data obtained by Malnoë et al. [START_REF] Malnoë | The Plastid Lipocalin LCNP Is Required for Sustained Photoprotective Energy Dissipation in Arabidopsis[END_REF] demonstrated a link between LCNP and qH a slowly reversible form of non-photochemical energy quenching (NPQ), a mechanism by which photosynthetic organisms harmlessly dissipate excess absorbed light energy. The authors suggested that the accumulation of peroxidized lipids observed in Arabidopsis lcnp mutants following abiotic stress is a consequence of the absence of the photoprotective NPQ mechanism enabled by the plastid lipocalin and, thus, that LCNP might function in preventing the formation of peroxidized lipids. In addition to or as part of its role in NPQ, it is suggested that LCNP might detoxify peroxidized lipids directly [START_REF] Malnoë | The Plastid Lipocalin LCNP Is Required for Sustained Photoprotective Energy Dissipation in Arabidopsis[END_REF].

Phylogenetic studies established that plant lipocalins share similarities with the bacterial lipocalin, the mammalian apoliprotein D (APOD) and the insect Lazarillo protein. As observed in plants, the correlation between lipid oxidation status and lipocalin abundance was reported for animal and human organisms. For instance in mice, loss of APOD increased lipid peroxidation in the brain [START_REF] Ganfornina | Apolipoprotein D is involved in the mechanisms regulating protection from oxidative stress[END_REF]. Conversely, human APOD overexpression reduced lipid peroxidation in mice and Drosophila [START_REF] Ganfornina | Apolipoprotein D is involved in the mechanisms regulating protection from oxidative stress[END_REF][START_REF] Muffat | Human ApoD, an apolipoprotein up-regulated in neurodegenerative diseases, extends lifespan and increases stress resistance in Drosophila[END_REF]. Increased APOD levels in the human brain were reported under conditions that promote lipid peroxidation such as aging or Alzheimer's disease [START_REF] Terrisse | Increased levels of apolipoprotein D in cerebrospinal fluid and hippocampus of Alzheimer's patients[END_REF]. At the biochemical level, APOD ligands are well-known [START_REF] Rassart | Apolipoprotein D[END_REF][START_REF] Eichinger | Structural insight into the dual ligand specificity and mode of high density lipoprotein association of apolipoprotein D[END_REF][START_REF] Vogt | Bacterially produced apolipoprotein D binds progesterone and arachidonic acid, but not bilirubin or E-3M2H[END_REF] and the molecular mechanism underlying the antioxidative properties of APOD has been established. Indeed, the ability for APOD to control lipid peroxidation by catalyzing the reduction of radicalpropagating lipid hydroperoxides has been characterized [START_REF] Bhatia | Selective reduction of hydroperoxyeicosatetraenoic acids to their hydroxy derivatives by apolipoprotein D: Implications for lipid antioxidant activity and Alzheimer's disease[END_REF].

In order to characterize the mechanism underlying the protective role of LCNP against abiotic stresses we investigated whether APOD whose biochemical properties involved in oxidative stress tolerance have been described, is functionally orthologous to the chloroplast lipocalin.

To do so, human APOD cDNA fused to a chloroplast transit peptide encoding sequence was introduced in an Arabidopsis LCNP KO mutant line. Transgenic plants expressing APOD were challenged with different abiotic stresses. The plants displayed partial tolerant phenotype suggesting that human APOD can replace in part, LCNP. These data provide a base of understanding the antioxidative mechanism of the plastid lipocalin and some testable experimental assumptions.

Results

Targeting the human APOD to Arabidopsis thaliana chloroplast: gene expression and protein localization.

To determine whether APOD is functionally orthologous to LCNP we took an in vivo complementary approach expressing the human APOD encoding sequence in Arabidopsis. In order to target APOD to the chloroplast where LCNP is located, the putative transit peptide encoding sequence of LCNP was fused to the cDNA coding for APOD (Fig. 1A). The chimeric construct under the control of the Cauliflower Mosaic virus (CaMV) 35S promoter was introduced in an Arabidopsis mutant line lacking LCNP [START_REF] Levesque-Tremblay | The chloroplastic lipocalin AtCHL prevents lipid peroxidation and protects Arabidopsis against oxidative stress[END_REF]. Primary transgenic Arabidopsis plants were identified following Basta treatment of one-week old seedlings. In the T2 generation, 5 independent lines containing a single insertion were selected by analyzing the segregation ratio of herbicide resistance. Seeds were collected from homozygous T3 plants and used for further experiments.

The 5 transformant lines a661, a351, a421, a551, and a212 showed no phenotypic changes compared to wild type. In the five transgenic lines, APOD expression was analyzed by reverse transcriptase polymerase chain reaction (RT-PCR, Fig. 1B). Two lines a661 and a421 exhibited a higher expression as compared to the three other lines. At the protein level, immunoblot analysis was conducted to detect the presence of APOD (Fig. 1C). A strong band was observed in the two transgenic lines containing the higher amount of APOD transcripts (a661 and a221) while the protein was hardly detectable in the three other lines. The molecular weight of the band was estimated at 23 KDa, which does not correspond to the molecular weight of APOD estimated at 19.3 KDa suggesting either that the transit peptide was not properly processed or that APOD was post-translationally modified.

Within chloroplasts, LCNP was localized in the thylakoid lumen using a cellular fractionation approach [START_REF] Levesque-Tremblay | The chloroplastic lipocalin AtCHL prevents lipid peroxidation and protects Arabidopsis against oxidative stress[END_REF]. To investigate APOD subcellular localization we performed western-blot experiments on plastid sub-fractions isolated from chloroplasts purified by Percoll gradient centrifugation (Fig. 2). APOD appeared to be of low abundance in the total cell extract. As expected its concentration slightly increased in purified chloroplasts where the human lipocalin was present in both compartments, thylakoids and stroma. However, in NaCl stripped thylakoids APOD could not be detected suggesting that the protein is ionically attached to the membranes. These results suggest that in chloroplasts, APOD remains soluble in the stroma and also ionically attaches to the thylakoids.

To investigate further the localization of APOD in transgenic plants, the protein was tagged with EGFP at its C-terminus and transiently expressed using the CaMV 35S promoter in Nicotiana benthamiana leaves (Fig. 3). Six days later, leaf epidermal cells were observed by confocal microscopy. EGFP fluorescence coincided strongly with chlorophyll autofluorescence (Fig. 3A) indicating that, as expected, APOD:EGFP fusion product was associated with the chloroplast. In particular, confocal imaging revealed a strong fluorescence within the guard cell chloroplasts (Fig. 3A). Furthermore, EGFP fluorescence was also detected in other subcellular structures throughout the cell. To identify EGFP labeled structures co-localization experiments were performed with different compartment markers (Fig. 3B and3C). Fluorescence was evident in the endoplasmic reticulum network (ER) and in Golgi as demonstrated by the co-localization of the tagged protein with the mCherry signal targeted either to the ER (Fig. 3B) or to the Golgi (Fig. 3C). In contrast co-localization experiments performed with a marker of plasmalemma and tonoplast excluded the presence of proteins in these membranes (data not shown).

Effect of APOD expression on abiotic stress tolerance

The chloroplast lipocalin plays an important role in tolerance to environmental constraints. In Arabidopsis it has been demonstrated that a high level of LCNP helped plants to cope with environmental abiotic stresses such as drought, high light, cold acclimation and oxidative stress.

Conversely plants lacking LCNP exhibited more damage under abiotic stresses [START_REF] Levesque-Tremblay | The chloroplastic lipocalin AtCHL prevents lipid peroxidation and protects Arabidopsis against oxidative stress[END_REF]. In order to investigate whether APOD could also confer stress tolerance and play a role similar to that of LCNP in protecting plants, the transgenic lines (a661 and a421) expressing APOD in the LCNP KO background were exposed to different stress conditions including drought (Fig. 4) and oxidative stress (Fig. 5).

Arabidopsis lcnp, wild-type and transgenic seeds were planted in a mix of soil and sand. Three weeks later, water was withheld for 10 days. lcnp mutants showed severe withering symptoms as compared to wild type plants, attesting of the great sensitivity of these plants to a water deficit. The plants over-expressing LCNP (LCNP O/E) and those expressing APOD were only slightly wilted (Fig. 4A). When plants were re-watered, none of the KO mutant plants survived (Fig. 4B). In contrast, Col0, LCNP O/E and the two lines containing APOD coped well with the stress conditions. These results suggested that APOD expression in the drought-sensitive lcnp genotype could restore tolerance to a mild water stress to the level recorded in wild type plants.

Previous studies indicated that LCNP is involved in the protection of the photosynthetic apparatus against ROS. Therefore we have tested resistance of plants against cellular oxidative damage by applying the herbicide paraquat (PQ, methyl viologen) to leaf disks segments from transgenic plants. The electrons produced during photosynthesis reduce PQ and free oxygen radicals are formed initiating an oxidative stress and quickly damaging the photosynthetic apparatus. Leaf disks were exposed to 1 µM PQ for 8 hours under a light source. Functional damage was monitored by measuring the variable fluorescence (Fv)/maximal fluorescence (Fm) values representing the maximal yield of photochemistry in the photosynthetic process (Fig. 5). The control and lcnp mutant completely lost their photosynthetic function. As expected the LCNP O/E line was significantly more resistant to PQ treatment. Interestingly, plants expressing APOD (a661 and a421) exhibited a moderate resistance while a212, a line exhibiting a very low expression of APOD displayed an oxidative stress-sensitive phenotype similar to the control (Col0). Considering these results we can conclude that the constitutive presence of APOD confers enhanced tolerance to PQ-induced oxidative stress. However oxidative tolerance did not appear as notable as that recorded in LCNP overexpressing plants.

Effect of APOD expression on sustained photoprotective energy dissipation

It has been demonstrated that LCNP contributed to qH, a NPQ component, which operates under stress conditions such as cold and high light and appeared to be photoprotective [START_REF] Malnoë | The Plastid Lipocalin LCNP Is Required for Sustained Photoprotective Energy Dissipation in Arabidopsis[END_REF].

Whether APOD can replace LCNP in this function was investigated. Detached leaves of transgenic plants were exposed to high light at low temperature for 6 hours and maximal fluorescence Fm was recorded. The suppressor of quenching 1 mutant (soq1) with enhanced qH whose phenotype is reversed when LCNP is lacking was included in the experiment. Before the stress, Col0 and the different mutant and transgenic lines displayed similar maximal fluorescence parameter (Fm, data not shown). After stress (Fig. 6), as expected since already described by Malnoë et al. [START_REF] Malnoë | The Plastid Lipocalin LCNP Is Required for Sustained Photoprotective Energy Dissipation in Arabidopsis[END_REF], soq1 displayed a lower Fm compared with Col0 while Fm in lcnp was higher than in the control plant leaves. The photoprotective function of LCNP was observed in the leaves of the transgenic line overexpressing LCNP with a maximal chlorophyll fluorescence parameter lower than in Col0. In comparison, a661 and a421 displayed a Fm similar to that recorded in lcnp indicating that APOD expression cannot protect the thylakoid membrane as LCNP does.

Discussion

Discussion

We report that expression of human APOD in Arabidopsis, by means of transgenic lines, can partially compensate for the lack of the plant lipocalin, LCNP. The results are consistent with a conserved function of APOD and LCNP under stressful conditions. However, if the results obtained with the drought and oxidative stresses point to the protective effect of constitutive expression of APOD in plants lacking LCNP, this effect is not as effective as that conferred by LCNP overexpression. Moreover, when investigating APOD function in NPQ after high light stress at low temperature, it appeared that APOD could not contribute to qH as described for LCNP [START_REF] Malnoë | The Plastid Lipocalin LCNP Is Required for Sustained Photoprotective Energy Dissipation in Arabidopsis[END_REF]. It is intriguing to notice that in Arabidopsis, APOD can fulfill only certain functions of LCNP.

Molecular phylogenetic analysis of lipocalins not only revealed that the chloroplastic LCNP shared significant structural homology with the mammalian APOD [START_REF] Charron | Identification, expression, and evolutionary analyses of plant lipocalins[END_REF][START_REF] Ganfornina | A Phylogenetic Analysis of the Lipocalin Protein Family[END_REF], but also suggested that both proteins might correspond to the same overall function in terms of general nature of binding ligands since both proteins appeared in the same clade [START_REF] Charron | Identification, expression, and evolutionary analyses of plant lipocalins[END_REF][START_REF] Lakshmi | Structure-Based Phylogenetic Analysis of the Lipocalin Superfamily[END_REF]. In vitro ligand binding assays have shown that APOD is able to bind a variety of lipids in its hydrophobic binding cavity [START_REF] Rassart | Apolipoprotein D[END_REF]. More recently, it has been clearly demonstrated that APOD displayed the highest binding affinity for arachidonic acid [AA] and progesterone [START_REF] Rassart | Apolipoprotein D[END_REF][START_REF] Eichinger | Structural insight into the dual ligand specificity and mode of high density lipoprotein association of apolipoprotein D[END_REF][START_REF] Vogt | Bacterially produced apolipoprotein D binds progesterone and arachidonic acid, but not bilirubin or E-3M2H[END_REF]. Partial functional complementation of LCNP deficiency by APOD cannot be related to the nature of APOD ligands since neither AA nor progesterone are present in Arabidopsis. However we cannot exclude the possible affinity of APOD to other structurally related lipids that may be involved in signaling cascades. Indeed numerous signaling lipids whose synthesis can rapidly be activated upon abiotic stress signals have been identified in plants [for a review see 36]. In Arabidopsis, APOD could exert a control of the signaling lipid molecules that decipher the stress response cascade and thus modulate transgenic plants tolerance. In plants, a similar function for LCNP cannot be excluded as well.

During last decade, several authors have established that APOD function extended well beyond the transport of lipophilic molecules. Therefore, an extensive work allowed to establish a correlation between lipid oxidation status and APOD abundance in human and mammals.

Increased APOD expression was reported in the human brain under conditions that promote lipid peroxidation and up-regulation of antioxidant gene expression such as aging or neuropathological conditions including schizophrenia, Parkinson's and Alzheimer's diseases [START_REF] Terrisse | Increased levels of apolipoprotein D in cerebrospinal fluid and hippocampus of Alzheimer's patients[END_REF][START_REF] Hou | Lipid signalling in plant responses to abiotic stress[END_REF][START_REF] Ordoñez | Apolipoprotein D expression in substantia nigra of Parkinson disease[END_REF]. Studies in mice showed that loss of APOD increased lipid peroxidation in the brain, that cerebral APOD expression is increased in response to Paraquat-induced oxidative stress and that human APOD expression reduced lipid peroxidation [START_REF] Ganfornina | Apolipoprotein D is involved in the mechanisms regulating protection from oxidative stress[END_REF][START_REF] Muffat | Human ApoD, an apolipoprotein up-regulated in neurodegenerative diseases, extends lifespan and increases stress resistance in Drosophila[END_REF]. In addition, expression of human APOD reduced the accumulation of aldehydic end-products of lipid peroxidation in an aged Drosophila model [START_REF] Muffat | Human ApoD, an apolipoprotein up-regulated in neurodegenerative diseases, extends lifespan and increases stress resistance in Drosophila[END_REF][START_REF] Walker | Overexpression of a Drosophila homolog of apolipoprotein D leads to increased stress resistance and extended lifespan[END_REF]. In plants, the protective role of LCNP in oxidative stress has been related to its ability to prevent or modulate lipid peroxidation caused by ROS action [START_REF] Levesque-Tremblay | The chloroplastic lipocalin AtCHL prevents lipid peroxidation and protects Arabidopsis against oxidative stress[END_REF]. Thus, a LCNP KO Arabidopsis line, which exhibited serious damage upon photo-oxidative stress induced by drought, high light or paraquat displayed a rapid accumulation of hydroperoxyl fatty acids. In contrast, in LCNP over-expressing lines, which coped better with these conditions, the lipid peroxidation level remains very low [START_REF] Levesque-Tremblay | The chloroplastic lipocalin AtCHL prevents lipid peroxidation and protects Arabidopsis against oxidative stress[END_REF]. Related to APOD antioxidative function, it has been demonstrated that the human lipocalin had the capacities to scavenge hydroxyl radicals [START_REF] Zhang | Antioxidant activities of recombinant amphioxus [Branchisostoma belcheri] apolipoprotein D[END_REF] and catalyze the reduction of hydroperoxide of AA to the non-reactive alcohols, thereby preventing lipid peroxidation chain reactions [START_REF] Bhatia | Selective reduction of hydroperoxyeicosatetraenoic acids to their hydroxy derivatives by apolipoprotein D: Implications for lipid antioxidant activity and Alzheimer's disease[END_REF]. This reaction involves a highly conserved methionine residue (Met93 in human APOD) located at the hydrophobic surface of the protein [START_REF] Bhatia | Selective reduction of hydroperoxyeicosatetraenoic acids to their hydroxy derivatives by apolipoprotein D: Implications for lipid antioxidant activity and Alzheimer's disease[END_REF] and results in oxidation of the methionine sulfhydryl to form methionine sulfoxide (MetSO). A similar molecular function could be involved to explain the protective role of APOD in Arabidopsis. Indeed, it can be speculated that APOD could exert its reductive activity on plant lipid hydroperoxides such as hydroperoxides of linolenic (18:3), linoleic (18:2) and roughanic (16:3) acids, the major polyunsaturated fatty acids in plants. In this context, APOD could inhibit lipid peroxidation propagation initiated by drought and paraquat treatment. A lower affinity of APOD for plant hydroperoxides or an insufficient amount of APOD in plant cells could account for the lower protective effect of the human lipocalin as compared to that provided by LCNP. By analogy, the protective function towards oxidative stresses conferred by LCNP could be due to a surface methionine residue, similar to APOD Met93 and able to reduce plant fatty acid hydroperoxides, as well. In that respect, we investigated whether LCNP could exhibit a methionine residue structurally homolog of APOD Met93. In the absence of an experimentally-determined protein structure, homology modeling using Phyre2 provided a three-dimensional structure for LCNP.

Structural alignment with the human APOD model is shown in Fig 7 . The APOD Met93 is not conserved in LCNP ruling out the contribution of the Met93 homolog in LCNP oxidative protective function. However, LCNP contains many other Met residues whose putative reductive activity should be investigated in future studies [START_REF] Charron | Identification, expression, and evolutionary analyses of plant lipocalins[END_REF].

The appropriate subcellular localization and translocation of lipocalins are crucial to their functionality. LCNP located into the lumen of thylakoids [START_REF] Levesque-Tremblay | The chloroplastic lipocalin AtCHL prevents lipid peroxidation and protects Arabidopsis against oxidative stress[END_REF] has been shown to be involved in qH, which has been localized to the peripheral antenna of photosystem II, suggesting that LCNP could either directly form a NPQ site or indirectly modify the light harvesting complex II (LHCII) membrane environment [START_REF] Malnoë | The Plastid Lipocalin LCNP Is Required for Sustained Photoprotective Energy Dissipation in Arabidopsis[END_REF]. Lumen targeting is a complex process, which may involve two different pathways able to recognize a thylakoid-targeting signal peptide [START_REF] Albiniak | Targeting of luminal proteins across the thylakoid membrane[END_REF]. The N-terminal bipartite signal for chloroplast import and thylakoid transfer that drives the import of LCNP from the cytosol to the lumen has not yet been characterized. In our construct, LCNP transit peptide extension was defined based on unpublished data [Malnoë, personal communication]. In Arabidopsis transgenic plants, APOD fused to the putative transit peptide of LCNP was clearly targeted to the chloroplast (Fig. 2 and Fig. 3). However we could not find evidence that APOD penetrated into the lumen of thylakoids. Inappropriate processing could explain the APOD unexpected size described in Fig. 1 and the lack of APOD in the lumen.

APOD is a glycoprotein and we cannot exclude that glycosylation occurred in cells of transgenic plants. The localization of APOD fused to GFP into Golgi and ER supports this hypothesis (Fig. 3). Moreover, this post-translational modification could explain the unexpected molecular weight of APOD in a plant cell extract (Fig. 2). Lastly, targeting of glycoproteins into chloroplasts is well known [START_REF] Faye | Novel pathways for glycoprotein import into chloroplasts[END_REF] but import into thylakoids lumen has never been described. The lack of lipocalin into the chloroplast sub compartment, although the protein is present in the stroma, may explain why APOD did not contribute to qH as LCNP does.

Conclusions

Our data indicate that despite structural homology and similar contribution to lipid oxidation status in cells, APOD and LCNP display molecular features that confer specific functions to them. We propose that functionality and specificity likely reside in a particular methionine residue and glycosylation modifications that affect enzyme activity and subcellular localization, respectively. The findings of this study provide new leads for future research. In particular, we are examining LCNP methionine residues to assess their contribution in lipid hydroperoxide reduction.

Methods

Plant Material, Growth Conditions and Treatments

Arabidopsis plants were grown in phytotrons with an 8/16 day/night photoperiod and a light intensity of 130 µmol m -2 s -1 . Day and night temperatures were 22 and 18 °C, respectively. The relative humidity was set at 60 % and plants were cultivated at 100% relative soil water content.

For drought stress, 9 plants of each line were grown in a single pot containing 120 g of a mix of soil and sand (2/1, w/w). Three week old plants were then subjected to water deprivation by withholding watering for 10 days. For the oxidative stress treatment, about 9 mm diameter leaf disks were excised from fully expanded leaf blades and floated onto a 1 µM solution of paraquat (PQ, N,N'-dimethyl-4-4'-bipyridinium dichloride, Sigma-Aldrich) dissolved in water for 8 hours.

Nicotiana benthamiana plants were grown in soil under controlled conditions in a growth chamber (14-h photoperiod at 25°C followed by a 10-h night at 20°C). Arabidopsis and N benthamiana plants were watered with a complete nutrient solution as previously described [START_REF] Rumeau | New subunits NDH-M, -N, and -O, encoded by nuclear genes, are essential for plastid Ndh complex functioning in higher plants[END_REF].

The lcnp KO mutant line (lcnp-1) was available through the European Arabidopsis stock Center at Nottingham University. The LCNP overexpressing line was provided by Dr F Ouellet from Université du Québec (Montréal, Canada).

Chlorophyll Fluorescence Measurements

Chlorophyll fluorescence from leaf disks was measured at room temperature using a PAM-2000 modulated fluorometer (Walz, http://www.walz.com) as previously described [START_REF] Malnoë | The Plastid Lipocalin LCNP Is Required for Sustained Photoprotective Energy Dissipation in Arabidopsis[END_REF]. The maximal quantum yield of photosystem II photochemistry was measured as

Fv/Fm = [Fm -F0] / Fm
where Fm is the maximal fluorescence level (measured using a 800 msec pulse of saturating light) and F0 is the initial fluorescence level (measured after a 2 sec pulse of far-red light).

For the comparison of the different experimental groups, statistical differences of means were analyzed with the Student's t or ANOVA tests. Letters have been used to mark statistically identical groups of means. P values are given in the legend of figure for each group of means.. Chlorophyll fluorescence imaging was done with a laboratory-built instrument as described by Johnson et al. [START_REF] Johnson | A new setup for in vivo fluorescence imaging of photosynthetic activity[END_REF].

RT-PCR

Total RNA extraction, cDNA preparation, and RT-PCR were performed as previously described [START_REF] Fabre | Characterization and expression analysis of genes encoding alpha and beta carbonic anhydrases in Arabidopsis[END_REF]. Expression of APOD was analyzed by RT-PCR with specific primers TepGW and GWapoD (Additional file 1). As control, RT-PCR was performed using actin gene-specific primers (At2g37620, Additional file 1).

Construction of Transgenic Plants

DNA sequences, which encode human APOD and the putative transit peptide of LCNP from Arabidopsis (At3g47860) were amplified by polymerase chain reaction (PCR). The primers used for PCR are indicated in Fig. 1A, their sequences are listed in Additional file 1. The final PCR product displayed attB recombination sites, which allowed cloning in pDONR201 vector using a Gateway BP Clonase enzyme mix (Invitrogen). One entry clone was fully sequenced before subsequent cloning in the binary Gateway destination vectors pB2GW7 and pB7FWG2 (Plant Systems Biology, VIB-Ghent University, Belgium; [START_REF] Karimi | GATEWAY vectors for Agrobacterium-mediated plant transformation[END_REF]) by using a Gateway LR Clonase enzyme mix (Invitrogen). The pB2GW7 vector allows expression of the cloned sequence and pB7FWG2 vector allows fusion of EGFP with the APOD C-terminus. For both vectors, the cDNA is placed under the control of the 35S CaMV promoter and of the 3′ untranslated transcribed region of a nopaline synthase gene. The binary constructs were introduced into the Agrobacterium tumefaciens strain C58. The resulting bacterial culture containing the recombinant pB2GW7 was used to transform Arabidopsis ecotype Columbia-0 (Col0) the standard flower dip method [START_REF] Clough | Floral dip: a simplified method for Agrobacterium-mediated transformation of Arabidopsis thaliana[END_REF]. Transgenic plants were selected in soil after Basta treatment. The recombinant Agrobacterium strain containing the pB7FWG2 construct was infiltrated in Nicotiana benthamiana leaves according to the protocol described by Leuzinger et al. [START_REF] Leuzinger | Efficient agroinfiltration of plants for high-level transient expression of recombinant proteins[END_REF].

Chloroplast Purification and Chlorophyll Measurement

Intact chloroplasts from Arabidopsis leaves were extracted and purified by isopycnic centrifugation according to Seigneurin-Berny et al. [START_REF] Seigneurin-Berny | Purification of intact chloroplasts from Arabidopsis and spinach leaves by isopycnic centrifugation[END_REF]. All operations were carried at 5°C. In brief, Arabidopsis leaves (100 g, from 5 week-old plants) were ground in 400 mL of grinding

Protein Extraction, SDS-PAGE and Western Blotting

Preparation of a total protein extract from Arabidopsis leaves was performed according to Rumeau et al. [START_REF] Faye | Novel pathways for glycoprotein import into chloroplasts[END_REF]. Following grinding in liquid nitrogen, the powder was resuspended in extraction buffer [10 mM Tris-HCl pH 8 containing 20 mM NaCl, 2 mM Na2-EDTA, 5 mM Dithiothreitol]. The extract was filtrated trough two layers of muslin and the proteins were acetone [80 %; v/v] precipitated. Denaturing SDS-PAGE was performed as described by Laemmli [START_REF] Fabre | Characterization and expression analysis of genes encoding alpha and beta carbonic anhydrases in Arabidopsis[END_REF] using 13 % acrylamide gels. Proteins were either stained with BioSafe Coomassie (BioRad) or electro transferred onto 0.45 µm nitrocellulose membrane (Biotrace Pall) and probed with antibodies. Antibodies against APOD were purchased at Sigma-Aldrich/Merck. Immunocomplexes were visualized with 1:5000 diluted anti-rabbit IRDye800 antibody (Invitrogen). Detection was enabled by the Odyssey infrared imaging system.

Confocal Microscopy Observation

Pieces of N benthamiana leaves were sampled and mounted in water. Imaging was performed using a Zeiss LSM 780 confocal laser scanning microscope and either a 20x (Plan-Apo) or a 40x (C-Apo Corr FCS) objective. For EGFP and mCherry fluorescence analysis, the 488 nm excitation line of an argon laser was used and the fluorescence signal was detected using an emission band width of 490-530 for EGFP and of 590-620 for mCherry.

3D Structure Prediction and Structural Alignment

Three-dimensional structure of LCNP was predicted and modeled from Phyre 2 database [START_REF] Kelley | The Phyre2 web portal for protein modeling, prediction and analysis[END_REF].

The top model with 100 % confidence was considered. Structural alignment with the human APOD model (PDB: 2HZR; [START_REF] Eichinger | Structural insight into the dual ligand specificity and mode of high density lipoprotein association of apolipoprotein D[END_REF]) was achieved using the Matchmaker tool in the UCSF Chimera package [START_REF] Pettersen | UCSF Chimera--a visualization system for exploratory research and analysis[END_REF].

  medium (20 mM, Tricine, pH 8.4 containing 0.4 M, Sorbitol; 10 mM, Na2-EDTA; 10 mM, NaHCO3; 0.1% [w/v] Bovine serum albumin) using a Waring blender. Chloroplasts were collected by low speed centrifugation at 2070 x g for 2 min. The chloroplasts in the pellet were re-suspended in washing buffer (20 mM, Tricine, pH 7.6; 0.4 M, Sorbitol; 2.5 mM, MgCl2; 1.25 mM, Na2-EDTA) and loaded on the top of a preformed continuous Percoll gradient. Intact chloroplasts, which concentrated in a green band at the lower part of the gradient were recovered and rinsed with washing buffer. The upper part of the gradient contains broken chloroplasts. Following low speed centrifugation (2070 x g for 2 min), chloroplasts were osmotically lysed in 10 mM Tris-HCl, pH 8. The thylakoid membranes were collected by centrifugation. Stromal proteins, which remained in the supernatant were acetone precipitated. Thylakoids were resuspended in hypotonic buffer containing 200 mM NaCl and finally concentrated by low speed centrifugation. Chlorophyll was extracted with 80 % [v/v] aqueous acetone and quantified by spectrophotometry following the Arnon's method.

  pore-K + -channel1.mCherry (TPK1-mCherry) plasmids and Dr François Ouellet for the LCNP over expressing line. We thank Nathalie Duong (Aix-Marseille University, Cadarache) for technical assistance with the Arabidopsis transformation, Michel Havaux for preliminary discussions and Jean Alric and Louis Dumas (UMR7265-CNRS-CEA-AMU, Cadarache) for help with Chlorophyll fluorescence imaging. We also thank the Phytotec platform (Cadarache) for growing plants under control and stress conditions. Support for the microscopy equipment was provided by the Région Provence Alpes Côte d'Azur, the Conseil Général des Bouches du Rhône, The French Ministry of Research, the CNRS and The Commissariat à l'Energie Atomique et aux Energies Alternatives. Finally we thank Xenie Johnson and Jean-Luc Montillet (CEA Cadarache) for critical reading of this manuscript.

Figure 1 .

 1 Figure 1. Molecular characterization of transgenic plants expressing a chimeric gene

Figure 2 .

 2 Figure 2. Subcellular localization of APOD. Intact chloroplasts were isolated from leaves of

Figure 3 .

 3 Figure 3. Co-localization of APOD-EGFP with compartmental markers. Nicotiana

Figure 4 .

 4 Figure 4. Tolerance of Arabidopsis APOD-containing transgenic lines to water deficit stress.

Figure 5 .

 5 Figure 5. Tolerance of Arabidopsis APOD-containing transgenic lines to oxidative

Figure 6 .

 6 Figure 6. Fluorescence phenotype in APOD-containing transgenic lines induced by cold and

Figure 7 .C r u d e c e l l e x t r a c t Figure 2 CFigure 6 LCNP 35 660Figure 7

 726357 Figure 7. Overlay of the protein structures of human ApoD (purple ribbon) and the homology

SDS: Sodium dodecyl sulfate; TIL: Temperature-Induced Lipocalin; VDE: Violaxanthin-de-Epoxidase; ZEP: Zeaxanthin Epoxidase

Declarations

Acknowledgements

We thank Pr Eric Rassart (Université du Québec, Canada) for the kind gift of APOD cDNA, Dr. Olivier Batistic (Universität Münster, Germany) for providing the mCherry.Calnexin1 (mCherry-CNX1), N-acetylglucosaminyl-transferase1.mCherry (GNT1-mCherry) and Two-

Funding

No funding was obtained for this study.

Availability of data and material

Not applicable

Ethics approval and consent to participate

Not applicable

Consent for publication

Not applicable

Competing interests

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Authors' contributions

DR conceived the project and designed the experiments. DR and PH performed the experiments. DR wrote the manuscript and PH read and approved it.


C Preux

I Malinouskaya

Reservoir simulation model calibration methodology with polymer flooding based on laboratory experiments

Keywords: polymer flooding, simulation, mobility reduction

Due to its high apparent viscosity, polymer is considered as an alternative to water as injection fluid in hydrocarbon reservoirs. The mobility of the waterpolymer mixture is much smaller than that of water alone, which helps prevent undesirable viscous fingering. Nevertheless, the selection of a polymer process involves a procedure for assessing the performance of the polymer in terms of mobility reduction. This procedure involves laboratory experiments in which the pressure losses are measured during polymer displacement, as well as during water displacement. In the next step of the procedure, a model is calibrated by simulation in order to predict the efficiency of the polymer. Although the polymer models traditionally used at this stage are relatively simple, their calibration from the results of the laboratory tests is much more intricate, insofar as the mobility reduction generated by the polymer depends not only on the concentration but also on the temperature, the salinity and the shear rate. In this paper, we put forward a new approach to this multivariate problem and propose a rigorous calibration methodology which allows for a fast physical representativity, thus limiting the number of laboratory experiments. This methodology is the subject of a pending patent application (FR 18/58547).

Introduction

In order to improve the oilfield recovery factor, many oil companies resort to adding polymer in injected water. The idea is to take advantage of the large viscosity of the polymer in order to reduce the mobility of the waterpolymer mixture. Numerous experimental studies show that the mobility of the latter strongly depends on various parameters such as the polymer concentration c, the shear rate due to the polymer flooding velocity γ, the temperature T and the salinity or ionic strength I µ in case of salted water [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF], [START_REF] Rodriguez | Monitoring Thermal and Mechanical Stability of Enhanced Oil Recovery EOR Acrylamide Based Polymers PAM Through Intrinsic Viscosity IV Determination Using a New Capillary Rheology Technique[END_REF], [START_REF] Smidsröd | Estimation of Relative Stiffness of the Molecular Chain in Polyelectrolytes from Measurements of Viscosity at Different Ionic Strengths[END_REF]. For example, a high impact of the low salinity on the polymer behavior, in particular, on the mobility reduction, was demonstrated by [START_REF] Alsofi | SmartWater synergy with chemical EOR: Effects on polymer injectivity, retention and acceleration[END_REF]. In [START_REF] Torrealba | Improved polymer flooding injectivity and displacement by considering compositionally-tuned slugs[END_REF], the study of the slug-based compositionally-tuned polymer flooding reveals that the polymer viscosity can be controlled through the mixture salinity. In most theoretical studies, summarized by [START_REF] Sorbie | Polymer-Improved Oil Recovery[END_REF], the analyses are carried out only for one or two parameters at a time. For example, in [1], the authors use calibrated model to consider polymer concentration and salinity effect, but neglect the temperature effects.

For the calibration of the reservoir simulation model to take into account all of the mentioned physical effects, we have to conduct a large number of experiments corresponding to all possible combinations of the chosen parameters {c, γ, T, I µ }. This turns out to be expensive and impractical to implement.

The present work addresses this calibration issue by introducing a methodology that requires as few laboratory experiments as possible. This methodology enables us to obtain a complete description of the mobility reduction as a function of any set of parameters {c, γ, T, I µ }. After describing a general polymer model in Section 2, we provide full details of the calibration methodology in Section 3. Step by step, the dependencies on the polymer concentration, the salinity/ionic force, the shear rate and the temperature are taken into account and the required measurements associated with these parameters are stated. Then, in Section 4, we apply the methodology to an experimental data set [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF][START_REF] Rodriguez | Monitoring Thermal and Mechanical Stability of Enhanced Oil Recovery EOR Acrylamide Based Polymers PAM Through Intrinsic Viscosity IV Determination Using a New Capillary Rheology Technique[END_REF] and compute the corresponding mobility reduction function. Finally, we draw some conclusions on the efficiency of this methodology and discuss some prospects of the present work in Section 5.

Polymer model formulation

The polymer displacement model is based on the assumption that the polymer-added water has a reduced mobility. The derivation of this model, used by many reservoir simulators, relies on a modification of the water viscosity which is expressed as

µ wp = R m (c, I µ , γ, T )µ w , (1)
where µ w is the viscosity of the injection water, µ wp the viscosity of the water containing polymer and R m (c, T, I µ , γ) the mobility reduction. The latter is a function of the polymer concentration c, the temperature T , the salinity/ionic strength I µ and the shear rate γ. The determination of the mobility reduction function R m (c, I µ , γ, T ) from experimental data is the object of model calibration for the purpose of predicting polymer displacements. Notice that the total mobility reduction from polymer injection is caused by two effects:

-the increase in polymer viscosity, -the reduction in permeability presumably by polymer retention, adsorption, and inaccessible pore volume.

In this paper, we consider only the increase in polymer viscosity. The permeability reduction R k , which appears in the phase velocity such as

u w = - Kk r R k R m µ w ( ∇p w + ρ w g) (2)
is not studied in this paper.

In practice, the laboratory measurements allow us to obtain the solvent viscosity µ wp and to deduce the corresponding value of R m as a relative viscosity µ wp /µ w . On the other hand, these results can be considered in terms of the reduced specific viscosity (µ wp -µ w )/(cµ w ), where µ w is the water viscosity. This gives rise to the intrinsic viscosity

[µ] = lim c→0 µ wp -µ w µ w c . (3)
The well-known formulation for the polymer concentration dependence proposed by Huggins [START_REF] Huggins | Solutions of long chain compounds[END_REF] is written as

R m (c) = 1 + [µ] c + k ([µ] c) 2 , (4)
where [µ] is defined by (3) and k is the Huggins coefficient. This formula is treating bulk polymer solutions but is used by default in different models of polymer flooding ( [START_REF] Yuan | Modeling multiphase non-Newtonian polymer flow in IPARS parallel framework[END_REF][START_REF] Sharma | A Practical Method to Calculate Polymer Viscosity Accurately in Numerical Reservoir Simulators[END_REF][START_REF] Li | Modeling and simulation of polymer flooding including the effects of fracturing[END_REF][START_REF] Ebaga-Ololo | Prediction of Polymer Flooding Performance with an Artificial Neural Network: A Two-Polymer-Slug Case[END_REF]). The intrinsic viscosity of the polymer mixture [µ] characterizes its so-called viscosifying power. This quantity represents the expansion volume of the polymer in a water of a given salinity at a given temperature and a shear rate, and is expressed in cm 3 /g or in L/g. The Huggins coefficient k is a dimensionless number; for the hydrolyzed polyacrylamide polymers (HPAM) in a good solvent, its value is about 0.4 ± 0.1 [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF]. Using definition (3) and formulation (4), [µ] and k result from the best fit of a linear part of the specific viscosity (reduced to low concentration values) as shown in Figure 1. The dependences on the temperature, the salinity and the shear rate will be managed through these two parameters. Finally, we search a formulation based on these dependances:

R m (c, I µ , γ, T ) = 1 + [µ] (I µ , γ, T )c + k (I µ , T ) ([µ] (I µ , γ, T )c) 2 , (5)
The principal assumption in this methodology is that the different parameters (c, I µ , γ, T ) are constant with respect to others (salt, shear rate and temperature). We search also the intrinsic viscosity and the Huggins coefficient in an uncoupling form:

[µ] (I µ , γ, T ) = [µ] Iµ (I µ ). [µ] γ ( γ). [µ] T (T ) (6)
k (I µ , T ) = k Iµ (I µ ).k T (T ) (7)
It remains a simplifying assumption to limit the number of experimental data and to limit the variables and we show in the last section that this assumption gives a correct match.

Calibration methodology of the mobility reduction function

In order to calibrate the polymer model described above, we can use polymeric water displacements results, obtained under given displacement conditions (polymer concentration, injection rates, temperature and salinity). Usually, these displacements are carried out on cores initially saturated with a liquid phase (water and/or oil), which is mobile or residual depending on the core history and the purpose of the measurements. The considered displacements are the drainage processes with the increasing water saturation.

For each displacement, the laboratory must provide the following measurements:

-the evolution of the pressure drop along the core: the ratio between the two stabilized pressure losses measured at the end of each displacement is the main result of a couple of displacements and is called mobility reduction;

-the liquid phase (water and/or oil) and gas productions.

Less accurate but faster, we can determine rheological properties of polymer by viscosimeter. In this case we treat only bulk solution polymer viscosity without taking into account the porous media effect. We obtain finally a correlation which can be considered as a starting point for the coreflood history match. The calibration of the mobility reduction of a given rock-fluids-additive system involves the displacement measurements for any possible combination of parameters {c, I µ , γ, T }.

As shown before, the dependence on the concentration is expressed by (4) and the dependence on the ionic strength I µ , the shear rate γ and the temperature T is contained in the intrinsic viscosity [µ] and the Huggins coefficient k . Therefore, the calibration methodology consists of the following steps. First, we choose a reference set of parameters {I µ,1 , γ1 , T 1 } and use them for the first series of measurements with various polymer concentrations c i (i = 1, . . . , N ) to deduce the intrinsic viscosity and Huggins coefficient as described before (Figure 1). Schematically,

S1 : {c i } N i=1 , I µ,1 , γ1 , T 1 → {[µ] 1 , k 1 } . (8)
It should be noted that for further application of the methodology, it is convenient to restrict the shear rate γ1 to very small values, namely, γ1 → 0.

Then, in order to study the influence of the ionic strength, we modify the reference value of I µ . We perform the same series of measurements as before to deduce the corresponding values of [µ] (I µ ) and k (I µ ). This procedure can be repeated several times to obtain the curves of the intrinsic viscosity and Huggins coefficient as functions of I µ . However, we will demonstrate that our methodology only needs one additional series of experiments with a new value of the ionic strength.

Similarly, changing the shear rate and the temperature in the reference set of parameters, we get the corresponding intrinsic viscosities [µ] ( γ) and [µ] (T ) and Huggins coefficients k ( γ) and k (T ), respectively. Again, we will demonstrate that only one additional series of measurements per parameter is required to obtain a full description of the intrinsic viscosity as a function of this parameter. Nevertheless, the dependence of the Huggins coefficient on the temperature in presence of a variable salinity appears to be more complex and requires one additional series of measurements with simultaneously modified values of I µ and T .

Determination of salinity/ionic strength dependence

We consider here the dependence on the ionic strength I µ , keeping in mind that the dependence on the salinity s can be determined exactly in the same way since I µ depends on the salinity s = T DS (for Total Dissolved Salts). For example, considering 3 pseudo-ions, we use modified ionic strength which can be estimated according to the composition of the salt as

I µ (s) = C + M + + α µ C ++ M ++ + β µ C ++ M ++ 2 , (9)
where the molar masses M +,++,-and the mass concentrations C +,++,-correspond to the 3 groups of pseudo-ions: 2 pseudo-cations I+, I++ and 1 pseudoanion I-. This formula ( 9) is based on the reservoir simulator COREP OL presented in [START_REF] Lecourtier | An Innovative Polymer Flooding Simulator Based on Advanced Concepts in Polymer Physics[END_REF] and [START_REF] Le | Physical and thermodynamical aspects in numerical model SARIP[END_REF]. Then, this modified ionic strength was adopted in the reservoir simulator SARIP CH and P umaF low T M . It take into account the fact that divalent ions have a strong influence on polymer viscosity. In [START_REF] Dang | Rheological Modeling and Numerical Simulation of HPAM Polymer Viscosity in Porous Media[END_REF], the author show that the effect of divalent ions is observed by a significant reduction in the viscosity of the polymer solution. It seems that the divalent ions are tightly bind to anions along the polyelectrolyte chain because it has a higher charge, causing the polymer chain to contract to its minimum size.

For polymer type HPAM, the reservoir simulator COREP OL adviced to choose α µ = 10.5 and β µ = 216. The intrinsic viscosity is prescribed as a linear function of

I -1/2 µ [14, 19], that is, [µ] (I µ ) = [µ 0 ] + b I I -1/2 µ , (10)
where [µ 0 ] is the intrinsic viscosity when I µ → ∞. There are two unknown parameters [µ 0 ] and b I . Therefore, one additional series of measurements is required. For this, we use the reference set S1 [START_REF] Douarche | Modeling Chemical EOR Processes: Some Illustrations from Lab to Reservoir Scale[END_REF], set an ionic strength to I µ,2 and proceed to a new series of measurements

S2 : {c i } N i=1 , I µ,2 , γ1 , T 1 → {[µ] 2 , k 2 } . ( 11
)
The results of S1 and S2 are inserted into formulation (10) and yield a system of two linear equations in two unknowns. The solution of this system provides the parameters [µ 0 ] and b I as

b I = [µ] 1 -[µ] 2 I -1/2 µ,1 -I -1/2 µ,2 , (12)
[µ 0 ] = [µ] 2 - [µ] 1 -[µ] 2 I -1/2 µ,1 -I -1/2 µ,2 I -1/2 µ,2 . (13)
Thus, the intrinsic viscosity as a function of the ionic strength is obtained by

[µ] (I µ ) = [µ] 2 + [µ] 1 -[µ] 2 I -1/2 µ,1 -I -1/2 µ,2 I -1/2 µ -I -1/2 µ,2 . ( 14
)
According to [START_REF] Wolff | Viscosité des solutions de polyelectrolytes[END_REF], the Huggins coefficient depends on the salt composition and the concentration range. The lineary dependance seems better adapted to weak ionic forces. For high ionic strengths, however, there is too few experimental results to be more affirmative. In this paper, at first approximation, we decide to consider the Huggins coefficient as a linear function of

I µ k (I µ ) = α I I µ + β I . ( 15
)
Similarly to the intrinsic viscosity, the results of the two sets of measurements S1 and S2 provide the parameters α I and β I in the form

α I = k 1 -k 2 I µ,1 -I µ,2 , β I = k 2 - k 1 -k 2 I µ,1 -I µ,2 I µ,2 . ( 16
)
The definition of the intrinsic viscosity [µ] (I µ ) and the Huggins coefficient k (I µ ) through analytical formulas involving two parameters ([µ 0 ] and b I for viscosity and α I and β I for k ) implies two series of experiments corresponding to the two different values of I µ for any ionic strength, but for the same conditions on the temperature and shear rate.

Determination of shear rate dependence

Very often the power law is used to fit the viscosity measurements as a function of the shear rate in the case of non-Newtonian fluids [START_REF] Medina-Torresa | Rheological properties of the mucilage gum (Opuntia ficus indica)[END_REF][START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF]. The Cross model [START_REF] Cross | Rheology of Non-Newtonian Fluids: a New Flow Equation for Pseudoplastic Systems[END_REF] was found more efficient for a wide range of the shear rates as well as in Newtonian fluids at small shear rate. The most generalized model proposed in the literature is the Carreau model [START_REF] Carreau | Rheological Equations from Molecular Network Theories[END_REF], which reads

µ( γ) -µ γ∞ µ γ0 -µ γ∞ = 1 + (λ γ) 2 (n-1)/2 , ( 17
)
where µ γ∞ and µ γ0 are the Newtonian viscosities corresponding to γ → ∞ and γ → 0, respectively; here, λ is a time constant that corresponds to a characteristic shear rate, i.e.,

λ = ( γc ) -1 , (18)
where γc depends on the type of media. For example, in [START_REF] Chauveteau | Rodlike Polymer Solution Flow Through Fine Pores: Influence of Pore Size on the Rheological Behavior[END_REF], it is defined as

γc = a 4u r h , ( 19
)
where a is a form factor that can be obtained experimentally, u = Q/φS with Q the flow rate, S the section area and φ the porosity, and r h is a specific pore size r h = 8K/φ with K the permeability. The viscosity µ γ∞ can be set to the viscosity of the solvent without polymer [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF]. Therefore, the Carreau model can be written in terms of intrinsic viscosity as

[µ] ( γ) = [µ γ0 ] 1 + ( γ γc ) 2 (1-n)/2 . ( 20
)
Since the imposed shear rate in the reference set of parameters S1 ( 8) is very small, we can write

[µ γ0 ] = [µ] (I µ ), (21)
where [µ] (I µ ) is obtained by [START_REF] Medina-Torresa | Rheological properties of the mucilage gum (Opuntia ficus indica)[END_REF]. A series of measurements, say S3, with a new value of the shear rate, γ2 , is required in order to determine the parameter n in [START_REF] Sorbie | Polymer-Improved Oil Recovery[END_REF]. In other words,

S3 : {c i } N i=1 , , I µ,1 , γ2 , T 1 → {[µ] 3 , k 3 } . ( 22
)
Similarly to the previous case, the results of S1 and S3 are plugged into formulation [START_REF] Sorbie | Polymer-Improved Oil Recovery[END_REF] and lead to a system of equations from which we obtain n under the form

n = 1 - 2 log [µ] 1 [µ] 3 log 1 + γ2 γc 2 . ( 23
)
Then, the intrinsic viscosity as a function of the shear rate and the ionic strength is inferred as

[µ] (I µ , γ) = [µ] (I µ ) 1 + ( γ γc ) 2 log [µ] 1 [µ] 3 log 1+ [ γ2 γc ] 2 . (24)
Let us remind that (21) has been obtained under condition γ → 0. However, it can be generalized for any value of γ as

[µ γ0 ] = [µ] (I µ ) 1 + ( γ1 γc ) 2 (1-n)/2 , ( 25
)
with the same value of n as found by [START_REF] Wolff | Viscosité des solutions de polyelectrolytes[END_REF]. Therefore, the generalized formulation will differ from [START_REF] Yuan | Modeling multiphase non-Newtonian polymer flow in IPARS parallel framework[END_REF] by multiplier [1 + ( γ1 / γc ) 2 ] (1-n)/2 . The Huggins coefficient as a function of shear rate is generally defined as a constant.

Determination of temperature dependence

The intrinsic viscosity dependence on the temperature can be expressed under the form

[µ] (T ) = [µ T 0 ] M α 1 T -1 T 0 , (26)
such as proposed by [START_REF] Amus | The Unperturbed Molecular Dimensions of Polyethylene Oxide in Aqueous Solutions from Intrinsic Viscosity Measurements and the Evaluation of the Theta Temperature[END_REF]. In (26), [µ T 0 ] is the intrinsic viscosity at reference temperature T 0 , M is the molecular weight and α is a constant. Setting the reference temperature T 0 = T 1 , the intrinsic viscosity [µ T 0 ] is determined as a function of the ionic strength and the shear rate obtained by [START_REF] Yuan | Modeling multiphase non-Newtonian polymer flow in IPARS parallel framework[END_REF], that is,

[µ T 0 ] = [µ] (I µ , γ). ( 27
)
Then, a new series of measurements is required to deduce the second parameter α. These measurements are performed using the reference set of parameters and setting a new temperature T 2 . In a manner similar to the previous ones, by varying the polymer concentration c we are in a position to obtain the intrinsic viscosity and the Huggins coefficient for this new set of parameters S4 :

{c i } N i=1 , I µ,1 , γ1 , T 2 → {[µ] 4 , k 4 } . ( 28
)
Solving the system of equations obtained substituting ( 8) and (28) into the formulation (26), we end up with the coefficient

α = log [µ] 1 [µ] 4 log M 1 T 1 - 1 T 2 -1 . ( 29
)
Consequently, the intrinsic viscosity [µ] is expressed as a function of the ionic strength, the shear rate and the temperature as

[µ] (I µ , γ, T ) = [µ] (I µ , γ) [µ] 1 [µ] 4 - T 2 (T -T 1 ) T (T 2 -T 1 ) , (30)
where [µ] (I µ , γ) was obtained by [START_REF] Yuan | Modeling multiphase non-Newtonian polymer flow in IPARS parallel framework[END_REF]. The Huggins coefficient dependence on the temperature is imposed as a linear function

k (T ) = α T T + β T . (31)
Similarly to the ionic strength dependence, two series of measurements, S1 and S4, are sufficient to determine the parameters α T and β T . However, they are found only for a given value of the ionic strength, I µ,1 . Taking into account the dependence on I µ indicated by [START_REF] Li | Modeling and simulation of polymer flooding including the effects of fracturing[END_REF], we can turn the Huggins coefficient into the form

k = (α 1 I µ + α 2 ) T + (β 1 I µ + β 2 ) . (32)
These cross-dependence on the ionic strength and the temperature implies a system of four equations in four parameters, while the measurements S1, S2 and S4 yield only three equations. Thus, one additional series of measurements is required with the ionic strength I µ,2 (as in series S2) and the temperature T 2 (as in series S4), namely,

S5 : {c i } N i=1 , I µ,2 , γ1 , T 2 → {[µ] 5 , k 5 } . (33)
Solving (31) for measurements S1 and S4, we obtain the two parameters α T = α T,1 and β T = β T,1 as

α T,1 = k 1 -k 4 T 1 -T 2 , β T,1 = k 4 - k 1 -k 4 T 1 -T 2 T 2 . (34)
Similarly, for S2 and S5, the parameters in (31) are α T = α T,2 and β T = β T,2 and they are obtained by

α T,2 = k 2 -k 5 T 1 -T 2 , β T,2 = k 5 - k 2 -k 5 T 1 -T 2 T 2 . (35)
Therefore, it follows from (31)-(32) a system of four equations in the four parameters α 1 , α 2 , β 1 and β 2 . Solving this system and invoking the results (34)-( 35), we end up with

α 1 = (k 1 -k 4 ) -(k 2 -k 5 ) (I µ,1 -I µ,2 ) (T 1 -T 2 ) , α 2 = I µ,1 (k 2 -k 5 ) -I µ,2 (k 1 -k 4 ) (I µ,1 -I µ,2 ) (T 1 -T 2 ) , β 1 = T 1 (k 4 -k 5 ) -T 2 (k 1 -k 2 ) (I µ,1 -I µ,2 ) (T 1 -T 2 ) , β 2 = I µ,1 T 1 k 5 -I µ,1 T 2 k 2 -I µ,2 T 1 k 4 + I µ,2 T 2 k 1 (I µ,1 -I µ,2 ) (T 1 -T 2 ) . (36)
Inserting the parameters (36) into (32), we can deduce the Huggins coefficient as a function of the ionic strength and the temperature.

Determination of mobility reduction

The five above-mentioned series of measurements ( 8), ( 11), ( 22), ( 28) and (33) allow us to obtain a full description of the intrinsic viscosity and the Huggins coefficient as functions of the ionic strength, the shear rate and the temperature, [µ] (I µ , γ, T ) and k (I µ , T ).

Dependence

Experiments

Results Parameters

[µ] k for [µ] for k Salinity/ S1 : {c i } N i=1 , I µ,1 , γ1 , T 1 [µ] 1 k 1 b I , α I , ionic strength S2 : {c i } N i=1 , I µ,2 , γ1 , T 1 [µ] 2 k 2 [µ 0 ] β I Shear rate S1 : {c i } N i=1 , I µ,1 , γ1 , T 1 [µ] 1 n - S3 : {c i } N i=1 , I µ,1 , γ2 , T 1 [µ] 3 Temperature S1 : {c i } N i=1 , I µ,1 , γ1 , T 1 [µ] 1 k 1 α α T , S4 : {c i } N i=1 , I µ,1 , γ1 , T 2 [µ] 4 k 4 β T Temperature S2 : {c i } N i=1 , I µ,2 , γ1 , T 1 k 2 α 1 , α 2 + Salinity S5 : {c i } N i=1 , I µ,2 , γ1 , T 2 k 5 β 1 , β 2
Table 1: The summary of the experiments required to determine each of dependence of [µ] on I µ , γ, and T and the parameters to deduce for this.

Using the formulations ( 14), ( 24) and ( 30), the complete dependence of the intrinsic viscosity on the ionic strength, the shear rate and the temperature can now be written as

[µ] (I µ , γ, T ) = [µ] 2 + ([µ] 1 -[µ] 2 ) I -1/2 µ -I -1/2 µ,2 I -1/2 µ,1 -I -1/2 µ,2 [µ] 1 [µ] 4 - T 2 (T -T 1 ) T (T 2 -T 1 ) 1 + ( γ γc ) 2 log [µ] 1 [µ] 3 log 1+ [ γ2 γc ] 2 . ( 37
)
As far as the Huggins coefficient is concerned, it becomes equal to

k (I µ , T ) = (k 1 -k 4 ) -(k 2 -k 5 ) (I µ,1 -I µ,2 ) (T 1 -T 2 ) I µ T + I µ,1 (k 2 -k 5 ) -I µ,2 (k 1 -k 4 ) (I µ,1 -I µ,2 ) (T 1 -T 2 ) T + T 1 (k 4 -k 5 ) -T 2 (k 1 -k 2 ) (I µ,1 -I µ,2 ) (T 1 -T 2 ) I µ + I µ,1 T 1 k 5 -I µ,1 T 2 k 2 -I µ,2 T 1 k 4 + I µ,2 T 2 k 1 (I µ,1 -I µ,2 ) (T 1 -T 2 )
(38) Plugging these expressions into formulation (4), the mobility reduction R m can be deduced for any set of parameters {c, I µ , γ, T } without performing any new laboratory experiment.

Table 1 summarizes the measurements required to determine the dependence on each parameter. It should be noted that depending on the consid-ered case, some measurements may turn out to be useless. For example, if there is no temperature variations in the reservoir, the series of experiments S4 and S5 are not necessary. The intrinsic viscosity and the Huggins coefficient are obtained directly from ( 24) and ( 15)-( 16), respectively, imposing in the reference set of parameters T 1 equal to the temperature of the reservoir.

Application to experimental data

We now apply our calibration methodology to the experimental data set provided by [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF], in order to obtain a full reduction mobility description as a function of the salinity (or ionic strength), the shear rate, the temperature and the polymer concentration.

For the experimental data, a hydrolyzed polyacrylamide (HPAM) referenced as FLOPAAM 3630S with molecular weight M w = 20 × 10 6 daltons is used. The measurements are performed in nano-filtered seawater (NF-SW) of salinity 10 g/L and synthetic seawater (SSW) of salinity 33.5 g/L at a temperature T = 20 • C.

In [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF], the author use viscosimeter to determine rheological properties of polymer. All bulk viscosities were measured using an Antan Paar MCR 301 rheometer. As detailed at the beginning of section 3, it would be better to have polymer viscosity obtained by coreflood but by default, we use these data available in the litterature.

The temperature dependence was not studied in [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF]. Therefore, we will use an analytical formula for the viscosity dependence on the temperature [START_REF] Vinogradov | Rheology of Polymers[END_REF] and apply it to the measurements of viscosity [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] at low shear rate for various polymer concentrations. Thus, we will obtain the so-called synthetic data that will then be used for the purpose of demonstration of our methodology.

The parameters for five series of measurements (S1-S5) required to determine R m are given in Table 2.

Ionic strength/salinity dependence

According to the composition of the saline water, NF-SW and SSW, the three groups of ions are determined: two pseudo-cations I+, I++ and one pseudo-anion I-. Their molar mass M +,++,-and mass concentration C +,++,-are calculated and inserted into (9) to deduce the corresponding ionic strength I µ . All data are given in Table 3.

The results of measurements performed with NF-SW by [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] provide the intrinsic viscosity [µ] 1 = 6.4 L/g and the Huggins coefficient k 1 = 0.32 and

s, [g/L] γ, [s -1 ] T , [ • K] c, [g/L] S1 10
C +,++,-[g/L], salinity s [g/L], ionic strength I µ [g/L].
the measurements with SSW provide [µ] 2 = 2.066 L/g and k 2 = 0.35. Thus, we can calculate b I and [µ 0 ] (13) and find [µ] (I µ ) by ( 14) as

[µ] (I µ ) = 13.21I -1/2 µ + 0.027. ( 39
)
If there is no thermal effects, then the Huggins coefficient k (I µ ) can be deduced from [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] as

k (I µ ) = 0.0008I µ + 0.32. (40)
Finally, we compare our correlation with other available experimental data from [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF]. For the two salinities (Nf-SW and SSW) and for a temperature T = 20 • C and a shear stress γ = 0.1s -1 we obtain a good agreement between analytical and experimental data as shown in Figure 2.

It should be noted that formulation ( 4) is done for a low polymer concentration. To adapt (4) for "semi-diluted" regime, we must add a fourth order term. Since this higher order term is not considered in our model, there are light differences for high polymer concentration, especially for NF-SW. However, the data match is pretty good for "diluted regime".

Shear rate dependence

Based on the results obtained by [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] for various shear rates, we could estimate the intrinsic viscosity [µ] 3 = 3.75 L/g. Since we consider that the Huggins coefficient is a constant as function of the shear rate, the estimation of k 3 is not required.

The characteristic shear rate γc is evaluated by [START_REF] Smidsröd | Estimation of Relative Stiffness of the Molecular Chain in Polyelectrolytes from Measurements of Viscosity at Different Ionic Strengths[END_REF] with the flow rate Q = 2.88 × 10 -4 m 3 /d and the shape factor a = 2.5 which corresponds to the experimental value obtained for granular media by [START_REF] Chauveteau | Rodlike Polymer Solution Flow Through Fine Pores: Influence of Pore Size on the Rheological Behavior[END_REF]. Thus, we obtain γc = 15.08 s -1 .

(41)

The power n in ( 20) is obtained by [START_REF] Wolff | Viscosité des solutions de polyelectrolytes[END_REF], which results in

n = 0.75. (42)
Thus, the intrinsic viscosity as a function of the shear rate and the ionic force can be expressed as

[µ] (I µ , γ) = [µ] (I µ ) (1 + 0.0044 γ2 ) 0.124 . ( 43
)
If in the case considered there is no variations of the salinity, then [µ] (I µ ) = [µ] 1 ; otherwise, using (39) we obtain

[µ] (I µ , γ) = 13.21I

-1/2 µ + 0.027 (1 + 0.0044 γ2 ) 0.124 .

(44) Let us notice that using I µ = I µ,2 = 41.96 g/L and γ = γ1 , which correspond to series of measurements S2, the expression (44) yields the intrinsic viscosity [µ] (I µ,2 , γ1 ) = 2.066 g/L. It is in agreement with the intrinsic viscosity obtained directly from the measurements S2, [µ] 2 = 2.066 g/L. Now, we compare our correlation with other available experimental data from [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF]. For the two salinities (NF-SW and SSW) and for a temperature T = 20 • C and different values of shear stress, we obtain a good agreement as shown in Figure 3.

This comparaison shows that the correlation presented in this paper yields the results close to the experimental values.

Temperature dependence

For the temperature dependence of the intrinsic viscosity and the Huggins coefficient, two series of measurements S4 and S5 are required (Table 2). However, we did not find the corresponding measurements for the same polymer and water salinity as used before. In order to demonstrate the relevance of our methodology for the temperature dependencies, we will use the measurements of series S1 and S2 for T 1 and transform them to the data for T 2 using the exponential dependence on T given by [START_REF] Vinogradov | Rheology of Polymers[END_REF], i.e.,

µ(T ) = µ 0 exp E a R 1 T - 1 T 1 , (45)
where E a = 9300 J•mol is an activation energy, R = 8.314 J•mol -1 •K -1 is the ideal gas universal constant and µ 0 is the viscosity obtained by [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] for γ → 0 for NF-SW (series S1) and SSW (series S2) at temperature T 1 . It should be pointed out that the temperature used in the formulas so far is expressed in • K, but for the sake of simplicity, we are switching to the more common unit • C. The viscosity computed by (45) for the temperature T 2 = 70 • C for the solvent NF-SW yields the data of series S4, and for the solvent SSW yields the data of series S5. Then, to obtain the intrinsic viscosity by (3), we use µ w equal to the viscosities measured by [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] for NF-SW and SSW and transformed by (45). Therefore, [µ] and k are deduced using the linear fit of µ(T 2 ) as function of the polymer concentration c (see Section 2). Thus, we obtain the results for series S4 and S5

[µ] 4 = 3.86 L/g, k 4 = 0.52,

[µ] 5 = 1.36 L/g, k 5 = 0.37.

Owing to formula (30), the intrinsic viscosity as function of the temperature, shear rate and the salinity/ionic strength is obtained as

[µ] (I µ , γ, T ) = [µ] (I µ ,
Finally, for the Huggins coefficient k as a function of the salinity and the temperature (32), we obtain α 1 = -0.00009, α 2 = 0.0044, β 1 = 0.0286, β 2 = -0.959.

(50) Since we assume that k does not depend on the shear rate, the Huggins coefficient k (I µ , T ) can be written as

k (I µ , T ) = -0.959 + 0.0286I µ + 0.0044T -0.00009I µ T. (51)
Finally, we compare our correlation with experimental data from [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] for a temperature T = 20 • C and with synthetic datas derived from [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] and obtained by the formula 45. For the two salinities (NF-SW and SSW) and for a shear stress γ = 0.1s -1, we obtain a good agreement as shown in Figure 4.

Let us notice that if there is no salinity dependence, the series S5 is not necessary, and the Huggins coefficient can be deduced by (31) with the coefficients found by (34).

Mobility reduction

The mobility reduction dependence R m (c, I µ , γ, T ) on the polymer concentration, the salinity/ionic strength, the shear rate and the temperature is expressed by the Huggins formula (4) using the intrinsic viscosity [µ] (I µ , γ, T ) (49) and the Huggins coefficient k (I µ , T ) (51). In Figure 5, R m (c, I µ , γ, T ) is shown for T = 20 • C and T = 90 • C, and for the ionic force I µ = 4.3 g/L and I µ = 41.96 g/L. Comparing R m for different values of T , it is obvious that the temperature variations play an important role in the mobility reduction and cannot be neglected. For example, increasing the temperature from 20 • to 90 • , the mobility reduction factor may decrease up to three times, as illustrated in Figure 5, panels (a) and (c).

Similarly, I µ , γ and c have a significant impact on the variations of R m . In Figure 5, panels (a) and (b), for instance, increasing the ionic strength from 4.3 g/L to 41.96 g/L, R m has the effect of decreasing nearly seven times at the most extreme case (c = 2 g/L, γ = 1 s -1 ). Therefore, when dealing with a reservoir that is supposed to have variable temperature, and/or salinity, and/or shear rate and/or polymer concentration, the mobility reduction function should be correctly calibrated according to each presented phenomenon as summarized in Table 1.

Conclusions

We have presented a new methodology that enables us to completely determine the mobility reduction function R m as a function of the salinity, the temperature and the shear conditions for various polymer concentrations. The most remarkable feature of this determination is that it is achieved by means of a limited number of laboratory experiments, more specifically by using at most five sets of measurements given in Table 1. Application to a literature experimental data set [START_REF] Moradi | Experimental Investigation of Polymer Flow Through Water-and Oil-wet Porous Media[END_REF] has testified to the relevance of this methodology was testified. This also served to highlight the importance of accounting for each of the mentioned-above dependences in estimating the mobility reduction. A future work can be to apply this methodology on coreflood experimental results instead of bulk viscosities, with complete thermal results.

Figure 1 :

 1 Figure 1: Reduced specific viscosity function of polymer concentration. (Reduced specific viscosity in blue and linear part of the specific viscosity in red.)

  {0.1, 0.25, 0.5, 0.75, 1, 1.5, 2}

Figure 2 :

 2 Figure 2: Reduction mobility R m c, I µ , γ = 0.1s -1 , T = 20 • C function of polymer concentration. The points are experimental data, the plain curve is the correlation presented in this paper.

Figure 3 :

 3 Figure 3: Reduction mobility R m (c, I µ , γ, T = 20 • C) function of polymer concentration for different salinities and different values of shear stress. The points are experimental data, the plain curve is the correlation presented in this paper.

Figure 4 :

 4 Figure 4: Reduction mobility R m (c, I µ , γ = 0.1s -1, T ) function of polymer concentration for different salinities and different values of temperature. The points at T = 20 • C are experimental data, the points at T = 70 • C are synthetic data and the plain curve is the correlation presented in this paper.

Figure 5 :

 5 Figure 5: Reduction mobility R m (c, I µ , γ, T ) for T = 20 • C (a,b) and 90 • C (c,d), and for the ionic force I µ = 4.3 g/L (a,c) and 41.96 g/L (b,d).

Table 2 :

 2 Series experiments used to determine R m (c, I µ , γ, T ) where I µ depends on the salinity s (9).

		M + M ++ M -	C +	C ++	C -	s	I µ
	NF-SW	23	28.17 35.75 3.85 0.05 6.10	10	4.3
	SSW	23.35 27.83 36.63 10.78 1.61 21.16 33.5 41.96

Table 3 :

 3 The saline water characteristics: molar mass M +,++,-[g/mol], concentration

Acknowledgements

The authors would like to thank Quang-Huy Tran for his valuable help and advice in writing this paper. [1] AlSofi, A.M., Liu, J.S., Han, M., 2013. Numerical simulation of surfactantpolymer coreflooding experiments for carbonates, Journal of Petroleum Science and Engineering, 111, pp.184-196