Kim Tremblay Parrado

Luc

Pr Khantutta-Kim Tremblay-Parrado

Luc Avérous
email: luc.averous@unistra.fr

Prof Luc Avérous

Renewable responsive systems based on original click and polyurethane crosslinked architectures with advanced properties

Keywords: Renewable Resources, Oleic acid, Biobased polymers, Polyurethane, Diels-Alder Cycloaddition

HAL is

Introduction

Nowadays, developing biobased and environmentally friendly materials is not enough. Science has to face developing advanced materials with extending lifetimes and with a controlled end of life. Polymer materials with innovative architectures allowing for cutting-edge properties such as self-healing and reprocessability, do not only extend the life of end-user materials, but also allow for the reduction of energy use, resources and waste. Polyurethanes (PUs) are of particular interest to integrate these novel properties due to their global versatility. PUs are used for a wide range of applications such as adhesives, coatings, automotive and construction industry. [START_REF] Haponiuk | Polyurethane Polymers[END_REF] Their production is ranked 6 th amongst all polymers, [START_REF] Carré | [END_REF] with a planned production of approximately 23 million tons in 2019. [START_REF] Akindoyo | Polyurethane types, synthesis and applications-a review[END_REF] Although the biobased PU market is expanding, limited work has explored the development biobased PUs with particular architectures to allow for dynamic and adaptable materials. Developing biobased materials and PUs is a starting point to a circular economy but by designing biobased PUs with dynamic architectures, this truly takes in to account the cradle-tograve reasoning of materials to facilitate recyclability and the extension of material lifetime.

To develop renewable materials, vegetable oils have a particular importance, as they are abundantly available at rather low costs with attractive chemical structures for a rich chemistry. [START_REF] Lligadas | [END_REF] Soybean oil is the most abundantly produced vegetable oil and it is has been particularly explored by academics and industrials for PU thermoset applications. [START_REF] Lligadas | [END_REF][5] Studies reporting the use of vegetable oils to synthesize diols for linear thermoplastic PUs (TPUs) include (i) the use of methyl oleate and ricinoleic acid through transesterification, epoxidation, ring-opening of epoxides and thiol-ene addition, [6] (ii) methyl ester of rapeseed oil through transesterification and thiol-ene coupling, [7] (iii) polyricinoleate diol by polycondensation [8] and (iv) the dimerization of rapeseed oil fatty acids at high temperature by a Diels-Alder (DA) mechanism. [START_REF] Ionescu | Chemistry and Technology of Polyols for Polyurethanes[END_REF] Consequently, the use of vegetable oils for the development of biobased PUs is a favorable starting point. Nevertheless, there is an apparent lack, particularly at the industrial level, in the development of advanced biobased PUs materials with dynamic architectures for a well-thought out reflection of the extension of polymer material lifetimes.

In order to extend the lifetime of polymers, many efforts have been made to develop dynamic and reversible polymers. Such a behavior is based on the development of dynamic covalent bonds that can be formed and cleaved by exposure to an external stimulus or autonomously. Crosslinked polymers containing reversible covalent bonds are defined as Covalent Adaptive Networks (CANs) [START_REF] Kloxin | [END_REF] and can be further subdivide in two categories [11] , dissociative and associative systems.

Associative CANs consists of bond exchange between polymer chains, where the overall crosslinking density is preserved as a new covalent bond can only be formed if another has been broken, such as vitrimers, [12] which can also be biobased. [13] Dissociative CANs consist of chemical bond exchanges where bonds are first broken and then formed again in another place. Reversible covalent bonds such as the DA [14] reaction or alkoxyamine [15] chemistry can be used. As highlighted by the work of Gandini and co-workers, the thermoreversible furan-maleimide [4+2] cycloaddition DA reaction is the most extensively studied reversible covalent bond of dissociative CANs used in the synthesis of macromolecular architectures. [16] The DA reaction between a furan and maleimide moiety is considered as a cycloaddition "click" reaction, generally occurring at moderate temperature (below 90 °C) and the reversion reaction denoted as the retro-DA (r-DA) reaction, occurs between 110 and 130 °C. [17] The reactivity and reversibility of the DA and r-DA reactions can be tuned significantly depending on the substituents located on a furan and maleimide moiety. [18] Several biobased CANs have been synthesized using the thermoreversible DA reaction derived from lignins [19] , tannins [20] , alginate [21] , chitosan [22] , starch [START_REF] González | [END_REF] or vegetable oils. [16a] Using the DA reaction in PUs is of particular interest as it is believed that the shape memory behavior of PUs facilitates the self-healing by DA reaction. [24] Consequently, reversibly crosslinked PUs have been studied through a multitude of different strategies. [25] Nevertheless, only very few studies explore the use of biobased PUs as a canvas to investigate reversibly crosslinked polymers. Gu et al. synthesized a biobased TPU backbone using a renewable polylactide copolymer diol and 2,5-furandimethanol as a soft segment and chain extender respectively. The reversible covalent bonds were located in the hard segments (HS) where the DA reaction took place between the furan ring of the chain extender and a bismaleimide. [26] Wilson et al. synthesized an original C11 difuranic diol from furfural and 5-hydroxymethylfurufral, which they used to synthesize polyesters and reversibly crosslinked polyurethanes networks with a bismaleimide. [27] In the present work, we produced reversibly crosslinked PUs based on vegetable oils using thermally reversible furan-maleimide DA reaction. We synthesized a new chemical architecture from oleic acid, consisting of a diol structure, containing pendent furan rings, which was thoroughly characterized and denoted as the Furan Oligomer (FO). As depicted in Scheme 1, FO was incorporated into the linear polymer backbone and further crosslinked with the use of a bismaleimide based on polypropylene oxide (PPO BMI). The corresponding materials with varying crosslinking contents were tested as remendable and self-healing PUs.

Results and Discussion

Analysis of the Synthesis of FO

The acid moiety and alkene site of octadec-9-enoic acid (OA) were used as reaction sites to yield an oligomer with a diol structure containing pendent furan rings, as depicted in Scheme 2. OA was esterified using furan-2-ylmethanol (FM) at ambient temperature to yield furan-2-ylmethyl-oleate (FMO). The advantage of this type of esterification reaction (Steglich esterification) is that no energy source is required, minimizing the global energy requirement, in agreement with one of the main principles for a green chemistry. [28] Nevertheless, this esterification reaction can also be accomplished using Candida Antarctic lipase B, an enzymatic catalyst, with greener conditions as explored by Sengupta et al. [29] . More recently, Muthusamy et al. used the same enzymatic catalyst to obtain furan based oligoesters. [30] The use of enzymes as green catalysts for macromolecular synthesis has become prevalent in the recent years as highlighted by Kobayashi and coworkers. [31] FTIR spectrum of FMO exhibits a predominant band at 1738 cm -1 , confirming the presence of an ester carbonyl group and the furan heterocycle bands at 1503, 1375, 1347, 1228, 1080, 920, 885, 739 and 599 cm -1 (Figure 1). The appearance of the resonances of the methylene protons of the ester moiety at 5.1 ppm and the furan ring protons at 6.36, 6.39 and 7.41 ppm on the 1 H NMR spectrum of FMO confirms the expected structure (Figure 2). The technical grade oleic acid received contains small amounts of ethyl ester impurities as the saponification of the triglycerides takes place in ethanol. This is evidenced by the quartet signal associated the CH2 of the ethyl ester at 4.21 ppm. The results of the analysis of 13 C NMR of FMO agreed with the assigned structure as depicted in Figure S1 in the supplementary information (SI). This type of epoxidation is nowadays routinely used on vegetable oils. [32] When compared to conventional academic epoxidation based on meta-chloroperoxybenzoic acid (mCPBA) as an oxidizing agent in solvent, [33] in situ epoxidation uses milder oxidizing agents and globally adheres better to green chemistry principles. The FTIR spectrum of FMOO (Figure 1c) shows the disappearance of the double bonds at 3009 cm -1 . By 1 H NMR analysis (Figure 2c) the structure is confirmed by the disappearance of the vinyl and allyl protons at 5.35 and 1.99 ppm respectively, and by the appearance of the epoxide and protons alpha to the epoxides at 2.90 and 1.45 ppm respectively. The results of the analysis of 13 C NMR of FMOO agreed with the assigned structure as depicted in Figure S2 in SI. An oligomer polyether diol was finally obtained by the acid-catalyzed ring-opening polymerization (ROP) of FMOO and denoted as Furan Oligomer (FO). This cationic polymerization was accomplished using fluoroantimonic acid, HSbF6, as it predominantly achieved oligoether species. [34] It was observed that 1.5 w.t. % of initiator was required to achieve complete oligomerization. The reaction was performed in bulk and followed by 1 H NMR analysis. The FTIR spectrum (Figure 1d) exhibits the broad stretch of the hydroxyl groups at 3440 cm -1 . The FO structure was further confirmed by [START_REF] Haponiuk | Polyurethane Polymers[END_REF] H NMR analysis (Figure 2d) by the disappearance of the epoxide protons and protons in alpha to the epoxides at 2.90 and 1.45 ppm, respectively, and the appearance of the formed polyether backbone between 3.00-3.60 ppm (for detailed integration see Figure S3). By 13 C NMR and 13 C NMR DEPT 135, as seen in Figure S4 and Figure S5 in the SI respectively, carbons associated to the polyether backbone corresponding to ether and carbon containing secondary hydroxyl groups clearly appear. Concurring with Lligadas et al., water was introduced in the oligomerization synthesis to suppress the occurrence of macrocyclization. This allowed for the nucleophilic attack of water to compete with intramolecular attacks, consequently favoring the proportion of linear species. Three different oligomers architectures were synthesized.

As highlighted in Table 1, introducing higher water equivalents limits the oligomerization and yields overall higher hydroxyl values (HV) of the synthesized oligomers. The oligomer hydroxyl functionality was determined by using the Mn and HV and remains very close to two for all oligomers synthesized. This was further confirmed by the MALDI-TOF MS analysis of Oligomer Sample B as seen in Figure S6 and Figure S7 in SI, where the repeating units, n, varied from 2 to 9. The most intense peaks belonging to the series of oligomers with n values of 3-5. The HV was determined using 31 P-NMR and quantification results can be found in Figure S8 and Table S2 in SI. [35] Quantitative 1 H NMR was used to determine the furan content (mmol furan/g) of FO and quantification results are available in Figure S9 and Table S3 in SI. Oligomer B was then used for the PU synthesis.

Analysis of DA of FO and N-Methylmaleimide

FO was reacted with monofunctional N-methylmaleimide in DMSO at 65 ºC, to study the DA adduct formation by [START_REF] Haponiuk | Polyurethane Polymers[END_REF] H NMR in absence of gelation. [START_REF] Haponiuk | Polyurethane Polymers[END_REF] H NMR attests the formation of the DA adduct (Figure 3a). Proton assignment was achieved using the 1 H NMR of initial compounds and comparison to the literature. [19b, 36] To quantify the conversion, proton signals, f1 and f4 (in the furan) and m2 (in the maleimide) and f1 ' , f4 ' and m2 ' (in the adduct) were used as they are free from overlap with other signals (Figure 3a). The conversion was calculated in accordance with Equation 1 for proton set f1 and f1 ' as follows; where Intf1 and Intf1' represent the integral values of their respective protons: Mn (SEC; g/mol) Ð [b] Functionality (Mn/EW) [c] Furan Content by [START_REF] Haponiuk | Polyurethane Polymers[END_REF] Similar results could be obtained with proton sets f4, f4 ' and m2, m2 ' . The evolution of the conversion with respect to time at 65 ºC is depicted on Figure 3b. An equilibrium conversion of 65% was attained after 121 hours of reaction time, as similarly reported in literature. [37] Furthermore, proton signals, f1 ' and m2 ' , were used to quantify the evolution of two diastereomers of the DA adduct (endo and exo) with respect to time (Figure 3b), as the chemical shift differs slightly for each diastereomer. The endo adduct which is kinetically favored, [36] is first formed at 62%. As the reaction time progresses, the thermodynamically stable exo diastereomer becomes the predominant entity (91 w.t. %) after 121 hours of reaction time.

Analysis of DA and r-DA Reactions between FO and a Bismaleimide

To test and simulate the formation of DA adducts between PU chains, the FO was reacted with PPO BMI. DSC was used to monitor the formation of the DA adducts and the r-DA reaction. A stoichiometric mixture of FO and PPO BMI was heated from -30 to 175 ºC. As controls, FO and PPO BMI were separately investigated under the same conditions. As seen in the thermograms reported in Figure 4a, no thermodynamic phenomena take place for each separate entity.

However, in the thermogram of the mixture (orange), an exothermic peak begins to appear between 50 to 60 ºC, indicating the DA reaction. Two endothermic peaks, corresponding to the r-DA, follow the exothermic peak. The first one, between 90-120 º C, corresponds to the r-DA of the endo diastereomer, while the second one refers to the r-DA of the exo diastereomer, between 120-150 °C. [36] The thermoreversibility of the network formed between the FO and the BMI was further qualitatively studied by dissolving the mixture in DMSO and exposing it to different heating cycles.

As depicted in Figure 4b-i, the mixture is first completely soluble in DMSO. Once exposed to 60 ºC for 24 hours, a gel is formed by cross-linkage (Figure 4b-ii). The mixture was then exposed to 120 ºC for 1h, causing the disassembly of the network due to the r-DA reaction (Figure 4b-iii).

When repeating the heating cycle, the same phenomenon took place (Figure 4b-iv). To further evidence the DA reaction, the mixture was studied prior and after gel formation by FTIR. Before crosslinking, the FTIR spectrum of the mixture contains signals of free furan (739 and 1503 cm -1 ) and maleimide rings (695 and 826 cm -1 ) (Figure 4c). After gel formation, the FTIR spectrum exposes a complete disappearance of the respective free furan and maleimide ring signals. To quantitatively study the DA reaction and r-DA between FO and BMI, the rheological behavior of gel formation between the two entities was evaluated by a dynamic experiment. As seen in Figure S10, the gelation reaction was analyzed by a time sweep test at 60 °C at 1% strain and 1 Hz, while recording the variation of the two dynamic moduli. Gelation occurs when the moduli cross over at 1740 s (29 mins, (G' ~ G'' ~ 650 Pa). Thereafter, heating steps were imposed to evaluate the thermoreversibility of the gel, 60 °C for the DA and 120 °C fore r-DA. During the r-DA heating steps, the gel would disassemble and the two moduli shoot back to zero. The gel was reassembled two more times, each time leading to quicker gelation times. This thus proves the thermoreversibility of the crosslinking points of the PU materials made using these two building blocks.

Analysis of the Synthesis of Biobased Crosslinked PU containing FO and BMI

The synthesis of crosslinked biobased PUs containing thermoreversible DA adducts was obtained by using FO and PPO BMI. Scheme 1 illustrates the route for the preparation of the control (linear PUsreaction 1) systems and crosslinked PUs (reactions 1 and 2). Different types of PUs were prepared by varying the proportion of FO in a PU system by the polyaddition between the biobased polyester polyol (chemical structure found in Figure 5) and hexamethylene di-isocyanate (HDI). The FO content in the PU systems was varied from 0 to 30% of the total hydroxyl moles required to synthesize PUs with a final NCO:OH ratio of 1.05. PUs were obtained without HS (segments synthesized with short diols and di-isocyanates) to study the effect of the FO on a model, a simple system. The nomenclature of all PU systems synthesized is summarized in Table 2 All PU systems post curing (linear and crosslinked) were insoluble in conventional solvents (CHCl3, DMSO, THF, DMF) and thus were examined by FTIR. It has been shown that symmetrical di-isocyanates such as HDI increase the crystallinity and hard segment packing of polyurethanes which may account for the lack of solubility that occurs in the presented networks. [START_REF] Das | [END_REF] A comparison of the FTIR spectra of FO, biobased polyester rapeseed polyol, PU-FO-30, PU-FO-30-XL and PPO BMI is displayed in Figure 6. In Figure 6c of the FTIR spectrum PU-FO-30, the disappearance the hydroxyl stretches (approximately at 3440 cm -1 ) associated to the hydroxyls of FO and rapeseed polyol is observed. Consequently, the appearance of vibrations at 3380 and 1550 cm -1 corresponding to -N-H stretching and bending respectively of urethane group, whereas the ester urethane stretch present itself at a shouldering located at 1700 cm -1 . The FTIR spectrum of PU-FO-30XL (Figure 6d) displays a similar spectrum than that of PU-FO-30. However, the addition of PPO BMI to yield a crosslinked system results in the appearance of the carbonyls of the maleimides at 1705 cm -1 . Moreover, as the PU-FO-30XL is crosslinked via DA reaction, the disappearance of furan heterocycle and C-H bending of maleimide ring (826 and 692 cm -1 ) is observed.

Characterization of Biobased Crosslinked PUs

The thermal stability of all PU materials was evaluated by TGA. Detailed TGA results are found in Figure S11 in SI. As exhibited in Table 3, the initial thermal degradation (T5%) for all materials begins at temperatures above that of the r-DA reaction. This is indicative that reprocessing and disassembly of crosslinked points is thus possible without thermal degradation, although the introduction of the FO seems to decrease the T5% (from 337 to 230 °C for PU-FO-0 to PU-FO-30).

Furthermore, the respective crosslinked material have lower T5%, with except of PU-FO-10XL. The glass transition temperatures (Tg) were measured by DSC and results are summarized in Table 3. The Tg of crosslinked materials were recorded using a single heating ramp, from -80 °C to 175 °C in order to avoid any significant degradation as indicated by TGA results. When comparing a control PU-FO-30 with its respective crosslinked PU counterpart (PU-FO-30XL), the Tg increases due to a high level of crosslinking. Furthermore, as seen in Figure 7, the thermograms of crosslinked PUs exhibit the presence of an endothermic peak between 90 and 150 ºC corresponding to the r-DA reaction, whereas, their control PU counterparts are flat-lined as no BMI is present. Materials containing increased amounts of FO, (and consequently increased crosslinking density) logically exhibit a more prominent endothermic r-DA peak between 90 and 150 ºC. A slight endothermic peak seems to appear in all crosslinked materials of Figure 7 between 30 to 60 ºC. As previously mentioned, crosslinked materials are evaluated by a single heating ramp form -80 to 175 ºC. These endothermic peaks were present as well in linear material upon first heating but disappeared upon the second heating when as the thermal history of the material has been erased. Thus, these slight endothermic peaks appear to be associated the thermal history of the material. Detailed DSC results are found in Figure S12. From Figure 8, which exhibits the stress-strain curves of PU-FO-0, PU-FO-10XL, PU-FO-20XL

and PU-FO-30XL, it can be deduced that by increasing the portion of FO in the PU structure and crosslinking by PPO BMI, the material properties vary from a thermoplastic elastomer (PU-FO-0) to a thermoset material (PU-FO-30XL). Furthermore, it was evidenced that mechanical properties can be further tuned by varying the equivalents of PPO BMI introduced into the material. This is evidenced by the mechanical properties of PU-FO-20XL-0.7BMI and PU-FO-20XL-0.5BMI, whereby varying the equivalents of PPO BMI between 0.7 and 0.5 to furan equivalents resulted in tensile strengths and Young's modulus in an intermediate range between PU-FO-20 and PU-FO-20XL (Table 3). All PU materials were lastly characterized by swelling test in DMF. As seen in Table 3, when first examining linear materials PU-FO-0, PU-FO10, PU-FO-20, and PU-FO-30, it can be observed that the polymer networks experience higher swelling when increasing the amount of FO in the polymer architecture. Contrastingly, due the crosslinking by DA reaction, the results of crosslinked PUs yield considerably lower swelling ratios in comparison to their respective control PU. The most considerable effect of DA crosslinking is seen for the highest crosslinked content, between PU-FO-30 to PU-FO-30XL, where the swelling ratio decreased from 146 to 70%, respectively, and the insoluble fraction increases from 78 to 93%, respectively. Except PU-FO-30, all PU material were relatively insoluble. Furthermore, the swelling ratio of PU material PU-FO-20XL-0.7BMI and

PU-FO-20XL-0.5BMI yields intermediate results between the crosslinked material, PU-FO-20XL

and the linear control PU, PU-FO-20. This is a testament of tunable properties when varying not only the FO content but as well the PPO BMI concentration.

Thermoreversibility of Biobased Crosslinked PU leading to Remendable and Self-healing

Behaviors

Preamble studies conducted to prove the thermoreversible between FO and the monofunctional N-methylmaleimide or PPO BMI permitted to test and simulate the thermoreversibility of crosslinked PU by the DA and r-DA reaction. To first evaluate the thermoreversibility of crosslinked marterials, PU-FO-20XL was examined by FTIR after exposing the material to different heating cycles. As depicted in Figure 9, when heating the material for 2h at 120 ºC, free maleimide ring peaks appear prevalent at 692 and 825 cm -1 due the occurrence the r-DA reaction. These free maleimide ring peaks disappear once the DA reaction takes place, when the material is reexposed to 70 ºC for 20h. This heating cycle was repeated and yields the same result, as seen in Figure 9. This indicates that the materials crosslinked by DA have the potential of being reprocessable and could be self-healing. Reprocessability was also studied on PU-FO-30XL. It was first processed by solvent casting and then characterized (Table 4). This material was then reprocessed by compression molding at 150 °C for 25 minutes. The material was cured at 60 °C in the oven for 48 hours and post cured for 24 hours at room temperature. The uniaxial tensile test and Tg were studied for this material labeled PU-FO-30XL-RP1. The material was then reprocessed, re-evaluated, and labeled PU-FO-30XL-RP2. The results are summarized in Table 4 and are conclusive that the material can indeed be reprocessed without significant material degradation. The Tg of the reprocessed material is stable, as is the Young's modulus. Nevertheless, reprocessing does induce only some light decrease in the material performances as the tensile strength and elongation at break slightly decreases.

As depicted in Figure 10a, self-healing was studied by cutting a dumbbell sample of PU-FO-30XL

in half and exposing it to 1 hour at 120 ºC and 2 days at 60 ºC and denoted PU-FO-30XL-HL1. By utilizing the thermo-reversibility of the DA bonds, the breaking and reforming of the crosslinks allows a severed dumbbell sample to be healed when exposed to such heating cycles. The initial heating at 120 °C for 1 hour allows to break the crosslinks, thus enhancing network mobility.

Consequently, the polymer chains can locally flow giving rise to adhesion to the two severed pieces. The final heating step at 60 ºC for 2 days allows the network to be reformed by the DA reaction. As seen Figure 10b, the healed materials show a novel integrity and recovery in terms of Young's modulus with around 0.5 MPa, a value equivalent to the initial material. Only at high elongation, a decrease of the performances is visible. This behavior is largely indicative that the thermoreversible DA crosslinking of PUs allows the material to have self-healing properties.

Conclusion

Thermo-reversible and self-healing crosslinked biobased systems based on urethane groups and clickable functions were successfully synthesized using vegetable oil derivatives and utilizing Diels-Alder reaction to control the design of their macromolecular architectures. A new biobased building block has been designed. The Furan Oligomer was developed from oleic acid in a sustainable way, largely in agreement with different green chemistry principles. As a general trend, crosslinked systems had characteristically higher Tg, insoluble ratio and enhanced stiffness and strength and lower swelling ratio when compared to their control PU counterparts. The crosslinked materials exhibit modular mechanical properties with respect to Furan Oligomer content. These biobased materials present a large range of behavior, from a typical thermoplastic elastomer-like material (least crosslinked material) to a thermoset (most crosslinked material). Reprocessing cycles were successfully performed and exhibited stable material properties. The systems also present self-healing properties. Dumbbell samples were cut in half, healed by joining the pieces together and exposing them to 120 °C, and cured at 60 °C. Both parts were able to recombine and considerably mend the material. To this end, this work is a stepping-stone in developing vegetable oil-based architectures with integrated dynamicity for evolution of high-performance and advanced biobased materials.

Future efforts will be dedicated to develop a greener chemistry with e.g. the reduction of the solvent consumption and the use of more sustainable solvents. In the near future, similar crosslinked vegetable oil based polyurethane formulation varying HS content can be studied for material application such as adhesives and coatings thus providing exciting perspectives for a greener future with advanced behavior to decrease the global impact of these materials, with a controlled end of life.

Experimental Section

Reagents and Materials

Octadec-9-enoic acid (OA) from sunflower oil was kindly supplied by ITERG. It is a technical grade fatty acid with an acid value of 185 mg KOH/g. Furfuryl alcohol (FA, 98%), 4-dimethylaminopyridine (DMAP, 99%), hydrogen peroxide (H2O2, 30% w/v aqueous solution), 2-chloro-4,4,5,5-tetramethyl-1,3,2-dioxaphospholane (CL-TDP, 95%), chromium (III) acetylacetonate (99.99%), cholesterol (>99%), dimethyl sulfoxide-d6 (DMSO, 100%, 99.96% atom D) were purchased from Sigma Aldrich; N,N'dicyclohexylcarboiimide (DCC, 99%), polyether catalysis HSbF6 (65% w/w aqueous solution) were purchased from Alfa Aesar; hexamethylebne diisocyanate (HDI, 99%), chloroform-d (CDCl3, 100%, 99.96% atom D) were purchased from Acros Organics; dichloromethane dried over molecular sieve (DCM, >99%, stabilized with amylene), diethyl ether (stabilized with BHT) were purchased from Carlo Erba; toluene (100%,) was purchased from VWR Chemicals; glacial acetic acid (GAA, >99%), N,Ndimethylformamide (DMF, 99.8%) was purchased from Fisher Scientific and amberlite IR-120 H from Fluka Chemicals. Biobased polyester polyol was purchased from Oleon is derived from dimeric fatty acids from rapeseed oil with purity greater than 98% and weight average molar mass (Mw) around 3 000 g/mol. Hydroxyl and acid values were 33 and 0.253 mg/g KOH, respectively. The poly(propylene oxide) based bismaleimide (DP = 3) cross-linker was purchased from Specific Polymers (Castries, France). 2,3,4,5,6-pentafluorobenzaldehyde (PFB,98%) was purchased from fluorochem.

Syntheses and Material Processing

Esterification of Furan-2-ylmethanol and Octadec-9-enoic Acid from Sunflower Oil

The esterification of Octadec-9-enoic acid (OA) was adapted from Vilela et al. and Rauf et al., [28,39] for an esterification conducted under mild conditions. A solution included one molar equivalent of OA, N,N'-dicyclohexylcarboiimide (DCC, 1.1 mol eq.), 4-dimethylaminopyridine (DMAP, 0.1 mol eq) in 25

w/v % of dried dichloromethane (DCM) was stirred magnetically for one hour in a two-necked round bottom flask at room temperature. Furfuryl Alcohol (FA, 1.1 mol eq) was then added dropwise through a dropping funnel. The reaction mixture was left stirring overnight at room temperature and under nitrogen. The N,N' -dicyclohexylurea (DCU) formed during the reaction precipitated and was filtered off. The filtrate was washed twice with water, twice with 5% acetic acid solution and finally again with water until a neutral pH was attained. The organic phase was dried over anhydrous sodium sulphate.

The solvent was evaporated under reduced pressure. The ester formed, Furan-2-ylmethyl-octadec-9enoate (FMO) was then further purified over a silica gel plug using petroleum ether/diethyl ether (95: The epoxidation of FMO was conducted by the in-situ formation of peracetic acid from glacial acetic acid and H2O2, and was adapted from Arbenz et al. In a two-necked round bottom flask, equipped with a magnetic stirrer and a dropping funnel, FMO (1 mol eq of unsaturations), amberlite IR-120 H (25 wt% of FMO), glacial acetic acid (0.2 mol eq) was inserted into the flask and dissolved in 12.5 wt% (m:v) of toluene. The solution was stirred to 70 °C and then 1.2 mol eq. of H2O2 was added dropwise. The mixture was stirred at 70°C for 7 hours. In comparison to current literature [32,40] , a lower acetic acid and H2O2 but increased toluene content is required to yield a high conversion of the alkene bond into an epoxy moiety, all the while minimizing the hydrolysis of the newly formed ester bond. The optimization trials of reactional components stoichiometry can be found in SI Table S1. Amberlite IR-120 H was filtered off and the solution was washed with distilled water until a neutral pH was attained.

The organic phase was dried with anhydrous sodium sulphate and then filtered.

Crosslinking of Polyurethanes containing of FO

The polyurethane was dissolved in dry DMF. To this, BMI was added in a 1:1 furan to maleimide ratio (mol/mol). The reaction mixture was stirred at 70 °C for 12 h to allow for the DA adducts to form. The solution was poured into a polyetrafluoroethylene mold and put into the oven to evaporate the solvent (70 °C, 48 h). Solvent traces were removed placing material under vacuum at 70 ºC for 24h.

Material Reprocessing of Crosslinked Polyurethanes

Polyurethane films are reprocessed by compression molded in tile mold (10 cm x 10 cm x 1 mm) at 150 °C in LabTech Scientific press hot press. First, the films were cut in several small pieces and placed in the center of the tile and left to undergo a 10 minutes preheating cycle to soften the material.

This was followed by several venting steps and the material was pressed with a constant applied force of 16 MPa between the two plates. The square materials were then cured at 60 °C in the oven for 48 hours and post cured for 24 hours at room temperature.

Study of the Self-Healing Behavior

The macroscopic self-healing ability of material PU-FO30XL was determined by observing how a dumbbell-shaped sample cut in half healed. To accomplish this, a dumbbell-shaped sample was cut in half in the middle and subsequently healed by placing the halves in contact for 1 hour at 120 º C, followed by 60 ºC overnight and 24 h at room temperature.

Methods

Acid Value

Acid value (AV) was determined according to ISO 660. Approximately 0.1 g of OA was dissolved in 50 mL of a 50:50 ethanol and diethyl ether solution. One drop of phenolphthalein solution used an indicator was added to the solution. Then titration is performed with 0.1M potassium hydroxide (KOH) aqueous solution. AV is express in mg of KOH.g -1 according to Equation 2. Where Veq (mL) is the titration equivalent volume, C (mol.L -1 ) is the KOH solution concentration and Ws (g) is the OA weight.

𝐴𝑉 = (𝑉

Iodine Value

Iodine value (IV), known as the measure of the double bond content, was determined using the Wijs method. This result is expressed in g of iodine (I2) per 100 g of sample. A 0.5g sample was dissolved in 20 mL of 50:50 mixture of cyclohexane and glacial acetic acid. Then, 25 mL of the Wijs solution (0.1M) composed of iodine chloride in glacial acetic acid solution was added; the corresponding solution was stirred for 1 h in the dark. Then, 20 mL of a 10 wt% potassium iodide aqueous solution and 150 mL of water was added to quench the reaction. The solution was then titrated with a 0.1 M sodium thiosulphate (Na2S2O3) solution using starch solution as indicator. IV in I2 g/100 g was determinate according to Equation 3: 𝐼𝑉 = (𝑉 𝑏𝑙𝑎𝑛𝑘 -𝑉 𝑠 ) * 𝐶 * 126.9 𝑊𝑠 * 100 (3) Vblank (mL) and VS (mL) are the volumes of Na2S2O3 solution required for blank and sample titrations, respectively. C (mol L -1 ) is the Na2S2O3 solution concentration and WS (g) is the sample weight. It is important to note that Wijs method does not react with the conjugated double bonds of the furan ring.

NMR Spectroscopies

All NMR spectra were recorded on a Bruker Ascend™ 400 or 500 MHz spectrometer. Sample were dissolved in deuterated chloroform (CDCl3) or deuterated dimethyl sulfoxide (DMSO-d6) with concentrations varying between 8-10 and 20-30 mg/mL for 1 H-NMR and 13 C-NMR, respectively. The number of scans was set to 128 for 1 H-NMR and 2048 for 13 C-NMR. The calibration of 1 H-and 13 C-NMR spectra was performed using the chloroform peak at 7.26 and 77.16 ppm, respectively and DMSO peak at 2.50.

31

P NMR analysis was performed on FO and polyester polyol samples phosphytilated with 2-chloro-4,4,5,5-tetramethyl-1,3,2-dioxaphospholane. Cholesterol was used as an internal standard, as described in standard protocols. [35a] 128 scans were recorded with a 15s delay and a spectral width of 80 ppm (180-100 ppm).

For quantitative 1 H NMR analysis, about 20 mg of FO were dissolved in 0.5 mL of DMSO-d6 before the addition of 100 mL of a standard solution of pentafluorobenzaldehyde in DMSO-d6, 32 scans were collected 10 s delay.

Fourier Transform Infrared Spectroscopy (FTIR)

Infrared spectroscopy was performed using a Fourier transformed infrared spectrometer, Nicolet 380 used in reflection mode equipped with an ATR diamond module (FTIR-ATR). An atmospheric background was collected before each sample analysis (32 scans, resolution 4 cm -1 ).

Mass Spectroscopy -MALDI-TOF

Mass spectra were acquired on a time-of-flight mass spectrometer (MALDI-TOF-TOF Autoflex Speed LRF, Bruker Daltonics, Bremen, Germany) equipped with a nitrogen laser (λ = 337 nm). An external multi-point calibration was carried out before each measurement. Scan accumulation and data processing were performed with FlexAnalysis 3.0 software. 2.5-dihydroxybenzoic acid (DHB) was obtained from Sigma-Aldrich. Matrix solutions was freshly prepared: DHB was dissolved to saturation

Scheme 1 .

 1 Scheme 1. Illustration of the general synthesis of biobased thermoreversible polyurethane-based systems using Furan Oligomer

Scheme 2 .Figure 1 .

 21 Scheme 2. Global reaction pathway of FO

Figure 2 . 1 H

 21 Figure 2. 1 H NMR spectra (a) of OA (b) FMO (c) FMOO and (d) FO

Figure 3 .

 3 Figure 3. (a) 1 H NMR spectra of mixture of FO and N-methylmaleimide after 0 and 24h, (b) conversion and percentage of endo structure with respect to time at 65 ºC.

Figure 4 .

 4 Figure 4. (a) DSC thermograms of PPO BMI, FO and the mixture of both, from 0 to 170 ºC, (b) i -Initial mixture of FO and PPO BMI, iimixture after 24h at 60 ºC, iiimixture after additional 1h at 120 ºC, ivmixture after additional 24h at 60 ºC, (c) FTIR spectra of initial mixture (green) and final dried crosslinked gel (orange).

  . The different control PU systems, with varying amounts of FO content were denoted as PU-FO-0, PU-FO-10, PU-FO-20, and PU-FO-30. Thermoreversible crosslinked PU systems were obtained by dissolving the corresponding linear PU in dry DMF, adding the required amount of PPO BMI and letting the solution stir overnight at 70 ºC, to allow the DA reaction to take place. These solutions then underwent solvent casting in a PTFE square mold for 48 h at 70 ºC followed by 24 h under vacuum at 70 ºC to yield a crosslinked PU. Crosslinked PUs containing the required PPO BMI mass to yield a 1:1 furan to maleimide ratio were denoted as PU-FO-10XL, PU-FO-20XL and PU-FO-30XL. Two additional crosslinked PUs were synthesized using the linear PU-FO-20 but with 1: 0.7 and 1: 0.5 furan and maleimide ratios and denoted as PU-FO-20XL-0.7BMI and PU-FO-20XL-0.5BMI, respectively.

Figure 5 .

 5 Figure 5. Chemical structure of the biobased polyester polyol supplied by the provider. Polyol based on dimeric fatty acids from rapeseed oil.

Figure 6 .

 6 Figure 6. FTIR spectra of (a) FO (b) Rapeseed Polyol (c) PU-FO-30 (d) PU-FO-30XL (e) PPO BMI

Figure 7 .

 7 Figure 7. Thermograms of four PUs: PU-FO-10, PU-FO-10XL, PU-FO-20XL and PU-FO-30XL from 0 to 175 °C.

Figure 8 .

 8 Figure 8. Stress-strain curves of PU-FO-0, PU-FO-10XL, PU-FO-20XL and PU-FO-30XL.

Figure 9 .

 9 Figure 9. FTIR spectra of original PU-FO-20XL followed by: PU-FO-20XL heated for 2h at 120 °C, PU-FO-20XL heated for 20h at 60 °C and PU-FO-20XL heated for 2h at 120 °C.

Figure 10 .

 10 Figure 10. Self-healing evaluation. (a) (i) Photo of PU-FO-30XL dumbbell sample cut in half, (ii) healed dumbbell sample of PU-FO-30XL after heating1 hour at 120 ºC and 2 days at 60 ºC, (iii) lifted healed dumbbell with zoom on healed section (b) Stress-strain curves of PU-FO-30XL and PU-FO-30XL-HL1

  13 C NMR (500 MHz, CDCl3, δ(ppm)): 174 (C=O), 150 (C-2 of 2-furan), 143 (C-5 of 2-furan), 111 (C-3 of 2-furan), 111 (C-4 of 2-furan), 85.0-70.0 Polyether backbone(C-O-C), 57.9 (OCH2-2-furan), 34.2 (CH2-C(=O)), 31.9-22.7 (aliphatic CH2), 14.1 (CH3) Model Reaction between FO and N-methylmaleimide 200 mg of FO (0.378 mmol of furan) was dissolved in 2 mL of DMSO-d6 in a 10 mL round bottom flask.N-methylmaleimide (0.378 mmol of maleimide) was added to obtain a furan to maleimide ratio of 1:1 (mol/mol). The solution was placed in an oil bath regulated to 65 ºC and samples of 200 µL were taken at regular intervals over 5 days. The samples were further diluted to 650 µL and analyzed by 1 H NMR to determine the reaction conversion and endo/exo ratio of the DA adducts.Diels-Alder and Retro-Diels-Alder Reaction between FO and a BismaleimideFO and a PPO-based BMI were mixed together in a small flask at a 1:1 furan to maleimide ratio (mol/mol) and dissolved in DMSO at 40 wt% concentration. The sample was heated in an oven at 60 ºC without stirring. Once gelation occurred (typically after 24 hours), samples were heated to 120 ºC for 1 hour in an oven to perform the r-DA reaction and the mixture returned to a liquid state. The sample was underwent the same cycle one more time. After crosslinking in solution, the gel was washed with water and dried in an oven at 40 ºC under vacuum. The gel was analyzed by FTIR compared to the initial mixture of FO and BMI.Synthesis of Polyurethanes containing FOAll glassware and reactants were previously dried. Four different kinds of polyurethanes films were prepared with varying amounts of FO (0%, 10%, 20% and 30% of total the hydroxyl value was derived from the FO, for example for FO-20%, 2.96 mmol OH from FO). Under nitrogen, in a three-necked round bottom flask, biobased rapeseed polyester polyol (11.8 mmol OH) and FO ( 2.96 mmol OH) was added to hexamethylene di-isocyanate (HDI, 15.5 mmol NCO) and the mixture was mechanically stirred at 70 °C for 3-4 h. The ratio -NCO/-OH functional groups were fixed to 1.05 in all cases.

Table 1 .

 1 Synthesis and properties of different FO obtained by cationic ROP of FMO

	Oligomer Sample	Initiator HSbF6 (w.t. %)	H20 (mol %)	Hydroxyl Value by 31 P NMR (mg	Equivalent Weight (g/equiv) [a]
				KOH/g)

Table 2 .

 2 Linear PU (control) and crosslinked systems: nomenclature and corresponding formulations.

			Furan	Rapeseed
	PU System	Type of	Oligomer (FO)	Polyol Content	PPO BMI
	Nomenclature	System	Content [a]	[a]	Equivalents [b]
			(%)	(%)
	PU-FO-0	Linear	0	100	0
	PU-FO-10	Linear	10	90	0
	PU-FO-10XL	Crosslink	10	90	1
	PU-FO-20	Linear	20	80	0
	PU-FO-20XL	Crosslink	20	80	1
	PU-FO-20-0.7BMI	Crosslink	20	80	0.7
	PU-FO-20-0.5BMI	Crosslink	20	80	0.5
	PU-FO-30	Linear	30	70	0
	PU-FO-30XL	Crosslink	30	70	1

Table 3 .

 3 Characterizations of synthesized biobased polyurethanes.

	PU System	TGA T5% (°C)	DSC Tg (°C)	Young's Modulus E (MPa)	Tensile Strength σ (Mpa)	Elongation ɛ (%)	Swelling Ratio in DMF (%)	Insoluble Fraction (%)
	PU-FO-0	337	-50	0.39 ± 0.03	0.58 ± 0.02	563 ± 36	36.9 ± 1.8	96.4 ± 0.3
	PU-FO-10	327	-51	0.14 ± 0.01	0.36 ± 0.02	368 ± 27	51.6 ± 2.3	94.4 ± 0.6
	PU-FO-10XL 333	-51	0.32 ± 0.01	0.51 ± 0.03	409 ± 54	44.3 ± 0.6	95.0 ± 0.9
	PU-FO-20	298	-51	0.11 ± 0.01	0.38 ± 0.03	288 ± 4	58.5 ± 3.4	92.9 ± 0.6
	PU-FO-20XL 267	-50	0.32 ± 0.01	0.50 ± 0.01	232 ± 2	46.9 ± 0.1	95.2 ± 2.9
	PU-FO-20-0.7BMI	287	-51	0.30 ± 0.01	0.40 ± 0.02	208 ± 2	48.2 ± 6.3	92.0 ± 2.9
	PU-FO-20-0.5BMI	293	-50	0.26 ± 0.01	0.32 ± 0.01	232 ± 11	55.7 ± 3.0	90.6 ± 1.2
	PU-FO-30	250	-55	NA	NA	NA	146.1 ± 7.4	77.5 ± 0.8
	PU-FO-30XL 234	-51	0.57 ± 0.09	0.49 ± 0.04	107 ± 9	70.4 ± 2.7	92.5 ± 0.6

Table 4 .

 4 Characterization of biobased PU, PU-FO-30XL reprocessed several times.

		DSC	Young's Modulus	Tensile Strength
	PU System	Tg	E	σ	Elongation (%)
		(°C)	(MPa)	(MPa)
	PU-FO-30XL	-51	0.57 ± 0.09	0.49 ± 0.04	107 ± 9
	PU-FO-30XL-RP1	-50	0.62 ± 0.03	0.47 ± 0.04	83 ± 5
	PU-FO-30XL-RP2	-50	0.58 ± 0.02	0.40 ± 0.03

-ylmethlyl 8-(3-octyloxiran-2-yl)octonoate (FMOO)

  The reaction was quenched by the addition of water and dissolved in diethyl ether. The organic solution was washed with a 5% sodium bicarbonate solution and distilled water until a neutral pH was attained. The solution was dried over anhydrous sodium sulfate, and filtered. The solvent was evaporated under reduced pressure and the oligomer was dried under vacuum overnight, at 50 °C. The oligomerization resulted in an orange clear viscous oil.

	The solvent was evaporated under reduced pressure. Epoxidized FMO (EFMO) was dried under vacuum overnight, at 50 °C. The reaction product, Furan-2-ylmethlyl 8-(3-octyloxiran-2-yl)octonoate (FMOO), was clear yellow oil. Furan-2-1 H NMR (400 MHz, CDCl3, δ (ppm)): 7.41 (1H, dd, 5-H of 2-furan)), 6.39 (1H, d, 3-H of 2-furan), 6.36 (1H, dd, 4-H of 2-furan), 5.06 (2H, s, OCH2), 2.90 (2H, m, CHOCH), 2.32 (2H, t, CH2C(=O)), 1.62 (2H, m, CH2CH2C(=O)), 1.37-1.18 (24H, m, aliphatic CH2), 0.88 (3H, t, CH3). 13 C NMR (500 MHz, CDCl3, δ(ppm)): 174 (C=O), 150 (C-2 of 2-furan), 143 (C-5 of 2-furan), 111 (C-3 of 2-furan), 111 (C-4 of 2-furan), 57.9 (OCH2-2-furan), 57.2 (HC-O-CH), 34.2 (CH2-C(=O)), 31.9-22.7 (aliphatic CH2), 14.1 (CH3) Oligomerization of Epoxidized Furan-2-lymethyl-octadec-9enoate (FMOO) For a molar equivalent of FMOO, H2O (0.7 mol eq) was added in a two-necked round bottom flask and stirred mechanically. A cationic initiator, HSbF6 (1.5 wt% of FMOO) was added dropwise. The mixture H of 2-furan), 6.36 (1H, dd, 4-H of 2-furan), 5.06 (2H, s, OCH2), 3.80 -2.82 (m, polyether backbone), 2.32 (2H, t, CH2C(=O)), 1.62 (2H, m, CH2CH2C(=O)), 1.37-1.18 (24H, m, aliphatic CH2), 0.88 (3H, t, was stirred at room temperature for 1 h. Furan Oligomer: 1 H NMR (400 MHz, CDCl3, δ (ppm)): 7.41 (1H, dd, 5-H of 2-furan)), 6.39 (1H, d, 3-CH3.

Acknowledgements

The authors would like to thank BPI-France for the funding of Trans'Alg Project and the corresponding consortium. Iterg for kindly providing technical grade oleic acid. The authors would like to thank Dr. Antoine Duval for his input, the R&D Team of Soprema (France) and Sophie Wendels for help with rheological measurements.

in a H2O/CH3CN/HCOOH (50/50,1%) solution. Typically, 0.5µL of a mixture containing the sample solution and the matrix (1/1) was deposited on the stainless-steel plate.

Rheological Measurements

Time required to form a gel between FO and BMI at 60 °C (gelation time, tgel) was determined by dynamic oscillatory rheology measurements performed on a a TA Instruments Discovery HR-3 using ETC parallel 25 mm plate geometry upper plate and 60mm lower plate. The viscometer was equipped with a Peltier system for temperature control. The evolution of storage (G´) and loss (G´´) modulus was measured at 60 °C at a constant shear strain (1%) and at 1Hz. The gel point was considered as the time where the curves of storage and loss modulus crossover at a fixed temperature and frequency.

Thermal Gravimetric Analysis (TGA)

Thermogravimetric analysis (TGA) was performed using a TA Instrument Hi-Res TGA Q5000 at a heating rate of 10 °C/min from room temperature to 700 °C under nitrogen atmosphere (flow rate 25 mL/min). Sample weighed between 1-3 mg.

Differential Scanning Calorimetry (DSC)

Differential scanning calorimetry (DSC) was performed using a TA Instrument Q200 under nitrogen flow (50 mL/min). Samples of 1-3 mg were sealed in standard aluminum pans. Depending on the information to be acquired, different heating runs were performed. For the qualitative studies on the DA adduct and r-DA, the FO and PPO-based BMI mixtures were heated up from -30 °C to 175 °C with a heating ramp of 2.5 °C/min. The glass transition temperature (Tg) of crosslinked materials were recorded using a single heating ramp, from -80 °C to 175 °C at 10 °C/min. The Tg of the non-crosslinked materials were recorded using cyclic procedure involving a heating ramp from -80 °C to 175 °C at 10 °C/min, followed by a cooling ramp down to -80 °C at 5 °C/min and finally a second heating to 175 °C at 10 °C/min. The temperature was kept constant for 2 minutes at the end of each temperature ramp.

Uniaxial Tensile Tests

Tensile properties of materials were determined with an Instron tensile testing machine (model 5567 H, USA), at a rate of 20 mm/min, using dumbbell specimens (dimensions: 30 X 5 X 1 mm 3 ). For each formulation at least five dumbbell samples were tested.

Swelling Index and Gel Content Determination

Swelling measurements of the polymer network were carried out in DMF. The swelling percentage was determined using the weight difference between dried and swollen samples. The sample (written mi and ranging from 10 to 20 mg) was immersed in 15 mL of solvent. After 24h, the sample was removed from the solution, the surfaces were dried superficially using a filter paper, and the weight was recorded (ms). Three repetitions were conducted for each material and each respective solvent. For each sample, a swelling ratio (SR) was calculated according to the Equation 4:

𝑆𝑅 =

𝑚 𝑠 -𝑚 𝑖 𝑚 𝑖 * 100 (4)

The sample was then dried in a vacuum oven at 40 ºC during 1 day. The resulting weight was recorded (mf) and the insoluble fraction (IF) was calculated according to the Equation 5: 𝐼𝐹 = 𝑚 𝑓 𝑚 𝑖 * 100 (5)


Siavash Ghabezloo
email: ghabezlo@cermes.enpc.fr

Jean Sulem

Sylvine Guédon

François Martineau

Effective stress law for the permeability of a limestone

Keywords: permeability, effective stress, limestone, microstructure, microhardness test

Introduction

The determination of the permeability of a rock and its variation with changes of total stress and pore pressure has a great interest in petroleum reservoir engineering. The pore pressure decrease during the oil or gas production from a reservoir causes the reduction of the permeability of the reservoir and consequently the reduction of the production rate. The variations of permeability with the state of stress is also important in geophysics in the studies of rapid fault slip events when shear heating tends to increase the pore pressure and to decrease the effective compressive stress and the shearing resistance of the fault material (Rempel and Rice [START_REF] Rempel | Thermal pressurization and onset of melting in fault zones[END_REF], Sulem et al. [START_REF] Sulem | Thermo-Poro-Mechanical Properties of Clayey Gouge and Application to Rapid Fault Shearing[END_REF]). The increase of the permeability of the fault material with the pore pressure increase can lead to a faster dissipation of the excess pore pressure and consequently can decelerate the weakening of the shearing resistance of the fault material.

If a single variable can be defined as a linear combination of the total stress and the pore pressure to express the variations of the permeability, then we can say that the permeability follows an effective stress law. The concept of effective stress was first introduced by Terzaghi [START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[END_REF] who defined it as the difference between the total stress σ and pore pressure f p and attributed all measurable effects of a change in stress exclusively to changes in the effective stress:

d f p σ σ = - (1)
In equation (1) d σ is the Terzaghi effective stress, which is also called the differential pressure. More generally, the effective stress ( )

, f p σ σ σ ′ ′ =
can be defined as a stress quantity which can be used as a single variable to express the stress dependency of a property of a porous material (Equation ( 2)). Reducing the number of independent variables from two to one by using the concept of effective stress greatly simplifies the analysis of total stress and pore pressure dependency of the porous material properties.

Q ( ) ( ) , f Q Q p Q σ σ ′ = = (2)
If the total stress and the pore pressure vary in such a way that the effective stress remains constant, then no variation in the corresponding property Q is expected. Thus the expression of the effective stress can be obtained from the evaluation of the isolines of ( , )

f Q p σ .
Since different material properties may depend on total stress and pore pressure in different ways, there is not a unique effective stress which would be appropriate for all properties of the material, and consequently different effective stress expressions should be defined for the different properties.

Assuming that the function ( , ) f Q p σ is smooth enough so that its derivatives can be defined, the incremental variation of can be written as:

Q f f Q Q dQ d dp p σ σ ∂ ∂ = + ∂ ∂ (3)
We can re-write equation [START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[END_REF] in the following form:

f f Q p Q dQ d dp Q σ σ σ ⎡ ⎤ ⎛ ⎞ ∂ ∂ ∂ = -- ⎢ ⎥ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎣ ⎦ (4)
The above expression shows that the variation of the property Q can be expressed as a function of a single incremental quantity dσ ′ :

Q dQ d σ σ ∂ ′ = ∂ (5)
with

Q f d d n dp σ σ ′ = - ( 6
)
The coefficient is the effective stress coefficient corresponding to the property Q defined as:

Q n ( , ) f Q f Q p n p Q σ σ ∂ ∂ = -∂ ∂ (7)
The isolines of Q are obtained by the integration of the following differential equation:

0 d σ ′ = (8)
In the close vicinity of a given state of stress ( , f p σ ), the isolines are generally approximated with parallel straight lines (e.g. Bernabé [START_REF] Bernabé | The effective pressure law for permeability in Chelmsford granite and Barre granite[END_REF]) which is equivalent to the assumption that is a constant. Under this assumption, equation ( 6) can be easily integrated and a linear expression for the effective stress is obtained:

Q n Q f n p σ σ ′ = - (9)
The linear expression presented in equation ( 9) is the most common expression for the effective stress as it is usually used in the mechanics of porous media. The effective stress coefficient is equal to one in Terzaghi's definition which means that the total stress and the pore pressure have similar, but inverse, effects on the variation of the property Q . A value of the coefficient smaller (respectively greater) than unity means that the effect of pore pressure change on the variation of the property Q is less (respectively more) important than the effect of a change in total stress.

Q n Q n
Zimmerman [START_REF] Zimmerman | Compressibility of sandstones[END_REF] and Berryman [START_REF] Berryman | Effective stress for transport properties of inhomogeneous porous rock[END_REF] have derived general effective stress rules for various physical properties of rocks and presented the expressions of the effective stress coefficients corresponding to different physical properties together with some bounds and general relations among these coefficients.

Q n

As mentioned earlier, the great advantage of the effective stress concept is the simplification of the analysis by using one single variable σ ′ instead of two independent variables, σ and f p . The resulted simplification is significant when the effective stress coefficient is constant, but the variability of this coefficient adds new difficulties to the problem. In this case, if the effective stress coefficient is a simple function of the pore pressure and the confining pressure, for example a linear function of the differential pressure d σ , then the use of the effective stress concept may be still advantageous. Obviously, if the effective stress coefficient involves a complicated relation between the pore pressure and the confining pressure, it may be more appropriate to abandon the effective stress concept and to analyse the problem using the two independent variables.

The effective stress corresponding to the variations of the permeability of the rocks has been studied theoretically and experimentally by different researchers. Berryman [START_REF] Berryman | Effective stress for transport properties of inhomogeneous porous rock[END_REF] has studied the problem theoretically and has derived the expressions of the effective stress coefficient for the permeability of the rocks constituted by one single and two different minerals. His approach resulted in an effective stress coefficient which is smaller than one for a microhomogeneous material made up of one single mineral and which may be greater than one for a porous material constituted by two different minerals. This is for example the case in clayrich sandstones as the grains modulus of the pore-filling material is much smaller than the one of the sand grains. Experimental studies on both sedimentary and crystalline rocks can be found in the literature. Most of the studies on sedimentary rocks have been conducted on clay-rich sandstones (Zoback [START_REF] Zoback | High pressure deformation and fluid flow in sandstone, granite, and granular materials[END_REF], Zoback and Byerlee [START_REF] Zoback | Permeability and effective stress[END_REF], Walls and Nur [START_REF] Walls | Pore pressure and confining pressure dependence of permeability in sandstone[END_REF], Nur et al. [START_REF] Nur | Effects of fluid saturation on waves in porous rock and relation to hydraulic permeability[END_REF],

Walls [START_REF] Walls | Effects on pore pressure, confining pressure and partial saturation on permeability of sandstones[END_REF], Al-Wardy [START_REF] Al-Wardy | Analytical and experimental study of the poroelastic behaviour of clean and clay-rich sandstones[END_REF], Al-Wardy and Zimmerman [START_REF] Al-Wardy | Effective stress law for the permeability of clay-rich sandstones[END_REF]) and resulted in effective stress coefficients which are greater than one and which increase with the clay content. The evaluated effective stress coefficient by Walls [START_REF] Walls | Effects on pore pressure, confining pressure and partial saturation on permeability of sandstones[END_REF] varies from 1.2 for clean sandstone to 7.1 for sandstone containing 20% clay. These results show that the permeability of the studied rock is more sensitive to variations of the pore pressure than to the changes of the total stress.

To explain these results, Zoback and Byerlee [START_REF] Zoback | Permeability and effective stress[END_REF] have proposed a conceptual clay-shell model in which the rock consists of quartz, permeated with cylindrical pores that are lined with an annular layer of clay which is more compliant than the outer quartz layer. Such a model is more sensitive to changes in pore pressure than to changes in total stress, and therefore results in an effective stress coefficient greater than one. Al-Wardy and Zimmerman [START_REF] Al-Wardy | Effective stress law for the permeability of clay-rich sandstones[END_REF] have proposed a different microstructural model in which the clay is distributed in the form of particles that are weakly coupled to the pore walls and have obtained results that are more compatible with the experimental results. Kwon et al. [START_REF] Kwon | Permeability of Wilcox shale and its effective stress law[END_REF] have studied the permeability of illite-rich shale with a clay content of about 45% and found an effective stress coefficient equal to one which differs from the results obtained for clay-bearing sandstones with low clay content. They concluded that this is due to the contiguity of clays throughout the rock which support the applied confining pressure as well as the pore pressure.

As opposed to the results obtained for porous sedimentary rocks, the experimental studies performed on crystalline and jointed rocks resulted in effective stress coefficients which are smaller than or very close to one. Walsh [START_REF] Walsh | Effect of pore pressure and confining pressure on fracture permeability[END_REF] examined the experimental results of Kranz et al. [START_REF] Kranz | The permeability of whole and jointed Barre granite[END_REF] for flow through joints with polished surfaces and joints with jagged surfaces formed by a tensile fracture. The estimated effective stress coefficients are equal to 0.9 for the polished surfaces and 0.56 for the tension fractures. Experimental studies of Coyner [START_REF] Coyner | Effects of stress, pore pressure, and pore fluids on bulk strain, velocity, and permeability in rocks[END_REF] and Morrow et al. [START_REF] Morrow | Effective Pressure Law for Permeability of Westerly Granite Under Cyclic Loading[END_REF] on two granites resulted in permeability effective stress coefficients very close to one. Bernabé [START_REF] Bernabé | The effective pressure law for permeability in Chelmsford granite and Barre granite[END_REF], [START_REF] Bernabé | The effective pressure law for permeability during pore pressure and confining pressure cycling of several crystalline rocks[END_REF] studied the permeability of several crystalline rocks and found effective stress coefficients which are smaller than one and which decrease with increasing confining pressure due to the changes in the geometry of the cracks during closure. He also reported the strong loading path dependency of the effective stress coefficient. This stress path dependency decreases rapidly with the number of cycles and the effective stress coefficient approaches one after a few cycles.

In this paper an experimental study for determining the effective stress law for the permeability of a limestone is presented. A drained hydrostatic compression test is performed in order to evaluate the drained bulk modulus of the rock. Constant head permeability tests are performed with different conditions of confining pressure and pore pressure and the effective stress coefficient for the variations of the permeability of the rock is determined. The results enable us to compare the sensitivity of the permeability of the rock to the variations of the total stress and of the pore pressure. Microscopic observations have been performed in order to study the microstructure of the limestone and to better understand the mechanism of permeability evolution with total stress and pore pressure variations. On the basis of these observations, a conceptual model is proposed for the microstructure of the material in order to determine the effective stress coefficient for the permeability of the rock and to interpret the experimentally measured values of this coefficient. This model accounts for the relative compliance of the constituents of the rock which is studied by performing microhardness tests.

Description of the studied rock

The rock studied in this paper is an oolitic limestone of Neocomian age (-135 to -127 million years), collected near Nîmes in south of France. This white limestone is homogeneous at the scale of the quarry, at the scale of the blocs and also at the scale of the samples observed under a microscope.

The observations are performed using a polarizing optical microscope and an environmental scanning electron microscope (ESEM). These two observation methods are complementary for this study, as they provide two different magnification scales.

The ESEM is equipped by two types of detectors: A detector of secondary electrons (SE) and a detector of backscattered electrons (BSE). The SE detector results in images which emphasize the topographical contrasts on the observed sample and have a three-dimensional appearance. In these images, steep surfaces and edges tend to be brighter than flat surfaces.

The BSE detector can detect the contrast between areas with different chemical compositions, especially when the average atomic number of the various regions is different. The brightness of the images taken by this detector tends to increase with the atomic number. The measured connected porosity of the studied limestone is between 13.4% and 17.6% with the average value equal to 15.7%. The measured apparent density of the skeleton is equal to 2.71g cm given by Bass [START_REF] Bass | Mineral Physics and Crystallography: A Handbook of Physical Constants[END_REF] for calcite. From these values the total porosity of the rock is evaluated equal to 17.7%. The difference between the connected and the total porosity reveals the presence of 2.0% occluded porosity in the rock matrix.

The porosity of the rock is studied using the optical microscope on a sample which was injected with a blue resin (Figure (3a)). The blue areas, which correspond to the pores, are mostly concentrated around the oolites in a layer with a thickness of 10 to 20 µm. The porosity of the rock can also be studied using the ESEM images taken with the BSE detector.

In these images the porous space appears as in black areas. An image is shown in Figure (3b) and shows clearly that the porosity is concentrated around the oolites. Comparing this porosity with the one observed in Figures (2a) and (2b) we can conclude that there are two types of porosity in the studied limestone: a macro-porosity and a micro-porosity. The macroporosity as seen in Figure (3) is mostly concentrated around the oolites while the microporosity corresponds to the intergranular voids between the grains of micrite in the oolite and between the grains of sparite in the cement. The analysis of several images showed that the blue areas, which represent the macro-porosity, fill about 13% of the total area of the images.

Comparing this value to the average connected porosity of the rock, 15.7%, one can deduce that most of the connected porosity is located around the oolites. Comparing the evaluated macro porosity, about 13%, and the total porosity, 17.7%, we can estimate the micro-porosity of the rock equal to 4.7%. As mentioned above, 2.0% of the porosity of the rock is occluded.

Comparing this value with the micro-porosity, we can infer that about 42% of the microporosity of the rock is occluded. The analysis of the image presented in Figure (1a) shows that about 28% of the area of this image is filled with the sparitic cement and thus the remaining 72% is filled with the oolites and the macro porosity. Knowing that the macro porosity is equal to 13% as presented above, the volumetric fraction of the oolites is found equal to 59%.

Laboratory tests

Laboratory tests in this study consist of a drained hydrostatic compression test for the evaluation of the compressibility of the rock and constant head permeability tests with different values of pore pressure and confining pressure in order to study their individual effect on the variations of the permeability. Note that all the tests are performed on the same sample. The drained compression test with a loading-unloading cycle is performed before the permeability tests. This guaranties that the sample behaves elastically during the permeability tests. The tests are performed on a 40mm diameter and 80mm length saturated sample in a triaxial cell as described in Ghabezloo and Sulem [START_REF] Ghabezloo | Stress dependent thermal pressurization of a fluid-saturated rock[END_REF]. The saturation procedure is the following: The sample is first dried at a temperature of 60°C until the measured mass remains constant and then it is put in a vacuum chamber during 24 hours to eliminate the remaining air bubbles. Water is then introduced in the chamber and the sample is kept immerged under water during 48 hours. The sample is then put in a rubber jacket with a grease layer inside the jacket and installed inside the triaxial cell. An initial confining pressure is applied and water is circulated using a pressure generator.

Drained hydrostatic compression test

The bulk modulus of the rock is evaluated in a drained hydrostatic compression test with a loading and unloading cycle. In this test, a fluid back pressure of 1.0MPa was maintained constant inside the sample. In Figure ( 4) the volumetric strain response is shown versus the applied total stress. The observed non-linear response reflects the stress-dependent character of the rock compressibility (Zimmerman [START_REF] Zimmerman | Compressibility of sandstones[END_REF], Sulem et al. [22][23]). Moreover this curve exhibits a different response in loading and unloading and irreversible deformation. Therefore the elastic tangent drained compression modulus is evaluated on the unloading curve. The slope of this curve is directly evaluated of the experimental data using the method used in Ghabezloo and Sulem [START_REF] Ghabezloo | Stress dependent thermal pressurization of a fluid-saturated rock[END_REF] where the slope of the tangent at each point is calculated as a difference quotient of N data points centred around the corresponding point. The evaluated bulk modulus is presented as a function of Terzaghi effective stress on Figure [START_REF] Zimmerman | Compressibility of sandstones[END_REF]. This curve can be approximated by the following linear expression:

d d d d K K σ σ = + (10)
This expression shows the linear increase of the drained bulk modulus with Terzaghi effective stress in the limit of the applied pressures. The non linear elastic response in hydrostatic compression can be attributed to the closure of the regions of imperfect bounding between grains (Zimmerman [START_REF] Zimmerman | Compressibility of sandstones[END_REF]) and the compaction of the porous space of the rock. Figure (3b) exhibits limited contact areas between the oolites and the surrounding cement, which can cause the compressibility of the microstructure of the rock. More contacts can be formed under stress leading thus to a strengthening of the rock. This strengthening effect will be limited at higher stress due to saturation of the network of contact points and possible grains breakage. Thus the linear increase of the bulk modulus is only valid at low stress.

Permeability tests

Constant head permeability tests have been performed in a triaxial cell on a water saturated limestone sample under isotropic loading. After the saturation, two pore pressure generators were used to keep a constant pore pressure difference of 300 kPa between the two ends of the sample. The confining pressure was always kept about 2MPa higher than the pore pressure to prevent the fluid infiltration between the sample and the jacket. The permeability tests have been performed with different combinations of pore pressure and confining pressure. The mean value of the two pore pressures imposed at the ends of the sample is taken as the sample average pore pressure for the evaluation of the effective stress inside the sample. After the saturation of the sample under the initial confining pressure, the pore pressure at each end of the sample was maintained constant and the confining pressure was increased by steps of 1.0MPa. At each step the flow rate was recorded after the stabilization of the fluid flow and the permeability of the sample was calculated using the Darcy's law. This procedure was repeated for three different pore pressure levels with eight confining pressure steps for each level. Thus, 24 permeability tests have been performed with various combinations of pore pressure and confining pressure. The results, presented in Figure [START_REF] Berryman | Effective stress for transport properties of inhomogeneous porous rock[END_REF], show the measured permeability as a function of the confining pressure for different average pore pressures.

These results show that the permeability decreases with the confining pressure increase for a constant average pore pressure. Moreover, under a constant confining pressure, the permeability increases with the pore pressure increase. These experimental results can be explained by the pore diameter changes under the effect of the confining pressure and the pore pressure variations. The shape of the permeability curves and its reduction with the confining pressure increase is similar to the results reported in the literature for porous sedimentary rocks (Coyner [17], Yale [START_REF] Yale | Network Modeling of Flow, Storage and Deformation in Porous Rocks[END_REF], David et al. [START_REF] David | Laboratory measurement of compaction-induced permeability change in porous rocks: Implications for the generation and maintenance of pore pressure excess in the crust[END_REF], Schutjens et al. [START_REF] Schutjens | Compaction-Induced Porosity/Permeability Reduction in Sandstone Reservoirs: Data and Model for Elasticity-Dominated Deformation[END_REF]) as well as for crystalline rocks (Kranz et al. [START_REF] Kranz | The permeability of whole and jointed Barre granite[END_REF], Coyner [START_REF] Coyner | Effects of stress, pore pressure, and pore fluids on bulk strain, velocity, and permeability in rocks[END_REF], Bernabé [START_REF] Bernabé | The effective pressure law for permeability during pore pressure and confining pressure cycling of several crystalline rocks[END_REF]). As mentioned by David et al. [START_REF] David | Laboratory measurement of compaction-induced permeability change in porous rocks: Implications for the generation and maintenance of pore pressure excess in the crust[END_REF], the compaction mechanism in crystalline rocks is related to the closure of microcracks and the pressure sensitivity of permeability decreases with confining pressure increase. This pressure sensitivity of the permeability is more important in crystalline rocks than in porous sedimentary rocks for which the compaction is related to the relative movement of grains.

Vairogs et al. [START_REF] Vairogs | Effect of rock stress on gas production from low-permeability reservoirs[END_REF] showed that the pressure sensitivity is more important for the rocks with lower permeability. On Figure [START_REF] Berryman | Effective stress for transport properties of inhomogeneous porous rock[END_REF], for the range of the applied confining pressures, the variation of the permeability with the confining pressure follows a power law. Even though this power law cannot be accepted as a general permeability-stress relationship due to the infinite value at zero stress, it can be accepted to express the permeability variations with the variations of the confining pressure in the limit of the applied pressures.

Analysis of test results

The easiest way to determine the effective stress coefficient for the variations of the permeability based on the obtained experimental results is to draw a contour plan of the permeability as a function of pore pressure and confining pressure. The slope of the isolines gives the effective stress coefficient corresponding to the variations of the permeability.

Parallel isolines would mean that the effective stress coefficient is constant. The contour plan is shown in Figure [START_REF] Zoback | High pressure deformation and fluid flow in sandstone, granite, and granular materials[END_REF]. In this figure, the same scale is used for the pore pressure and the confining pressure axes so that the slope of the isolines can be calculated directly on the graph. We can clearly see that the isolines are not parallel and that the effective stress coefficient calculated as the average slope of the isolines decreases with increasing values of the permeability. The effective stress coefficients evaluated on this graph varies between 0.9 and 2.4. Considering the limited number of data points on Figure [START_REF] Zoback | High pressure deformation and fluid flow in sandstone, granite, and granular materials[END_REF], it should be mentioned that most of the fine details of the isolines on this figure are just artefacts. Moreover, considering the small difference between the measured permeabilities at higher confining pressures, the evaluated effective stress coefficients should be seen as approximate values.

Nevertheless, these data clearly show the general trend for the evolution of the effective stress coefficient as described above.

In order to represent the effect of total stress and pore pressure on the permeability and on the effective stress coefficient, a power law for the variation of the permeability with the effective stress as defined by equation ( 9) is assumed:

b k aσ ′ = (11)
The effective stress coefficient is assumed to vary linearly with the differential stress (i.e.

Terzaghi effective stress):

k n k d n c d σ = + (12)
The coefficients and b in equation ( 11) and c and in equation ( 12) are evaluated using a least square algorithm by fitting the experimental data. The following expressions for the effective stress coefficient and for the permeability-effective stress dependency are found: a d The results of the curve fitting procedure are shown on Figure [START_REF] Zoback | Permeability and effective stress[END_REF]. We can observe that the experimental permeability measurements are well superposed when plotted against the obtained effective stress. Figure [START_REF] Walls | Pore pressure and confining pressure dependence of permeability in sandstone[END_REF] shows the simulation of the permeability tests using the above permeability law. This graphs show that the experimental data are well reproduced and that the contour plans of the measured and computed permeability are very well superposed.

As mentioned earlier, the main advantage of the effective stress concept is its simplicity.

Thus, one could simplify the stress-dependent permeability law given by equations ( 13) and ( 15) by neglecting the variations of the effective stress coefficient and take a mean value for , equal to 1.3.

k n

An effective stress coefficient which is greater than unity means that the effect of pore pressure variation on the permeability of the limestone is more important than the effect of the variation of the total stress. This result is similar to the ones presented by Zoback and Byerlee [START_REF] Zoback | Permeability and effective stress[END_REF], Walls and Nur [START_REF] Walls | Pore pressure and confining pressure dependence of permeability in sandstone[END_REF], Walls [START_REF] Walls | Effects on pore pressure, confining pressure and partial saturation on permeability of sandstones[END_REF] and Al-Wardy and Zimmerman [START_REF] Al-Wardy | Effective stress law for the permeability of clay-rich sandstones[END_REF] for clay-rich sandstone. In the clay-shell and clay-particle models proposed by Zoback and Byerlee [START_REF] Zoback | Permeability and effective stress[END_REF] and Al-Wardy and Zimmerman [START_REF] Al-Wardy | Effective stress law for the permeability of clay-rich sandstones[END_REF], an effective stress coefficient greater than one is interpreted by the presence of the clay in the rock and the great difference between the elastic moduli of the clay and the quartz. For the studied limestone a simple conceptual model for the microstructure is proposed in the following section in order to understand the mechanism of the variation of the permeability and of the corresponding effective stress coefficient with the confining pressure and the pore pressure.

Pore-shell model

Based on the microscopic observations of the limestone, a simplified model for the microstructure of the studied rock is proposed in this section. This model is presented in Figure [START_REF] Nur | Effects of fluid saturation on waves in porous rock and relation to hydraulic permeability[END_REF] and consists of a central oolite circular core that is covered by a pore shell and in the external part by a cement shell. Following the observation that an important portion of the connected porosity of the studied limestone is concentrated around the oolites, it is assumed that the porosity of the rock is entirely situated around the central oolite core. Consequently the micro-porosity of the rock observed between the micrite and sparite grains is neglected and oolite and cement are considered as non-porous elastic materials. This assumption is supported by the fact that about 42% of the micro-porosity of the rock, situated in the oolites and in the sparitic cement, is occluded. This model is in its principles similar to the clay-shell model and to the clay-particle model proposed respectively by Zoback and Byerlee [START_REF] Zoback | Permeability and effective stress[END_REF] and Al-Wardy and Zimmerman [START_REF] Al-Wardy | Effective stress law for the permeability of clay-rich sandstones[END_REF]. Similarly to the assumption of non porous cement and oolites in the proposed pore-shell model, a mechanism is needed to prevent the penetration and distribution of the pore pressure inside the clay phase in the clay-shell and clay-particle models. Indeed, as mentioned by Coyner [START_REF] Coyner | Effects of stress, pore pressure, and pore fluids on bulk strain, velocity, and permeability in rocks[END_REF], the compressibility of clay particles and of the rock solid grains are comparable so that if the pore pressure freely infiltrates into the clay pore space, the resulting effect will be equivalent of an 'unjacketed' compression test and will lead to a very small deformation caused by the compressibility of clay particles, and consequently the permeability change will be much less sensitive to the pore pressure variations. The oolites in the studied limestone are quasi-spherical. Moreover, as can be seen in Figure (3), the macro-porosity situated around the oolites is interconnected. Therefore we can consider a spherical geometry for the proposed pore-shell model. The problem can also be solved using a cylindrical geometry, as it is derived for the clay-shell and clay-particle models. In order to compare the results obtained with these two geometries, the solution for cylindrical geometry is presented in Appendix 1. As shown by Al-Wardy [START_REF] Al-Wardy | Analytical and experimental study of the poroelastic behaviour of clean and clay-rich sandstones[END_REF], it is expected that the results obtained from cylindrical and spherical geometries do not differ much.

Model formulation

The permeability of the proposed model is a function of the pore-shell geometry and thus a function of the pore-shell thickness:

( )

2 1 k f r r = - (16)
Assuming the permeability as the property in equation ( 7) and using equation ( 16), the expression of the effective stress coefficient for the permeability of the model is written as:

k Q k n ( ) ( ) 2 1 2 1 f f k p r r p n r r σ σ ∂ - ⎛ ⎞ ⎜ ⎟ ∂ ⎝ = -∂ - ⎛ ⎞ ⎜ ⎟ ∂ ⎝ ⎠ ⎠ (17)
The radius of the central oolite core ( ) is independent of the confining pressure 1 r σ , thus the following expression is obtained for :

k n 2 1 1 2 0 f f k r r r n r σ σ p p ∂ ∂ - + ∂ ∂ = ⇒ = ∂ ∂ ∂ ∂ (18)
The confining pressure is applied on the external boundary of the model and the pore pressure is applied inside the pore-shell. The elastic solution of the problem and the variations of the model geometry parameters with the applied pressures are obtained from the solution of a hollow sphere problem.

Considering a hollow sphere with the inner radius and the outer radius , the radial

The integration constants A and are obtained from the boundary conditions: B ( )(

)

3 3 3 3 3 2 i o p a p b A b a λ µ - = + - (20) ( ) ( ) 3 3 3 3 4
i o p p a b B b a µ - = - (21)
where λ and µ are the Lamé coefficients of the elastic cylinder.

For the central oolite core , and 0 = a 19), ( 20) and ( 21) the expression of the displacement is found as:

( )

3 2 f o o p u r r λ µ - = + (22)
The term 1 f r p ∂ ∂ in equation ( 18) is thus given by:

1 1 3 2 f o r r p o λ µ ∂ - = ∂ + (23)
For the outer cement shell ,

)

3 3 3 3 2 3 2 3 3 3 3 3 2 3 2 3 2 3 2 4 f f c c c p r r p r r u r r r r r r r σ σ λ µ µ - - = + + - - (24) ( ) ( ) ( ) ( )
3 1 2 3 2 2 1 2 1 1- -1 3 1 c c k c o K r n r r K r ν ν - ⎛ ⎞ = + ⎜ - ⎝ ⎠ ⎟ (28)
Note that only the Poisson's ratio of the cement appears in the above relation, whereas the Poisson's ratio of both the oolites and the cement appear for the cylindrical geometry as shown in equation ( 40) in Appendix 1.

The range of variations of the Poisson's ratio of rocks is relatively narrow. In order to simplify equation ( 28), we assume that 0.3

c ν =
which is the value of the Poisson's ratio of calcite presented by Bass [START_REF] Bass | Mineral Physics and Crystallography: A Handbook of Physical Constants[END_REF]. Using this assumption equation ( 28) is reduced to the following form:

( ) ( )

3 1 2 3 2 8 1 1- - 21 c k o K r n r r K r ⎛ ⎞ = + ⎜ ⎝ ⎠ 1 ⎟ (29)
Based on the microscopic observations and the porosity measurements presented in the previous section, the following values are considered for the geometry of the model:

(30) 1 2 3 125µm , 135µm , 150µm r r r = = =
With these dimensions, the porosity of the model is equal to 15.0% which is approximately equal to the average measured connected porosity of the studied limestone, equal to 15.7%.

Moreover, the volumetric fractions of different constituents of the microstructure of the rock, evaluated by image analysis and presented in the previous section, 59% oolite and 28% cement, are approximately retrieved with these values. In doing so, the following simple linear variation of the effective stress coefficient with the c o K K ratio is obtained: 0.9 0.1 (spherical geometry)

c k o K n K = + (31)
In case of a microscopically homogeneous material, c K K o = , the coefficient is found equal to one. Note that for the cylindrical geometry, as presented in Appendix 1, we obtain a very close expression for the effective stress coefficient:

k n 0.92 0.07 (cylindrical geometry) k c o n K K = + (32)
We can see that the proposed pore-shell model, which is based on a detailed study of the microstructure of the limestone, is able to produce permeability effective stress coefficients that are greater than one and vary with the relative bulk moduli of the different constituents of the rock. The observations of the microstructure of the cement and of the oolites have shown that the interlocking of the particles in the cement can indeed lead to higher bulk modulus than for the oolites ( ). The /

c o K K > 1 c o
K K ratio will be studied experimentally in the following section by performing microhardness tests.

Microhardness testing of the limestone

In the previous section it was shown that the permeability effective stress coefficient obtained The Vickers hardness is defined with reference to the area of the permanent impression (Das [START_REF] Das | Vickers hardness concept in the light of Vickers impression[END_REF]), so it measures in some lumped way a combination of elastic, plastic, and fracture properties. There is, however, no unique relation between the hardness number and the primary parameters, but empirical relations may be obtained for certain classes of materials.

For example, for a wide range of metals, the ratio of the hardness to uniaxial yield strength is a constant value between 2.7 and 3.0, but this relation does not hold for frictional materials, for which the hardness is a function of at least two variables: the cohesion and the friction angle of the material (Ganneau [START_REF] Ganneau | From nanohardness to strength properties of cohesive-frictional materialsapplication to shale materials[END_REF]). Only few empirical relations can be found in the literature between the Vickers hardness and the macroscopic elastic and strength properties of rocks. Das and Hucka [START_REF] Das | Laboratory investigation of penetration properties of the complete coal series[END_REF] performed penetration tests on coal and showed a linear relation between the penetration strength and penetration modulus. They also presented a result from Honda and Sanada [START_REF] Honda | Hardness of coal[END_REF] which shows that the ratio of the Vickers hardness of coal to its Young's modulus is a constant. In absence of such data for limestone, we assume that the same type of relation holds for the constituents of the studied rock. Assuming also that the Poisson's ratio of the oolite and of the cement are equal, we obtain c o Vc Vo K K H H = .

The observation of the indentations of several performed microhardness tests resulted in a

Vc Vo H H ratio which is greater than one and varies between 1.6 and 4.0. Figure [START_REF] Walls | Effects on pore pressure, confining pressure and partial saturation on permeability of sandstones[END_REF] shows an image of an indentation situated on an oolite. Figure [START_REF] Al-Wardy | Analytical and experimental study of the poroelastic behaviour of clean and clay-rich sandstones[END_REF] shows two adjacent indentations situated on the sparitic cement. Both images are taken with the same magnification, so that the size of indented zones can be directly compared. This comparison clearly shows that the indentation on the oolite has a greater area and consequently that the hardness of the oolite is smaller than the hardness of the cement.

Discussion of model results

Inserting the values of c o K K ratio evaluated by microhardness tests in equation ( 31), we obtain that the effective stress coefficient is between 1.0 and 1.2. This is compatible with the result obtained using equation ( 13) for small differential pressures.

It should be noted that the c o K K ratio is obtained in the microhardness tests for zero confining pressure. It is expected that the bulk moduli and increase with the confining pressure due to the compaction of the pore space. This increase of the bulk moduli should be more important for the cement due to the interlocking of the particles and the greater coordination number than for the oolites. The ratio

c K o K c o
K K is thus expected to increase with the confining pressure which results in an increase of the permeability effective stress coefficient (equation ( 31)). In the absence of experimental data on the stress dependency of the bulk moduli and , we can try to have a rough estimation from the drained bulk modulus of the rock (equation ( 10)) and considerations from homogenization theory. It is reasonable to assume that only the bulk modulus of the cement increases with confining pressure whereas the bulk modulus of the oolites remains constant. In the proposed poreshell model, the confining pressure is applied only on the outer cement shell and has no effect on the oolite core. This may be realistic to some extent for low confining pressures. As can be surrounded in a matrix of sparitic cement is given by:

d o o c K f K f K c ≤ + (35)
where o f and c f are the volume fractions of the oolites and of the cement ( ).

Considering the macro-porosity of the rock as a part of the oolite fraction, we get results in an effective stress coefficient which varies between 1.2 and 2.0. This is comparable to the experimental results where was found to vary between 0.9 and 2.4.

k n k n
It should be emphasized that this simple pore-shell model for the studied limestone is based on strong assumptions and is only used here for a better understanding of the mechanism of the evolution of the permeability with the pore pressure and the confining pressure without trying to obtain quantitative predictions.

Conclusion

The effective stress law for the permeability of a limestone is studied in an approach which combines laboratory testing, microstructural observations and modelling. 24 constant head permeability tests have been performed in a triaxial cell with various conditions of confining pressure and pore pressure. The experimental results have shown that the pore pressure increase and the confining pressure decrease both result in an increase of the permeability, but the effect of the pore pressure change on the variation of the permeability is more important that the effect of a change in the confining pressure. It is shown that an effective stress law can be defined for the permeability and that the corresponding effective stress coefficient increases linearly with the differential pressure and is greater than one as soon the differential pressure exceeds few bars. A power law is proposed for the variation of the permeability with

-Figure 1 -

 1 Figure 1-Microstructure of the studied limestone observed with (a) optical microscope, (b) ESEM Figure (2a) presents the details of the micrite crystals forming the oolites. We can see that the microstructure of the oolites is similar to a granular material with grains diameters varying from 2 to 5 µm and with a fine porosity between the grains. The sparite crystals which constitute the cement connecting the oolites are shown in Figure (2b). We can see clearly the difference between the microstructure of the oolites and the one of the sparitic cement. This latter is composed of interlocked grains with a bigger size than the micrite grains of the oolites. Both sparite and micrite are entirely composed of calcite.

Figure 2 -

 2 Figure 2-Details of (a) an oolite and (b) the sparitic cement observed with the ESEM

3 2 .

 2 22 g cm and the measured absolute density of the grains is equal to 3 2.70 g cm which is compatible with the value

3

 3

Figure 3 -

 3 Figure 3-Porous space of the limestone-concentration of the main porosity around the oolites, (a) optical microscope (b) ESEM

Figure 4 -

 4 Figure 4-Drained hydrostatic compression test with a loading-unloading cycle

Figure 5 -

 5 Figure 5-Drained bulk modulus of the studied limestone as a function of Terzaghi effective stress

Figure 6 -

 6 Figure 6-Results of permeability tests performed with different conditions of pore pressure and confining

Figure 7 -Figure 8 -

 78 Figure 7-Permeability contour plan and the approximate effective stress coefficient corresponding to each isoline

Figure 9 -

 9 Figure 9-Comparison of the contour plans of the experimental results (dashed lines) and the proposed permeability-effective stress model (solid lines)

Figure 10 -

 10 Figure 10-Pore-Shell model

stresses at the inner and outer boundaries are respectively

  stresses are taken negative. The well-known Lamé solution holds for the radial displacement :

  s denote by o λ and o µ the Lamé coefficients of the oolite core. Using equations (

=

  and let's denote by c λ and c µ the Lamé coefficients of the cement shell. Using equations (19), (20) and (21) the expression of the displacement is found as:

Figure 11 -Figure 12 -

 1112 Figure 11-Microhardness test: view of an indentation situated on an oolite Figure 12-Microhardness test: view of two adjacent indentations situated on the sparitic cement

  [START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[END_REF], the oolites are connected to the surrounding cement with a few contact areas. The Voigt upper bound of a homogenized medium constituted by spherical oolites

σ

  unloaded state. From equation (35) we obtain an upper bound for in the unloaded state equal to 4GPa. Keeping this value constant, the variation of is found from equation (35) to be between 12 and 34 GPa when increases from 0 to 8.95MPa. Using equation[START_REF] Honda | Hardness of coal[END_REF], this

  can be re-written using the bulk modulus and of the Poisson's ratio K on the microscopic observations and the porosity measurements presented in the previous section, the following values are considered for the geometry of the cylindrical model: With these dimensions, the porosity of the model is equal to 15.6% which is approximately equal to the average measured connected porosity of the studied limestone, equal to 15.7%. Moreover, the volumetric fractions of different constituents of the microstructure of the rock, evaluated by image analysis are approximately retrieved with these values. Inserting these dimensions in equation (41), the following linear variation of the effective stress coefficient with the c o K K ratio is obtained:

The terms 2 f r p ∂ ∂ and 2 r σ ∂ ∂ in equations [START_REF] Morrow | Effective Pressure Law for Permeability of Westerly Granite Under Cyclic Loading[END_REF] are given by: ( )( ) ( )

) ( )

Introducing ( 23), ( 25) and [START_REF] Schutjens | Compaction-Induced Porosity/Permeability Reduction in Sandstone Reservoirs: Data and Model for Elasticity-Dominated Deformation[END_REF] in equation ( 18) with some algebraic re-arrangements, the following expression is obtained for the effective stress coefficient corresponding to the variations of the permeability of the model:

Equation ( 27) can be re-written using the bulk modulus and of the Poisson's ratio

) of the material: the effective stress. The test results are well reproduced using the proposed permeabilityeffective stress law.

The microscopic observations of the microstructure of the rock shows that the studied limestone is entirely composed of calcite and is constituted of quasi spherical oolites cemented by sparitic calcite. The observations have shown that the main porosity of the rock is concentrated around the oolite cores. Based on the microscopic observations, a conceptual pore-shell model is proposed to study the effective stress law for the permeability of the studied limestone. The proposed model is able to reproduce permeability effective stress coefficients that are greater than one and which vary with the ratio of the bulk moduli of the two constituents of the rock. The ratio of the bulk moduli of the sparitic cement to the one of the oolites is studied experimentally by performing microhardness tests and is found to be greater than one. According to the proposed pore-shell model, this results in an effective stress coefficient which is greater than one. The microstructural approach is thus consistent with the experimental results of the permeability tests.

Appendix 1

In this appendix we present the equations of the pore-shell model for cylindrical plane strain geometry. All of the parameters are the same as the ones used in section 5. With the integration constants A and given by: B

Following the same procedure presented in section 5.1 for spherical geometry, the effective stress coefficient for the permeability of the cylindrical geometry is obtained as: