Damian Paliwoda

Marco Fabbiani

Frederico Alabarse

Patrick Hermet

Jérôme Rouquette

Francesco Di Renzo

Julien Haines
email: julien.haines@umontpellier.fr

Fran- Cesco Di Renzo

Compressibility

Keywords:

come

Introduction

Zeolites are microporous materials with pore diameters of less than 1 nm, built up of rings and cages of corner-sharing TO4 (T=Si, Al) tetrahedra. Due to their porosity, zeolites have a large number of applications in catalysis, as molecular sieves and sorbents, for example. A consequence of porosity is that such structures can potentially collapse under the influence of high pressure giving rise to phase transitions and pressure-induced amorphization (PIA). There are several examples (ANAlcime-ANA, zeolite-RHO-RHO, Linde Type A-LTA, Virgina Polytechnic Institute-FIve-VFI, Theta-ONe-TON) of high pressure phase transitions involving distortion and partial collapse of the pores of zeolites [START_REF] Gatta | Elastic Behavior, Phase Transition, and Pressure Induced Structural Evolution of Analcime[END_REF][START_REF] Lee | Phase Transition of Zeolite Rho at High-Pressure[END_REF][START_REF] Jorda | Synthesis of a Novel Zeolite through a Pressure-Induced Reconstructive Phase Transition Process[END_REF][START_REF] Alabarse | AlPO4-54-AlPO4-8 Structural Phase Transition and Amorphization under High Pressure[END_REF][START_REF] Thibaud | High-Pressure Phase Transition, Pore Collapse, and Amorphization in the Siliceous 1D Zeolite, TON[END_REF] . At higher pressures, PIA is widespread, giving rise to novel amorphous materials [START_REF] Alabarse | AlPO4-54-AlPO4-8 Structural Phase Transition and Amorphization under High Pressure[END_REF][START_REF] Thibaud | High-Pressure Phase Transition, Pore Collapse, and Amorphization in the Siliceous 1D Zeolite, TON[END_REF][START_REF] Greaves | The rheology of collapsing zeolites amorphized by temperature and pressure[END_REF][START_REF] Greaves | Zeolite Collapse and Polyamorphism[END_REF][START_REF] Huang | IR Spectroscopic Study of the Effects of High Pressure on Zeolites Y, a and Sodalite[END_REF][START_REF] Isambert | Amorphization of Faujasite at High Pressure: An X-Ray Diffraction and Raman Spectroscopy Study[END_REF][START_REF] Readman | Pair Distribution Function Analysis of Pressure Treated Zeolite Na-A[END_REF][START_REF] Haines | Topologically Ordered Amorphous Silica Obtained from the Collapsed Siliceous Zeolite, Silicalite-1-F: A Step toward "Perfect" Glasses[END_REF][START_REF] Vezzalini | High-Pressure-Induced Structural Changes, Amorphization and Molecule Penetration in MFI Microporous Materials: A Review[END_REF] .

Guest insertion under pressure, in particular, in the case of noble gas elements in high silica zeolites, results in complete filling of the pores of the host zeolite. This can strongly modify the high pressure behavior of zeolites, by resisting the phenomenon of pore collapse and resulting phase transitions and PIA. Noble gases have been inserted in several small pore zeolites such as Mobil-FIve-MFI [START_REF] Haines | Deactivation of Pressure-Induced Amorphization in Silicalite SiO2 by Insertion of Guest Species[END_REF] , NATrolite-NAT [START_REF] Lee | Pressure-Induced Argon Insertion into an Auxetic Small Pore Zeolite[END_REF] , LTA [START_REF] Niwa | Pressure-Induced Noble Gas Insertion into Linde-Type A Zeolite and Its Incompressible Behaviors at High Pressure[END_REF] and TON [START_REF] Thibaud | Saturation, of the Siliceous Zeolite TON with Neon at High Pressure[END_REF] . In TON, for example, the pores are filled gradually by Ne or Ar with pressure increase. The amount of Ar inserted is about a factor of 2 less than for Ne, and the kinetics of desorption are slow with argon remaining in the pores many hours after pressure release. In the case of Xe-filled LTA [START_REF] Seoung | Irreversible Xenon Insertion into a Small-Pore Zeolite at Moderate Pressures and Temperatures[END_REF] , the observed insertion is irreversible on pressure release. Another guest molecule of particular interest is water, which has been shown to give rise to energy storage properties in pure silica zeolites 18- 20 . These zeolites can also be used as host materials for the polymerization of organic and inorganic molecules to form polymer/zeolite nanocomposites [START_REF] Santoro | High-Pressure Synthesis of a Polyethylene/Zeolite Nano-Composite Material[END_REF][START_REF] Scelta | High Pressure Polymerization in a Confined Space: Conjugated Chain/Zeolite Nanocomposites[END_REF][START_REF] Santoro | High Pressure Synthesis of All-Transoid Polycarbonyl [-(C=O)-]n in a Zeolite[END_REF][START_REF] Santoro | Synthesis of 1D Polymer/Zeolite Nanocomposites under High Pressure[END_REF][START_REF] Paliwoda | Anomalous Volume Changes in the Siliceous Zeolite Theta-1 TON Due to Hydrogen Insertion under High-Pressure, High-Temperature Conditions[END_REF][START_REF] Alabarse | High-Pressure Synthesis and Gas-Sensing Tests of 1-D Polymer/Aluminophosphate Nanocomposites[END_REF] .

Zeolite Socony Mobil-twelve (ZSM-12) or Mobil TWelve (MTW) is high silica zeolite with a 1-D pore system [START_REF] Lapierre | The Framework Topology of ZSM-12 -a High-Silica Zeolite[END_REF] . The structure is monoclinic with a C2/c space group [START_REF] Fyfe | Crystal-Structure of Silica-Zsm-12 by the Combined Use of High-Resolution Solid-State Mas Nmr-Spectroscopy and Synchrotron X-Ray-Powder Diffraction[END_REF] . MTW has slightly elliptical pores along the [010] direction based on 12-membered rings of SiO4 tetrahedra. The elliptical pore diameter is 5.7 Å x 6.8 Å 29 . This system is of interest as a model, medium pore zeolite with empty 1-D pores due to the almost complete absence of charge compensating cations due to the high Si/Al ratio. This zeolite has been found to be particularly promising for energy storage upon water intrusion/extrusion at high pressure with a stored energy of 15 J/g for an intrusion pressure of 132 MPa [START_REF] Tzanis | High Pressure Water Intrusion Investigation of Pure Silica 1D Channel AFI, MTW and TON-Type Zeolites[END_REF] .

The goal of the present study is to investigate the stability of the structure of this 1-D medium pore zeolite under high pressure by x-ray powder diffraction and also to study guest insertion of argon and its effect on the high pressure behavior of this material.

Experimental and theoretical methods

Microcrystalline MTW was prepared by sol-gel techniques using methyltriethylammonium chloride as the structure directing agent (SDA) with a molar composition of 0.2 Na : 0.2 SDA : 0.007 Al : Si : 20 H2O. The crystallization was carried out at 140°C under hydrothermal conditions for 11 days and the obtained powder was calcined at 600°C for 5h. The Si:Al ratio of the crystals was found to be 41.9 by EDS analysis. The remaining sodium was exchanged using ammonium chloride, followed by a second calcination to obtain the protonic form of the zeolite. The nitrogen adsorption analysis at 77K showed a classical type I isotherm with an exposed specific surface area of 316 m 2 /g and the total pore volume of 0.14 cm 3 /g that corresponds to 100% of the total available porosity. The amount of adsorbed N2 in the micropores was close to 2.9 mmol/g for the sodium form after the first calcination and close to 3.4 mmol/g for the protonic form after the second calcination.

High pressure synchrotron powder X-ray diffraction (=0.4957 Å) measurements in membrane-driven diamond anvil cells were performed with an 80 µm beam spot on the sample on the Xpress beamline equipped with a PILATUS3 S 6M (DECTRIS) detector at the ELETTRA Sincrotrone Trieste (Trieste, Italy). The XRD images (see supporting information) were converted to 1-D diffraction profiles using Dioptas [START_REF] Prescher | Dioptas: A Program for Reduction of Two-Dimensional X-Ray Diffraction Data and Data Exploration[END_REF] based on a calibration using NIST SRM 674b CeO2 (a = 5.411651 Å [START_REF] Prescher | Dioptas: A Program for Reduction of Two-Dimensional X-Ray Diffraction Data and Data Exploration[END_REF] ) as a standard. The sample to detector distance was determined to be 634.38 mm based on this calibration. The hole in the preindented stainless steel gasket was loaded with the submicronic MTW powder under an argon atmosphere in a glove box to avoid the presence of any H2O molecules in the pores. A ruby microsphere was added as pressure calibrant. Pressure was measured before and after every x-ray exposure based on the shift of the ruby R1 fluorescence line [START_REF] Mao | Calibration of the Ruby Pressure Gauge to 800 Kbar under Quasi-Hydrostatic Conditions[END_REF] . The standard deviations of the reported pressures are based on the values obtained before and after each exposure. In the first experiment, non-penetrating liquid DAPHNE7474 [START_REF] Murata | Pressure Transmitting Medium Daphne 7474 Solidifying at 3[END_REF] oil was used as the pressure-transmitting medium. In a second independent experiment, argon was loaded cryogenically in the liquid phase. Le Bail fits to obtain the unit cell parameters and Rietveld refinements were performed using the program Fullprof [START_REF] Rodriguez-Carvajal | Magnetic Structure Determination from Powder Diffraction Using the Program Fullprof[END_REF] . The quoted estimated standard deviations for the lattice parameters and the fractional atomic coordinates are those obtained directly from these refinements. Fractional atomic coordinates for the framework from previous work were used as the starting model for MTW [START_REF] Fyfe | Crystal-Structure of Silica-Zsm-12 by the Combined Use of High-Resolution Solid-State Mas Nmr-Spectroscopy and Synchrotron X-Ray-Powder Diffraction[END_REF] . Soft constraints were applied to the Si-O and O-O distances. An overall isotropic atomic displacement parameter (ADP) was used for all Si, O and Ar atoms as no improvement was obtained using individual ADPs. Crystal structures were plotted using the program VESTA [START_REF] Momma | Vesta 3 for Three-Dimensional Visualization of Crystal, Volumetric and Morphology Data[END_REF] . EosFit [START_REF] Gonzalez-Platas | EosFit7-Gui: A New Graphical User Interface for Equation of State Calculations, Analyses and Teaching[END_REF] was used to fit the P-V data.

Structure relaxation (atomic positions and lattice parameters) was performed within the density functional theory as implemented in the SIESTA package [START_REF] Sanchezportal | Density-Functional Method for Very Large Systems with LCAO Basis Sets[END_REF] . No constraint was imposed during the relaxation.

The structure was considered as relaxed when the maximum residual atomic force and the pressure are smaller than 0.03 eV/Å and 2x10 -5 eV/Å 3 , respectively. We used the generalized gradient approximation (GGA) to the exchange correlation functional as proposed by Perdew, Burke and Ernzerhof [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] . Core electrons are replaced by nonlocal norm-conserving pseudopotentials. The valence electrons are described by a double-zeta singly polarized basis set. The localization of the basis is controlled by an energy shift of 50 meV. Real space integration is performed on a regular grid corresponding to a plane-wave cutoff of 350 Ry and using a 3x9x3 k-points mesh. Van der Waals corrections (DFT-D3) are included in our calculations using the semi-empirical dispersion potential parametrized by Grimme [START_REF] Grimme | Semiempirical GGA-Type Density Functional Constructed with a Long-Range Dispersion Correction[END_REF] .

Results and discussion

MTW powder was first studied in the nonpenetrating DAPHNE7474 pressure transmitting medium to determine the intrinsic high-pressure behavior of this framework (Figure 1). The structure of MTW in the diamond anvil cell at close to ambient pressure was refined by the Rietveld method based on the structural model of MTW with a C2/c space group from the literature [START_REF] Fyfe | Crystal-Structure of Silica-Zsm-12 by the Combined Use of High-Resolution Solid-State Mas Nmr-Spectroscopy and Synchrotron X-Ray-Powder Diffraction[END_REF] . The initial unit cell parameters are: a=24.935(2) Å, b=5.0162(3) Å, c=24.349(1) Å, =107.854(6)° (Table 1).

Based on Fourier difference maps, some electron density was identified in the pores, which may arise from residue of the SDA or from trace argon atoms in the pores as the MTW was stored and manipulated in a glove box under an argon atmosphere. This electron density was modelled using nitrogen atoms with partial occupancy placed on the maxima of electron density. The partial occupancies correspond to close to 3 N atoms or 1 Ar atom per unit cell. This structural model was used for the data up to a pressure of 1.53 GPa. Very good fits were obtained (Figures 2 and3) and with excellent agreement factors (Rpro-file=12.7%, RBragg=8.4%). The principal changes are in the pore dimensions (Figures 4 and5), which decrease from 5.6 Å x 6.9 Å at ambient pressure to 5.0 Å x 6.7 Å at 1.53 GPa. The pores thus become more elliptical (Figure 5). This is linked to much greater decreases in the intertetrahedral Si-O-Si bridging angles in and linked to the four-membered rings of tetrahedra along the pore walls, that is involving Si1 and Si2. In particular, the Si1-O2-Si2 angle decreases by 11.5(9)° between 0.25 and 1.53 GPa (Table 2). This is consistent with the compressibility along c being slightly higher than along a (Table 1 and Figure 6). The highest compressibility is observed along b, corresponding to an accordion-like deformation of the zigzag chains parallel to the pore axis.

Table 1. Unit cell parameters of the monoclinic C2/c and P2/n (in italics) phases of MTW in DAPHNE7474 and Ar as a function of pressure.

P(GPa) a(Å) b(Å) c(Å) (°) V(Å
Above 1.53 GPa, a series of new reflections, such as  01, 101, 01,  03 and 301 for example, are observed in the diffraction pattern of MTW. These reflections violate the reflection conditions of the C2/c, space group, and imply the loss of C-centering, and the c glide planes. The reflections are consistent with a primitive monoclinic cell with slightly decreased a and c lattice parameters (Figure 6) and a volume change at 2 GPa of around 4% (Figure 7). The b parameter is relatively unaffected. The monoclinic angle also decreases confirming that the changes occur principally in the ac plane, which also becomes more compressible.

The possible monoclinic subgroups of the C2/c space group, taking into loss of the C-centering and the 5) -6.1 (10) The P-V data were fitted to a second-order Birch-Murnaghan equation of state [START_REF] Birch | Equation of State and Thermodynamic Parameters of NaCl to 300-Kbar in the High-Temperature Domain[END_REF] . The fit for the lowpressure C2/c phase yielded a bulk modulus B0 of 15.5(2) GPa, which is typical of empty siliceous zeolites, such as MFI [START_REF] Haines | Topologically Ordered Amorphous Silica Obtained from the Collapsed Siliceous Zeolite, Silicalite-1-F: A Step toward "Perfect" Glasses[END_REF][START_REF] Vezzalini | High-Pressure-Induced Structural Changes, Amorphization and Molecule Penetration in MFI Microporous Materials: A Review[END_REF] and TON [START_REF] Thibaud | High-Pressure Phase Transition, Pore Collapse, and Amorphization in the Siliceous 1D Zeolite, TON[END_REF] and other empty porous materials, such as metal-organic frameworks 40- 42 . The corresponding value for the P2/n phase is 9.4(4) GPa with an initial relative volume of 1.007 (6).

c
If the initial relative volume is fixed to 1, a B0 of 9.95(4) GPa is obtained, but the fit is poorer. The higher compressibility of the P2/n phase is due to its high flexibility arising from its lower symmetry, which allows for additional compression mechanisms and more flexibility of the zeolitic framework. This additional flexibility in the lower symmetry phase allows for progressive pore collapse as a function of pressure. Similar behavior with increased compressibility in high pressure, low symmetry phases due to additional compression mechanisms has been observed in zeotypes, such as ANA [START_REF] Gatta | Elastic Behavior, Phase Transition, and Pressure Induced Structural Evolution of Analcime[END_REF] , including the minerals wairakite [START_REF] Ori | Pressure-Induced Structural Deformation and Elastic Behavior of Wairakite[END_REF] and leucite [START_REF] Gatta | Leucite at High Pressure: Elastic Behavior, Phase Stability, and Petrological Implications[END_REF] , and TON [START_REF] Thibaud | High-Pressure Phase Transition, Pore Collapse, and Amorphization in the Siliceous 1D Zeolite, TON[END_REF] , and also in other framework solids, such as zinc cyanide [START_REF] Collings | Homologous Critical Behavior in the Molecular Frameworks Zn(Cn)2 and Cd(Imidazolate)2[END_REF] . The pore diameters are 4.97Å x 6.70 Å at 1.53 GPa in the C2/c phase and 3.54 Å x 9.30 Å and 3.83 Å x 8.23 Å at 2.16 GPa in the P2/n phase. This mechanism is similar to that reported for TON [START_REF] Thibaud | High-Pressure Phase Transition, Pore Collapse, and Amorphization in the Siliceous 1D Zeolite, TON[END_REF] . This transition was found to be reversible with some degree of hysteresis and trace amounts of the P2/n phase persists even at 0.5 GPa. Further compression of the P2/n phase resulted in a gradual broadening of the diffraction lines and at pressures above 10 GPa, only broad features remain characteristic of an amorphous form as has been found for many other zeolites at high pressure 4-7, 9-12 .

Distinct behavior is observed when MTW is compressed in argon up to 1.8 GPa, the pressure at which

Ar was observed to be solid. All other points correspond to Ar in the liquid phase. At low pressures, the unit cell parameters and volume are significantly greater due to argon filling (Figures 6 and7) and many, intertetrahedral Si-O-Si angles increase in particular the angles principally in the ac plane along the pore walls (Table 2). In addition, the diffraction lines become broader and less symmetrical with increasing pressure. Similar broadening due to guest insertion has been observed in TON and can be related to strain induced by the filling process [START_REF] Thibaud | Saturation, of the Siliceous Zeolite TON with Neon at High Pressure[END_REF] . In the non-penetrating DAPHNE7474, the diffraction lines remain sharp and symmetrical in the same pressure range.

In order to determine the structure of Ar-filled MTW, the MTW structure in the non-penetrating pressure medium was used as a starting model and Fourier difference maps were used to locate the Ar atoms.

Refinements (Figure 2) were performed with the Ar atoms partially occupying 3-5 sites in a disordered arrangement (Figure 3, CIF file in Supporting Information). The argon content was found to be 9(1) Ar/atoms per unit cell and was found to be independent of pressure over the range from 0.29 to 1.52 GPa and is slightly lower than the value obtained for nitrogen in adsorption isotherms, which is close to 11 N2 molecules per unit cell at 77 K. This indicates that in this medium-pore zeolite, complete filling readily occurs at very low pressure, in contrast to small-pore zeolites such as TON [START_REF] Thibaud | Saturation, of the Siliceous Zeolite TON with Neon at High Pressure[END_REF] , which has 4.6 Å x5.6 Å diameter pores, for which filling is gradual over this pressure range. Pressure thus is less critical for optimal pore filling of argon in this medium-pore zeolite. In MTW and TON, there are essentially the same number of Ar atoms per pore just below the solidification of Ar near 1.5 GPa [START_REF] Errandonea | Structural Transformation of Compressed Solid Ar: An X-Ray Diffraction Study to 114 GPa[END_REF] . Initially, in the larger MTW pores at low pressure, there is more available space for the number of argon atoms inserted. In the case of argon filling in MTW, the short pore diameter increases strongly with respect to MTW at ambient pressure (Figure 5) and is close to 50% less compressible than when the pores are empty. This indicates that the structure has a degree of flexibility to accommodate the argon atoms. Another difference with respect to TON is that the pore ellipticity only increases slightly in Ar-filled MTW under pressure, whereas a very strong increase in ellipticity is observed in Ar-filled TON [START_REF] Thibaud | Saturation, of the Siliceous Zeolite TON with Neon at High Pressure[END_REF] in a similar manner to empty TON [START_REF] Thibaud | High-Pressure Phase Transition, Pore Collapse, and Amorphization in the Siliceous 1D Zeolite, TON[END_REF] . Above 0.6 GPa, TON already adopts a high pressure structure with collapsed pores. Pressure is a factor in ensuring the filling with argon of these pores in TON, which are progressively collapsing. The short pore diameter in MTW lies along c, which becomes the most incompressible direction in the filled system (Figure 6). This can be correlated with the much smaller decrease in the intertetrahedral Si1-O2-Si2 bridging angle in the four-membered rings of tetrahedra along the pore walls (Table 2). The presence of Ar in the pores greatly reduces the compression (a,c) and distortion () in the ac plane, which correspond to the most affected parameters prior to and at the phase transition in empty MTW. The overall compressibility of Ar-filled MTW is 35% lower than that of empty MTW. The bulk modulus is 23.8 (6) GPa with an initial relative volume of 1.009 (1). Upon decompression from 1.8 GPa down to 0.27 GPa, the observed behavior is reversible in terms of unit cell volume, but the diffraction lines remain strongly broadened and asymmetrical.

Conclusions

The siliceous medium-pore zeolite MTW exhibits an increase in the ellipticity of its pores with compression. At close to 1.5 GPa, a phase transition from the C2/c to the P2/n structure is observed with a 4% volume change due to pore collapse. Further compression leads to gradual amorphization. Argon readily fills the pores of MTW with a constant value of 9(1) Ar atoms/unit cell over the pressure range investigated. This contrasts with the gradual filling observed for the small-pore zeolite TON. The short pore diameter is close to 50% less compressible due to argon filling and this is the principle origin of the 35% lower compressibility of the structure when compressed in argon. The present results on phase stability and pore filling of MTW at high pressure will be of great value for designing and synthesizing nanocomposite materials based on MTW and also for the use of MTW in energy storage applications. V / V 0 P (GPa)

Figure 1 .

 1 Figure 1. X-ray diffraction profiles of MTW in non-penetrating DAPHNE7474 oil (=0.4957 Å). Ar-

Figure 3 .

 3 Figure 3. Refined crystal structure of C2/c structure of MTW at 0.81 GPa in DAPHNE7474 (left) and

Figure 4 .

 4 Figure 4. Detail of the refined crystal structure of C2/c phase of MTW at 0.25 GPa in DAPHNE7474

Figure 5 .

 5 Figure 5. Minimum and maximum pore diameters of MTW in DAPHNE7474 (black symbols) and in

Figure 6 .

 6 Figure 6. Relative cell parameters of MTW in DAPHNE7474 (black and red symbols) and in argon

Figure 7 .

 7 Figure 7. Relative volume (V/V0) of MTW in DAPHNE7474 (black and red symbols) and in argon (blue

Table 2 .

 2 glide plane are P2/n, P21/n, P2 and P21. Possible shoulders could correspond to 203 and 300 reflections indicating that the n glide might also be lost. Some of the low pressure C2/c phase remained untransformed amounting to close to 10% at 2.16 GPa and 8% at hydrostatic limit of the DAPHNE7474 fluid near 4 the new phase in the highest symmetry monoclinic subgroup P2/n starting with atomic positions of the low pressure phase. During the refinement cycles the agreement factors improved rapidly leading to a structural model with collapsed pores. In order to have a less arbitrary configuration, the resulting model was relaxed by DFT. The resulting structure yielded very good agreement with the ex-Intertetrahedral

	perimental data with a Bragg R-factor of 8.7% using the experimental lattice parameters. Similar agree-
	ment factors were obtained using this relaxed structure as a starting model in Rietveld refinements. The
	experimental lattice parameters, a=23.537(5) Å, b=4.834(1) Å, c=22.868(5) Å, =107.05(2)° are com-
	pared to the relaxed values a=23.8995 Å, b=4.8929 Å, c= 23.5694 Å, = 106.727°. This slight overesti-
	mate is commonly observed at the GGA level. Attempts using the P21/n and P21 space groups resulted in
			DAPHNE7474		Argon
		0.25 GPa 1.53 GPa	difference 0.29 GPa	 GPa	difference
	Si1-O2-Si2	152.1(5)	140.6(5)	-11.5(9)	154.2(5)	148.9(5)	-5.3(10)
	Si3-O3-Si1	143.3(5)	136.8(5)	-6.5(10)	143.6(5)	139.5(5)	-4.1(10)
	Si3-O12-Si1 143.3(5)	135.5(5)	-7.8(10)	143.6(5)	136.5(5)	-7.1(10)
	Si7-O6-Si3	140.0(4)	134.2(5)	-5.8(9)	140.8(4)	136.3(4)	-4.5(8)
	Si7-O11-Si6 155.3(5)	151.2(6)	-4.1(11)	155.3(5)	152.8(6)	-2.5(11)
	Si4-O9-Si6	136.1(6)	131.1(6)	-5.0(12)	138.6(6)	136.4(6)	-2.2(12)
	Si4-O13-Si2 145.1(5)	137.3(6)	-7.8(11)	146.3(5)	139.5(5)	-6.9(10)
	Si2-O5-Si4	151.3(5)	142.6(6)	-8.7(11)	153.4(5)	147.3(

GPa. The high pressure phase is clearly a low symmetry distortion of the starting C2/c phase, based on the large number of common reflections with similar intensities. Attempts were thus made to refine the structure of strong distortion to the framework and poor agreement values. It is possible that the structure is in fact P2 with a many more degrees of freedom, but the present two-phase data cannot clearly be used to support such a distortion with three different types of pores. In addition, the DFT calculations were performed with no symmetry constraints and converged readily to give P2/n symmetry. Si-O-Si bridging angles (°) along the pore walls as function of pressure. Angles lying principally out of the ac plane are in italics. Atom labels correspond to the CIF files in SI.

ACKNOWLEDGMENT

We acknowledge funding from the Agence Nationale de la Recherche in the framework of the contracts ANR-19-CE08-0016 and ANR-19-CE09-0019. The research leading to this result has been supported by the project CALIPSOplus under Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020. The synchrotron X-ray diffraction experiments were performed at the Xpress beamline from Elettra Sincrotrone Trieste (proposal number: 20205124).

ASSOCIATED CONTENT

Supporting Information. Selected 2-D powder x-ray diffraction patterns (PDF file) and crystallographic information files (CIF) files containing the structural data for the MTW-DAPHNE7474 at 0.25, 0.81 and 1.53 GPa, the MTW-Ar at 0.29, 0.83 and 1.52 GPa and the relaxed crystal structure of P21/n MTW at 2.2GPa have been supplied as supporting information. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION

Corresponding Author

*Julien Haines

Julien.Haines@umontpellier.fr

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interests.


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Keywords: permeability, porosity, transient method, creep, hardened cement paste

Introduction

The evaluation of the permeability in the laboratory is basically an inverse problem. The two most widely used methods for evaluation of the permeability of geomaterials are the steady state method and the transient pulse method. The steady state method consists in applying a constant pressure gradient to the sample and measuring the resulted flow rate. The permeability of the sample is then calculated using Darcy's law. This method is particularly appropriate for high permeability materials. The transient pulse method is based on the evaluation of the decay of a small step change of pressure imposed in one end of a sample. This method, which was pioneered by Brace et al. [START_REF] Brace | Permeability of Granite under High Pressure[END_REF], is appropriate for low permeability materials and is used extensively to evaluate the permeability of different geomaterials (e.g. Bernabé [START_REF] Bernabé | The effective pressure law for permeability in Chelmsford granite and Barre granite[END_REF] [3] on granite, Escoffier et al. [START_REF] Escoffier | Under stress permeability determination of the Meuse/Haute-Marne mudstone[END_REF] on mudstone). In the original setting of Brace et al. [START_REF] Brace | Permeability of Granite under High Pressure[END_REF], a cylindrical sample is connected to two fluid reservoirs with equal pressures. A sudden increase of the fluid pressure in the upstream reservoir unbalances the system and causes a fluid flow from this reservoir, through the sample to the downstream reservoir in order to equilibrate the pressures in the sample and in the reservoirs. The permeability of the sample is then back analysed from the kinetics of the decay of the pressure in the upstream reservoir. Some alternative methods to evaluate the permeability of low permeability materials have been recently developed by monitoring the response of a saturated body to mechanical and thermal strains. Among them, the beam bending method [START_REF] Scherer | Bending of gel beams: method of characterizing mechanical properties and permeability[END_REF] [6] [START_REF] Scherer | New methods to measure liquid permeability in porous materials[END_REF] is based on the fact that bending of a saturated porous beam creates a pressure gradient in the pores, as the top half is in compression and the half bottom is in tension. This pressure gradient causes the fluid flow to equilibrate the pressure and consequently the force required to sustain a fixed deflection decreases. The permeability and also the elastic modulus of the body are evaluated by analysing the kinetics of force relaxation. Thermopermeametry is another method that is based on the analysis of thermal expansion kinetics [START_REF] Scherer | Characterization of saturated porous bodies[END_REF] [7] [START_REF] Ai | Thermal expansion kinetics: method to measure permeability of cementitious materials: II, application to hardened cement paste[END_REF]. The thermal expansion of water is higher than the one of solids, so a rapid temperature increase and then isothermal hold of a saturated porous body causes an initial dilation and pore fluid pressurization due to the greater expansion of the pore fluid. This initial dilation is then followed by a contraction of the body caused by the dissipation of the excess pore pressure and the fluid outflow. Analysis of the kinetics of thermal dilation yields the permeability of the body. Beam bending and thermopermeametry methods were originally developed for gels and later extended to more rigid materials like cement paste and mortar. Dynamic pressurization [START_REF] Scherer | Dynamic pressurization method for measuring permeability and modulus: I. Theory[END_REF] [10] is another permeability evaluation method which is performed on a sample that is enclosed in a vessel full of fluid under a constant pressure. In the initial state, the pressure in the vessel and the sample pore pressure are in equilibrium. A sudden increase of the pressure in the vessel causes an initial contraction of the sample. This contraction, when the pressure is kept constant, is followed by a time dependent dilation due to the progressive increase of the sample pore pressure to reach equilibrium with the pressure inside the vessel. The permeability is then evaluated by analysing the kinetics of the dilation of the sample.

One can see that these methods used for evaluation of transport properties of low permeability materials are based on the analysis of the kinetics of diffusion of pore pressure or the strains induced by pore pressure diffusion. The same principle is used in this work for the evaluation of the permeability of a hardened cement paste. The test method used here is similar to the one presented by Hart and Wang [START_REF] Hart | A single test method for determination of poroelastic constants and flow parameters in rocks with low hydraulic conductivities[END_REF] as a single test method for evaluation of poroelastic constants and flow parameters of low permeability rocks. These authors used the excess pore pressure generated in a sample due to an undrained loading for the evaluation of the permeability. The pressure at one side of the sample is connected to a reservoir with a constant volume and the kinetics of the pore pressure variations at the other end of the sample and in the reservoir is analysed for evaluation of the permeability.

Our experimental study for the evaluation of the permeability of a hardened cement paste is a part of a larger study on the thermo-poro-mechanical behaviour of this material for petroleum applications. Indeed, in oil wells, a cement sheath is placed between the rock and the casing for support and sealing purpose. The cement lining is submitted to various thermal and mechanical loadings during the life of the well from the drilling phase to the production phase and finally in the abandonment phase when the well must seal the subsurface from the surface, as for instance for storage and sequestration of greenhouse gas. In due course of these actions, the cement can be damaged and the mechanical and transport properties can be degraded, this degradation being detrimental to its functions. Moreover, the determination of the permeability of this cement over time is essential for the prediction of the sealing performance of the well when CO 2 storage and sequestration is planned. The results of our previous experimental study including drained and undrained hydrostatic compression tests, unjacketed tests and drained and undrained heating tests have been presented in Ghabezloo et al. [START_REF] Ghabezloo | Poromechanical behaviour of hardened cement paste under isotropic loading[END_REF] [START_REF] Ghabezloo | The effect of undrained heating on a hardened cement paste[END_REF].

By applying an undrained hydrostatic loading, an excess pore pressure is generated inside the sample, related to the applied hydrostatic stress by the Skempton coefficient of the material. At the end of the hydrostatic loading phase, a constant pore pressure much lower than the existing pore pressure inside the sample is applied at one end of the sample. Under this pressure boundary condition, the pore fluid flows out of the sample and the pore pressure decreases. The pore pressure change at the other end of the sample and the radial strain in the middle of the sample are both measured as functions of time and back analysed to evaluate the permeability. During the test, the pore pressure in the sample varies greatly from one end to another. This variable pore pressure keeping the confining pressure constant, induces a variable effective stress in the sample and results in a heterogeneous strain field. The stress-dependent character of the poroelastic parameters of the hardened cement paste (Ghabezloo et al. [START_REF] Ghabezloo | Poromechanical behaviour of hardened cement paste under isotropic loading[END_REF]) and also the creep of the material during the test add some particular aspects to the back-analysis, which makes this problem different from the classical solutions of transient permeability evaluation tests. The porosity changes due to the increase of the effective stress and also the presence of additional creep deformations induce a decrease of the permeability during the test. The coupled measurement of pore pressure and deformations during the test is back analysed to evaluate the permeability-porosity relationship using the results of a single transient test.

It is well-known that there is no unique permeability-porosity relationship that can be applied to all porous materials. As mentioned by Bernabé et al. [START_REF] Bernabé | Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes[END_REF], one reason is that the porosity is a scale invariant material property; if the material and its pore space could be uniformly expanded or contracted everywhere, there would be no change in the porosity, but the permeability would change in this case. Moreover, different pores in a given material have different contributions to the permeability of the material according to their size and shape. The porosity takes into account only the relative volume of the pores to the total volume, and not the shape and the distribution of the size of the pores. Thus two porous materials with the same porosity, but different pore shape and pore-size distributions, must have different permeabilities. For a given evolution process that changes both permeability and porosity of a porous material, for example elastic or plastic compaction, microcracking or chemical alteration, it is usually assumed that there is a power-law relationship k α φ ∝ between these parameters (Bernabé et al. [START_REF] Bernabé | Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes[END_REF]). The exponent of this relation may be integer or non-integer, constant or variable, according to the properties of the material and of the evolution process. Based on the experimental data of Bernabé et al [START_REF] Bernabé | Permeability, porosity and pore geometry of hot-pressed calcite[END_REF], Walder and Nur [START_REF] Walder | Porosity reduction and crustal pore pressure development[END_REF] postulated a slightly different power-law in which the permeability vanishes at a critical non-zero porosity. 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] presented a compilation of published data on the permeability-porosity relationships for different geomaterials. One can see that the reported values of exponent α vary between 1.1 and 25.4 for different materials and the higher values correspond in general to rocks with a high porosity. However, no clear correlation between α and the petrophysical properties could be found. Experimental study of Zhu and Wong [START_REF] Zhu | The transition from brittle faulting to cataclastic flow: Permeability evolution[END_REF] on Berea and Boise sandstones resulted in a variable α which increases when pressure increases. Meziani [START_REF] Meziani | Gas permeability measurements of cement-based materials under hydrostatic test conditions using a low-transient method[END_REF] performed an experimental study on the gas permeability of mortar under hydrostatic loading and found a strongly non-linear permeability-porosity relationship. The exponent α is found to be a decreasing function of the loading level, decreasing from 36 to 26 when the confining pressure is increased up to 57 MPa.

Based on an idea presented by Bernabé et al. [START_REF] Bernabé | Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes[END_REF], the total porosity of the rock can be separated into two categories of effective and non-effective porosity according to the contribution of the pores in the fluid transport. These two quantities are not purely geometrical and depend on the fluid velocity field.

These authors propose that the exponent α is related to the ratio of the effective to non-effective porosity and its variation during an evolution process so that α is dependent upon the particular physical process of porosity evolution and the pore geometry of the material. Consequently as emphasized by Bernabé et al. [START_REF] Bernabé | Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes[END_REF], searching a permeability-porosity relationship is meaningful either for a given rock sample during a porosity evolution process, for example elastic compaction, or for different rock samples only if these samples correspond to different stages of the same evolution process.

Poromechanical background

We present here the framework used to describe the response of the sample during the performed test. The equations are written for a porous material which is not necessarily homogeneous and isotropic at the micro-scale. The theoretical basis of the formulation has been presented in many earlier studies. Among them, one can refer to the milestone papers and textbooks of Biot and Willis [START_REF] Biot | The elastic coefficients of the theory of consolidation[END_REF], Brown and Korringa [START_REF] Brown | On the dependence of the elastic properties of a porous rock on the compressibility of the pore fluid[END_REF], Rice and Cleary [START_REF] Rice | Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents[END_REF], Zimmerman [START_REF] Zimmerman | Compressibility of sandstones[END_REF], Detournay and Cheng [START_REF] Detournay | Fundamentals of Poroelasticity[END_REF], Vardoulakis and Sulem [START_REF] Vardoulakis | Bifurcation Analysis in Geomechanics[END_REF], Wang [START_REF] Wang | Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology[END_REF], Coussy [START_REF] Coussy | [END_REF]. This framework is recalled here in a comprehensive manner in order to clarify the mathematical and physical significance of the different parameters used in the analysis and also to take into account the effect of non-elastic strains in the evaluation of the permeability.

Poroelastic formulation

The porosity φ of a porous material is defined as the ratio of the volume of the porous space V φ to total volume V in the actual (deformed) state:

V V φ φ = (1)
We consider a saturated sample under an isotropic state of stress σ (positive in compression) and we define the differential pressure d σ (i.e. Terzaghi effective stress) as the difference between the confining pressure σ and the pore pressure f p .

d f p σ σ = - (2)
The variations of the total volume V and of the pore volume V φ are given in the following expressions as a function of variations of two independent variables, d σ and f p :

f d d s f d p dp dV d V K K dp dV d V K K φ φ φ σ σ = - - = - - (3)
Where d K , s K , p K and K φ are four elastic moduli defined below:

1 1 1 1 , f f d d p d p p V V K V K V φ φ σ σ ∂ ⎛ ⎞ ⎛ ⎞ ∂ = - = - ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ (4) 1 1 1 1 , d d s f f V V K V p K V p φ φ φ σ σ ⎛ ⎞ ⎛ ⎞ ∂ ∂ = - = - ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ (5)
Equation ( 4) corresponds to an isotropic drained compression test in which the pore pressure is kept constant inside the sample. The variations of the total volume of the sample V and of the volume of the pore space V φ with the applied confining pressure give respectively the drained bulk modulus d K and the modulus p K . Equation ( 5) corresponds to the unjacketed compression test, in which equal increments of confining pressure and pore pressure are simultaneously applied to the sample. The differential pressure d σ in this condition remains constant. The variation of the volume of the sample with respect to the applied pressure gives the unjacketed modulus s K . The variation of the pore volume of the sample in this test could in principle be used to evaluate the modulus K φ . However experimental evaluation of this parameter is very difficult as the volume of the fluid exchanged between the sample and the pore pressure generator has to be corrected for the effect of fluid compressibility, and also for the effect of the deformations of the pore pressure generator and of the drainage system in order to access to the true variation of the pore volume of the sample. In the case of a porous material which is homogeneous and isotropic at the micro-scale

s m K K K φ = =
, where m K is the bulk modulus of the single solid constituent of the porous material. In the case of a porous material which is composed of two or more solids and therefore is heterogeneous at the micro-scale, the unjacketed modulus s K is some weighted average of the bulk moduli of solid constituents [START_REF] Berryman | Effective stress for transport properties of inhomogeneous porous rock[END_REF]. The modulus K φ for such a material has a complicated dependence on the material properties. Generally it is not bounded by the elastic moduli of the solid components and can even have a negative sign if the bulk moduli of the individual solid components are greatly different one from another [START_REF] Berge | Realizability of Negative Pore Compressibility in Poroelastic Composites[END_REF] [START_REF] Berge | Pore Compressiblity in Rocks[END_REF]. From Betti's reciprocal theorem, the following relation holds between the elastic moduli [START_REF] Brown | On the dependence of the elastic properties of a porous rock on the compressibility of the pore fluid[END_REF] [33]:

1 1 1 1 p d s K K K φ ⎛ ⎞ = - ⎜ ⎟ ⎝ ⎠ (6)
Using equation ( 6), the number of the required parameters to characterize the volumetric poro-elastic behaviour of a porous material is reduced to three, and among them, the experimental evaluation of the modulus K φ is very difficult. Using the definition of the porosity presented in equation (1), the following equation is obtained for the variation of the porosity:

dV d d V V V φ φ φ φ = - (7)
Replacing equation ( 3) and then equation [START_REF] Scherer | Characterization of saturated porous bodies[END_REF] in equation ( 7), the expression of the variation of porosity is found:

1 1 1 1 1 d f d s s d d d p K K K K φ φ φ σ φ φ ⎛ ⎞ ⎛ ⎞ - = - - + - ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (8)
As shown by Hart and Wang [START_REF] Hart | Poroelastic effects during a laboratory transient pore pressure test[END_REF], for the case of the axisymmetric triaxial test, the transient pore pressure can be approximated by the solution obtained for the one-dimensional diffusive flow except at early times for which the fully coupled poroelastic response includes also radial flow. Thus, for the analysis of the test results, the one-dimensional solution is used in the following.

Consider now an elementary volume with the z axis parallel to flow direction, q the fluid mass flux per surface area S (orthogonal to z ) and f M the total fluid mass inside the volume. Fluid mass conservation implies that:

1 0 f dM q V dt z ∂ + = ∂ (9)
Let's define f m as the fluid mass per unit volume of the porous material,

f f f m M V ρ φ = = .
The variations of f m is written as:

1 f f f dm dM M dV V dt V dt V dt = - (10)
Replacing equation [START_REF] Grasley | Dynamic pressurization method for measuring permeability and modulus: II. cementitious materials[END_REF] in equation ( 9) we obtain:

0 f f dm dV V q m dt dt z ∂ + + = ∂ (11)
Knowing that

f f m ρ φ = we can write f f f dm d d φ ρ ρ φ = + . Replacing f f f f d dp K ρ ρ
= and equation [START_REF] Scherer | New methods to measure liquid permeability in porous materials[END_REF] in this relation we obtain:

f f f f f dp dV dV dm K V V φ φ φρ φρ ⎛ ⎞ = + - ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (12)
Inserting equation ( 12) and f f m ρ φ = in equation ( 11) the following expression is obtained:

1 0 f f f dp dV V q K dt dt z φ φ φρ ⎛ ⎞ ∂ + + = ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ⎠ (13)
Replacing equation ( 3) and then equation ( 6) in equation ( 13) the following relation is found for the case of 0 dσ = :

1 1 1 1 0 f f f d s dp q K K K K dt z φ ρ φ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ∂ - + - + = ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ⎠ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ (14)
The fluid mass flux q in equation ( 14) is given by Darcy's law:

f f f p q k z ρ µ ∂ = - ∂ ( 15
)
where k is the permeability and f µ is the fluid viscosity. Replacing equation (15) in equation (14) the following expression is obtained:

f f f u f f dp p k dt z z ρ β ρ µ ⎛ ⎞ ∂ ∂ = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ (16)
where:

( ) ( ) 1 1 1 1 1 u f d s K K K K φ β φ = - + - (17)

Effect of non-elastic strains

The above framework can be extended to account for the effect of non-elastic strains which can be produced during the test. These strains can be plastic, viscoelastic or viscoplastic and induce non-elastic porosity changes. The non-elastic changes of the total volume, pore volume and solid volume are defined by: ;

;

ne
φ φ φ = - = - = - (18)
The non-elastic increment of pore volume ne dV φ can be calculated from the definition of the porosity (equation ( 1)) and knowing that

s V V V φ = -.
(

) 1
ne e ne ne s dV dV dV V d d φ φ φ ε φ ε ⎡ ⎤ = - = - + - ⎣ ⎦ (19)
From ( 19) we obtain:

1 ( 1
)
ne ne ne s dV d d V φ φ ε φ ε φ -⎡ ⎤ = -- ⎣ ⎦ (20)
Using equation [START_REF] Sulem | Hydromechanical Behaviour of Fontainebleau Sandstone[END_REF], equation ( 3) is re-written with the additional contribution of the non-elastic volume changes:

1 f ne d d s ne f ne d s p dp dV d d V K K dp dV d d d V K K φ φ φ σ ε σ ε φ ε φ φ - = + + - - = + + - (21)
Using equations ( 7) and ( 21) the following relation is obtained for the variations of the porosity:

( )

1 1 1 1 1 1 ne ne d f s d s s d d d p d d K K K K φ φ φ φ σ ε ε φ φ φ ⎛ ⎞ ⎛ ⎞ - - = - - + - - - ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (22)
Replacing equation ( 21) and then equation ( 6) in equation ( 13) the following relation is found for the case of 0 dσ = :

( )

1 1 1 1 1 0 ne ne s f f f f d s d d dp q dt K K K K dt z φ ε φ ε ρ φ ρ ⎡ ⎤ ⎛ ⎞ -- ⎛ ⎞ ∂ - + - - + = ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ⎠ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ (23)
Replacing equations ( 15) and [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] in equation ( 23) the following expression is found:

( )

1 ne ne s f f f u u f f d d dp p k dt dt z z ε φ ε ρ β β ρ µ ⎛ ⎞ -- ∂ ∂ - = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ (24)
If the non-elastic strains of the solid phase are neglected with respect to the total non-elastic strains, which is a common assumption for most geomaterials, equation ( 24) can be simplified in the following form:

ne f ff u u f f dp p d k dt dt z z ρ ε β β ρ µ ⎛ ⎞ ∂ ∂ - = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ (25)

Experimental program

A class G oil well cement is used to prepare the cement paste with a water to cement ratio 0.44 w c = . The fresh paste is conserved in 14cm cubic moulds for four days in lime saturated water at 90°C temperature. This temperature is chosen in order to reproduce the curing conditions of a cement lining installed in a deep oil well. After four days the temperature is gradually reduced in order to prevent any cracking of the blocs due to a sudden temperature change. Then, the blocs are cored and cut to obtain cylindrical samples with 38 mm diameter and 76 mm length. The two ends of the cylindrical samples are rectified to obtain planar surfaces perpendicular to the axis. After the sample preparation phase, the samples are cured for at least three months in a bath containing a chemically neutral solution towards the cement paste under a controlled temperature of 90°C. Before performing a test, the temperature of the sample is reduced slowly to prevent any thermal cracking. The porosity of the studied cement paste was evaluated by mercury intrusion porosimetry to be equal to 0.26 φ =

. More details about the preparation of the samples can be found in Ghabezloo et al. [12].

The triaxial cell used in this study can sustain a confining pressure up to 60 MPa. It contains a system of hydraulic self-compensated piston. The loading piston is then equilibrated during the confining pressure build up and directly applies the deviatoric stress. The axial and radial strains are measured directly on the sample inside the cell with two axial transducers and four radial ones of LVDT type. The confining pressure is applied by a servo controlled high pressure generator. Hydraulic oil is used as confining fluid. The pore pressure is applied by another servo-controlled pressure generator with possible control of pore volume or pore pressure. More details on this triaxial equipment are found in Sulem and Ouffroukh [START_REF] Sulem | Hydromechanical Behaviour of Fontainebleau Sandstone[END_REF] and in Ghabezloo and Sulem [START_REF] Ghabezloo | Stress dependent thermal pressurization of a fluid-saturated rock[END_REF].

After the curing period the samples which are stored in a fluid at 90°C can be considered as completely saturated. The process of installing the sample inside triaxial cell may cause a partial desaturation of the sample. For this reason a seven days re-saturation phase is performed inside the triaxial cell. During this period, the sample is maintained under a confining pressure equal to 1.2MPa and a back fluid pressure equal to 1.0MPa is applied to the sample while the volume of the fluid injected in the sample is monitored.

As explained earlier, the present permeability test is performed after an undrained hydrostatic compression test and uses the induced excess pore pressure inside the sample for the evaluation of the permeability. One can see several advantages in this method. First it gives access to the permeability of the sample under different states of stress (including high levels of stress and pore pressure) with a single test and gives thus an estimate of the porosity permeability relationship. Moreover, in creeping materials with low permeability, the standard tests cannot give accurate information on the short-term behaviour but only allows an estimation of the long-term permeability. An other marginal advantage is that one high capacity pressure generator for the application of the confining pressure and a low capacity one for the saturation process and the application of the pore pressure at one end of the sample are needed, while, in a conventional permeability test two additional high capacity pressure generators are needed for application and control of the pore pressure at the ends of the sample. The undrained loading is performed using a loading rate equal to 0.1MPa/min. This loading rate is slow enough to ensure the homogeneity of the generated pore pressure inside the sample. After the undrained loading phase, the pore pressure at one end of the sample is reduced instantaneously to a low value (1.0MPa) which causes a fluid flow out of the sample. The other end the drainage system is closed and the pore pressure is monitored using a pore pressure transducer. The volume of the drainage and pressure measurement system at the closed side of the sample influences the measurement of the pore pressure. In our triaxial cell the total volume of the drainage system for an undrained loading is equal to 2300mm 3 which is a relatively small volume. 400mm 3 of this volume is connected to the lower side of the sample and the remaining 1900mm 3 is connected to the upper side. In order to minimize the effect of the dead volume of the drainage system, the fluid flow is established at the upper side which has a greater dead volume and the pressure measurement is performed at the lower side which has a smaller dead volume. Comparing this small dead volume (400mm 3 ) to the volume of the pore space of the sample (22410mm 3 ), one can see that the correction of the pore pressure measurement as derived in Ghabezloo and Sulem [START_REF] Ghabezloo | Stress dependent thermal pressurization of a fluid-saturated rock[END_REF] can be neglected.

The instantaneous reduction of the pore pressure at the upper end of the sample causes a high pressure gradient and a flow of the pore fluid out of the sample. The gradual decrease of the pore pressure inside the sample, while the confining pressure is kept constant, increases the effective stress in a non-uniform manner along the height of the sample which results in non-uniform deformations. Depending on the nature of the tested material and on the magnitude of effective stress change, these deformations may include a non-elastic part. Non-elastic strains have a significant effect on the response of the sample and should be taken into account in the analysis of the test results. During the test, the radial deformations are measured using four LVDT transducers in the middle of the sample. These measured deformations along with the pore pressure measurements are used in the back analysis of the permeability as shown in the next section.

Three permeability tests (A, B and C) were performed. The test results are presented on Figure [START_REF] Brace | Permeability of Granite under High Pressure[END_REF] where the pore pressure measured at the lower end of the sample and the radial strain measured in the middle of the sample are plotted as functions of time. The three tests correspond to different confining pressures and different initial pore pressure inside the sample. Tests B and C have been performed on the same sample. At the end of the test B the sample is unloaded in drained condition and then reloaded in undrained condition up to 50MPa, then test C is performed. One can see in Figure [START_REF] Brace | Permeability of Granite under High Pressure[END_REF] that for all the tests the rate of pore pressure decrease reduces significantly after about 7 hours. One can also observe that despite the small change of effective stress after 7 hours, the measured strains continue to increase significantly. This phenomenon may be attributed to the creep of the hardened cement paste under the applied effective stress.

Analysis of the results

The results obtained in the performed test are back analysed to evaluate the permeability of the sample and its changes during the test. The analysis is performed using equation ( 25) which takes into account the effect of non-elastic strains produced during the test. As one can see in this equation, the poroelastic properties of the tested material must be known to evaluate the u β parameter (equation ( 17)

). The results of a previous experimental study on the poromechanical behaviour of the considered hardened cement paste are presented in Ghabezloo et al. [12]. The unjacketed modulus s K was evaluated equal to 21GPa. Several drained hydrostatic compression tests revealed the stress dependency of the drained bulk modulus d K of the hardened cement paste. The test results showed the degradation of the elastic bulk modulus with the effective stress increase:

( )

8.69 0.087 : GPa , : MPa

d d d d K K σ σ = - (26)
The microscopic observation of the samples after the drained tests showed that this damage phenomenon is caused by the micro-cracking of the hardened cement paste, even under hydrostatic loading, which can be attributed to the heterogeneity of the microstructure of this material. The parameter K φ , as explained above, is very difficult to evaluate and is commonly taken equal to the unjacketed modulus s K . It can be shown that the deviation of this parameter from the unjacketed modulus s K does not have a great influence on the poromechanical formulations (Zimmerman et al., [START_REF] Zimmerman | Compressibility of porous rocks[END_REF]). The physical parameters of pore fluid, f ρ , f µ and f K are taken equal to the ones of pure water and their variations with pore pressure are taken into account (Spang [START_REF] Spang | Excel Add-In for Properties of Water and Steam in SI-Units[END_REF]). The elastic strains due to the variations of the pore pressure can be calculated easily using equation (3). The LVDTs measurements give the radial deformations in the middle of the sample. The measured radial strains are decomposed into an elastic and a non-elastic part. Due to the non-uniformity of the pore pressure, different points of the sample undergo different deformation. In order to estimate these strains in all points of the sample a constitutive law must be assumed for the non-elastic part of the strains. The parameters of this model are calibrated from the strains measured in the middle of the sample. Here we assume that the non-elastic part of the strains is of viscoelastic type and we choose a simple model, composed of a system of parallel spring and dashpot. We also assume that the creep rate is controlled by Terzaghi effective stress (i.e. differential pressure d σ ).

(

)

ne ne d d a b dt ε σ ε = - (27)
In equation ( 27 The variations of the stress state and also the deformations of the sample during the test can modify its permeability. In order to evaluate and take into account the variation of the permeability during the test, we can assume that the permeability varies either with the stress state or with the porosity changes. Assuming that the permeability is controlled by the stress state is appropriate if the deformation process is elastic. In that case, an appropriate effective stress law for the variation of the permeability has to be established, as explained in Ghabezloo et al. [START_REF] Ghabezloo | Effective stress law for the permeability of a limestone[END_REF]. On the other hand, for a material which exhibits timedependent deformation, such an assumption is not appropriate as for example, the permeability changes during a creep test due to the deformation process whereas the stress state remains constant. Thus, for complex deformation process, empirical permeability-porosity laws are commonly used. We choose here a power law ( )

0 0 k k α φ φ =
, so that we can write:

dk d k φ α φ = (28)
As mentioned before, the coefficient α may be constant or variable, integer or non-integer, according to the properties of the material and of the evolution process. For the sake of simplicity, in the back analysis of the results a constant integer coefficient is assumed. The exponent α and the initial permeability 0 k are evaluated in the back analysis of the experimental results. Before the beginning of the permeability evaluation test, the permeability of the sample is first modified by the applied undrained loading. The undrained test performed on this sample with three unloading-reloading cycles at different levels of confining pressure, as presented in Ghabezloo et al. [START_REF] Ghabezloo | Poromechanical behaviour of hardened cement paste under isotropic loading[END_REF], does not show any irreversible strains in unloading. Consequently the variations of the sample porosity caused by the applied undrained loading can be easily evaluated using equation ( 8) by knowing the final values of confining pressure and pore pressure. The initial porosity is equal to 0 0.26

φ =
and the drained bulk modulus d K is calculated using equation [START_REF] Detournay | Fundamentals of Poroelasticity[END_REF] for the average effective stress during the undrained loading. Using these parameters, the porosity change and the resulting permeability change caused by the initial undrained loading can be evaluated (equation ( 28)).

The back analysis of the test results using equation ( 25) is performed using a finite difference numerical scheme. The first analysis is performed on test A. The parameters a and b , the exponent α and the initial permeability 0 k are evaluated using the least square method. The error between the computed results and the measured data is calculated as the sum of errors corresponding to the pore pressure and the radial strain. For each iteration, the parameters are changed manually and the sum of square of the difference between the measured and the calculated values is calculated on a set of points.

The best fitting parameters are then evaluated by minimizing the calculated error. For the test A the following parameters are found: 1.03 10 m -× under the effect of the initial undrained loading followed by a progressive reduction to 20 2 6.34 10 m -× at the end of the test. We observe that the undrained loading phase does not modify much the permeability of the sample. By doing this analysis we obtain a permeability porosity relationship for the tested sample using the results of a single transient test under the following form: The above computation takes into account both the effect of non-elastic deformation and the effect of permeability change with porosity. In order to investigate separately these two effects, two additional simulations are performed. In the first one, the analysis is performed by neglecting the effect of nonelastic strains during the test and by considering a constant permeability (case 1). In the second one, the effect of non-elastic strains is taken into account, keeping a constant permeability (case 2). For each case the required parameters are evaluated using the least square algorithm. The results are presented on the Figure ( 4) and compared with the results of the complete model (case 3). For case 1, significant differences between the experimental results and the computed ones are observed. Especially the important difference between the measured and the calculated radial strain shows that it is essential to take into account the effect of non-elastic strains in the analysis of the test results. The results of case 2 show that an acceptable response can be obtained by considering the effect of non-elastic strains even with a constant permeability. Case 3 is identical to the analysis presented on Figure [START_REF] Bernabé | The effective pressure law for permeability in Chelmsford granite and Barre granite[END_REF] and results in an even better compatibility with the experimental results in particular for the permeability curve in the last part of the test. For case 2, a constant permeability is evaluated equal to Two main physical mechanisms occur in the course of the loading process, micro-cracking as mentioned above and compaction. The reduction of the permeability during the test shows that the compaction process is dominant and that the net result of these two mechanisms is a reduction of the permeability. Moreover, one can infer that the induced micro-cracks remain closed under stress and do not influence the permeability significantly.

-⎛ ⎞ = × ⎜ ⎟ ⎝ ⎠ (30)
At the end of this test, the permeability of the node at the centre of the sample is 6) where a good compatibility between the simulation and the test results can be observed. In Figure [START_REF] Scherer | Characterization of saturated porous bodies[END_REF] a small pore pressure increase can be observed in the beginning of the test C in the results of the numerical simulation. Due to the high value of the initial effective stress in this test (32.4MPa) and also the low permeability of the sample, in the beginning of the test the effect of the creep deformations is more important than the effect of the pore pressure diffusion. This explains the small initial pore pressure increase obtained in the simulation. The evaluated value of the porosity sensitivity exponent α , equal to 11, is a relatively high value comparing with the range of variations of α for other geomaterials. Such a high value of α is usually attributed to a pore structure containing relatively large, pressure-insensitive nodal pores connected to each other by narrow pressure-sensitive throats (Bernabé et al. [START_REF] Bernabé | Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes[END_REF]) or to a pore structure in which pocket-like pores contribute much to the porosity and little to the permeability (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]). This latter description seems to be mostly appropriate for the case of a permeability-porosity relationship which is established for different samples of a porous material with different porosities. Based on the idea of Bernabé et al. [14], a high value of α means that as a result of the compaction process, the reduction of the effective porosity is more important than the reduction of the non-effective porosity, or that some part of the effective porosity is transformed into non-effective porosity. We have shown that for two samples of a given material, the permeability evolves with the porosity with the same value of α , mostly independently of the level of effective stress or creep deformations in the three performed tests. This shows that in the absence of a difference between the pore shape and geometry of the samples, the permeability variation is controlled by the process of porosity evolution, which is the same in the different tests. This result is in accordance with the idea of Bernabé et al. [START_REF] Bernabé | Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes[END_REF], as explained here above.

The obtained permeability-porosity relationship with an exponent α equal to 11 should not be seen as a general permeability-porosity law for cement paste but specific to the cement used in this study and to the specific hydrostatic compaction process. This relation could in principle be combined with other empirical relationships found in the literature which describe the evolution of the permeability of the cement paste with other parameters such as w c (water to cement ratio), the pore size distribution etc… For example Breysse and Gérard [START_REF] Breysse | Modelling of permeability in cement-based materials: Part 1-Uncracked medium[END_REF] performed a statistical analysis on the values of permeability of different cement pastes prepared with different w c ratios as reported in the literature and found a power law for the relationship between the permeability and w c . Higher values of w c result in a cement paste with a higher initial porosity and thus a higher initial permeability. Breysse and Gerard relationship can be used to evaluate the initial permeability to be put in our permeability evolution law. On the other hand empirical relationships can be found in the literature to estimate the initial porosity as a function of w c (Taylor [38]).

Conclusions

The permeability-porosity relationship for a hardened class G oil well cement paste is evaluated experimentally using a single transient test. The excess pore pressure generated in the sample after an undrained hydrostatic compression test is used for the evaluation of the permeability. This pressure is released at one end of the sample and the variations of the pore pressure at the other end and also the radial deformations in the middle of the sample are measured during the dissipation of the pore pressure. The major advantage of the proposed method as compared to the classical steady state or transient permeability measurement methods, which are performed at a given level of stress, is that one can directly access to the permeability-porosity relationship in a single test. The test results are back analysed to evaluate a power-law permeability-porosity relationship. The creep of the sample during the test and its effect on the measured pore pressure and deformations are introduced in the formulation and are taken into account in the back analysis of the results. For a non creeping material, the same type of analysis can be used to assess a permeability-effective stress relationship. The exponent of the powerlaw is evaluated equal to 11 in one of the tests and is shown to be the same for the other tests with different initial stress and pore pressure levels. Consequently we can conclude that this coefficient is mostly dependent upon the compaction process which modifies the porosity of the material in accordance with the idea presented by Bernabé et al. [START_REF] Bernabé | Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes[END_REF].

It should be emphasized that the obtained power-law relationship describes the variations of the permeability of a given hardened cement paste with a given initial porosity submitted to a hydrostatic loading. In order to extrapolate this law to other cement pastes, prepared with different / w c ratio, one can use empirical relationships existing in the literature which relate the initial permeability and the initial porosity to this parameter.

  Evaluation of a permeability-porosity relationship in a low permeability creeping material using a single transient test Siavash Ghabezloo, Jean Sulem, Jérémie Saint-Marc To cite this version: Siavash Ghabezloo, Jean Sulem, Jérémie Saint-Marc. Evaluation of a permeability-porosity relationship in a low permeability creeping material using a single transient test. International Journal of Rock Mechanics and Mining Sciences, 2009, 46 (4), pp.761-768. 10.1016/j.ijrmms.2008.10.003. hal-00336706v2

Figure 1 -

 1 Figure 1-Results of the permeability tests: Evolution of the pore pressure measured at the lower end of the sample and of the radial strain measured at the middle of the sample during the test.

  ), a and b are two model parameters; ( ) 1 ab is the viscosity and b is the compressibility of the underlying dashpot and spring model. It should be noted at this point that, as the tested samples have experienced different loading paths before the permeability tests, their creep behaviour may be different so that a and b should be calibrated for each test.

  the numerical simulation together with the experimental data are presented in Figure[START_REF] Bernabé | The effective pressure law for permeability in Chelmsford granite and Barre granite[END_REF]. One can see that the measured radial strains are well reproduced in the simulation using the assumed viscoelastic model and the calibrated parameters. The measured pore pressure at the lower end of the sample is also well reproduced by the model.The variation of calculated permeability at the centre of the sample during the test is presented in Figure[START_REF] Bernabé | The effective pressure law for permeability during pore pressure and confining pressure cycling of several crystalline rocks[END_REF]. One can see an initial permeability reduction from

Figure 2 -

 2 Figure 2-Test A: Test results and numerical simulation.

Figure 3 -

 3 Figure 3-Test A: Evolution of the calculated permeability at the centre of the sample.

  seen as an average permeability of the deforming sample during the test. The relatively small difference between the results of case 2 and case 3 is due to the small reduction of the porosity and consequently of the permeability of the sample during the test (Figure (3)).

Figure 4 -

 4 Figure 4-Test A: Effect of non-elastic strains and porosity dependent permeability: Case 1: Elastic solution, constant permeability; Case 2: Non-elastic solution, constant permeability; Case 3: Non-elastic solution, porosity dependent permeability. For the analysis of the results of the tests B and C we assume that the coefficient α is only a function of the compaction process and consequently is the same for all tests, equal to 11. For each of the tests B and C, the coefficients a and b of equation (27) and the initial permeability 0 k are evaluated to find the best accordance between the simulation and the test results. For test B the following parameters are found: 5 -1 4.1 10 s a -= × ,

  this sample is then unloaded in drained condition and test C is performed on the same specimen. For test C the evaluated parameters are: , the initial permeability of the sample calculated for test C is very close to the final permeability of the sample in test B which gives confidence in the approach. The results are presented in Figures[START_REF] Scherer | Bending of gel beams: method of characterizing mechanical properties and permeability[END_REF] and (

Figure 5 -

 5 Figure 5-Test B: Test results and numerical simulation.

Figure 6 -

 6 Figure 6-Test C: Test results and numerical simulation.

Acknowledgments

The authors gratefully acknowledge TOTAL for supporting this research. They wish also to thank Yves Bernabé and Ahmad Pouya for fruitful comments and discussions and François Martineau for his assistance in the experimental work.