Laurent Perrier

Frédéric Plantier

David Grégoire

A novel

Keywords: numbers: 68.43.-hbra Adsorption, deformation, microporous materials, experiment, manometric technique, digital image correlation

I. INTRODUCTION

Following the IUPAC recommendation 1,2 , the pore space in porous materials is divided into three groups according to the pore size diameters: macropores of widths greater than 50 nm, mesopores of widths between 2 and 50 nm and micropores (or nanopores) of widths less than 2 nm. Zeolites, activated carbon, tight rocks, coal rocks, source rocks, cement paste or construction materials are among these materials. In recent years, a major attention has been paid on these microporous materials because the surface-to-volume ratio (i.e., the specific pore surface) increases with decreasing characteristic pore size. Consequently, these materials can trap an important quantity of fluid molecules as an adsorbed phase. This is important for applications in petroleum and oil recovery, gas storage, separation, catalysis or drug delivery.

For these microporous materials, a deviation from standard poromechanics, which was introduced by Biot 75 years ago 3,[START_REF] Coussy | Poromechanics[END_REF] , is expected. In very small pores, the molecules of fluid are confined. Interaction between molecules is modified. This effect, denoted as molecular packing, includes fluid-fluid and fluid-solid interactions and has significant consequences at the macroscale, such as instantaneous swelling deformation. In different contexts, these deformations may be critical. For instance, in situ adsorption-induced coal swelling has been identified [START_REF] Larsen | [END_REF]6 as the principal factor leading to a rapid decrease in CO 2 injectivity during coal bed methane production enhanced by CO 2 injection. Generally, natural a) Corresponding author: david.gregoire@univ-pau.fr and synthesised porous media are composed of a double porosity: the microporosity where the fluid is trapped as an adsorbed phase and a meso or a macro porosity required to ensure the transport of fluids to and from the smaller pores. If adsorption in nanopores induces instantaneous deformations at a higher scale, the matrix swelling may close the transport porosity, reducing the global permeability of the porous system or annihilating the functionality of synthesised materials. Conversely, gas desorption can lead to matrix shrinkage and microcracking, which may help oil and gas recovery in the context of unconventional petroleum engineering 7 . The effects of adsorbent deformation on physical adsorption has also been identified 8 as the major challenge concerning gas porosimetry in nano-porous non-rigid materials (e.g. metal organic framework). In conclusion, there is now a consensus in the community that major attention has to be focused on the coupled effects appearing at the nanoscale within microporous media because they may have significant consequences at the macroscale.

Experimentally, different authors tried to combine gas adsorption results and volumetric swelling data. Pioneer works of Meehan 9 showed the effect of carbon dioxyde sorption on the expansion of charcoal but only mechanical deformations were reported and no adsorption quantities were measured. Later on, different authors 7,[10][11][12][13][14][15] performed tests on bituminous coal, because it is of utmost importance in the context of CO 2 geological sequestration and coal bed reservoirs exploitation. However, most results in the literature are not complete in a sense that adsorption and swelling experiments are not measured simultaneously 9,[START_REF] Robertson | 4th Annual DOE/NETL Conference on Carbon Capture and Sequestration[END_REF] or performed on the exact same coal samples 12 . Other authors presented simultaneous in situ adsorption and swelling results but the volumetric strain is extrapolated from a local measurement -using strain gauges 7,[START_REF] Harpalani | SPE Annual Technical Conference and Exhibition[END_REF][START_REF] Battistutta | [END_REF] or LVDT sensors 15,19 -or by monitoring the silhouette expansion 11 .

Recently, important efforts have been involved in molecular simulations [20][21][22][23] or in poromechanical modeling [START_REF] Perrier | Continuum Mechanics and Thermodynamics[END_REF][START_REF] Vermorel | [END_REF][26] in order to predict adsorption-induced deformation in nanoporous materials. However, there is still a need for detailed experimental data presenting simultaneous adsorption-induced deformation measurements for validation purpose, especially for heterogeneous materials.

In this context, this paper aims to present a novel apparatus allowing the simultaneous in situ measurement of both the adsorption quantities and the deformations induced on the same sample. This apparatus is composed of a manometric technique coupled with a digital image correlation set-up for full-field displacement measurements. In section II, the novel apparatus is described and the experimental method is fully detailed by presenting both the manometric technique and the digital image correlation set-up. In section III, this new apparatus is used to characterise in situ adsorption-induced deformation on a nanoporous activated carbon saturated with pure methane and carbon dioxyde.

II. EXPERIMENTAL APPARATUS : PRINCIPLES AND METHODS

A. Device description

A new experimental set-up is proposed to measure insitu and simultaneous adsorption-induced deformation in microporous materials. The set-up couples a home-built manometric apparatus for gas adsorption measurements and a digital image correlation set-up for full-field displacement measurements.

Figure 1 shows a sketch of the experimental set-up. It is composed of:

• a manometric set-up for gas adsorption measurements:

a dosing volume (V D ) composed of a dosing cell and a connecting network. The dosing volume is (V D ± ∆V D ) = (95.95 ± 0.09) ml, with a relative error of ∆V D V D = 0.09%, a measuring volume (V M ) composed of a measuring cell where the sample takes place on a sapphire window. The volume cell is (V 0 M ± ∆V 0 M ) = (188.8 ± 0.3) ml with a relative error of 0.2%, a digital manometer (Wiika, measuring range:

[0, 250] bar, precision: ±0.005 bar),

an electrical heat regulator (Horst GmbH HT31) coupled with a PT100 temperature probe,

two in situ thermocouples IEC, type K, (RS thermocouple, precision: ±0.1 K) ;

• a digital image correlation set-up for full-field measurements:

a digital camera (PixeLINK, PL-B781U),

an optical lens (Edmundoptics, focal length : 25 mm),

a standard LED lighting unit, a computer.

B. Gas adsorption measurements

Excess adsorption isotherms are built step by step from gas adsorption measurements performed using an homebuilt manometric set-up. The manometric technique has been widely used by the community for gas adsorption measurements and different studies may be found in the literature (see e.g. Rouquerol et al. [START_REF] Rouquerol | Adsorption by powders and porous solids: principles, methodology and applications[END_REF] , Battistutta et al. [START_REF] Battistutta | [END_REF] , Badalyan et al. [START_REF] Badalyan | [END_REF] ). The principle of the technique consists in balancing the gas quantities before and after adsorption thanks to temperature and pressure measurements at the equilibrium.

Carrying out an adsorption isotherm measurement at a given temperature requires different steps:

1. A porous sample, of a known mass m, is introduced in the measuring cell.

2. The porous sample is purified under given vacuum and heating conditions (typically 24h at 343.15 K).

3. An accurate estimation of the measuring volume V M is performed using a non adsorbing gas (typically helium is chosen).

4. The gas adsorption isotherm is finally built step by step by estimating the exceeding adsorbate moles of gas for incremental increasing pressure at thermal equilibrium. At each step, an amount of gas is introduced in the dosing volume V D . After equilibrium, pressure and temperature in V D are measured. Then, the gas is allowed to fill the whole system V D + V M and a part of it is adsorbed on the adsorbent sample and a part of it remains nonadsorbed. When the new equilibrium is reached, the new pressure and temperature in the whole system V D +V M are measured and the amount of nonadsorbed gas is deduced. The amount of exceeding adsorbed gas is then estimated by difference following Eq. (1).

n ex i = n ex i-1 + δn ex i , δn ex i = VM M ρ M i-1 + VD M ρ D i -VD+VM M ρ M+D i . (1)
In Eq. ( 1), VM M ρ M i-1 is the amount of remaining nonadsorbed gas in the measuring volume V M at the end of step (i-1), VD M ρ D i is the amount of gas introduced in the dosing volume V D at the beginning of step i, VD+VM M ρ M+D i is the amount of non-adsorbed gas in the whole system at the end of step i, M is the gas molar mass and ρ is the gas density. The subscript represents the pressure step and the superscript represents the volume from which the gas density is estimated through an equation of state for given pressure and temperature. Typically, the equations developed by Span and Wagner Since the excess adsorption isotherm is built in an incremental way through Eq. ( 1), the relative errors accumulate step by step.

Typically, for one excess adsorption amount, the measurement uncertainty is given by Eq. (2).

∆n ex i = ∆n ex i-1 + ρ M i-1 -ρ M+D i ∆VM M + ρ D i -ρ M+D i ∆VD M + (∆ρ M i-1 + ∆ρ M+D i ) VM M + (∆ρ D i + ∆ρ M+D i ) VD M .
(2) In Eq. ( 2), ∆α is the measurement uncertainty of the quantity α.

The manometric part of the device has been validated by comparing the excess adsorption quantities measured on an Ecosorb activated carbon with the ones obtained with a reference gravimetric technique [START_REF] Khaddour | [END_REF] . Figure 2 shows the comparison in term of excess adsorption isotherms (T = 303.15 K) obtained for pure CH 4 (P ∈ [0-130] bar) and pure CO 2 (P ∈ [0 -60] bar). A good agreement is clearly observed between the two set of data which validates the manometric technique for adsorption measurements.

Note that different test have been performed to ensure that the temperature is properly stabilised during the whole measurement duration. A typical temperature shift of 0.1 K is observed after 12 hours (which corresponds to the thermocouple sensor uncertainty) and the typical duration of a full adsorption isotherm measurement is 6 hours for the material tested here.

C. Swelling measurement by digital image correlation

Swelling strain is estimated by full-field displacement measurements performed by digital image correlation (DIC) using Icasoft software 33 . The general framework of the digital image correlation (DIC) algorithm used in this paper is based on the formulation presented by Grégoire et al. 34 . The general principle is briefly recalled in Appendix A. Details may be found in Grégoire et al. 34 .

After purification, a reference image of the surface sample is acquired. At each pressure step, a deformed image is acquired. The two digital images corresponding to the reference and the deformed state are described by a discrete function representing the grey level of each pixel (Eq. A1). The optimal displacement field determination consists in the minimization of a cross-correlation coefficient on a set of image pixels (Eq. A2). The use of a cross-correlation allows to correct a possible difference in brightness between the two images 34 . Thanks to this technique, the full-field displacement is estimated with a typical resolution of 1/100 of a pixel 33 .

Fig. 3 presents a typical result of the full-field displacement estimation preformed by the DIC technique for an activated carbon saturated with pure CO 2 at P b = 50.8 bar. Fig. 3.a, 3.b and 3.c present respectively the reference image and the full-field displacements in the horizontal and vertical directions. A typical adsorptioninduced swelling displacement field is obtained.

From the DIC estimation of the displacement field (u = U x e x + U y e y ), the deformation field is calculated based on a continuum mechanics framework with small displacement-gradient assumption (ε = 1 2 (∇(u)+ t ∇(u))) as well as the volumetric swelling strain (ε v = tr(ε), where tr() is the trace of the matrix). For an isotropic homogeneous materials, the volumetric swelling strain is directly equal to three times the ortho-radial strain (ε v = 3 × ε θθ ). See Appendix B for details on the volumetric swelling strain estimation. For an anisotropic, heterogeneous or cracked material, the volumetric swelling strain is not the quantity of interest and the DICbased method provides the full displacement field on the surface of the material from which the local deformation can be estimated. Figure 4 presents a typical result of the full-field ortho-radial strain estimation for an isotropic and homogeneous activated carbon saturated with pure CO 2 at P b = 50.8 bar. A typical adsorption-induced homogeneous swelling strain field is obtained.

III. APPLICATION TO THE DETERMINATION OF ADSORPTION-INDUCED SWELLING ON ACTIVATED CARBON.

A. Adsorbent and adsorbates description

An activated carbon (Chemviron) is used as adsorbent material in this study. The sample is a cylinder and its main characteristics are collected in Table I. The geometrical dimensions have been measured with a caliper, the mass has been measured with a Precisa scale (XT 2220 M-DR), the specific pore surface has been measured with a gas porosimeter (Micromeretics ASAP 2020) according to the BET theory 35 . The specific micropore volume has been estimated according to the IUPAC classification 2 (pore diameter down to 2 nm) based on a pore size distribution deduced from a low-pressure adsorption isotherm (N 2 at 77 K from 8.10 -8 to 0.99 P/P 0 in relative pressure range) measured with the same gas porosimeter according to the HK theory 36 . The adsorbates, CO 2 and CH 4 , as well as the calibrating gas, He, are provided by Linde Gas with a minimum purity of 99.995%, 99.995% and 99.999% respectively.

B. Equilibration time, adsorption kinetics and deformation kinetics

C. Simultaneous adsorption and induced deformation measurements

The main advantage of the proposed method is to provide simultaneous in situ measurements of both adsorption and deformation for the same sample in the exact same conditions. Fig. 6 presents the results of these simultaneous measurements for an activated carbon filled with pure CO 2 and pure CH 4 at T = 318.15 K and T = 303.15 K respectively. Full-field deformation maps are presented as supplementary material (see section V). Collected experimental data are reported in appendix D. Fig. 6.a presents the results in term of excess adsorption/desorption isotherms. CO 2 and CH 4 gas sorption in activated carbon is a reversible phenomenon and no hysteresis is observed between adsorption and desorption paths as previously reported in the literature [START_REF] Khaddour | [END_REF] . Noting that adsorbed quantity amount increases when temperature decreases, Fig. 6.a shows that CO 2 is preferentially adsorbed in carbon compare to CH 4 as previously reported in the literature 12,[START_REF] Battistutta | [END_REF] . This is the reason why CO 2 injection is used to increase CH 4 recovery in Enhanced Coal Bed Methane production. Fig. 6.b presents the results in term of adsorptioninduced volumetric strain. CO 2 and CH 4 gas adsorptioninduced deformation is a reversible phenomenon but a small hysteresis is observed between the adsorption and the desorption paths. This hysteresis is not linked to the adsorption-deformation couplings but is due to an elastic compaction of the carbon matrix grains [START_REF] Perrier | Coupling between adsorption and deformation in microporous media[END_REF] . Cycling effect and material compaction are detailed in appendix C. For a given pressure, CO 2 adsorption produces more volumetric deformation than CH 4 adsorption, which is the source of the rapid decrease in CO 2 injectivity during coal bed methane production enhanced by CO 2 injection. Fig. 7 presents the evolution of the volumetric strain versus the excess adsorbed quantity for both CO 2 and CH 4 at T = 318.15 K and T = 303.15 K respectively. The two evolutions for the two different gases are close together showing that the volumetric swelling is directly linked to the excess adsorbed quantity.

IV. CONCLUDING REMARKS

A new experimental set-up for the study of adsorptioninduced deformation has been presented. The main advantage of the proposed method is to provide simultaneous in situ measurements of both adsorption and deformation for the same sample in the exact same conditions, which can be directly used for model validation. Gas adsorption measurements are performed using an home-built manometric apparatus and deformation measurements are performed using a digital image correlation set-up. This set-up allows full-field displacement measurements, which may be crucial for heterogeneous, anisotropic or cracked samples. For homogeneous and isotropic samples, homogeneous swelling strain deformation fields are observed and the volumetric swelling strain may be deduced. For an anisotropic, heterogeneous or cracked material, the volumetric swelling strain is not the measurable quantity of interest but the DIC-based method still provides the full displacement field on the surface of the material from which the local deformation can be estimated.

The novel apparatus and the experimental method have been fully detailed and used to characterise in situ adsorption-induced deformation on a nanoporous acti- vated carbon saturated with pure methane and carbon dioxyde. For a given pressure, CO 2 is preferentially adsorbed in this activated carbon compare to CH 4 and CO 2 adsorption produces more volumetric deformation than CH 4 adsorption. However, the simultaneous measurements of both adsorption and strain show that for a given amount of excess adsorbed quantity, a similar induced deformation is observed for CH 4 and CO 2 showing that the macroscopic swelling is intrinsically linked to the amount of excess adsorbed quantities within the nanopores.

V. SUPPLEMENTARY MATERIAL

See supplemental material at [URL will be inserted by AIP] for the full-field deformation maps used to estimated the volumetric swelling evolution presented in Figure 6 based on the procedure presented in appendix B. phenomenon is elastic. In Figure 6 are presented the data acquired during the third cycle for both CH 4 and CO 2 gas adsorption.

Appendix D: Collected experimental data

This appendix collects the experimental data presented in Figure 6.

:

  FIG. 1. Sketch of the experimental set-up.

2 FIG. 2 .

 22 FIG. 2. Comparison of the excess adsorption isotherms measured by the manometric technique (developed in this study)and a reference gravimetric technique (Khaddour et al.[START_REF] Khaddour | [END_REF] ) for an Ecosorb activated carbon saturated with pure CH4 and pure CO2 at 303.15 K.

Fig. 5

 5 Fig. 5 presents a typical results of the kinetics of sorption and deformation for an activated carbon saturated with pure CO 2 at T = 303.15 K for both adsorption (Fig. 5.a) and desorption (Fig. 5.b) for a bulk pressure varying respectively from P b = 0 bar to P b = 20 bar and from P b = 20 bar to P b = 15 bar. The thermodynamical equilibrium is reached after a time duration, which is identified equal to 15 min. This duration represents the minimum time increment between two constitutive experimental acquisitions for this material.

FIG. 3 .

 3 FIG. 3. Typical result of the full-field displacement estimation preformed by the DIC technique for an activated carbon saturated with pure CO2 (T = 303.15 K, P b = 50.8 bar, pixel size: 17.9 µm): (a) reference image; (b) full-field displacement in the horizontal direction; (c) full-field displacement in the vertical direction.

FIG. 5 .

 5 FIG. 5. Typical kinetics of sorption and deformation for an activated carbon saturated with pure CO2 at T = 303.15 K: (a) adsorption from P b = 0 bar to P b = 20 bar; (b) desorption from P b = 20 bar to P b = 15 bar.

FIG. 6 .FIG. 7 .

 67 FIG. 6. Simultaneous adsorption and induced swelling measurements for an activated carbon filled with pure CO2 and pure CH4 at T = 318.15 K and T = 303.15 K respectively: (a) excess adsorption isotherm; (b) volumetric induced swelling.

FIG. 10 .

 10 FIG. 10. Adsorption-induced deformation for a activated carbon saturated with pure CO2 at T = 303.15 K, P b = 50.8 bar: (a) norm of the displacement field; (b) ortho-radial strain.

  29 , Setzmann and Wagner 30 and Arp and McCarty 31 are used to estimate respectively the densities of CO 2 , CH 4 and He.

TABLE I .

 I Main characteristics of the adsorbent material.

	Property	Unit		Value
	Height	(cm)	h = 1.922 ± 0.004
	Diameter	(cm)	d = 2.087 ± 0.002
	Volume	(ml)	V ech = 6.57 ± 0.03
	Mass	(g)	m = 4.137 ± 0.001
	Specific pore surface	(m 2 .g -1 )	SBET = 1090 ± 10
	Specific micropore volume (cm 3 .g -1 )		v φµ = 0.51
	Total micropore volume	(ml)		V φµ = 2.11
	Microporosity	(%)	φµ =	V φµ V ech = 32.1

TABLE II .

 II Collected experimental data in term of relative bulk pressure (P b ), excess adsorbed quantities (n ex ) and swelling strain (εv) for an activated carbon saturated with pure CH4 at T = 303.15 K ± 0.1 K.

	P b	n ex	εv
	(bar)	(mmol.g -1 )	(%)
	2.850±0.005	0.846±0.003	0.046±0.006
	11.280±0.005	2.316±0.011	0.174±0.022
	24.045±0.005	3.485±0.025	0.404±0.029
	39.285±0.005	4.338±0.045	0.673±0.063
	55.915±0.005	4.886±0.071	0.909±0.083
	74.190±0.005	5.333±0.104	1.145±0.073
	92.870±0.005	5.627±0.144	1.365±0.036
	105.230±0.005	5.853±0.185	1.503±0.064
	72.140±0.005	5.301±0.215	1.226±0.077
		4.635±0.235	0.953±0.089
	32.930±0.005	3.931±0.248	0.712±0.100
	21.810±0.005	3.156±0.257	0.542±0.094
	14.090±0.005	2.474±0.263	0.376±0.074
	8.805±0.005	1.815±0.267	0.254±0.056
	5.230±0.005	1.201±0.269	0.185±0.045
	2.795±0.005	0.643±0.271	0.098±0.026
	1.115±0.005	0.200±0.272	0.079±0.016

TABLE III .

 III Collected experimental data in term of relative bulk pressure (P b ), excess adsorbed quantities (n ex ) and swelling strain (εv) for an activated carbon saturated with pure CO2 at T = 318.15 K ± 0.1 K.

	P b	n ex ± ∆n ex	εv ± ∆εv
	(bar)	(mmol.g -1 )	(%)
	4.230±0.005	2.004±0.004	0.160± 0.111
	12.505±0.005	4.160±0.013	0.529± 0.088
	24.230±0.005	6.039±0.027	1.133± 0.273
	37.795±0.005	6.962±0.047	1.762± 0.168
	45.685±0.005	7.328±0.068	2.110± 0.176
	32.570±0.005	6.662±0.082	1.696± 0.147
	22.575±0.005	5.692±0.091	1.246± 0.241
	15.235±0.005	4.694±0.097	0.873± 0.224
	10.005±0.005	3.734±0.101	0.641± 0.134
	6.385±0.005	2.753±0.104	0.449± 0.112
	3.855±0.005	1.913±0.106	0.327± 0.117
	2.095±0.005	1.187±0.107	0.261± 0.044
	0.875±0.005	0.545±0.108	0.259± 0.039
	BIBLIOGRAPHY

ACKNOWLEDGMENTS

Financial supports from the Région Aquitaine through the grant CEPAGE (20121105002), from the Conseil Départemental 64 through the grant CEPAGE2 (2015 0768), from the Insitut Carnot ISIFoR and from the Université de Pau et des Pays de l'Adour through the grant Bonus Qualité Recherche are gratefully acknowledged. We also gratefully acknowledge Pr. Gilles Pijaudier-Cabot for him advices and our different discussions.

Appendix A: General framework for digital image correlation

The general framework of the digital image correlation (DIC) algorithm used in this paper is based on the formulation presented by Grégoire et al. 34 . Here it is now applied for adsorption-induced deformation measurements. The principle of DIC was advanced in the context of experimental mechanics by Sutton et al. 38 ,39 . Two digital images corresponding to a reference and a deformed state are described by discrete functions representing the grey level of each pixel and related by:

where f and f * are the discrete functions of the reference and the deformed state respectively, and d(x) is the displacement field vector (Fig. 8). The optimal displacement field determination consists of the minimization of a cross-correlation coefficient on a set of initial image pixels, called a subset:

where ∆M is the surface of the subset in the reference image (see Fig. 9). The displacement field is typically decomposed on an appropriate functional basis on a subset: where u j are the unknown displacement vectors, P is a point of the subset, d(P ) is its displacement vector given by the minimization of Eq. (A2) and n j are interpolating functions to be chosen.

According to the numbers of unknowns introduced in the decomposition, Eq. (A3) is written as many times as needed to have a problem well-posed. Finally, the problem consists of inverting a matrix given by:

where D is the vector of the optimal displacement obtained by the minimization of Eq. (A3), [M ] is the matrix of the functional basis and U is the vector of the unknown displacements.

Typically, a bilinear continuous displacement field decomposition ({n j } = {x, y, xy, 1}) and a cubic spline interpolation are used as in Touchal Mguil 33 . Figure 9 shows a subset before (ABCD centered on P ) and after (A * B * C * D * centered on P * ) deformation on the same coordinate system. The displacement field is estimated on each point by means of: Finally, the algorithm is carried out on each subset of the initial image in order to obtain the full field displacement u(x, y) = U x (x, y) U y (x, y) (e x ,e y ) with a resolution of 1/100 of a pixel.

Appendix B: Full-field deformation estimation and swelling calculation

Theoretical background

Traditionally, the deformation field is calculated based on a continuum mechanics framework with small displacement-gradient assumption (ε = 1 2 (∇(u) + t ∇(u))). For adsorption-induced deformation in an isotropic homogeneous medium, we expect a swelling displacement field and cylindrical coordinates (e r , e θ , e z ) are more appropriate to estimate the volumetric swelling. One gets:

where:

B2) and:

For a perfect swelling deformation field in an isotropic homogeneous medium, one may obtain:

Numerical errors may be introduced during the estimation of a radial strain field based on the numerical differentiation of a discrete radial displacement field (Eq. B4). Therefore, the ortho-radial strain field is directly linked to the norm of the displacement field assuming that the ortho-radial displacement field is negligible

where (x 0 ; y 0 ) is the origin of the cylindrical coordinate system. Therefore the volumetric swelling is given by:

Example and validation

The method proposed for the estimation of the adsorption-induced volumetric swelling is applied on an activated carbon saturated with pure CO 2 at T = 303.15 K, P b = 50.8 bar. The resulting full-field displacement estimation performed by the DIC technique is presented in Figure 3.a.

In order to estimate the volumetric swelling through Eq. B5, the origin of the cylindrical coordinate system has to be identified. For this purpose, the norm of the displacement field is interpolated by cubic spline using Scilab software (the interpolation area is represented by a red square in Figure 3.a) and the origin is defined as the point (x 0 = 563.5 px; y 0 = 577.2 px) where the interpolated field reaches its minimum. Figure 10.b presents the ortho-radial strain estimated in the interpolation area by Eq. B5. A typical constant ortho-radial strain field is obtained in all the interpolation area except in the vicinity of the cylindrical coordinate origin point where a peak artefact is observed. Finally, the volumetric strain is estimated by averaging the ortho-radial strain field in the interpolation area where a mask (represented in white in Figure 10.b) is applied in the vicinity of the cylindrical coordinate origin point to minimize the influence of the artefact peak. The mask dimensions are 150 × 150 pixels, which represents 5.6 % of the interpolation area. A volumetric swelling of ε v = 2.766 ± 0.003 % is obtained.

The validation of the proposed method is performed by comparing the strain field estimated based on classical continuum mechanics (Eq. B2-B3).

Based on continuum mechanics (Eq. B2-B3), one obtains: ε rr = 0.947 % ; ε θθ = 0.920 % ; ε rθ = 0.003 % Therefore, one indeed observes that ε rr ≈ ε θθ , ε rθ ≈ 0 and ε v ≈ 3 × ε θθ .

Appendix C: Cycling effect and material compaction

This appendix and Figure 11 present the influence of three different loading cycles on both adsorption/desorption isotherms and volumetric swelling strain for an active carbon firstly saturated with pure CH 4 at T = 303.15 K ± 0.1 K and then saturated with pure CO 2 at T = 318.15 K ± 0.1 K. The process to produce the active carbon composes three phases: first the carbon is grinded, then it is activated and finally it is compacted to obtain a cylindrical sample. During the first cycle of gas adsorption, there is a competition between a grain compaction shrinkage and the adsorption-induced volumetric swelling and a large hysteresis is observed because of the material compaction. This compaction is mostly irreversible and after the first cycle, the second and the third cycles are superimposed. Note that a small hysteresis remains because a small part of the compaction


Christophe Simon

Jean-Marc Thiriet

Pavol Barger

Jean-François Aubry

Reliability and credibility evaluation of networked control systems

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

INTRODUCTION

Networked Control Systems (NCS) are an extension of classical distributed control systems. The main difference is the presence of a communication network. It could be considered as a particular component interacting with all the other components of the system. In this way, the communication network takes a central place in control systems. This strategic position is not without consequences for the safety point of view. Thus, meticulous studies have to be done with a set of appropriate criteria [START_REF] Juanole | Quality of service of communication networks and distributed automation: models and performances[END_REF]. Mainly, studies have been done on the communication network itself and are part of the performances evaluation of the communication. The dependability study is not really taken into account.

The study of the reliability of an NCS is so a particularly complex problem and need the use of appropriate evaluation methods. A solution to this problem thanks to Monte-Carlo simulations and colored Petri nets is proposed in [START_REF] Barger | Safety analysis and reliability estimation of a networked control system[END_REF].

The study of reliability is generally achieved under the frame of the probability theory. This formal and rigorous frame is mastered when we dispose of reliable failure probabilities for the various components. More, the reliability index is a probability value which does not provide any information on the sensitivity or credibility of the index. The possibility theory is then a formal frame which is more open because it takes into account the uncertainty on the initial probability precision and the result credibility. These points interest our study.

In this paper, we describe in the first section the NCS application process to which we have imple-mented the study. In a second section, we recall the frame of the study of this NCS thanks to colored Petri nets in order to get a probabilistic reliability index. In the third section, we propose the formal frame of the possibility theory and this frame is used for the study of the reliability of the NCS. The last section concerns our study of the NCS reliability with the possibility theory and we propose some analysis.

NETWORKED CONTROL SYSTEM

In order to show the interest of probabilistic and possibilistic methods to study the reliability of NCSs, we propose to focus on a simple control loop showing all the complexity of the study.

We define a small NCS which is designed to control the liquid level in a tank. In order to achieve this, the NCS is composed of (Fig. 1): -a sensor, -a controller, -an actuator, -a communication network.

The mission of the NCS is to maintain the required liquid level in the tank. For this reason an analogue liquid level value is provided by the sensor. Its measure is communicated via the network to the controller which determines the new control value and sends it to the actuator which represents a pump on the entry of the tank and which conditions the input flow of the tank. A small additional pertur-Reliability and credibility evaluation of networked control systems C. Simon, J.M. Thiriet, P. Barger ESSTIN,2,France, Lorraine, Nancy, France, e-mail: Jean-Francois.Aubry@ensem.inpl.fr ABSTRACT: The evaluation of system reliability is often based on classic probabilistic tools such as fault trees or reliability diagrams. But with few information or with low probability events, probability distributions are weakly defined and the probability theory is not well fitted. The idea is to use the possibility theory applied to classic tools of the reliability evaluation [START_REF] De Cooman | A possibilistic uncertainty model in classical reliability theory, Fuzzy Logic and Intelligent Technologies in Nuclear Science[END_REF][START_REF] De Cooman | On modelling possibilistic uncertainty in two-state reliability theory[END_REF] to obtain the system reliability with some information on the credibility of results. Similarly, the control of systems evolves to distributed architectures with the integration of communication systems [START_REF] Ligusova | Effect of Element Initialization in Synchronous Networked Control System to Control Quality[END_REF]). The proposed study concerns the evaluation of the reliability of a networked control system by a possibility theory approach with the evaluation of the results credibility. bation (a leak or evaporation) affects the controlled system.

J.F. Aubry

INPL -Polytechnic National Institute of

In the initial state the tank is empty and has to be filled. The required final liquid level is fixed at 500 height units. All the components are represented as discrete systems with Ts as the sampling period. All through this work Ts is considered constant and equal to 10 time units. The sensor, the controller and the actuator are time triggered. The communication network is an event triggered element. In order to combine these two different triggers, input and output buffers are placed at the junctions between the communication network and the other components.

Different failures chosen arbitrarily can occur on any of these components. No reparations of component failures are considered. A detailed description of each component of the NCS follows.

Sensor

In this example a periodic sensor is considered. An hypothesis of measuring the true value is done but does not affect the scope of the proposed method. Every Ts a new measure is obtained and prepared to be sent. Two different functionalities can be distinguished on the sensor: 1 measurement, 2 communication.

The sensor failure does not affect the communication capacity. The consequence is that the sensor is unable to send the measured value and sends instead a frame informing of the sensor failure. The probability (uniform distribution) of this failure is fixed as a constant equal to 1/100 Ts where Ts is the sampling period of the system. This measuring element failure is referred to as the Sensor_failure in the following.

Controller

The design of a controller for an NCS is a problem beyond the scope of this communication. A proposition can be found in [START_REF] Walsh | Scheduling of networked control systems[END_REF]. The controller implemented is a Smith predictor with a PI algorithm. The supposed pure delay is 2*Ts which corresponds to one Ts delay for transmission for each information from the sensor to the controller and from the controller to the actuator. All over this study the communication network respects these delays.

During the failure of the controller, the communication functionality is stopped. Thus no message is sent to the actuator. The corresponding failure rate is constant and placed arbitrarily as 1/100 Ts. The failure described is referred to as the Controller_failure in the following.

If the controller does not obtain a measure from the sensor, either because the later has failed or because the information is lost during the transmission, it continues to determine the control value using the last measure obtained.

Actuator

A discrete 1 st order system represents the actuator inspired by a pump. Two basic states are distinguished: idle and acting. The actuator is declared to be idle if its input (a value from the controller) and its output (actual pumping throughput) are equal. Otherwise, its state becomes acting.

Two different failures are considered with the actuator: a start failure and a functioning failure both of them with constant, uniform probability distributions. The first one is caused by the blocking of the system while initiating an action after the idle state of the actuator. The second one is a typical wear-out problem for which the occurrence probability is increased by the continuous action.

The start failure (referred to as Actuator_blocked) is an error which is not dependent on the time of run but rather on the number of solicitations. The failure probability is defined as one blocking over 100 start requests. Thus it is dependent on the chosen control algorithm. This failure is more important when a binary control law is applied (pump either on 100% or closed) than with a PI algorithm where starts and stops are less frequent.

The functioning failure (or Actuator_wearout) is represented with a stochastic failure appearance while the pump is functioning. A uniform distribution of the probability is considered and the failure rate is fixed as 1/50 Ts. This failure rate is also a function of the control algorithm, but its dependence varies from the Actuator_blocked failure function.

The co-existence of these two different kinds of failure rates is rather problematic in traditional safety evaluation methods but can be resolved easily in a CPN model.

Communication network

The communication network is the only event triggered component in the modeled example. The transport delay of a transmission of each communication frame is considered constant and is fixed as 1/10 Ts. In each macrocycle two communication frames are transmitted: one containing the measured value, sent from the sensor to the controller and the second one with the control value from the controller to the actuator. In case of need, this model can be modified to represent a random delay (such as a jitter). The medium access protocol is of little importance for this study, as the network bandwidth is guaranteed to be highly superior to the bandwidth needed.

The interest of a communication network in this example is in the errors appearing during the transmission. Two errors are integrated and studied in this case: communication Frame_loss, -Frame_alternation.

In this study it consists in erasing the data field in the frame.

The probabilities are constant and identical for both errors and fixed at 1 Frame_loss out of 20 frames transmitted and 1 Frame_alternation out of 20 frames. Thus statistically every 10 th frame transmitted is not received correctly.

All occurrence rates of previously defined failures are presented in table 1.

Evaluating failure rate

The study of the reliability of an NCS from the failures of its components is a complex problem, which can be considered as an hybrid system. Table 1 shows indeed some failure rates expressed as a function of time or as a function of events. Moreover, the determination of the global reliability of the system does not depend only on the time but also on the system state, the nature of the network failure, and the type of lost information. We want to add that the type of information is not taken into account in the present study.

For this study, it is not possible to use the traditional reliability tools such as functional diagrams or failure trees … We consider the failure of the system as its inability to accomplish its mission, i.e. to keep the level of the liquid close to the set plus or minus 10% as it is shown in figure 2. Two behaviors are considered: 1 the tank is not filled and the liquid height is not closed to the desired value, 2 the liquid level has approached the desired valued but leaves it. The study is essentially centered on the reliability analysis according to an information point of view and the mission. So, events like medium interferences or electrical spikes are not taken into account because it overcomes the information point of view of the study. Now the context is defined, the aim is to find the relationship between the elementary failures defined in table 1 and the global failure of the system. To reach this end, we have used a quantitative approach thanks to simulation [START_REF] Barger | Safety analysis and reliability estimation of a networked control system[END_REF]. Considering the system as a hybrid one, we have chosen to use colored Petri nets for a functional and malfunctioning model of the NCS. The Petri net-based model of the system is built from the models of the components with a modular architecture (producerconsumer type) because of the existence of the communication network. The Petri net models can integrate any network behaviors even for low level protocols. But the more complex is the behavior, the more complex the model is.

The functional and malfunctioning models of the subparts having a stochastic nature, the analysis of the complete system can be achieved only through Monte-Carlo simulations [START_REF] Labeau | Procedures of Monte Carlo transport simulation for applications in system engineering[END_REF]. It consists in repeating a single simulation a certain number of times. This number has to be rather high in order to be able to consider the results as statistically reliable and has to be closed to the curse of dimensionality expresses by [START_REF] Bellman | Adaptive Control Processes: A Guided Tour[END_REF]. So the number of simulation increases exponentially with the number of events and the time when they occurred. Moreover, as other reliability analysis based on models simulations, the complexity of the model is linked to the complexity of the system modeled. Then, for a reliable analysis, the number of simulation should tend to infinity!! As the simulation is the way we choose for reliability analysis and as a simulation is time consuming, our study of the reliability of the NCS is based on a finite number of simulations. In [START_REF] Barger | Safety analysis and reliability estimation of a networked control system[END_REF], we operate 6734 simulations where each event code was stored with its occurrence time that defines a scenario. Over all simulations, 1114 scenarios induce a global system failure. For this study, we increase the number of simulations to 10000 and obtain 1782 failed scenarios. We note that with 32% of more simulations, the malfunctioning states are more than 1% up as in the previous study. As a conclusion, the number of simulation is too small to obtain a reliable and credible rate of malfunctioning. So, analyzing scenarios in the normative framework of probability theory becomes difficult due to the lack of Monte-Carlo simulations. Then, we choose to investigate the normative framework of possibility theory. [START_REF] Zimmerman | An application-oriented view of modeling uncertainty[END_REF] classified the causes of uncertainty as: lack of information, abundance of information, measurement and subjective belief. In our case, simulations are missing and do not allow a reliable analysis of scenarios thanks to the probability theory; it is not possible to take into account the uncer-tainties of the event occurrences on the evaluation of their influence on the global failure.

Other normative theory can be used as interval theory [START_REF] Moore | Interval Analysis[END_REF], certainty theory [START_REF] Kanal | Uncertainty in artificial intelligence[END_REF], possibility theory [START_REF] Dubois | Possibility theory. An approach to computerized processing of uncertainty[END_REF], evidence theory (Shafer 1976) and fuzzy set theory [START_REF] Zimmerman | Fuzzy Set Theory-and its applications[END_REF].

We have chosen to investigate the possibility theory in this context of missing information. Let's recall some principles of this theory we want to exploit here. X is a variable which is not completely known, and Ω is the set of all the possible values for X. We can then define a distribution of possibilities on ω such as πx(ω):Ω→[0,1] that expresses the degree with which it is possible that the current value of a poorly known variable X is ω. If A is an event, in other words a subset of ω, the possibility measure that A be x is: Π(A)=sup ω∈A π X (ω). The dual measure of the possibility is the necessity measure defines by: N(A)=1-Π(¬A). It allows characterizing the uncertainty on the decision attached to A in specifying with what possibility degree the opposite proposal is possible. Possibility distributions are always normalized to 1 in order to always define a possible event from two opposite ones.

These differences are important compared to the probability theory where the definition of the probability of an event defines immediately the value of its contrary. However, a certain amount of constraints is associated to the measurement of possibility or necessity:

( ) ( ) ( ) ( ) 1 0 0 1 = Π ⇒ > = ⇒ < Π A A N A N A (1)
These constraints express the fact that one event which is not completely possible is not certain, and that an event which is at least a little necessary is completely possible. This duality allows associating credibility to the decision taken from the couple possibility -necessity.

Composition of measure

The operators for conjunction and disjunction get different properties for the probability theory and the possibility theory. In the possibility theory, we should respect the duality of possibility and necessity measures. This duality is part of the inference rules:

-Disjunction: Π(A∨B)=max(Π(A), Π(B)) is the possibility that at least one of the two hypotheses be correct is equal to the greatest possibility. In probability theory, the disjunction inference is defined as: p(A∨B)=p(A)+ p(B)-p(A∧(B). -Conjunction: Π(A∧B)≤min(Π(A),Π(B)). Two events are possible but their intersection is less possible than the lowest possibility.

N(A∧B)=min(N(A),N(B)). The certainty that the two hypotheses be correct is equal to the certainty of the less certain hypothesis.

NCS RELIABILITY ANALYSIS WITH POSSIBILITY THEORY

There is not a sufficient amount of Monte-Carlo simulations to allow us having a frequency view of the reliability study of the NCS. Moreover, probability theory does not allow us to define if a proposition on the system state according to elementary events is credible.

Analysis of stories: events/time

During Monte-Carlo simulations of the NCS, elementary events that define scenarios are time recorded. Our first analysis of these scenarios has consisted in searching a relation between event type, event occurrence time and the final state of the system {Fail, Work}.

All events with their code in ordinates as defined in table 1 and their occurrences time in abscissa are shown in figure 4. The system final state is represented by a '*' for the Work state or by a 'o' for Fail state. Each event is coded as previously defined in order to show if it contributes or not to the Fail state of the system.

The simulation time does not exceed 900uT and 1 st order system reach the steady state after 120uT where uT is the simulation quantum time. A first analysis of figure 4 is that events 9 and 10 seem to conduct to a general system failure when they occur after 500uT. The event 4 occurrence did not conduct to a malfunctioning when it occurs after 300uT. Moreover, we note that events 7 and 8 are the most met events and their occurrences include working scenarios as well as failed scenarios with similar occurrence rate.

From figure 4, it is difficult to define a credible calculus probability structure. But, one can use the possibility theory by defining two possibility distributions for each event. These distributions explain how much it is possible to an event E i to induce or not a general failure of the NCS.

Without introducing experts' knowledge in the decision process on the reliability analysis, we have proposed a couple of possibility distributions π Fail (E i ,t), π Work (E i ,t) for each event as shown in figure 5. Each distribution has been heuristically proposed from event occurrences and by taking account the final state of the NCS. Moreover, if no data permits a precise definition of the distribution, no risk is made and we consider that all can occur {Fail or Work}.

From proposed possibility distributions, one can compute the possibility of a general failure of the NCS S or not. For each scenario, one should infer the possibility measure of event E i at its occurred time t as defined in equation 2.

( ) ( ) ( ) ( ) t E S t E S i Work story E Work i Fail story E Fail i i , , π π ∈ ∈ ∧ = Π ∧ = Π
(2)

The two propositions of equation 2 define the degree of possibility that the NCS is in failure or not from the conjunction of possibility measures of events.

As defined in the previous paragraph, the possibility theory is a more open frame than the probability theory and the calculus of necessity measures by equation 3 allows us to evaluate the credibility of propositions given by equation 2.

( ) ( ) ( ) ( ) t E S N t E S N i Fail story E Work i Work story E Fail i i , 1 , 1 π π ∈ ∈ ∧ - = ∧ - = (3)
The calculus of necessity measures is dual to possibility measures and allows computing the degree of possibility of the opposite proposition in this context of binary states. Then, the necessity measure defines how a proposition is credible by taking into account how its opposite is possible.

If the proposition Π Work (S)≈1 which explains it is very possible that the NCS is functioning and if Ν Work (S)≈0 then the proposition Π Work (S)≈1 is not credible.

Let's taking an example with the following scenario:

{(E 1 ,50uT), (E 7 ,200uT), (E 8 ,300uT), (E 7 ,400uT), (E 6 ,600uT), (E 10 ,700uT)}. If we compute the possibility and necessity measures from the distributions shown in figure 5 This example shows how it is important to use both the possibility and necessity. As one can see according to distributions of figure 5, it is fully possible that the system goes to the Work state but it is not credible because the necessity measure is 0.

This analysis of simulation results in figure 4 remains too general. One can note that each event can induce a general failure and that the possibility theory gives complementary information on the credibility of results. But, the chosen representation of events doesn't show the elementary contribution of each event to the general failure. Moreover, the importance of the contribution depending on the occurrence order of events in a story is hidden. The im-portance of an elementary is context dependant and it induces a more complex analysis.

Analysis of the scenarios: scheduled events vs. time

To make a better analysis and begin to answer to the previous note, we choose to represent events according to their occurrence time and their order positions in a story. The goal of this new representation is to show that events positions in a scenario according to the occurrence time have a great influence on the general failure. Moreover, one can observe in figure 6 that the event code '1' (Sensor_low_failure) doesn't induce a general system failure if it is the first event and if it occurs after the beginning of the story. When it occurs in the second position (see Fig. 7), the mission failure is possible if its occurrence time is less than 50uT. The same analysis can be done for event code '2' (Sensor_high_failure). One can note that this analysis is more precise than the one of the previous paragraph. It is clear that the type of events, the events order and the occurrences times play an important role in the general failure. Computing the system failure possibility Π Fail (S) is conditioned by the schedule of events. So, we should define possibility distributions with the following procedure but by taking into account the event order in the story. Figures 8 and9 show the possibility distributions for the two first events. where i is the event code and j is the index of the event in the story.

CONCLUSION

In this paper, we have investigated a NCS reliability analysis by the possibility theory with Monte-Carlo simulations and an information point of view. By using Monte-Carlo simulations we eliminate the formal definition of a structure function which is difficult to obtain with this kind of NCS. We have analyzed simulations story by the possibility theory which find here a good application because there is not a sufficient amount of simulations for a probabilistic point of view.

The proposed analysis remains imperfect. Stories are made of a large number of events and it is impossible to condition possibilities when this number tends to infinity. Moreover, some sequences can be identified in many stories and our analysis does not show neither them nor their relation to the possibility of failure. To take into account this information, we should analyze stories with the new equation 6 where a possibility distribution of an event is conditioned by the previous event as in Bayesian networks [START_REF] Weber | Reliability modelling with dynamic Bayesian networks[END_REF] For this kind of analysis, we should take a more comfortable tool like possibilistic graphs [START_REF] Borgelt | Learning Possibilistic Graphical Models from Data[END_REF] where conditional possibilities are well manipulated. Moreover, we are interested in cycles of events that seem to be classical in our simulations.

We should also integrate in our analysis the system dynamic, its robustness and its state. In this work the system dynamic is characterized by time because all the simulations are step responses of a 1 st order (low time value → high dynamic; large time value → low dynamic).

Finally, we should integrate in our analysis the nature of the corrupted information because the kind of corruption plays a role in maintaining or not the NCS in function.

Figure 1 .

 1 Figure 1. NCS application process

Figure

  Figure 2. 4 simulation examples

Figure 3 .

 3 Figure 3. Sensor part of Petri net simulation

Figure

  Figure 6. First events

Figure 8 .

 8 Figure 8. First events possibility distributions

Figure 4 .

 4 Figure 4. Unsequenced event representation

Table 1 .

 1 Failure and errors review

	Event name	Event code	Failure probability
	Sensor_low_failure	1	1/100 Ts
	Sensor_high_failure	2	1/100 Ts
	Sensor_analog_failure 3	1/100 Ts
	Controller_failure	4	1/100 Ts
	Actuator_blocked	5	1/100 requests
	Actuator_wearout	6	1/50 Ts
	Frame_alternation	7	1/20 frames
	Frame_loss	8	1/20 frames
	Alarm_activated	9
	Alarm_applied	10

  or Markov chains.

	( ) ( ) S Work S Fail	E E	story story	( Work E ( E i Fail	E E i i	i	t	) t	)