M Azabou
email: mejda.azabou@mines-paristech.fr

Ahmed Rouabhi

Laura Blanco Martín

F Hadj-Hassen

M Karimi-Jafari

G Hévin

L Blanco-Martín

Hévin

Rock salt behavior: From laboratory experiments to pertinent long-term predictions

Keywords: rock salt, constitutive modeling, long-term creep, laboratory tests, time 10 extrapolation

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

Introduction

Solution mined caverns in salt formations have proven to be among the most efficient techniques for storage of different energy vectors since the 1960s [START_REF] Bays | Use of salt cavities for underground storage[END_REF][START_REF] Ozarslan | Large-scale hydrogen energy storage in salt caverns[END_REF][START_REF] Bai | An overview of hydrogen underground storage technology and prospects in China[END_REF][START_REF] Iordache | Hydrogen underground storage in Romania, potential directions of development, stakeholders and general aspects[END_REF][START_REF] Lord | Geologic storage of hydrogen: Scaling up to meet city transportation demands[END_REF][START_REF] Michalski | Hydrogen generation by electrolysis and storage in salt caverns: Potentials, economics and systems aspects with regard to the German energy transition[END_REF][START_REF] Tarkowski | Salt domes in Poland -potential sites for hydrogen storage in caverns[END_REF][START_REF] Firme | Salt caverns history and geomechanics towards future natural gas strategic storage in Brazil[END_REF][START_REF] Soubeyran | Thermodynamic analysis of carbon dioxide storage in salt caverns to improve the power-to-gas process[END_REF][START_REF] Lankof | Assessment of the potential for underground hydrogen storage in bedded salt formation[END_REF]. Today these facilities have an important role in the energy transition context. Their flexible and high deliverability makes them the most promising solution to respond to peak demand for intermittent clean energies. The feasibility studies and the design of these caverns should account for the exploitation phase and for the decades and sometimes centuries of the abandonment phase. The optimal design of these structures rests upon the use of a constitutive model with pertinent short and long-term predictions.

There cannot be a unique rheological model able to describe the behavior of a geological material such as rock salt, under all kinds and within all ranges of thermo-mechanical loadings.

What is sought instead, are approximate laws that can provide a close enough description to the real behavior of the rock within a given range of loading conditions. This explains the abundance of rock salt constitutive models in the literature. All the existing models, despite the differences in the phenomena they account for (dilatancy, healing, inverse creep, transient, steady state and tertiary creep, damage, extension, strain under tensile loading. . . ), agree on two facts: first, rock salt has a strongly time dependent behavior and second, laboratory tests are essential for their parameters calibration. The multitude of models proves their limitations and justifies the need for the research on rock salt rheology to continue.

Long-term behavior is one of the main challenges for the rheological modeling of rock salt.

In fact, it is technically possible to conduct tests under all desired loading conditions (different strain rates, temperatures, mechanical and thermal cycling. . . ). However, contrary to the thermo-mechanical variables, we do not have control over the time of laboratory experiments, which is negligible compared to the lifetime of the underground facilities (a few months or years in the best case scenario, compared to few decades or even centuries in the field). When performing a time-extrapolation of the existing models, three kinds of responses are obtained: (i) a transient response where the strain rate continuously decreases [START_REF] Tijani | Le sel gemme en tant que liquide visqueux[END_REF][START_REF] Labaune | Dilatancy criteria for salt cavern design: A comparison between stress-and strain-based approaches[END_REF][START_REF] Rouabhi | Phenomenological behavior of rock salt: On the influence of laboratory conditions on the dilatancy onset[END_REF], (ii) a purely steady-state response where the strain rate remains constant and the strain increases linearly over time [START_REF] Odqvist | Mathematical theory of creep and rupture[END_REF] and (iii) a transient response that stabilizes into a steady-state regime where the strain starts increasing linearly over time [START_REF] Munson | Salt constitutive modeling using mechanism maps[END_REF][START_REF] Devries | Salt damage criterion proof-of-concept research[END_REF][START_REF] Heusermann | Nonlinear finite-element analysis of solution mined storage caverns in rock salt using the Lubby2 constitutive model[END_REF][START_REF] Bérest | Transient behaviour of salt caverns. Interpretation of mechanical integrity tests[END_REF][START_REF] Hampel | The CDM constitutive model for the mechanical behavior of rock salt: Recent developments and extensions[END_REF][START_REF] Nazary Moghadam | Modeling time-dependent behavior of gas caverns in rock salt considering creep, dilatancy and failure[END_REF][START_REF] Wolters | Evaluation of rock salt barriers with respect to tightness: Influence of thermomechanical damage, fluid infiltration and sealing/healing[END_REF][START_REF] Günther | Steady-state creep of rock salt: Improved approaches for lab determination and modelling[END_REF][START_REF] Khaledi | Stability and serviceability of underground energy storage caverns in rock salt subjected to mechanical cyclic loading[END_REF][START_REF] Deng | A viscoelastic, viscoplastic, and viscodamage constitutive model of salt rock for underground energy storage cavern[END_REF][START_REF] Wu | Viscoelastic-plastic damage creep model for salt rock based on fractional derivative theory[END_REF]. since in terms of laboratory tests both models give satisfactory interpretations, the most conservative predictions should be selected. However in practice, the optimal design of salt caverns cannot be based on overly conservative approaches.

The purpose of this paper is to propose a model with pertinent predictions for long-term creep behavior. We propose an enrichment to the distorsion part of a model that has been detailed in [START_REF] Rouabhi | Phenomenological behavior of rock salt: On the influence of laboratory conditions on the dilatancy onset[END_REF]. The modification consists in adding to the transient component, a second transient mechanism inspired by the [START_REF] Munson | Salt constitutive modeling using mechanism maps[END_REF] transient creep. This allows to have a compromise between the steady state models and the transient ones in terms of the strain rate and the characteristic time. The added mechanism allows also to describe inverse creep [START_REF] Julien | Une modélisation constutive et numérique du comportement rhéologique du sel gemme[END_REF][START_REF] Heusermann | Nonlinear finite-element analysis of solution mined storage caverns in rock salt using the Lubby2 constitutive model[END_REF][START_REF] Karimi-Jafari | Comportement transitoire des cavités salines profondes[END_REF][START_REF] Günther | Steady-state creep of rock salt: Improved approaches for lab determination and modelling[END_REF][START_REF] Labaune | Comportement thermomécanique du sel gemme : Application au dimensionnement des cavités[END_REF][START_REF] Gharbi | Determining upper and lower bounds for steady state strain rate during a creep test on a salt sample[END_REF] and links it to hysteresis loops of the unloading-loading cycles of triaxial compression tests. The authors are aware that in-situ data that covers years of exploitation activity, is needed to fully validate the model and truly assess the pertinence of its time extrapolation. However, acquiring such data and using it for research work is difficult, since -even if they exist-these measurements are destined for exploitation purposes.

This paper is structured as follows. In the first section two existing models and the new proposed one are presented. In section 3, we compare these constitutive laws on the basis of five experimental campaigns conducted at Mines ParisTech on rock salts from different geographic locations. The comparison concerns the capability of the three models to fit long-term (creep) and short-term (conventional tiaxial compression) tests with the same parameter set, and the pertinence of their long-term predictions. In section 4 the new model is used to analyze a comprehensive test campaign and the inverse creep phenomenon is discussed in section 5.

Constitutive models

All these constitutive laws agree on the partition of the total strain rate tensor ε into elastic εel , thermal εth and viscoplastic parts εvp :

ε = εel + εth + εvp ; εel = 1 + ν E σ - ν E tr( σ)1 ; εth = α th Ṫ 1 (1)
where E, ν are the elastic constants, σ is the Cauchy stress tensor, 1 is the unit tensor, α th is the linear thermal expansion coefficient and T is the temperature. The viscoplastic strain tensor is generally given as a function of the thermodynamic state (σ, T ) and internal variables ξ with a rate that depends on σ, T and the internal variables:

εvp = F (σ, T, ξ), ξ = G (σ, T, ξ) (2)
it is on the definitions of ξ, F and G that those models differ. In this section we present the following models: Munson-Dawson, Lemaitre and the model developed in this work that we shall call RTL.

The Munson-Dawson constitutive model

This model has been detailed in [START_REF] Munson | Salt constitutive modeling using mechanism maps[END_REF] and it states the following εvp = F εss ; εss = εss J = εss (σ, T )J ;

J = σ /||σ || (3)
where σ is the deviatoric part of the Cauchy stress tensor σ = σ -1 3 tr(σ)1 , εss denotes the steady state strain rate tensor and F is defined as follows

F =      exp ∆(1 -ζ/ε * t ) 2 if ζ ≤ ε * t exp -δ(1 -ζ/ε * t ) 2 if ζ ≥ ε * t (4) ζ = (F -1) εss (5)
In this work we assume the following

εss (σ, T ) = A exp A M ( 1 T r - 1 T ) G n ; ε * t = K 0 e cT q m (6)
where G is a loading function chosen as G = q, with q the von mises equivalent stress, • are the Macaulay brackets, i.e., x = (x + |x|)/2, T r is the reference temperature, A, A M , n, c, K 0 , m, ∆ and δ are material constants. We note that this model accounts for transient creep (via the function F ) that saturates at ε * t and the steady state flow εss takes over.

The Lemaitre constitutive model

This model assumes that rock salt creep is a purely transient flow, the viscoplastic strain rate decreases asymptotically to zero. It states the following

εvp = 3/2 γvp J (7)
where γ vp is the viscoplastic distorsion defined as

γvp = γ vp (σ, T, γ vp ) = α exp(A L ( 1 T r - 1 T ) G K β/α γ 1-1 α vp (8)
α, β, A L and K and positive rheological constants and T r the reference temperature. Similarly to the previous model, G is a loading function G = G(σ, T, γ vp ) and is chosen as

G = q.

The RTL2020 constitutive model

Similar to the model detailed in [START_REF] Rouabhi | Phenomenological behavior of rock salt: On the influence of laboratory conditions on the dilatancy onset[END_REF], for the viscoplastic strain rate tensor, RTL assumes the following

εvp = 3 2 γvp N - 1 √ 3 ζvp I (9)
where ζvp = -tr(ε vp ) and γvp = 2/3||ε vp || are the rates of the volumetric strain and the viscoplastic distorsion, respectively, N is a unit tensor defining the deviatoric flow direction and I = 1/ √ 3. The general formalism of the viscoplastic distorsion is defined as γvp = γL + γR ; with

     γL = ψ L (σ, T, γ L ) γR = ψ R (σ, T, γ R ) (10
ψ L (σ, T, γ L ) = α exp A R ( 1 T r - 1 T ) G A 2 n 2 γ 1-1 α L (11)
where α, A R , A 2 and n 2 are material parameters, G is a loading function. The added distorsion mechanism γ R is defined as

ψ R (σ, T, γ R ) =              A(1 -γ R / γR ) n R(σ, T ) if γ R ≤ γR ; γR = γR (σ, T ) -B(γ R / γR -1) m R(σ, T ) if γ R ≥ γR (12)
here A, B, m and n are material parameters such that A > 0 ,B > 0 and m > 1 ,n > 1.

The model offers enough flexibility in the choice of the functions ψ L , γR and R. In this work we have made the following considerations:

• γL corresponds to the viscoplastic distorsion of Lemaitre model γvp ;

• γR is considered to be analogue to the saturation value ε * t in Munson-Dawson model, it is given by γR

= G A 1 n 1 (13)
where A 1 and n 1 are material parameters.

• As for the definition of the function R, it is considered to be the same as ψ L in order to link the two mechanisms and ensure that their contributions have the same order of magnitude

R(σ, T ) = exp A R ( 1 T r - 1 T ) G A 2 n 2 (14)
• We chose the loading function G such that G = q.

As for the volumetric viscoplastic strain, it remains unchanged from the definition given in RTL [START_REF] Rouabhi | Phenomenological behavior of rock salt: On the influence of laboratory conditions on the dilatancy onset[END_REF] as

ζvp = φ σ, T, γ vp γvp = z p/N z nz -γ vp p/M z mz + γ vp γvp ( 15
)
where z, N z , n z , M z and m z are material parameters and p is the mean pressure p = -tr (σ)/3 .

We note that the RTL model is equivalent to the Munson Dawson model when the following considerations are made:

• the parameter α from equation 11 is set to 1;

• the parameter A from equation 12 is taken as A = exp(∆);

• the parameter n from equation 12 is linked to the parameter ∆ via the following equation

n =      -0.0098∆ 3 + 0.2040∆ 2 + 0.5622∆ + 2.0252 when ∆ ≤ 6 1.9987∆ -1.4567 when ∆ > 6 (16)
• B = δ = 0 (equations 4 and 12).

The advantage of the mechanism γ R in this model is that it can be easily integrated compared to the exponential formulation used in Munson-Dawson. Under constant stress and temperature, the analytical solution for equation 11 can be written as

γ R (t) = γ R 0 + γR -γ R 0 ξ ( 17
)
where ξ is given by

ξ(t) = 1 - 1 1 + V (t -t 0 ) 1 k-1 (18) with V = (k -1)U A 1 n 1 A 2 n 2 q n 2 -n 1 (19) When γ R ≤ γR we have: k = n, U = A(1 -γ R 0 / γR ) n-1 and if γ R ≥ γR then k = m, U = B(γ R 0 / γR -1) m-1 .
In Figure 2 we illustrate the role of each of the parameters A, B, n, m, A 1 , n 1 , A 2 and n 2 . We start from the parameter set given in Table 1 and in each case one parameter is varied, the simulated scenario is one of a uniaxial creep test were the axial stress (q in this case) is held constant at 10 MPa during 15 days and then dropped to 5 MPa for another 15 days. In the plots we present the evolution of the distorsion γ R in time and the value of γR in each deviatoric stage ( recall that γR = q A 1 n 1 ).

A B n m A 1 n 1 A 2 n 2 0.5 0.05 5 5 0.3 3 0.5 4
Table 1: Initial parameter set used to study the role of the parameters of the distorsion mechanism γ R ( the unit system is such that strain is in µm/m, stress is in MPa and time in days).

Time (days)

A 1 =0.25

A 1 =0.35 γ R (%)
Time (days) stresses such that the mean pressure (Q + 2P )/3 is held constant at 10 MPa throughout the test.

n 1 =2.8 n 1 =3.2 γ R (%) Time (days) γR A 2 =0.3 A 2 =0.9 A 2 =1.5 γ R (%) Time (days) γR n 2 =2 n 2 =3 n 2 =5

Parameters fitting and time extrapolation

For each salt, the results are analyzed with the three models (Munson-Dawson, Lemaitre and RTL) and the best fitting obtained (with the same parameter set for each model) is provided. Afterwards, these parameters are used to perform a time extrapolation of the models on a virtual scenario of a uniaxial creep test of 10000 days.

In the three models we are studying, the temperature effect is given by the Arrhenius law, and all the creep tests we are analyzing in this section are carried out at constant temperature. For this reason, we have set the parameters A M = A L = A R = 1725 K, also the reference temperature T r was set to 289 K. In addition, since the stages of deviatoric stress in all the creep tests are ascending, the parameter B from eqautions 12 was set to 0 and therefore the parameter m was not adjusted. 2.

We can see that the three models give satisfactory interpretations to both the short-term and the long-term tests, they are all able to describe the behavior of rock salt at the laboratory scale for this specific type of tests (time and loading ranges). As for the long-term predictions, the transient creep phase described by Munson-Dawson model is practically inexistent compared to the steady state flow and it has the same time span of laboratory tests durations (a few days), the time extrapolation of this model is rather conservative since the steady state flow is dominant and the strain increases linearly over time. However, the Lemaitre model is overly optimistic as it assumes that the strain rate keeps on decreasing infinitely and the strain almost tends to stabilize. On the other hand, in all the studied cases RTL gives predictions that are between these two extremes, the strain increases in time but neither linearly nor with an almost-zero rate. Besides, in RTL time extrapolation we distinguish a characteristic time (for the mechanism γ R ) that is comparable to the duration of the prediction.

Analysis of a comprehensive testing campaign with RTL

In order to show the capabilities of RTL to fit simultaneously short and long-term tests, with different and relatively complicated loading paths, we use this constitutive model to analyze a comprehensive testing campaign performed on specimens from salt 5 at Mines ParisTech.

Laboratory tests

In addition to the triaxial compression and the creep tests shown in Figure 7, we conducted the following tests on specimens from salt 5:

• a typical triaxial compression test under a confining pressure P = 5 MPa and an constant axial strain rate of εax = 10 -5 s -1 , performed at room temperature;

• a constant mean stress triaxial compression test, where the confining and the axial pressures (respectively P and Q) vary with the respective rates of Ṗ = -1 MPa/d and Q = 2 MPa/d in order to maintain a constant mean stress of (Q + 2P )/3 = 17 MPa (at room temperature);

• a multistage creep test at room temperature, where the effect of the mean pressure is investigated: during some of the stages the axial and lateral stresses are changed in order to maintain a constant deviatoric stress and change the mean pressure;

• a multistage creep test similar to the last one, but performed at T = 70 °C.

This testing campaign is comprehensive as it is includes tests with different and non-classic loading paths, also all the creep tests last over 200 days.

Parameters fitting

Since some of the tests have different temperature stages, the parameters A R and α th were fitted. The elastic parameters E and ν were fitted using the unloading/reloading cycles of the triaxial compression tests. The remaining parameters were fitted in two independent steps: those related to the deviatoric part and those related to the volumetric part. The experimental measurements of the distorsion were used to fit the volumetric strain part; this ensures the independence of the two fittings. The fittings of the analyzed tests are given in Figure 8 and the corresponding single parameter set in Table 3. As can be seen, RTL allows a satisfactory matching to the experimental measurements and it describes correctly the deviatoric and the volumetric behavior. The quality of the fittings shown in Figure 8 proves the robustness of RTL as a rock salt constitutive law.

Inverse creep

When the deviatoric stress level decreases during a classic creep test, the specimen's length increases for a brief duration before it starts decreasing again. This phenomenon assumes that the viscoplastic strain rate can be negative following a loading drop (inverse creep). Observing this during classic creep tests is delicate due to its small magnitude and brief duration. However this phenomenon is exhibited during the unloading-loading cycles of classic triaxial compression tests, in fact we observed that during these tests the decrease of the deviatoric stress comes with an irreversible strain and the creation of a hysteresis loop.

We select the first triaxial compression test and the third creep test presented in Figure 8, we fit the two data sets with RTL and a single parameter set twice: the first fitting does not allow a change in the direction of the viscoplastic strain (i.e., the parameter B is set to 0) and the second fitting is performed with B = 0. The results are respectively shown in Figure 9 and 10 and the corresponding parameters are listed in Table 4.

Both fittings qualities are satisfactory, in Figure 9 the loading drop during the two analyzed tests is purely elastic as it can be seen in the zoomed plots. However, activating the inverse creep mechanism (B = 0) allows to describe the hysteresis loop observed during the shortterm test; it characterizes correctly the viscoplastic flow that comes with this unloading-loading cycle. Also, following the load drop in the creep test, the model describes a very small and hardly distinguishable inverse creep: the strain rate decreases and then starts increasing again. We would like to highlight the fact that in the experimental measurements inverse creep was not observed, which is almost always the case during creep tests. This phenomenon has a very small order of magnitude (as predicted by the model in Figure 10), yet this fitting proves that it is directly linked to the loops of the unloading-loading cycles of triaxial compression tests. In Figure 11 we detail the fitting of the creep test in Figure 10 and we explicit the contributions of the distorsion mechanisms of RTL. We can see that following the load drop γ R decreases and tends to the new γR but with a very small rate, which is why γ L (steadily increasing) prevails and this results in a very discreet inverse creep phenomenon.

Conclusion

In this paper we discussed the aptitude of rock salt constitutive models to provide pertinent short and long-term predictions, since they are crucial for the optimal design of salt caverns. Our discussion was based on the confrontation of existing models with five experimental campaigns conducted on salts from different geographic locations. We found that the studied models gave satisfactory interpretations to laboratory tests. However, their long-term predictions were either too conservative or very optimistic. A new model was developed and included in this confrontation. It also allowed correct interpretations for laboratory tests but unlike the existing models, it gave pertinent predictions for the long term.

This new model was also tested against a comprehensive testing campaign that comprises short-term, long-term and non-classic creep tests. It provided a valid fitting for the campaign with a single parameters set.

We note that an absolute validation of the model and the pertinence of its long-term predictions, requires confrontation with in-situ data over years of the cavern's lifetime. This can be done when a continuous monitoring of the underground facilities is performed and in-situ measurements are used to re-calibrate the model parameters and update the predictions.

Acknowledgments

This work has been performed in the framework of the projects CITEPH and STOPIL-H2, the authors thank the sponsors and partners of the projects. The authors would also like to acknowledge Michel Tijani from Mines ParisTech and Nicolas Gatelier from Geostock, for their constructive discussions.

Figure 1 :

 1 Figure 1: Time-extrapolation of three models with different creep responses on a uniaxial creep test.

Figure 1

 1 Figure 1 illustrates this point where we show a qualitative comparison of the three responses based on the simulation of a uniaxial creep test over 30 years. As shown, the transient model gives an overly optimistic prediction while the two other models are rather conservative. Bérest et al. (2008) studied the differences in the long-term behavior predicted by these types of models. The study confirmed that the transient models and the steady state ones predict significantly different behaviors on the long term, and suggested that,

  ) here γL describes a transient creep mechanism of Lemaitre's type, and γR is an added transient mechanism inspired by the Munson-Dawson model. The first transient mechanism is defined by

Figure 2 :

 2 Figure 2: Effect of the parameters of the distorsion mechanism γ R in RTL model, predictions of a multistage creep test, note that in the sub-figures describing the effects of parameters A 1 and n 1 , the value of γR corresponding to each parameter is plotted in the same color in dashed line.

  Figures 3 to 7 show the fittings of the three models on the tests from the five campaigns and the results of the time extrapolation. In each of these figures, the top graph represents the analysis of a triaxial compression test (at a constant confining pressure and a constant strain rate that are specified on the graph), the middle graph corresponds to a multistage creep test (the test temperature and conditions are indicated on the graph) and the bottom figure is a time extrapolation of the three models for a uniaxial creep test at T = 40°C. The fitting parameters are provided in Table

Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 34567 Figure 3: Fittings of tests on salt 1 and time extrapolation of the three fitted models.

Figure 8 :

 8 Figure 8: Fitting of three short-term tests (top) and three creep tests (bottom) with a single parameter set. Note that for the second creep test, an isothermal (room temperature), isotropic phase (at (Q + 2P )/3 = 12 MPa) was applied.

Figure 9 :Figure 10 :

 910 Figure 9: Fitting of a conventional triaxial test (left) and a multistage creep test (right) with a single parameter set without accounting for inverse creep.

Figure 11 :

 11 Figure 11: Fitting of a the multistage creep test shown in Figure 10 and detailing of the contributions of the distorsion mechanisms in RTL.

Table 2 :

 2 Parameters used to fit the experimental data in Figures 3 to 7. The unit system is such that strain is in µm/m, stress is in MPa and time in days.

	Parameters Salt 1	Salt 2	Salt 3	Salt 4	Salt 5
	Elasticity
	E	29518	27499	28567	32172	22126
	ν	0.32	0.32	0.30	0.31	0.2
	Munson-Dawson
	A	4.636 × 10 -8 2.713 × 10 -3 0.5882	0.5513	4.46 × 10 -4
	n	8.0975	3.7	1.	1.01	3.936
	m	1.387	2.3	2.8075	2.6922	2.7062
	K 0	159.5602	69.6769	31.2385	18.5017	5.8505
	∆	14.0464	9.3	11.1551	11.6182	15.3698
	Lemaitre
	α	0.1363	0.2288	0.1746	0.1801	0.1904
	β	2.4189	2.6356	2.0518	2.265	3.4854
	K	0.4938	0.4832	0.16	0.3034	1.4091
	RTL
	α	0.5072	0.3601	0.2601	0.3201	0.6709
	A 1	0.4896	0.061	0.0181	0.04107	0.2937
	n 1	2.5093	1.473	1.162	1.3711	2.2839
	A 2	0.1916	1.0945	0.3986	0.868	4.5478
	n 2	2.7967	9.2482	9.6768	9.1676	6.3023
	A	36.055	6.81×10 -3	0.01	2.94 × 10 -2	684
	n	11.5263	9.2644	13.5	12.11	8.8926

Table 3 :

 3 Parameters used to fit both short-term and creep tests on Landes salt ( the unit system is such that strain is in µm/m, stress is in MPa and time in days).

	Elasticity and Temperature effect
	E	ν	A R	α th
	22126	0.2	4457	3.27 × 10 -5
	Distorsion
	a	A 1	n 1	A 2	n 2	A	n	B	m
	0.5812	0.1854	2.1012	3.7009	6.7562	155.6582 11.989	0.01918 2.2195
	Dilatancy
	z	n z	N z	m z	M z
	0.4523	1.2644	0.0241	1.028	0.024

Table 4 :

 4 Parameters used for the fittings shown in Figures 9 to 11 ( the unit system is such that strain is in µm/m, stress is in MPa and time in days)

	Elasticity and Temperature effect
	E	ν	A R	α th
	22126	0.2	4457	3.27 × 10 -5
	Distorsion without accounting for inverse creep
	a	A 1	n 1	A 2	n 2	A	n	B	m
	0.5812	0.1854	2.1012	3.7009	6.7562	155.6582 12.11	0	-
	Distorsion accounting for inverse creep
	a	A 1	n 1	A 2	n 2	A	n	B	m
	0.3840	0.2136	2.0581	2.4199	9.9698	0.2223	12.11	3.72 × 10 -5 4.5795