Hai Hoang

Khac Hieu Ho

Anne Battani

Magali Pujol

Guillaume Galliero
email: guillaume.galliero@univ-pau.fr

On Elemental and Isotopic Fractionation of Noble Gases in Geological Fluids by Molecular Diffusion

Keywords: Noble gas, isotopes, fractionation, diffusion, molecular simulation

Noble gases are ideal natural tracers to characterize the storage, migration and origin of fluids in geological environments. They are chemically and biologically inert and therefore are only partially separated by physical processes including the one related to the molecular diffusion phenomenon, so called kinetic fractionation. However, its precise quantification is difficult to achieve and its modeling is still open for debates. Thus, in this work, we have investigated the capability of simple and predictive models to estimate elemental and isotopic fractionation of noble gases in various geological fluids. To do so, molecular dynamics simulations on noble gases in water, gas and oil under sub-surface conditions were performed. These numerical results were found to be consistent with previous ones including experiments and molecular simulations when available, i.e. in water. Interestingly, it appeared that the widely used square-root law is able to provide a good prediction for elemental fractionation between noble gases in all solvents, except for helium. Such unexpected result from the theoretical point of view is explained by a peculiar relationship between the molecular mass and the molecular size of noble gas elements. Regarding the noble gas isotopic fractionation, the square-root law

Introduction

Noble gases have been widely used as natural tracers to characterize the storage, migration and origin of fluids in geological environments [START_REF] Ozima | Noble gas geochemistry[END_REF][START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF][START_REF] Burnard | The noble gases as geochemical tracers[END_REF]. Interestingly, they are chemically and biologically inert and therefore are only fractionated, i.e. partially separated, by physical processes [START_REF] Ozima | Noble gas geochemistry[END_REF].

Among the existing physical processes, molecular diffusion is expected to significantly induce noble gases kinetic fractionation in geo-fluids [START_REF] Marty | On the noble gas, isotopic fractionation in naturally occurring gases[END_REF][START_REF] Prinzhofer | Gas geochemistry of the Macuspana Basin (Mexico): thermogenic accumulations in sediments impregnated by bacterial gas[END_REF][START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF]. Such a kinetic fractionation, simply called fractionation in the following, emerges from the relative differences between molecular diffusion of noble gases when migrating into a fluid. However, its precise quantification and its modeling are still open for debates [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF][START_REF] Tyroller | Fractionation of Ne and Ar isotopes by molecular diffusion in water[END_REF]Tyroller et al., 2018;[START_REF] Seltzer | Precise determination of Ar, Kr and Xe isotopic fractionation due to diffusion and dissolution in fresh water[END_REF], wider than the noble gas community [START_REF] Watkins | Kinetic fractionation of non-traditional stable isotopes by diffusion and crystal growth reactions[END_REF][START_REF] Wanner | Isotope fractionation due to aqueous phase diffusion-What do diffusion models and experiments tell? -A review[END_REF]. The purpose of this article is to provide new insights on the quantification of noble gases fractionation by molecular diffusion in various geo-fluids, in particular in oil and gas.

One of the main reasons for the debate to still be ongoing is the difficulty to experimentally quantify the elemental and even more the isotopic fractionation of noble gases due to molecular diffusion in geo-fluids [START_REF] Jähne | Measurement of the diffusion coefficients of sparingly soluble gases in water[END_REF][START_REF] Tempest | Kinetic isotopic fractionation of argon and neon during air-water gas transfer[END_REF][START_REF] Tyroller | Fractionation of Ne and Ar isotopes by molecular diffusion in water[END_REF], Tyroller et al., 2018;[START_REF] Seltzer | Precise determination of Ar, Kr and Xe isotopic fractionation due to diffusion and dissolution in fresh water[END_REF]. Furthermore, these experiments are mostly limited to what occurs in water at standard conditions. As an alternative to experiments, molecular dynamics (MD) simulations [START_REF] Allen | Computer simulation of liquids[END_REF][START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF] proved to be an effective tool for diffusion coefficients estimation in fluids. In particular, this numerical tool was already used to successfully estimate the elemental and isotopic fractionation of noble gases by molecular diffusion in water [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]. With progresses in molecular modeling, MD simulations are now able to provide accurate results for the equilibrium and transport properties of various geo-fluids under subsurface conditions, including noble gases, natural gases and oil [START_REF] Leach | Molecular Modelling: Principles and Applications[END_REF][START_REF] Martin | Transferable potentials for phase equilibria. 1. United-atom description of n-alkanes[END_REF][START_REF] Siu | Optimization of the OPLS-AA force field for long hydrocarbons[END_REF][START_REF] Marrink | Perspective on the Martini model[END_REF][START_REF] Hoang | Simultaneous description of equilibrium, interfacial, and transport properties of fluids using a Mie chain coarse-grained force field[END_REF][START_REF] Hoang | Elemental and isotopic fractionation of noble gases in gas and oil under reservoir conditions: Impact of thermodiffusion[END_REF].

A significant number of theoretical and empirical models are already proposed in the literature to quantify mass diffusion coefficients in various environments (Chapman and Cowling, 1981;[START_REF] Poling | The Properties of Gases and Liquids, Fifth Edition[END_REF][START_REF] Cussler | Diffusion: Mass Transfer in Fluid Systems[END_REF]. However, quite surprisingly, the most common approach to quantify the fractionation by molecular diffusion, whether isotopic or elementary, is simply based on the evaluation of the inverse of the square root of the mass ratios of the two species considered whatever the geo-fluids and the pressure and temperature conditions [START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF][START_REF] Watkins | Kinetic fractionation of non-traditional stable isotopes by diffusion and crystal growth reactions[END_REF][START_REF] Wanner | Isotope fractionation due to aqueous phase diffusion-What do diffusion models and experiments tell? -A review[END_REF]. This simple relationship (called in the following the square root law) although theoretically correct for isotopic fractionation under low-density gas condition or in the limit of ultra-tight porous medium [START_REF] Marty | On the noble gas, isotopic fractionation in naturally occurring gases[END_REF][START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF] only, consistently describes noble gases elemental fractionation in aqueous fluids under standard conditions [START_REF] Jähne | Measurement of the diffusion coefficients of sparingly soluble gases in water[END_REF].

Nevertheless, recent experimental studies and numerical simulations showed that the square root law tends to strongly overestimate noble gas isotopes fractionation in aqueous fluids [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF][START_REF] Tempest | Kinetic isotopic fractionation of argon and neon during air-water gas transfer[END_REF][START_REF] Tyroller | Fractionation of Ne and Ar isotopes by molecular diffusion in water[END_REF]Tyroller et al., 2018;[START_REF] Seltzer | Precise determination of Ar, Kr and Xe isotopic fractionation due to diffusion and dissolution in fresh water[END_REF]. Similar findings have been noticed for other solute elements such as alkali and alkali earth metals, halides and CO2, in both liquid water and silicate melts [START_REF] Richter | Kinetic isotopic fractionation during diffusion of ionic species in water[END_REF][START_REF] Zeebe | On the molecular diffusion coefficients of dissolved CO2, HCO j and CO j and their dependence on isotopic mass[END_REF][START_REF] Bourg | Isotopic mass dependence of metal cation diffusion coefficients in liquid water[END_REF][START_REF] Watkins | Kinetic fractionation of non-traditional stable isotopes by diffusion and crystal growth reactions[END_REF][START_REF] Wanner | Isotope fractionation due to aqueous phase diffusion-What do diffusion models and experiments tell? -A review[END_REF].

Thus, to rationalize all the results on noble gases, we propose in this work to systematically study the limits and capabilities of the various fractionation models applied to them in various geo-fluids, including not only water as already looked after in the seminal work of [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF], but also gas (methane) and oil (n-hexane), using molecular simulations data. As it will be shown, this approach allowed us to explain why the square root law holds well for elemental fractionation and led us to propose a generic semi-empirical model able to provide a good estimate of noble gas isotopic fractionation by diffusion whatever the geo-fluids and the thermodynamic conditions considered.

The article is organized as follows: in section II, we provide details on existing fractionation models and on numerical details of molecular simulations. The results obtained from molecular simulations and theoretical models are presented and discussed in sections III and IV, respectively. Finally, the main results of this study are summarized in section V, which is the conclusion.

Theory and Simulation

Kinetic Fractionation by molecular diffusion

Molecular diffusion processes are expected to induce both isotopic and elemental kinetic fractionation of trace elements in geo-fluids [START_REF] Marty | On the noble gas, isotopic fractionation in naturally occurring gases[END_REF][START_REF] Watkins | Kinetic fractionation of non-traditional stable isotopes by diffusion and crystal growth reactions[END_REF][START_REF] Wanner | Isotope fractionation due to aqueous phase diffusion-What do diffusion models and experiments tell? -A review[END_REF]. Such a fractionation is simply due to a difference in the relative migration velocity, by molecular diffusion, of the components when advection is absent. It is a complex and coupled transient phenomenon occurring only when the system is out of equilibrium, i.e. when the compositions of the considered components are not constant over space [START_REF] Cussler | Diffusion: Mass Transfer in Fluid Systems[END_REF]. As noble gases are present under trace in geo-fluids, the problem is simpler than the general one covering all ranges of concentration. For a diluted component i, the migration velocity by molecular diffusion is simply characterized by its self-diffusion coefficient, Di. As a consequence, the relative fractionation between two diluted species i and j in a solvent, is fully defined by the fractionation coefficient α, which writes as:

⁄ = (1)
Thus, the problem of quantifying the noble gases fractionation by molecular diffusion reduces to the knowledge of the self-diffusion coefficients of noble gases in the considered geo-fluids under the studied pressure and temperature conditions. However, self-diffusion measurements are difficult to achieve [START_REF] Jähne | Measurement of the diffusion coefficients of sparingly soluble gases in water[END_REF][START_REF] Tempest | Kinetic isotopic fractionation of argon and neon during air-water gas transfer[END_REF][START_REF] Tyroller | Fractionation of Ne and Ar isotopes by molecular diffusion in water[END_REF]Tyroller et al., 2018;[START_REF] Seltzer | Precise determination of Ar, Kr and Xe isotopic fractionation due to diffusion and dissolution in fresh water[END_REF], even more in subsurface conditions (i.e. high pressures). Hence, an accurate model of noble gas self-diffusion in geological fluids under subsurface conditions would be more than welcome.

Molecular diffusion theory

Many theoretical and empirical models are proposed in the literature to estimate the molecular diffusion of gas molecules in a fluid (Chapman and Cowling, 1981;[START_REF] Poling | The Properties of Gases and Liquids, Fifth Edition[END_REF][START_REF] Cussler | Diffusion: Mass Transfer in Fluid Systems[END_REF][START_REF] Watkins | Kinetic fractionation of non-traditional stable isotopes by diffusion and crystal growth reactions[END_REF][START_REF] Wanner | Isotope fractionation due to aqueous phase diffusion-What do diffusion models and experiments tell? -A review[END_REF]. However, their ability to provide accurate results is often limited to a specific range of thermodynamic conditions which can be classified into four main regimes: free molecular regime, kinetic theory regime, intermediate density regime, and liquid regime. Figure 1 shows these diffusion regimes and the models which are, a priori, the most suitable for each regime. In the following sections, these models are briefly presented and their ability to predict the noble gas elemental and isotopic fractionation in geo-fluids is discussed.

Figure 1: The various regimes of molecular diffusion for different fluid conditions and the corresponding theoretical models.

Free-molecular regime

The "free-molecular" regime corresponds to the situation where the mean-free path of the molecules in the fluid phase is larger than the size of the confining solid structure (pore, tube, chamber…). Fluid molecules in this regime collide with the walls of the solid structure more frequently than with other fluid molecules. Such a regime is typically observed at near vacuum conditions when the confining structure is large, as in cosmic environment. Under sub-surface conditions it may occur only in extremely tight porous medium as the mean free path of gas under surface conditions is about tenth of nanometers, and this value strongly decreases with pressure. Under these very specific conditions, the molecular diffusion of a species i is proportional to the thermal velocity of the molecules, and therefore is inversely proportional to the square root of its molecular mass, Mi, i.e.

∝ / (2)
This relationship is often referred to as Graham's law. In such conditions, the fractionation coefficient between component j and i, is simply given by:

⁄ = / (3)
This square root law is widely used to predict both elemental and isotopic fractionation of noble gases due to molecular diffusion in geological environments [START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF], despite its very limited applicability in sub-surface environments from the theoretical point of view as pointed out previously.

Rather surprisingly, experiments and molecular simulations showed that noble gas elemental fractionation in liquid water is well described by equation (3) [START_REF] Jähne | Measurement of the diffusion coefficients of sparingly soluble gases in water[END_REF][START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]. However, recent studies, including experiments [START_REF] Tempest | Kinetic isotopic fractionation of argon and neon during air-water gas transfer[END_REF][START_REF] Tyroller | Fractionation of Ne and Ar isotopes by molecular diffusion in water[END_REF]Tyroller et al., 2018;[START_REF] Seltzer | Precise determination of Ar, Kr and Xe isotopic fractionation due to diffusion and dissolution in fresh water[END_REF] and simulations [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF], evidenced that this relation strongly overestimates noble gases isotope fractionation in water.

Gas regime

The "gas" regime corresponds to the cases where the average distance between the fluid molecules is much larger than the size of the molecules, while their mean free paths are smaller than the size of the confining solid structure. Such conditions typically correspond to gases at standard conditions. In this regime, the molecular diffusion is controlled by binary collisions between molecules, and is consistently described by the kinetic theory (Chapman and Cowling, 1981). In such conditions, the self-diffusion coefficient of a noble gas i depends on both its own size and its own mass, but also on those of the solvent, in the following way:

∝ × / (4)
where is the radius of the noble gas molecule, is the radius of the solvent molecule,

= × + ⁄
is the reduced mass where is the molecular mass of solvent.

In such a regime, the fractionation coefficient between two noble gas isotopes i and j is given by:

⁄ = " " / = / × # $ $ % / (5)
It is interesting to notice that such a relation tends to equation (2) when the molecular mass of the solvent is large compared to the one of the studied noble gas isotopes as the second term in the right-hand side of equation ( 5) tends to unity. The elemental fractionation coefficient between noble gases i and j can be deduced as well from equation ( 4) which leads to:

⁄ = " " / × $ $ (6)
Although the kinetic theory provides a simple and explicit expression to compute both elemental and isotopic fractionation of noble gases in gases, it is less applied than equation ( 2) in subsurface environment [START_REF] Marty | On the noble gas, isotopic fractionation in naturally occurring gases[END_REF][START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF].

Dense gas regime

The "dense" gas regime is defined as the regime for which the fluid molecules mean free path is of the order of the molecule size. Such conditions typically correspond to what occurs in natural gases under reservoir conditions, i.e. under high pressures. In this regime, the molecular diffusion of gas molecules is rather complex as the collision dynamics is strongly correlated. Many theories exist to estimate molecular diffusion in such a regime, like the Enskog theory and its variants [START_REF] Kincaid | Modified Enskog theory for fluid mixtures[END_REF][START_REF] Kincaid | A test of the modified Enskog theory for self-diffusion[END_REF] or like the more recent entropy scaling law [START_REF] Samanta | Universal scaling laws of diffusion in a binary fluid mixture[END_REF][START_REF] Samanta | New universal scaling laws of diffusion and Kolmogorov-Sinai entropy in simple liquids[END_REF]. Both lead to relations similar to equation ( 4) for the molecular diffusion with a correction term depending on the local molecular structure, e.g. given by the radial distribution function (Allen and Tildesley, 1987), around the species of interest.

Thus, the Enskog theory and the entropy scaling law both lead to an isotopic fractionation coefficient equal to the one obtained by the kinetic theory, i.e. described using equation ( 5), whereas the elemental fractionation coefficient is provided by a relation similar to equation ( 6) modulated by a structural correction:

⁄ = " " / × $ $ × & (7)
where C is a correction term which depends both on thermodynamics conditions and on species involved.

Liquid regime

The "liquid" regime corresponds to the situation in which the gas molecules are solubilized in a dense fluid, i.e. the molecules are in close contact and strongly interact. These conditions are typically those found in a liquid. In such a regime, the diffusion coefficient of a dilute component i can be modeled by the generalized Stokes-Einstein relation [START_REF] Bhattacharyya | Power law mass dependence of diffusion: a mode coupling theory analysis[END_REF][START_REF] Ali | Mode coupling theory of self and cross diffusivity in a binary fluid mixture: Application to Lennard-Jones systems[END_REF]:

= ' ( ) * ( 8
)
where + is the friction coefficient of the liquid acting on the gas molecules. As shown by the mode coupling theory, this friction coefficient can be expressed as a combination of various contributions due to the binary collision dynamics, the continuum collision dynamics and the collective relaxation of the liquid. It typically leads to an isotopic fractionation coefficient which can be approximated by:

⁄ = , (9)
in whichis a parameter varying from 0 to 0.2, as confirmed by molecular simulations results [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF][START_REF] Bourg | Isotopic mass dependence of metal cation diffusion coefficients in liquid water[END_REF][START_REF] Zeebe | On the molecular diffusion coefficients of dissolved CO2, HCO j and CO j and their dependence on isotopic mass[END_REF]. The case of -= 0 corresponds to the Brownian dynamics limit, i.e. no isotope fractionation.

Regarding elemental fractionation, the situation is even more complex. As a first approximation, one can rely on the widely used Wilke-Chang correlation for estimating the diffusion coefficient of noble gases [START_REF] Wilke | Correlation of diffusion coefficients in dilute solutions[END_REF]. By assuming so, the elemental fractionation coefficient becomes:

⁄ = / / 0.2 (10)
where Vi is the molar volume of species i at a boiling point under normal conditions. It is interesting to point out that / / 0.2 ≈ , because the molar volume is closely related to the cube of the molecular size.

It is worth pointing out that equation ( 9) predicts a small isotopic fractionation of noble gases in liquids which is partially confirmed by recent experiments and simulations in water [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]Tyroller et al., 2018;[START_REF] Seltzer | Precise determination of Ar, Kr and Xe isotopic fractionation due to diffusion and dissolution in fresh water[END_REF]. In addition, elemental fractionation predicted by equation ( 10) indicates that fractionation is induced by the difference between the molecular sizes of the noble gases and not by the difference in their masses.

Molecular dynamics Simulations

Molecular Modeling

Noble Gases in Gas and Oil

To model the noble gases (Helium, Neon, Argon, Krypton and Xenon) and the nalkanes (Methane and n-Hexane) studied in this work, we have used the Mie Chain Coarse-Grained (MCCG) force field developed by [START_REF] Hoang | Simultaneous description of equilibrium, interfacial, and transport properties of fluids using a Mie chain coarse-grained force field[END_REF]. This coarse-grained force field have exhibited in previous works an excellent capability to predict the equilibrium and transport properties of these compounds and their mixtures under sub-surface conditions [START_REF] Hoang | Simultaneous description of equilibrium, interfacial, and transport properties of fluids using a Mie chain coarse-grained force field[END_REF][START_REF] Galliero | Thermodiffusion in multicomponent n-alkane mixtures[END_REF][START_REF] Hoang | Elemental and isotopic fractionation of noble gases in gas and oil under reservoir conditions: Impact of thermodiffusion[END_REF][START_REF] Hamani | Thermophysical properties of simple molecular liquid mixtures: On the limitations of some force fields[END_REF].

A fluid molecule modeled by the MCCG consists on a simple homo-nuclear chain composed of N spheres which are freely and tangentially bonded, N being equal to one for all noble gases. Interactions between two non-bonded spheres i and j are described by using the Mie λ-6 potential:

4 5 67 8 9 = # : : 2 % : 2 2 6: 29 ⁄ ; 8 <# = > % : -# = > % 2 @ (11)
where 7 8 is the distance between two spheres, ; 8 is the potential well depth, A 8 is the collision diameter, and B 8 is the exponent characterizing the repulsive interactions between non-bonded spheres. Two adjacent particles in a chain are linked by a rigid bond with a bond length equal to A. Hence, the molecular representation with the MCCG force field requires four parameters C, B, ;, A to fully describe a compound. These parameters were directly and univocally determined from the critical temperature, one saturated liquid density, the acentric factor and one saturated liquid reduced viscosity thanks to an extended corresponding states principle [START_REF] Hoang | Simultaneous description of equilibrium, interfacial, and transport properties of fluids using a Mie chain coarse-grained force field[END_REF], except for Helium due to the quantum effect on the critical properties [START_REF] Gunn | Prediction of thermodynamic properties of dense gas mixtures containing one or more of the quantum gases[END_REF]. The MCCG parameters of the studied species are reported in Table S1 in the Supplementary Material (SM).

For the non-bonded interaction parameters between different species 6; 8 , A 8 , B 8 9, the classical Lorentz-Berthelot combining rules were used to determine ; 8 and A 8 thanks to:

A 8 = 6= = 9 (12)
; 8 = E ; ; 88

and, an arithmetic average was used to obtain B 8 [START_REF] Hoang | Simultaneous description of equilibrium, interfacial, and transport properties of fluids using a Mie chain coarse-grained force field[END_REF]:

B 8 = : : (14)

Noble Gases in Water

The MCCG model is too simple to appropriately describe water molecules, particularly hydrogen bonding. Thus, water molecules were modeled by using the extended simple point charge (SPC/E) model, as it provides reasonable equilibrium and transport properties under the studied temperature, T=298.15K [START_REF] Berendsen | The missing term in effective pair potentials[END_REF]. A water molecule represented by the SPC/E model is composed of two positively charged hydrogen atoms and one negatively charged oxygen atom, in which the O-H bond and the H-O-H angle are fixed at 0.1nm and at 109.47°, respectively. The noble gases used in combination with water were modeled by the simple Lennard-Jones (LJ) sphere model, as usually used in literature [START_REF] Hirschefelder | Molecular theory of gases and liquids[END_REF][START_REF] Panagiotopoulos | Exact calculations of fluid-phase equilibria by Monte Carlo simulation in a new statistical ensemble[END_REF][START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF][START_REF] Warr | Optimizing Noble Gas-water interactions via Monte Carlo simulations[END_REF], to be consistent with the SPC/E model. Details related to the interaction parameters of these classical force fields are provided in Table S2 in the SM. The non-bonded interaction parameters between unlike atoms ; 8 and A 8 were determined by using the classical Lorentz-Berthelot combining rules as given in equations ( 12) and ( 13). As already reported in [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF], these molecular force fields yield noble gases selfdiffusion coefficients in water in excellent agreement with experiments [START_REF] Jähne | Measurement of the diffusion coefficients of sparingly soluble gases in water[END_REF], see table S3 in the SM.

Numerical Details

Molecular dynamics simulations were performed in the isobaric-isothermal ensemble by using an in-house code [START_REF] Hoang | Simultaneous description of equilibrium, interfacial, and transport properties of fluids using a Mie chain coarse-grained force field[END_REF] for systems composed of noble gases in gas/oil and using the LAMMPS code [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] for the noble gases in water. Simulation systems consisted of cubic boxes composed of 1000 to 2000 molecules. Equations of motion were integrated using the velocity-Verlet algorithm [START_REF] Verlet | Computer" experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules[END_REF], with a time-step of 1fs for the noble gases in water and a value varying from 1fs to 3fs, depending on the mass of noble gas isotopes, for the noble gases in gas and oil. Classical periodic boundary conditions were applied in all directions. Temperature and pressure were kept constant using a Berendsen thermostat and barostat, respectively [START_REF] Berendsen | Molecular dynamics with coupling to an external bath[END_REF]. To constrain the bond length and the bending angle, the classical RATTLE algorithm was employed [START_REF] Andersen | Rattle: A "velocity" version of the shake algorithm for molecular dynamics calculations[END_REF].

Non-bonded interactions were truncated at a cut-off radius 7 F = 3.5A 8 . Classical long-range corrections were included for the Mie and LJ interaction [START_REF] Allen | Computer simulation of liquids[END_REF][START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF]. Long-range electrostatic interactions were computed by using the Particle-Particle and Particle-Mesh (PPPM) method [START_REF] Deserno | How to mesh up Ewald sums. II. An accurate error estimate for the particle-particle-particle-mesh algorithm[END_REF]. To produce the results, the MD simulations were performed in two successive steps. First, systems were equilibrated for 3×10 6 time steps. Then, samplings were carried out during at least 6×10 6 time steps.

Noble gases self-diffusion coefficients were computed by using the usual Einstein relation [START_REF] Allen | Computer simulation of liquids[END_REF]):

= lim L→N 2L 〈 P Q -P 0 〉 (15)
where, t is the time, 〈… 〉 is the average over molecules and T Q is the position of molecule i at the time t. As noble gases are usually highly diluted in geo-fluids (of the order of 10 -3 to 10 - 5 in mole fraction [START_REF] Ross | A literature survey of noble gas solubility measurements in formation brines to interpret tracer experiments[END_REF]), equation ( 15) may provide simulation results with a poor statistics and so to very large error bars. So, to improve the statistics, we used simulation systems containing about 1%, in mole fraction, of noble gases, i.e. there are from 10 to 20 noble gas atoms in the simulation box. It was previously shown that such a concentration is consistent with diluted conditions for the computation of self-diffusion coefficient [START_REF] Hoang | Elemental and isotopic fractionation of noble gases in gas and oil under reservoir conditions: Impact of thermodiffusion[END_REF]. To further improve the statistic, the self-diffusion coefficients provided here correspond to averages over 10 independent runs, and associated error bars correspond to standard deviations.

Results

Molecular simulations were performed on noble gases (He, Ne, Ar, Kr and Xe) in: i) water under ambient conditions at T=298 K and P=0.1 MPa, ii) methane (gas) and iii) nhexane (oil) under reservoir conditions at T=423 K and P=50 MPa and T=323 K and P=10

MPa, respectively. To compute the fractionation coefficient from equation ( 1), the selfdiffusion coefficients of the noble gases were computed using equation (15) as described in section 2.3.2.

Elemental Fractionation of Noble Gases

Elemental fractionation coefficients of the major noble gases isotopes ( 4 He, 20 Ne, 40 Ar, 84 Kr and 132 Xe) in water, gas and oil are displayed in Fig. 2 and data are provided in table S4 of the SM. 20 Ne is used as the reference [START_REF] Jähne | Measurement of the diffusion coefficients of sparingly soluble gases in water[END_REF][START_REF] Holz | Self-diffusion of neon in water by 21Ne NMR[END_REF] and the fractionation coefficient of each noble gas is displayed relatively to the corresponding inverse square root of its molecular mass, i.e. (MNG/M 20 Ne) -1/2 .

Figure 2a clearly shows a good agreement between the results obtained from our simulations and the available experiment and simulation data for the noble gases in water (Jähne et al., 1987, Bourg and[START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]. This confirms the consistency of our molecular simulation approach. Elemental fractionation coefficients of noble gases in the gas and the oil obtained from the MD simulations are shown on Figs. 2b and2c. Even if small differences appear, it is interesting to notice that the results are very similar in gas and in oil and closely resemble to those obtained in water. This is particularly striking as the gas, oil and water studied significantly differ from each other in terms of molecular characteristics, pressure and temperature conditions and diffusion regime. [START_REF] Jähne | Measurement of the diffusion coefficients of sparingly soluble gases in water[END_REF]. Red open squares correspond to our simulation data. Blue open diamonds correspond to the molecular simulation data of [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]. Lines correspond to equation ( 3).

Another interesting, but surprising, result is that it appears clearly that the square root law, equation ( 3), yields a good estimate of the fractionation coefficient for 40 Ar, 84 Kr and 132 Xe, relatively to 20 Ne, not only in water as previously pointed out [START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF][START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF], but also in gas and oil. However, such a relationship overestimates [ 4 He/ 20 Ne].

Isotopic Fractionation of Noble Gases

The accessible accuracy on diffusion coefficients using molecular dynamics simulations is of the order on 7%, which is not precise enough to capture well the expected isotopic fractionation in most cases. So, to reduce the statistical uncertainties, the selfdiffusion coefficients of noble gas isotopes were computed by using a fitting procedure based on a strategy similar to the one employed by [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]. More precisely, first the self-diffusion coefficients of the studied noble gas for a wide range of masses (much larger than that of real isotopes) were computed. This allows to enhance the dependence of self-diffusion on the mass. Second, these values were fitted, as shown in Fig. 3 and Figs. S1-S4, to deduce the self-diffusion coefficients of noble gas isotopes of interest.

Interestingly, the results of the simulations indicate that the dependence of selfdiffusion coefficient on the mass can be quantitatively described by a power law in water and oil, consistently with previous simulation data in water [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF], whereas it is not the case in gas, as illustrated in Fig. 3 for Argon and in Figs. S1-S4 in the Supplementary Material for the other noble gases. It is interesting to notice as well that our molecular simulation results in water are in good agreement with the ones provided by [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF], see Fig. 3 and table S5 in the SM. Blue open diamonds correspond to the simulation data of [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]. Red solid circles correspond to the major isotope of Argon. Black dashed lines correspond to eq. ( 3), blue dasheddotted-dotted lines to eq. ( 5) and green dashed-dotted line to eq. ( 17 Isotopic fractionations in water obtained from the simulation results are shown in Fig. 4a and 4b with data provided in Table S6 in the SM. Results indicate that the MD simulations are consistent with the experiments [START_REF] Tempest | Kinetic isotopic fractionation of argon and neon during air-water gas transfer[END_REF][START_REF] Tyroller | Fractionation of Ne and Ar isotopes by molecular diffusion in water[END_REF]Tyroller et al., 2018;[START_REF] Seltzer | Precise determination of Ar, Kr and Xe isotopic fractionation due to diffusion and dissolution in fresh water[END_REF], except for [ 3 He/ 4 He], see Fig. 3a, and for [ 36 Ar/ 40 Ar], see Fig. 3b, obtained by [START_REF] Jähne | Measurement of the diffusion coefficients of sparingly soluble gases in water[END_REF] and [START_REF] Tyroller | Fractionation of Ne and Ar isotopes by molecular diffusion in water[END_REF] respectively. Deviation between MD simulation and experiment for [ 3 He/ 4 He] was already noticed by [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]. It may be related to a quantum isotopic effect not taken into account in the MD simulations as pointed out by previous authors [START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF]de Magalhaes et al., 2017) or an experimental problem. Regarding the experiment data of The isotopic fractionations of noble gases in the gas and the oil obtained from the MD simulations are shown in Figs. 4a, 4c and4d, respectively, and the data are provided in Tables S7 andS8 in the SM. Interestingly, the trends and the order of magnitude of the fractionation coefficients in both gas and oil are similar to what obtained in water. More precisely, the noble gas isotope fractionation is more pronounced in gas, and even more in oil, than in water, see Figs 4b, 4c and 4d, except for He, see Fig. 4a.

As can be deduced from Fig. 3 and Figs. S1-S4, the square root law is not appropriate to capture the dependence of self-diffusion on noble gases masses. Hence, as shown in Fig. 4, the square root law strongly overestimates the isotopic fractionation of noble gases in water, gas and oil compared to MD simulations results. For the heaviest noble gases in gas and in water, this overestimation can reach about one order of magnitude which confirms the limited applicability of the square root law to quantify the noble gas isotope fractionation in geofluids [START_REF] Ballentine | Tracing fluid origin, transport and interaction in the crust[END_REF][START_REF] Bourg | Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologic applications[END_REF][START_REF] Watkins | Kinetic fractionation of non-traditional stable isotopes by diffusion and crystal growth reactions[END_REF][START_REF] Wanner | Isotope fractionation due to aqueous phase diffusion-What do diffusion models and experiments tell? -A review[END_REF].

Discussion

Elemental Fractionation of Noble Gases

It has been shown that the square root law, equation (3), is able to provide a quantitative estimate of the elemental fractionation of 40 Ar, 84 Kr and 132 Xe, relatively to 20 Ne, in all geo-fluids studied in this work (water, gas, oil). This result is rather surprising as such a relation should only hold in the free-molecular diffusion regime, see section 2.2, whereas the studied solvent corresponds to the dense gas and the liquid diffusion regimes. Indeed, from equations ( 7) and ( 10), one would expect that elemental noble gases fractionation depends on their molecular radius ratio, and, to a less extent (in particular in a liquid solvent), to their mass ratio. More precisely, from equations ( 7) and ( 10), it is expected that the fractionation coefficient should be linked to the square of the inverse of the molecular radius ratio, i.e. to The fact that the mass ratio alone is sufficient to capture well the elemental fractionation of noble gas whatever the geo-fluids probably indicates that noble gases molecular radius and mass are strongly correlated. Such a link is confirmed in Fig. 5, which displays (RNG/R 20 Ne) -2 against (MNG/M 20 Ne) -1/2 for the major noble gas isotopes (data are provided in Table S9 in the SI). This figure, which is very similar to Fig. 2, clearly shows that, for Ne, Ar, Kr and Xe, there is a nearly linear link between the square of the inverse of the molecular radius and the square root of the inverse of the molecular mass. For helium, (R 4 He /R 20 Ne) -2 is below this linear line, which is consistent with the simulation results on elemental fractionation, see Fig. 2. Thus, because of this simple relation between size and mass for noble gases, the influence of the molecular radius on noble gases elemental fractionation, although dominating in dense fluids, see equation ( 10), can be relatively well captured by considering the square root law on mass only. This explains why, whatever the fluid regime, the square root law is well suited to quantify noble gas elemental fractionation in geo-fluids, except for He.

Isotopic Fractionation of Noble Gases

In the following, to avoid an over-interpretation, the discussion is limited to Ne, Ar, Kr and Xe isotope fractionation as the molecular simulations result is not consistent with the experimental point for He in water, as indicated previously.

As can be deduced from Fig. 3 as well as from Figs. S2-S4 and clearly exhibited on Fig. 4, the simulations results, and most previous experiments, showed that the square root law is unable to correctly predict the isotopic fractionation of noble gases in water, as well as in gas and oil. Interestingly, if one uses the kinetic theory relation, equation ( 5), suitable a priori for both gas and dense gas regimes, the results are far better than those deduced from the square root law, see Figs 4 and6. More precisely, as shown on Figs. 6a and 6b and in tables S6 and S7 (Supplementary Material), the kinetic theory relation is able to provide a reasonable estimate of the noble gases isotopic fractionation in water and in gas, showing nevertheless an increasing overestimation with the decrease of the studied noble gas molecular mass. When considering the results in oil, see Fig. 6c and in table S8 (Supplementary Material), the kinetic theory relation tends to systematically overestimate the fractionation by a factor of two or three. From the previous results, it seems that the deviations between the kinetic theory and molecular simulations data on isotope fractionation are related to the mass ratio between the noble gas and the solvent. More precisely, the deviations tend to increase when the molecular mass ratio between the considered noble gas and the solvent decreases. Thus, this indicates that a relation not dependent on the solvent molecular mass, such as equation ( 9), would fail to describe the noble gas isotopic fractionation in different geo-fluids. A more suitable relation would be of the following type:

⁄ = " " , = , × # $ $ % , ( 16
)
where β is a numerical parameter that may vary from ½ to 0 (i.e. from the kinetic theory value, ½, down to the Brownian dynamics one, 0) depending on both the diffusion regime and the molecular mass ratio between the considered noble gas and the solvent. However, from our results, the parametrization of β appears to be a complex function of the fluid regime and the solvent to noble gas molecular mass ratio preventing the possibility of building a simple predictive model based on eq. ( 16). So, to provide a practical and simple predictive model, the following semi-empirical approach is proposed to estimate fractionation between two noble gas isotopes, Iso1 and Iso2, Iso1 being the lightest one, in geo-fluids.

From Figs. 6a and 6b, it appears that, if the molecular mass of Iso1 is larger than about 3 times that of the solvent, the kinetic theory (eq. 5) holds well. So, in such cases, equation ( 5) can directly be used. When UVW ≤ 3 , the idea is to define two isotopes of a "theoretical" noble gas element, named Iso1* and Iso2*, which presents the same molecular characteristics as a real noble gas (except He), as exhibited on Finally by choosing UVW * = 3 _ < a_b * , it is then possible to apply the kinetic theory (eq. 5) to determine Iso1 * Iso2 * ⁄ and so to deduce Iso1 Iso2 ⁄ . This leads to the following final relation:

Iso1 Iso2 ⁄ = cdef * cdeg * g f × cdeg * d cdef * d g f = cdef cdeg / × h i j× k $lf k $lg m / (17)
For the sake of clarity, Fig. 6 provides a flowchart of the proposed scheme to estimate the isotopic fractionation of noble gases in geo-fluids. Noble gas isotopic fractionations predicted from the proposed scheme in water, gas and oil are provided respectively in Figs.

-6a, 6b and 6c (data are provided in Tables S6,S7 and S8 in the SM). Results clearly show that this scheme is able to yield results in good agreement with the simulation data whatever the geo-fluids. Compared to the kinetic theory relation, and even more to the square root law, the results are noticeably improved while keeping a purely predictive and a relatively simple scheme. This is further emphasized by the values of the self-diffusion coefficients deduced from the different relations compared to the simulation results, see Fig. 3 and Figs S1-S4. In this work, we investigated the capability of theoretical models to predict the elemental and isotopic kinetic fractionation of noble gases due to molecular diffusion in geological fluids. First, a brief review of the available theoretical models depending on the fluid conditions was presented. It was pointed out that, among them, the square-root law, which is the most widely used approach, is theoretically valid only under the free molecular regime, i.e. far from sub-surface conditions. To test the applicability domains of the different fractionation models, we performed molecular dynamics simulations to compute the elemental and isotopic fractionations of noble gases in water, gas (methane) and oil (nhexane) under subsurface conditions. Such simulations, which were found to provide results in good agreement with experiments and previous simulations in water, were performed to enlarge the geological fluids type on which the models could be tested.

Conclusions

Interestingly, the molecular simulation results evidenced that the noble gases elemental fractionation can be quantitatively estimated by the square-root relation for the major noble gas ( 20 Ne, 40 Ar, 84 Kr and 132 Xe) not only in water, as previously pointed out in the literature from experiments and molecular simulations, but also in oil and in gas, except for 4 He. This unexpected applicability of the square root relation is explained by a peculiar relationship between the molecular mass and the molecular size for noble gases, thanks to the kinetic theory. More precisely, the reciprocal of the square root of the molecular mass linearly scales with the reciprocal of the square of the molecular size for ( 20 Ne, 40 Ar, 84 Kr and 132 Xe), whereas it is not the case for 4 He.

Regarding the noble gas isotopic fractionation, it is confirmed, as recently shown in the literature in water, that the square-root law is not able to predict good results in all studied solvents. On the opposite, it was found that the kinetic theory relation has good predictive capability in gas that deteriorates for denser solvents like oil and water.

Finally, to be able to quantify the noble gas isotopic fractionation in geological fluids, we have proposed a predictive scheme yielding results in good agreement with the simulation data and most of the experimental results. Compared to the square-root law and the kinetic theory relation, this scheme is noticeably better. As a summary of the previous findings, all fluid regime domains and corresponding relations to be applied are summarized in table 1.

(Figure 2 :

 2 Figure 2: Relationship between elemental fractionation [NG/ 20 Ne] and the inverse of the

Figure 3 :

 3 Figure 3: Dependence of the logarithm of self-diffusion coefficient of Argon isotopes relatively to the

  ). Red full lines correspond to the linear (Figs. a and c) and second order polynomial (Fig. b) fit on the simulation data obtained from this work.

  [START_REF] Tyroller | Fractionation of Ne and Ar isotopes by molecular diffusion in water[END_REF] for [ 36 Ar/ 40 Ar], it is worth noticing that this data is not consistent with the experimental results obtained by[START_REF] Tempest | Kinetic isotopic fractionation of argon and neon during air-water gas transfer[END_REF] and[START_REF] Seltzer | Precise determination of Ar, Kr and Xe isotopic fractionation due to diffusion and dissolution in fresh water[END_REF], the two latter being in good agreement with the MD simulations data, see Fig.4band tableS5in the SM.

Figure 4 :

 4 Figure 4: (a) Isotopic fractionation of helium in various geo-fluids. (b) to (d) Isotopic

  , to a less extent, to the square root of the inverse of the molecular mass ratio,

Figure 5 :

 5 Figure 5: Relationship between molecular size and molecular mass for major noble gas

Figure 6 :

 6 Figure 6: Isotopic fractionation of noble gases (without Helium) for classical noble gas

Fig. 4 .

 4 It is then possible to apply the square root law to estimate the "elemental" fractionation between the two isotopes of the noble gas of interest, Iso1 and Iso2, and those of the theoretical noble gas. This leads to Iso1

Figure 6 :

 6 Figure 6: Flowchart of the proposed scheme to compute the isotopic fractionation of noble

Table 1 :

 1 Applicability domain for 8 prediction of noble gases in geo-fluids.

Acknowledgements:

We gratefully acknowledge TOTAL S.A. for the post-doctoral grant awarded to one of us (HH) and for letting us publish these results. We also thank the Pau University and the MCIA for providing computational facilities. Prof. Khac Hieu Ho would like to acknowledge