A Cherubini
email: aurelien.cherubini@ifpen.fr

GeoFluids B Garcia

A Cerepi

A Revil

Vadose Zone Journal | Advancing Critical Zone Science Streaming Potential Coupling Coefficient and Transport Properties of Unsaturated Carbonate Rocks

We measured the streaming potential coupling coefficient of natural saturated and unsaturated carbonate rocks. Saturation was achieved with NaCl brines with salinities ranging from 2 ´ 10 -3 to 2.0 mol L -1 . The magnitude of the coupling coefficient increased with decreasing salinity, similarly to the trend observed for sandstones. The permeability had a low impact on the values of the streaming potential coupling coefficient at high and low salinity. The zeta potential was calculated at full saturation using a modified version of the Helmholtz-Smoluchowski equation that accounts for surface electrical conductivity. Under atmospheric conditions, the magnitude of the zeta potential decreased with the increase in salinity. We also explored the relationships between the streaming potential coupling coefficient and water saturation in three partially saturated limestones using a steady-state flow experiment. We found good agreement between the van Genuchten approach and experimental data, and fitted both the relative permeability and capillary pressure curves with the same value of the van Genuchten exponent m v . We validated the predictive water relative permeability model described by Revil in water-wet rocks when the second fluid phase is non-polar.

Self-potential monitoring is a passive geophysical method based on the measurement of natural electrical fields in the subsurface of the Earth and provides data that are sensitive to water flow [START_REF] Thony | Field characterization of the relationship between electrical potential gradients and soils water flux[END_REF]. Among the different processes involved in the variation of these electric fields, electrokinetic processes often tend to dominate in the vadose zone. They have already been studied in unsaturated clayey [START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF] or sandy [START_REF] Mboh | Coupled hydrogeophysical inversion of streaming potential signals for unsaturated soil hydraulic properties[END_REF] soils and dolomites [START_REF] Revil | Streaming potentials in two-phase flow conditions[END_REF]Revil et al., 2007), but there exist few experiments in unsaturated intact limestones [START_REF] Cerepi | Streaming potential coupling coefficient in unsaturated carbonate rocks[END_REF]. Streaming potential signals are due to the presence of an electrical double layer at the charged interface of the grains in porous media [START_REF] Stern | Theory of the electrical double layer[END_REF][START_REF] Overbeek | Electrochemistry of the electrical double layer[END_REF][START_REF] Dukhin | Electrokinetic phenomena[END_REF].

The water-calcite interface has been increasingly studied in the last decade due to its use in a large range of applications including CO 2 sequestration [START_REF] Kaszuba | Carbon dioxide reaction processes in a model brine aquifer at 200°C and 200 bars: Implications for geologic sequestration of carbon[END_REF][START_REF] Xu | Reactive geochemical transport simulation to study mineral trapping for CO 2 disposal in deep arenaceous formations[END_REF][START_REF] Lerouge | Comparative EPMA and m-XRF methods for mapping microscale distribution of iodine in biocarbonates of the Callovian-Oxfordian clayey formation at Bure, eastern part of the Paris Basin[END_REF] and oil extraction from carbonate reservoirs [START_REF] Zullig | Interaction of organic acids with carbonate mineral surfaces in seawater and related solutions: I. Fatty acid adsorption[END_REF]. Complexation models can be used in concert with electrical double layer theory to compute the electrochemical properties of the water-calcite interface [START_REF] Wolthers | The surface chemistry of divalent metal carbonate minerals: A critical assessment of surface charge and potential data using the charge distribution multi-site ion complexation model[END_REF][START_REF] Wolthers | Calcite surface structure and reactivity: Molecular dynamics simulations and macroscopic surface modelling of the calcite-water interface[END_REF]. Recently, [START_REF] Li | Influence of surface conductivity on the apparent zeta potential of calcite[END_REF] focused their work on the surface conductivity effects on the zeta potential of calcite. They showed that the classical Helmholtz-Smoluchowski equation underestimates the zeta potential of calcite at low salinities and that surface conductivity should be included in the expression of the streaming potential coupling coefficient. This coefficient relates the fluid and streaming potential gradients when the total current density is zero (e.g., [START_REF] Sill | Self-potential modeling from primary flows[END_REF]. In unsaturated porous media, Revil et al. (1999b) established a relationship between the texture of porous media and this coefficient, later exploited by different researchers [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF][START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF][START_REF] Cerepi | Streaming potential coupling coefficient in unsaturated carbonate rocks[END_REF].

Our goal in this study was to provide new experimental evidence for the dependence of the streaming potential coupling coefficient of carbonate rocks with the saturation, comparing our data with the predictive model developed by [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF] the Corey or the van Genuchten exponent, often used to describe multiphase flow in the vadose zone. The water relative permeability has been calculated and successfully compared with the prediction of the model given by [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF]. [START_REF] Mboh | Coupled hydrogeophysical inversion of streaming potential signals for unsaturated soil hydraulic properties[END_REF] showed that the capillary pressure curves, the relative permeability, and the relative streaming potential coefficient could be described in a consistent way for a silica sand.

A new core flooding system based on the study of [START_REF] Cerepi | Streaming potential coupling coefficient in unsaturated carbonate rocks[END_REF] allowed us to provide the first experiments to date that could measure simultaneously the relative permeability and the streaming potential coupling coefficient for unsaturated carbonates. We show that the relative permeability, the relative streaming potential coupling coefficient, and the capillary pressure can be described in a consistent way for carbonates using the van Genuchten approach.

Theory

Streaming Potential Coupling Coefficient [START_REF] Revil | Coupled hydromechanical and electromagnetic disturbances in unsaturated porous materials[END_REF] defined the generalized cross-coupled L(w) (of electrokinetic nature) Ohm and Darcy constitutive laws for the Darcy velocity w w (m), which is the filtration displacement of the fluid with respect to the solid, and the electrical current density J (A m -2 ) under unsaturated conditions:

( ) ( ) ( ) ( ) w w w w w w S L S k S p L S é ù s ê ú é ù é ù ê ú ê ú ê ú = ê ú ê ú ê ú -Ñ ê ú ë û ë û h ê ú ë û J E w [1] ( ) ( ) ( ) w V w w k S Q S L w = h [2]
where s(S w ) (S m -1 ) is the electrical conductivity, k(S w ) (m 2 ) is the permeability, V Q (C m -3 ) is the dynamic excess of charge dragged by the electrolyte (given by a volume average of the local current density given by the local charge density of the diffuse layer times the local fluid flow velocity), h w (Pa s) is the dynamic viscosity of the water phase, E (V m -1 ) is the electrical field, p (Pa) is the pore fluid pressure, and S w (dimensionless) is the water saturation. The streaming potential coupling coefficient is given by [START_REF] Linde | Streaming current generation in two-phase flow conditions[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF] ( )

w 0 C S p = ae ö ¶j ÷ ç ÷ =ç ÷ ç ÷ ç ¶ è ø J [3] ( ) ( ) ( ) w w w L S C S S =- s [4] ( ) ( ) ( ) ( ) w V w w w w k S Q S C S S =- h s [5]
where j is the electrostatic potential of electrokinetic nature.

According to [START_REF] Revil | Coupled hydromechanical and electromagnetic disturbances in unsaturated porous materials[END_REF], the permeability and effective charge density depend on saturation:

( ) ( ) w r w 0 k S k S k = [6] ( ) 0 v v w w Q Q S S = [7]
where 0 v Q and k 0 are the effective charge density and the permeability at saturation, respectively.

Saturation Dependence of the Streaming Potential Coupling Coefficient

The resistivity index I r is defined thanks to the classical second Archie's law [START_REF] Archie | The electrical resistivity log as an aid in determining some reservoir characteristics[END_REF], for high ionic strengths (³0.1 mol L -1 ), when grain surface conductivity can be neglected compared with bulk water conductivity, as

( ) w r w w n I S S F - s º = s [8]
where F is the formation factor and n is the second Archie's exponent. One of the most popular models to describe the influence of saturation on the relative permeability is the Brooks andCorey (1964, 1966) where S wi and S e are the irreducible water saturation and the effective water saturation, respectively, and N w and N g are the Corey exponents for the water and gas phases, respectively. Alternatively, the water relative permeability can be explained from the equation of van [START_REF] Van Genuchten | A closed form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF] with the fitting parameter m v:

( ) v v 2 1/ rw e e 1 1 m m k S S é ù = -- ê ú ê ú ë û [12]
The index m v can also be derived from the capillary pressure curve [START_REF] Van Genuchten | A closed form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF] as

( ) v v 1 1/ c e e 1 m m P P S - - = - [ 13
]
where P c and P e denote the capillary pressure and the capillary entry pressure of water in the rock, respectively. [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF] proposed a relationship from modeling (also tested by [START_REF] Cerepi | Streaming potential coupling coefficient in unsaturated carbonate rocks[END_REF] in an experimental approach) between the relative permeability and the saturation using the streaming potential coupling coefficient, Archie's exponent, and the water Corey exponent from [START_REF] Brooks | Hydraulic properties of porous media[END_REF] as

( ) ( ) 1 r w w r w n k S S C S + = [14]
with a relative streaming potential coupling coefficient (ratio of the coupling coefficient at saturation S w to the value of the coupling coefficient at saturation) given by ( ) ( )

w w w wi r w 1 sat wi w 1 1 N n C S S S C S C S S + ae ö - ÷ ç ÷ = = ç ÷ ç ÷ ç - è ø [15]
Alternatively, using the van Genuchten exponent, we have

( ) ( ) ( ) v v 2 1 1/ r w w e e 1 1 m n m C S S S S -+ é ù = -- ê ú ê ú ë û [16]

Zeta Potential

Under steady-state conditions, the zeta potential is calculated from the streaming potential coefficient using the well-known Helmoltz-Smoluchowski equation in the absence of surface conduction in the electrical double layer (e.g., [START_REF] Hunter | Zeta potential in colloid science[END_REF]:

w sat w w C e z = h s [17]
where e (F m -1 ) is the dielectric permittivity, z (V) is the zeta potential, and s w (S m -1 ) is the electrical conductivity of the brine saturating the rock. When the surface conductivity is non-negligible (e.g., Revil et al., 1999a;[START_REF] Guichet | Modification of streaming potential by precipitation of calcite in a sand-water system: Laboratory measurements in the pH range from 4 to 12[END_REF][START_REF] Alroudhan | Zeta potential of intact natural limestone: Impact of potential-determining ions Ca, Mg and SO 4[END_REF][START_REF] Li | Influence of surface conductivity on the apparent zeta potential of calcite[END_REF], the zeta potential is related to the streaming potential coupling coefficient using the formation factor, F, which is the ratio of the conductivity of the electrolyte to the conductivity of the saturated rock sample when surface conductivity is negligible, and the electrical conductivity of the saturated porous material, s(S w = 1):

( )

w sat w w 1 C S F e z = h s = [18]
At low salinity, the evolution of the zeta potential was predicted by theoretical models, based on the Nernst-Planck-Poison-Boltzmann transport equation (Revil et al., 1999a). [START_REF] Pride | Electrokinetic dissipation induced by seismic waves[END_REF] found a logarithmic relationship in silica materials that relates the zeta potential with brine salinity, and [START_REF] Jaafar | Measurement of streaming potential coupling coefficient in sandstones saturated with high salinity NaCl brine[END_REF] suggested that the zeta potential is constant at high salinity (>0.4 mol L -1 ) also in silica-based materials:

10 f f log , 0.4 mol L a b C C z = + < [19] 1 f , 0.4 mol L c C - z = ³ [20]
where a, b, and c are constant values and C f is the brine salinity (mol L -1 ).

6 Experimental Methodology

Apparatus

The experimental setup used to measure the streaming potential coupling coefficient, the electrical conductivity, and the permeability under unsaturated conditions is shown in Fig. 1. The experiments are performed on cylindrical limestone core samples, wrapped in a rubber sleeve and confined within a core holder with a confining pressure of 3 ´ 10 6 Pa. This pressure is regulated by a pump to avoid temperature variations of the room and was constant throughout the experiment. Experiments were controlled with X-ray imaging to verify that there was no brine flowing along the external face of the sample and the rubber sleeve. The stainless steel body of the core holder does not come into contact with the sample or fluids and is electrically connected to the Earth.

The electrical and relative permeability experimental measurements were performed using a steady-state flooding system, allowing us to inject brine and gas simultaneously into the sample (rocks are water wet). One pump was used to inject brine into the core at a constant flow rate, while a gas flow regulator (Bonkorst F-201-CV) regulated the flow of gas (only N 2 was used in this study to avoid any chemical reaction with carbonates and gas). The brine used for electrical measurements during the multiphase flow experiments was comprised of only 10 g L -1 (0.17 mol L -1 ) NaCl in deionized water and had a conductivity of 1.58 S m -1 . A separator allows separation of the brine and gas at the outlet of the sample by gravity. The gas goes into the atmosphere while the brine is pumped again by the pump. The brine saturation decreases, whereas the gas saturation increases until the irreducible water saturation. The brine/gas flow rate ratio decreases during the drainage phase. The pressure difference measured for each brine/ gas ratio permits measurement of the relative permeabilities using Darcy's law when steady state is reached and the water saturation in the sample no longer varies. For each saturation step, the electrical conductivity of the sample (at a frequency of 1 kHz) and the streaming potential coupling coefficient are measured. The stabilization process between brine and gas in the sample is extremely fast and controlled by a scanner. This process allows calculation of the irreducible water saturation with an accuracy of 1% and avoids measurement saturation errors due to the drying of the porous medium after the passage of the gas. At the same time, the capillary curves of the samples are investigated with a porous plate apparatus.

This setup gave us the possibility to measure the streaming potential coupling coefficient under saturated and unsaturated conditions thanks to two non-polarizable Hastelloy (superalloy nickel-based, highly resistant to corrosion) circular electrodes placed on the cross-section of the sample. These electrodes have strictly the same diameter as the samples. The non-polarizable electrodes present a spiral shape and allow injection of the gas at the center of the sample cross-section and the brine at two other positions in order not to create preferential paths. The gas migration and the saturation were controlled with a scanner in real time.

Core Samples

Nine carbonate core samples (Table 1) were investigated in this study at full brine saturation and different brine concentrations: two algal rhodolith packstones (ESTA1 and ESTA2) from Provence (southeastern France), three ooid grainstones (BRAU1, BRAU2, and BRAU3) from Meuse (eastern France), two lithoclast packstones (RFF1 and RFF2), one asteries grainstone (STE1) from Saint-Emilion (southwestern France), and one wackestone (LAV1) from Vienne (western France). Only three of them were investigated at partial brine saturation (BRAU2, ESTA2, and RFF2) during multiphase flow experiments at constant brine concentration (10 g L -1 NaCl).

All the core samples have a length of 80 mm and a crosssectional diameter of 39 mm. They were drilled parallel to the stratification and dried in an oven (60°C) before each experiment. The petrophysical properties of each core sample, including water porosity (j w , calculated from the mass difference between saturated and dry samples), scanner porosity (j scan , calculated from computerized tomography), absolute permeability (k), formation factor (F), Archie's cementation (m) and saturation (n) exponents, the Corey exponents for water and gas (N w and N g ), the van Genuchten exponent (m v ), irreducible water saturation (S wi ), and the entry pressure of water (P e ) are reported in the Table 1. The samples have been chosen based on their large range of permeability values (0.002-0.6 mm 2 or 2-600 mD) with the aim to determine the influence of permeability on the electrokinetic properties of carbonate rocks.

Measurements of the Streaming Potential Coupling Coefficient

The streaming potential is created by injections of different brine flow rates in the sample, inducing a pressure and a potential difference between the inlet and the outlet of the sample (Fig. 2). The experiment for each sample at full saturation begins with an electrical potential reference measurement across the sample for a given fluid flow rate. The main parameter is the stability of the voltage between the inlet and the outlet of the sample. Then, another brine flow rate is applied, which induces instantaneous pressure and voltage differences across the sample. Data are recorded at a sampling frequency of 1 Hz. Different flow rates are induced to have a better accuracy in the calculation of the streaming potential coupling coefficient. The same process is Table 1. Petrophysical characteristics of the limestone samples studied, including water porosity (j w ), scanner porosity (j scan ), absolute permeability (k 0 ), formation factor (F), Archie's saturation (n) exponent, Corey exponent for water (N w ), irreducible water saturation (S wi ), van Genuchten exponent (m v ), and entry pressure of water (P e ). performed for each brine concentration. During the multiphase flow experiments, the streaming potential coupling coefficient is also measured at partial brine saturations. As mentioned above, the drainage phase of the sample is achieved by changing the flow rate ratio between brine and gas. The gas flow rate remains unchanged, while that of the water decreases. The streaming potential coupling coefficient is recorded when the brine flow rate is modified. In the experiment, we did not impose backpressure (used to maintain constant pressure downstream) at the outlet of the sample. The outlet pressure remained equal to 100 kPa during all the experiments, under saturated and unsaturated conditions. To validate our protocol, we needed to test streaming potential measurements with different backpressures. The streaming potential coupling coefficient was measured on one sample (BRAU1) at a salinity of 10 g L -1 NaCl with three different backpressures of 100 kPa, 1 MPa, and 1.3 MPa (Fig. 3a). The value of the streaming potential coupling coefficient remained constant and was not affected by the backpressure. The same test was done for two other brine concentrations (30 and 40 g L -1 NaCl) with the same result (Fig. 3b). In conclusion, the streaming potential coupling coefficient is not affected by the backpressure whatever the brine NaCl concentration.

Measurements of Archie's Saturation Exponent

A two-electrode configuration was used to measure the conductivity of the brine-saturated and unsaturated sample. The electrical resistance of the system was measured at the frequency of 1 kHz between the two Hastelloy electrodes. The conductivity of the sample was calculated using ( )

w 2 L S R r s = p [21]
where R (W) is the sample resistance, L (m) is the length, and r (m) is the radius of the cylindrical core sample.

Calculation of the Zeta Potential

The zeta potential is calculated from the measured streaming potential coupling coefficient under fully saturated conditions using Eq. [18]. The formation factor (F) was calculated as the ratio of the resistivity of the sample filled with brine [s(S w = 1)] to the resistivity of the brine (s w ), at high ionic strength (0.17 mol L -1 ), when grain surface conductivity can be neglected compared with bulk conductivity: Fig. 2. Lavoux limestone (LAV1; permeability of 0.002 mm 2 ) saturated with a NaCl brine at 1.0 g L -1 : (a) electrical potential and pressure difference along the core sample at (t + 1) -t with a step measurement of 1 s (1 Hz)-the values are stabilized when they reach 0; an increase in the flow rate leads to an increase in the pressure difference and a decrease in the electrical potential difference between the inlet and the outlet of the core (note that all the peaks are in the opposite direction, which means that the streaming potential coupling coefficient has a negative value); and (b) an example of measured streaming potential vs. applied pressure difference for the same sample (LAV1) for a brine concentration of 1.0 g L -1 . ( )

w w 1 F S s = s = [22]
The brine relative permittivity and the brine viscosity were adjusted as functions of temperature and salinity (e.g., [START_REF] Saunders | Fluid flow monitoring in oil fields using downhole measurements of electrokinetic potential[END_REF]. Measurements were performed under atmospheric conditions, without gas partial pressure.

6

Results

Relative Permeability

In Fig. 4a to 4c, we fit the relative permeability data calculated with a steady-state apparatus as a function of the saturation using the Brooks and Corey and the van Genuchten approaches for three samples: BRAU2, RFF2, and ESTA2. Corey exponents are equals to 9, 5, and 8, respectively, whereas van Genuchten exponents are 0.54, 0.66, and 0.52, respectively. The Corey exponents are in the same range as in previous studies that investigated relative permeabilities in a brine-gas system [START_REF] Krevor | Relative permeability and trapping of CO 2 and water in sandstone rocks at reservoir conditions[END_REF]. The van Genuchten exponents are also consistent because they are smaller than unity [START_REF] Brooks | Hydraulic properties of porous media[END_REF].

Streaming Potential Coupling Coefficient

Effective Volumetric Charge Density Dependence

According to [START_REF] Jardani | Detection of preferential infiltration pathways in sinkholes using joint inversion of self-potential and EM-34 conductivity data[END_REF], recently discussed by [START_REF] Jougnot | Monitoring of saline tracer movement with vertically distributed self-potential measurements at the HOBE agricultural test site, Voulund, Denmark[END_REF] and validated by a theoretical model [START_REF] Guarracino | A physically based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media[END_REF], the volumetric charge density The samples studied have an absolute permeability in the range 0.002 to 0.6 mm 2 (Table 1) and streaming potential coupling coefficient values as reported in Table 2. Our values of volumetric charge densities for carbonates are slightly below the predictions of Eq. [23] (Fig. 5) and give an empirical law in the 0.002-to 0.6mm 2 permeability range:

( ) ( ) 0 10 v 10 0 log 11.37 0.94 l ˆog Q k =- - [24]
In Fig. 6, data for five samples are represented depending on the salinity of the electrolyte, and compared with the theoretical value calculated from the model of [START_REF] Guarracino | A physically based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media[END_REF]. The excess charge density was calculated using the empirical zeta potential law that we define in this study (see Eq. [26] below) and the tortuosity formula from [START_REF] Winsauer | Resistivity of brine-saturated sands in relation to pore geometry[END_REF]. This model has never been applied to carbonates, but our data are consistent with other values calculated on limestone samples [START_REF] Pengra | Determination of rock properties by low-frequency AC electrokinetics[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF].

Permeability Dependence

The streaming potential coupling coefficient was measured on different carbonate rock samples for three different NaCl concentrations (0.4, 1.0, and 10.0 g L -1 ). The goal was to determine which parameter between absolute permeability and brine concentration mainly controls the magnitude of the streaming potential coupling coefficient in carbonate rocks (Fig. 7). The streaming potential coupling coefficient seems independent of the permeability at saturation whatever the brine concentration (with a factor of 3 between low and high permeabilities). That said, its magnitude depends significantly on the brine concentration (factor of ?3000 between the measurements at 0.4 and 10 g L -1 NaCl). The brine concentration has therefore a much stronger influence on the magnitude of the streaming potential coupling coefficient than the permeability at saturation (Fig. 7).

Salinity Dependence

We establish now a relationship between the salinity of the NaCl electrolyte and the streaming potential coupling coefficient in the carbonate rocks. The streaming potential coupling coefficients were measured for the salinity range 0.002 to 2 mol L -1 . Nine samples were investigated for different brine concentrations (Fig. 8). The magnitude of the measured coupling coefficient decreased with increasing salinity. The trend between the magnitude of the streaming potential coupling coefficient and the salinity C f (mol L -1 ) is a power law (R 2 = 0.96):

9 0.862 f 1.41 1 0 C C - - =- ´ [25]
Vinogradov et al. ( 2010) also established an empirical relationship between the streaming potential coupling coefficient and brine salinity in sandstones and found a trend close to Eq. [24] (dashed line in Fig. 8).

Saturation Dependence

Measurements of the streaming potential coupling coefficient during the drainage phase of the samples by N 2 were made on three limestone cores: BRAU2, RFF2, and ESTA2. The relative streaming potential coupling coefficient is determined from the ratio between its value under unsaturated and saturated conditions (Fig. 4d-4i). The three samples exhibit a non-monotonic decrease of the relative streaming potential coupling coefficient with the decrease in the saturation. At the reference state (C r = 1), the streaming potential coupling coefficient is equal to -5.79, -3.97, and -4.47 nV Pa -1 for the RFF2, BRAU2, and ESTA2 samples, respectively. In each case, measurements were compared with the laws established by [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF], based on the Brooks and Corey and the van Genuchten approaches, which predict a decrease in the streaming potential when the saturation decreases also ). Models were fitted with the same Corey and van Genuchten exponents used above and predicted that the streaming potential coupling coefficient falls to zero at the irreducible water saturation. The same behavior was already observed by different researchers (e.g., [START_REF] Revil | Streaming potentials in two-phase flow conditions[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF][START_REF] Vinogradov | Multiphase streaming potential in sandstones saturated with gas/brine and oil/brine during drainage and imbibition[END_REF]. (d-f ) relative streaming potential coupling coefficient for the three samples-at the reference state (C r = 1), the streaming potential coupling coefficient is -3.97, -5.79, and -4.47 nV Pa -1 for BRAU2 (gray squares), RFF2 (black triangles), and ESTA2 (black diamonds) samples, respectively, and the NaCl concentration of the electrolyte is 10 g L -1 -these data are fitted (plain lines) with a model from Revil et al. (2007) (Eq. [15]) with the initial Corey exponent, while dashed lines represent Eq. [15] with variable N w ; (g-i) relative streaming potential coupling coefficient fitted (plain lines) with a model from Revil et al. (2007) (Eq. [16]) with the initial van Genuchten exponent; and (j-l) water relative permeability of the three samples (same markers as above), calculated with the Revil et al. (2007) model (Eq. [14]). Data are compared with the Brooks and Corey model (plain lines) and van Genuchten models (dashed lines) with a quite good correlation.

Capillary Pressure

Capillary pressure curves were fitted with the van Genuchten exponent, m v , (Fig. 9). We observe a good correlation between the model and capillary pressure data recorded from porous plate experiments. The entry pressure of water, P e , which is a characteristic constant of the medium, has a value of 12, 33, and 16 kPa for the ESTA2, RFF2, and BRAU2 samples, respectively. In spite of the different experiments (steady-state flooding and porous plate), the van Genuchten exponent used to fit the capillary pressure model and the relative permeability model (above) described by van Genuchten (1980) is the same. This shows the consistency of the measurements in steady-state and porous-plate experiments.

Zeta Potential

Figure 10 shows the zeta potential (calculated from the Eq.

[18]) as a function of brine salinity for the samples investigated in this study. The magnitude of the zeta potential decreases with increasing salinity under atmospheric conditions, as observed in other studies that focused only on silica materials (e.g., [START_REF] Vinogradov | Measurement of streaming potential coupling coefficient in sandstones saturated with natural and artificial brines at high salinity[END_REF]. [START_REF] Vinogradov | Measurement of streaming potential coupling coefficient in sandstones saturated with natural and artificial brines at high salinity[END_REF] found that the zeta potential reaches a constant value at salinity approximately above 0.4 mol L -1 . We don't have enough data to verify this point.

According to the studies of [START_REF] Jaafar | Measurement of streaming potential coupling coefficient in sandstones saturated with high salinity NaCl brine[END_REF] and Vinogradov et al. (2010) (Eq. [19]), the trend of the zeta potential in the carbonate samples investigated in this study is given by Fig. 5. Charge density at electrical conductivity S w = 1 vs. the absolute water permeability for eight carbonate core samples studied: BRAU1, BRAU2, ESTA1, ESTA2, LAV1, RFF1, RFF2, and STE1. The relationship in carbonate rocks is also represented (solid line) as well as data from the literature: clayrocks [START_REF] Revil | Characterization of transport properties of argillaceous sediments: Application to the Callovo-Oxfordian argillite[END_REF], glass beads [START_REF] Bolève | Streaming potentials of granular media: Influence of the Dukhin and Reynolds numbers[END_REF][START_REF] Pengra | Determination of rock properties by low-frequency AC electrokinetics[END_REF], and limestones [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF][START_REF] Pengra | Determination of rock properties by low-frequency AC electrokinetics[END_REF]. The relationship in silica materials from Jardani et al. ( 2007) is also shown (dashed line).

Fig. 6. Charge density at electrical conductivity S w = 1 vs. the absolute water permeability for five carbonate core samples. Data were calculated for different brine salinities (C f ) and compared with the model of [START_REF] Guarracino | A physically based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media[END_REF]. We used the zeta potential trend of this study (Eq. [26]) and the tortuosity formula from [START_REF] Winsauer | Resistivity of brine-saturated sands in relation to pore geometry[END_REF] in this model. VZJ | Advancing Critical Zone Science p. 9 of 12 10 f 9.13 log 6.97

C z = - [26]
where C f is the brine salinity (mol L -1 ). The logarithmic equation has a theoretical argument in sandstones (Revil et al., 1999a) and seems applicable to limestone under atmospheric conditions without CO 2 partial pressure.

6 Discussion

Streaming Potential Coupling Coefficient

Influence of Saturation

The relative streaming potential coefficient obtained from the data does not decrease monotonically with decreasing water saturation, as expected considering Eq. [5]. This effect is commonly accepted and has been shown in other studies [START_REF] Revil | Streaming potentials in two-phase flow conditions[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF][START_REF] Jackson | Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model[END_REF][START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF][START_REF] Cerepi | Streaming potential coupling coefficient in unsaturated carbonate rocks[END_REF] when water is the wetting phase and the second phase (gas) is non-polar, and may be due to counterbalancing effects at partial water saturation between the conductivity, the permeability, and the excess charge density. Moreover, the excess charge density at partial saturation (Fig. 11a) was calculated from Eq. [5] using relative streaming potential coupling coefficient data and compared with Eq. [7] with a quite good correlation. This can confirm the non-monotonic trend between the relative streaming potential coefficient and the saturation.

Due to the complexity of measuring the streaming potential coupling coefficient during a drainage experiment, only a few data are available for the three samples investigated. Models imply a direct relationship between the relative streaming potential coupling coefficient and the saturation. This also includes petrophysical parameters, as the Corey exponent (N w ) (Brooks and Corey approach,Eq. [15]), the van Genuchten exponent (m v ) (van Genuchten approach, Eq. [16]), or the second Archie exponent (n) and therefore takes into account wettability effects. As shown in Fig. 4d to 4i, the Corey and van Genuchten exponents have a huge influence on the models and imply the need to realize drainage experiments under rigorous conditions. The solid lines (Fig. 4d-4i) were fitted using Corey and van Genuchten exponents calculated from experimental relative permeability data and were used in the relative permeability model (see below). Although the Corey and van Genuchten exponents vary in magnitude at N w ± 2 and m v ± 0.2, respectively, the predicted and measured relative streaming potential coupling coefficients seem to follow the same trend (R 2 = 0.83 and R 2 = 0.79, respectively). Nevertheless, these models have some limits. When another nonaqueous or nonpolar phase (oil) is the wetting phase, the relative streaming potential coupling coefficient increases with decreasing water saturation [START_REF] Jackson | Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model[END_REF] and Eq. [15-16] cannot be applied.

Relative Permeability Model

The relative permeability model proposed by Revil et al. (2007) (Eq. [14]) was tested for the first time (to our knowledge) in this study with experimental data. It relies on the streaming potential measurements to predict water relative permeability, also using Archie's saturation exponent. Figure 4 shows the water relative permeability calculated from Eq. [14] (Fig. 4j-4l) and the water relative permeability curves fitted with experimental data, using both the Brooks and Corey (Eq. [9]) and the van Genuchten (Eq. [12]) models (Fig. 4a-4c). Measurements predicted from Eq.

[14] and the experimental data are close (Fig. 11b, R 2 = 0.95), which confirms that this model gives a quite good approximation of the relative permeability when water is the wetting phase and the second phase is nonpolar. [START_REF] Jaafar | Measurement of streaming potential coupling coefficient in sandstones saturated with high salinity NaCl brine[END_REF] estimated that the diffuse layer collapses to zero when the salinity is >0.4 mol L -1 in silica materials. Therefore, we only took into account zeta potential values <1 mol L -1 of magnitude because the rock mineralogy does not seem to influence the streaming potential coupling coefficient values. A logarithmic relationship can connect the zeta potential and the salinity (Eq.

Zeta Potential Dependence with Salinity

[26]) (up to approximately 1 mol L -1 ) (see [START_REF] Jaafar | Measurement of streaming potential coupling coefficient in sandstones saturated with high salinity NaCl brine[END_REF][START_REF] Vinogradov | Measurement of streaming potential coupling coefficient in sandstones saturated with natural and artificial brines at high salinity[END_REF], validating the model proposed by Revil et al. (1999a). The logarithmic trend seems applicable to limestone samples [START_REF] Chen | Zeta potential of limestone in a large range of salinity[END_REF] under atmospheric conditions without CO 2 partial pressure. We expect that this result would be different under reservoir conditions, with CO 2 partial pressure, due to chemical reactions and catalysts as pressure and temperature. The magnitude of the zeta potential at salinity <1 mol L -1 increases with decreasing salinity (Fig. 10), as observed in previous studies, but is lower than that observed in silica materials.

Excess Charge Density Dependence with Salinity

Guarracino and Jougnot (2018) argued that a higher salinity leads to a lower excess charge density in silica materials. They related 0 v Q from hydraulic (porosity, absolute permeability, tortuosity) and pore water (electrolyte concentration, Debye length, zeta potential) parameters. The only parameter affected by the rock mineralogy is the zeta potential, hence the interest in defining a specific law for the zeta potential in carbonates (Eq. [26]). Their model seems more robust than that of [START_REF] Jardani | Detection of preferential infiltration pathways in sinkholes using joint inversion of self-potential and EM-34 conductivity data[END_REF] because it takes into account the pore water chemistry. We can see the huge influence of the salinity on the saturated excess charge (Fig. 6), with a saturated excess charge approximately five times higher at a concentration of 1 mol m -3 than 1000 mol m -3 . The data are quite well correlated with the model of [START_REF] Guarracino | A physically based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media[END_REF], which allows us at first sight to validate this model for carbonates with a monovalent electrolyte.

Conclusion

A core-scale experimental investigation was undertaken to investigate the relationships among the capillary pressure, the relative permeability, and the streaming potential coupling coefficient in carbonate rocks. We used pure NaCl brines and N 2 to perform Fig. 9. Fit of the capillary pressure data (dashed lines) using the van Genuchten exponent m v (van Genuchten, 1980) and capillary entry pressure of water (P e ). Note that the exponent m v is the same as that used for the relative permeability model (0.52, 0.66, and 0.54 for the ESTA2, RFF2, and BRAU2 samples, respectively), showing the accuracy of the experimental measurements. measurements, using the steady-state flow method. The following conclusions were reached:

1. The magnitude of the streaming potential coupling coefficient decreases with increasing salinity as predicted by current models. The zeta potential in carbonate rocks can be related to salinity with a logarithmic law in the same way as for silica materials for salinity <1 mol L -1 under atmospheric conditions.

2. The volumetric charge density can be directly estimated from the quasi-static permeability at full saturation. Our values of volumetric charge density are in the same range as the empirical equation proposed by [START_REF] Jardani | Detection of preferential infiltration pathways in sinkholes using joint inversion of self-potential and EM-34 conductivity data[END_REF] on sandstones and are consistent with other values calculated on limestone samples [START_REF] Pengra | Determination of rock properties by low-frequency AC electrokinetics[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF]. The shift between the curves is partly attributed to the electrolyte salinity.

3. We were able to fit experimental data of the relative permeability, the relative streaming potential coupling coefficient, and the capillary pressure in a consistent way for carbonates using the van Genuchten approach and exactly the same value of the van Genuchten exponent.

4. The relative streaming potential coupling coefficient is strongly dependent on the saturation and decreases non-monotonically with it. The Brooks and Corey approach (R 2 = 0.83) provides a slightly better fit with experimental data than the van Genuchten approach (R 2 = 0.79).

5. The relative permeability model proposed by [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF], based on the relative streaming potential coupling coefficient, was tested under water-wet conditions and presented a quite good correlation with experimental measurements performed on three limestones (R 2 = 0.95).

  model. The Brooks and Corey relationships for the relative wetting and non-wetting fluid permeabilities are

Fig. 1 .

 1 Fig. 1. Experimental apparatus for measuring the relative permeability and the streaming potential coupling coefficient: (a) simplified sketch of the apparatus with fluid flow lines (solid lines) and electrical connections (dashed lines); and (b) sketch of the non-polarizable Hastelloy electrode used to measure the streaming potential coupling coefficient and the sample resistivity with a frequency equals to 1 kHz. Note that the electrode has the same diameter as the sample wrapped in the sleeve and allows injection of gas and brine simultaneously in the sample.

Fig. 3 .

 3 Fig. 3. Measurement of the streaming potential coupling coefficient for the BRAU1 sample: (a) measurements at 10 g L -1 NaCl and three different backpressures (100 kPa, 1 MPa, and 1.3 MPa); and (b) measurements of the streaming potential coupling coefficient at three brine concentrations (10, 30, and 40 g L -1 ) and different backpressures.

Fig. 4 .

 4 Fig. 4. Modeling of water drainage relative permeability on three carbonated rocks: (a-c) experimental data of water relative permeability for three limestone cores: BRAU2, RFF2, and ESTA2-data have been fitted with the Brooks and Corey model and the van Genuchten model; (d-f ) relative streaming potential coupling coefficient for the three samples-at the reference state (C r = 1), the streaming potential coupling coefficient is -3.97, -5.79, and -4.47 nV Pa -1 for BRAU2 (gray squares), RFF2 (black triangles), and ESTA2 (black diamonds) samples, respectively, and the NaCl concentration of the electrolyte is 10 g L -1 -these data are fitted (plain lines) with a model from Revil et al. (2007) (Eq. [15]) with the initial Corey exponent, while dashed lines represent Eq. [15] with variable N w ; (g-i) relative streaming potential coupling coefficient fitted (plain lines) with a model from Revil et al. (2007) (Eq. [16]) with the initial van Genuchten exponent; and (j-l) water relative permeability of the three samples (same markers as above), calculated with the Revil et al. (2007) model (Eq. [14]). Data are compared with the Brooks and Corey model (plain lines) and van Genuchten models (dashed lines) with a quite good correlation.

Fig. 7 .

 7 Fig. 7. Absolute value of the streaming potential coupling coefficient dependence on permeability. Measurements were performed with three different NaCl salinity rates: 0.4, 1.0, and 10.0 g L -1 .

Fig. 8 .

 8 Fig. 8. Absolute value of the streaming potential coupling coefficient (C) as a function of salinity (C f ) in the range 0.002 to 1.5 mol L -1 for nine sample of carbonate rocks: BRAU1, BRAU2, BRAU3, ESTA1, ESTA2, LAV1, RFF1, RFF2, and STE1. The fit of these data (Eq. [24]) (plain line, R 2 = 0.94) is quite the same as the fit found by Vinogradov et al. (2010) for silica materials (dashed line, R 2 = 0.92).

Fig. 10 .

 10 Fig. 10. Zeta potential vs. salinity, showing only our data (intact limestones, black circles), and the corresponding model (solid line) in the range 10 -3 -10° mol L -1 (R 2 = 0.69).

Fig. 11 .

 11 Fig. 11. (a) Excess charge density data (Q V ) at partial saturation (S w ) for three samples (ESTA2, BRAU2, and RFF2) compared with their respective models, and (b) predicted water relative permeability (k rw ) using Eq. [14] vs. experimental data (R 2 = 0.95).

, including Core Ideas • Streaming potential coupling coef- ficient is dependent on saturation in carbonate rocks. • Salinity influences the streaming potential coupling coefficient. • Water relative permeability can be estimated from streaming potential measurements. • Transport properties of carbonate rocks can be modeled using the van Genuchten approach.

Table 2 .

 2 Calculated streaming potential coupling coefficient (C sat ) at a given NaCl concentration (C f ).

	Sample	C f	C sat
		mol L -1	nV Pa -1
	BRAU1	0.24	-5.00
	BRAU2	0.17	-3.97
	ESTA1	0.17	-7.20
	ESTA2	0.17	-4.47
	LAV1	0.17	-4.95
	RFF1	0.17	-7.31
	RFF2	0.17	-5.79
	STE1	0.017	-51.7

Acknowledgments

This research has been supported by IFP Energies Nouvelles, Rueil-Malmaison, France. The contribution of A. Revil is supported by Université Savoie Mont-Blanc and Université Grenoble Alpes. Different experimental data and petrophysical analysis used in this study and shown in Fig. 1 to 11 are available to the public from the authors on request (aurelien.cherubini@ifpen.fr). We also thank the two referees for their very constructive comments and their time.


Pierre Bernhard
email: pierre.bernhard@inria.fr

Marc Deschamps
email: marc.deschamps@univ-fcomte.fr

Arrow's (im)possibility theorem

Arrow's (im)possibility theorem is one of the most famous and important contributions in economics. It concerns the difficulty to aggregate a set of individual preferences, given as rankings of a set of available alternatives, into a unique social preferences ranking via a social welfare function, or into a unique social choice. Arrow proves that in a specific framework, it is impossible to find a social welfare function which simultaneously satisfies four conditions: universal domain, weak Pareto principle, independence of irrelevant alternatives, and no dictator. Our notice presents this theorem, one of its proofs, and, we hope, invites the reader to discover social choice theory.

Introduction

Kenneth Arrow (1921Arrow ( -2017) ) was one of the greatest economists of the twentieth century. In 1972, with J. Hicks, he received the Sveriges Bank Prize in Economic Sciences in Memory of Alfred Nobel aged 51 and remains its youngest recipient.

He made fundamental contributions in the fields of general equilibrium, mathematical programming, risk and insurance economics, organizational theory, and social choice and welfare. By his own admission ( [Stotsky, 2014]) if he had not existed "it wouldn't have made that much difference" to general equilibrium theory, whereas in social choice theory his contribution is not questioned, a fact that made him say: "So that, I'm proud of". On the origins of his work on social choice, one can refer to [Kelly, 1987, Suppes, 2005], and [Arrow, 2014].

Before formally explaining and proving this theorem, we will in turn discuss the question it originated from, the framework and the method followed by Arrow, as well as the scope of this result.

As the English poet Donne pointed out in 1624: "No man is an island". Indeed, it is essential to note that in every human society life is lived in the company of others, and that we must constantly make decisions in common that will affect us all. Obviously, in the highly unlikely case where everyone agrees unanimously there is no problem. The question arises in case of disagreement between individuals about choices that will affect them all. Taking up the idea of Knight's tripartition [Knight, 1944], Arrow considers that human societies have developed three alternative solutions to the problems of collective choice: authority, custom, and consensus. Within the later category, Arrow distinguishes between voting (in the political field) and the market (in the economic field). The question that Arrow asks is: how is it possible to arrive at aggregate judgements at the level of society in a transparent and indisputable way? In other words, how can we guarantee the sovereignty and rationality of individuals to make a collective decision? One can reformulate the question more specifically as follows: from individual preferences P i dealing with different alternatives (i.e a ranking) how does one construct a function of social preference (i.e a ranking function) P = f (P 1 , P 2 , ..., P n ) which is faithful to these preferences and rational? This question concerns as much the choice of a program on television for a family as the determination of the objectives of a company from the preferences of its shareholders, or the choice of a common policy for a set of countries.

To answer this question, Arrow did not resort to a historical or comparative approach but rather retained an analytical approach. His idea was to adopt an axiomatic to simultaneously study all possible aggregation methods in a unified framework. His objective was thus to study all the social preference functions, also called, with Arrow himself, social welfare functions (when society only wants the best option -and not a ranking-it is a social choice function). Arrow started his reasoning with the ideas that individuals are rational, that their preferences are ordinal (i.e the intensity of intrapersonal comparisons is meaningless), not comparable, and that they are not modified by the aggregation process. Arrow then defines what he calls "reasonable conditions" for the construction of the social welfare function, while noting that these can naturally be disputed. The number of these conditions may vary according to the way in which Arrow's construction is exposed, they are usually presented as four: universal domain, weak Pareto principle, independence of irrelevant alternatives, and no dictator. We will naturally return hereafter to the meaning of each of these principles. It is based on them that Arrow provides the answer to the question he posed (and which we call since "Arrow's (im)possibility theorem"): if there are at least three alternatives and two individuals then there is no social welfare function that satisfies these four principles. This is a result of major importance because of its universal scope. It includes all possible and imaginable forms of social welfare function in this framework. The four principles that form Arrow's "reasonable conditions" appear undemanding and natural. The result indicates, in particular, that all the voting rules are imperfect if Arrow's framework and axioms are considered satisfactory. For example, the majority rule respects the four principles, but is not a social choice function since it can lead to incoherent, because cyclical, social choices as Condorcet had already pointed out in the eighteenth century. As a consequence, the impact of this theorem on economic analysis is considerable since it opens both a new field of analysis with the axiomatic approach of social choice, and renews the foundations of the theory of welfare. But beyond that, this result influences part of political philosophy as well as political science.

Arrow's theorem

There are a large number of different demonstrations of Arrow's theorem. One even gets the impression that finding a new one has become a recurrent exercise in the theory of social choice. Today, the two main routes taken are those based on the notion of "decisive coalition" following [Arrow, 1951], [Arrow, 1963], and [Sen, 2017], and that of "pivotal player" following [Barberá, 1980]. As [Sen, 2014] notes, what matters is not so much the speed with which this theorem is obtained but the ease with which, without thorough mathematical knowledge, it can be understood. In its initial version, Arrow's theorem only assumes basic knowledge in logic. The presentation we retain does the same and will distinguish the formal setup (which is composed of the data, the problem and the principles) and the proof of the theorem.

The formal setup

Data and problem

We are given a finite set of N ≥ 3 alternatives A = {a, b, c . . .} (denoted here by lower case letters of the beginning of the alphabet), and a finite set of n players or voters V = {1, 2 . . . , n} denoted here by (small) integers or lower case letters beyond h in the alphabet (i, j, . . .).

Each voter has a complete set of preferences among the alternatives, i.e. a total strict (non reflexive) order. Thus "a is preferred to b by player i" is denoted a i b. Player i's complete preferences order is called P i . In this short presentation, we restrict players to strict orders for simplicity. Thus, "a and b are indifferent to player i" (which would be denoted a ≈ i b or a i b and b i a) is not allowed. It is not difficult to extend the results to either strict or non strict player orders. It makes statements and proofs longer, but not more difficult. The set of all possible (strict) orders over A is denoted P s ⊂ P the set of total orders.

A profile is a set of preferences for each voter, i.e. an ordered set

Π = {P 1 , P 2 , . . . , P n } ∈ P n s = P .
The objective is to construct a social welfare function, SWF, (or social preferences function) f : P → P from profiles to preferences assigning a complete system of social preferences P Π = f (Π) ∈ P to every profile Π. The social preferences will be simply denoted ≥ if no ambiguity results, or with a subscript such as ≥ Π to denote the social preferences under the profile Π, for a given SWF. If a ≥ b and b ≥ a, we write a b. To mean that a ≥ b but b ≥ \ a, we write a > b.

Two remarks are in order:

1. While it is possible, although unnecessary, to restrict individual preferences to being strict, it is very desirable to allow for non-strict social preferences, as purely symmetric players' preferences, such as in Condorcet's paradox, should lead to a tie in the social preferences.

2. This way of posing the problem hides what is sometimes considered as a separate axiom called Universal domain, i.e. the fact that the domain of the SWF be all of P as opposed to some subset of it. Restricting this domain was at the root of a slight mistake in Arrow's original publications [Arrow, 1950, Arrow, 1951], but also a possible path to obviating the (im)possibility theorem (see [START_REF] Sen | Necessary and sufficient condition for rational choices under majority decision[END_REF]).

Desirable properties

We describe here a set of properties that a SWF might exhibit. While all of them seem desirable, the first two will always be part of the axioms, i.e. properties that we definitely assume the SWF to share.

1. Unanimity rule (Or "Pareto efficiency"): a pair of alternatives being ordered the same way by all players should be ordered accordingly by the social preferences:

∀a, b ∈ A , [∀i ∈ V , a i b] ⇒ a > b .
2. IIA (For Independence of Irrelevant Alternatives): the relative preference between two alternatives in the social preferences should only depend on the preferences between these two alternatives in the players' preferences. If the relative rankings of two alternatives a and b are the same in a profile Π as in a profile Π , then a and b must have the same social relative rankings in P Π and P Π , irrespective of the rankings of any other alternatives.

3. Non-dictatorship A player is a dictator if the social order reflects his own preferences, irrespective of other players' preferences. Non-dictatorship says that there should not be a dictator.

What is really wanted is the stronger assumption of anonymity: all voters should be equal, swapping the preferences among them should not change the resulting social preferences. But it turns out that the weaker requirement of non dictatorship suffices to prove the impossibility theorem.

Neutrality

The conclusions of the social choice should not depend on which alternatives are considered. If the preferences between a and b in Π are the same as the preferences between a and b in Π , then the social preference between a and b in P Π should be the same as that between a and b in P Π . Stated otherwise: all alternatives are equal While some authors have posited neutrality as an axiom, this has been attacked by [Samuelson, 1977] where an example is given where neutrality violates the common sense of ethics. Yet we will show that it is implied by the unanimity rule and IIA.

Theorem and proof

We aim to prove Arrow's (im)possibility theorem:

Theorem 1 ( [Arrow, 1951, Arrow, 1963]) The three axioms of Pareto efficiency, IIA, and non-dictatorship are inconsistent.

We shall proceed via the equivalent statement:

Proposition 1 If a social welfare function satisfies the axioms of Pareto efficiency and IIA, it necessarily is a dictatorship of one of the players.

The proof goes in three steps: 1/ The neutrality lemma (demonstration that Pareto efficiency and IIA imply neutrality), 2/ use of the definition of decisive coalitions, and strongly decisive coalitions which are, in essence "joint dictatorship" of a coalition, and of Blau's lemma which states that a decisive coalition is necessarily strongly decisive, 3/ Sen's partitioning lemma which shows that a decisive coalition of two or more players always have a decisive strict sub-coalition, leading to the conclusion.

Lemma 1 (Neutrality) Unanimity rule and IIA imply neutrality.

Proof (This proof essentially follows [Geanakoplos, 2005].) Let Π be such that Lemma 2 ( [Blau, 1957]) Under Pareto efficiency and IIA, decisive coalitions are strongly decisive. Lemma 3 (Partitioning lemma. [Sen, 1970]) Let S be a decisive coalition of cardinal greater than one, and S = S 1 ∪ S 2 be a non-trivial partition of it. (i.e., S 1 = ∅, S 2 = ∅, S 1 ∩ S 2 = ∅.) Then either S 1 or S 2 is decisive.

Proof Let the premises of the lemma hold. Let a = b = c be three alternatives, and a profile Π be such that, that b > Π a. Then S 2 is decisive in profile Π. But by IIA, the rankings of c relative to a and b is irrelevant. Therefore the conclusion holds in any profile.

• ∀i ∈ S 1 , a i b i c, • ∀j ∈ S 2 , b j c j a, • ∀k / ∈ S , c k a k b.
Corollary 1 A SWF enjoying Pareto efficiency and IIA is necessarily dictatorial.

Proof The grand coalition V is decisive by Pareto efficiency. Partition it, and recursively partition the resulting decisive sub-coalitions until it remains a single player in this sub-coalition. Any preference it has results in the same preference in the social preferences. Hence it is a dictator.

Conclusion

Arrow's spectacular result, which is his doctoral dissertation, must result in "engagement rather than resignation," as Sen argued in his speech when he received the Sveriges Bank Prize in Economic Sciences in Memory of Alfred Nobel in 1998.

Arrow defended this idea in the conclusion of his 1972 speech when he himself received this award. We follow this idea throughout this notice by using parentheses for the term "(im)possibility". Arrow himself following an advice of Koopmans on this point, calls "possibility theorem" a result generally known as "impossibility theorem". Thus, since this result of 1951 two paths have mainly been followed: some have modified the Arrovian framework (for illustration [Sen, 1969] proved that if the social choice function is only quasi-transitive -i.e the indifference is no longer transitive-there is no more dictator but at least one player with a veto) and others have modified the principles (as e.g. [Black, 1948] who proved that when individual preferences are unimodal and there is an odd number of individuals, the majority rule leads to a transitive order at the collective level). Moreover, in recent years, other economists have been studying the different properties of all possible rules of social choice by trying to demonstrate whether some dominate them. All these contributions are extremely interesting and important However to finish this conclusion we wish to draw the reader's attention to one of the most beautiful spin-off of Arrow's theorem, namely the Gibbard-Satterthwaite theorem, which is at the heart of mechanism design. The first is related to the second in the sense that on the one hand Arrow from the beginning posed the question of the veracity of the individual preferences announced by the agents; and on the other hand, in the sense that Gibbard used it in its original proof.

Social choice and manipulation

The problem of social choice is concerned with choosing one best "social alternative" given the preferences of the players. Therefore, a social choice function (SCF) is a mapping F : P → A from profiles to alternatives. Typically the problem of electing a committee chairman or a political President.

A result closely related to Arrow's theorem, another impossibility theorem, is credited to Gibbard [Gibbard, 1973] and Satterthwaite [Satterthwaite, 1975]. It is concerned with preventing "strategic playing". A player i has an opportunity to play strategically if, by pretending another preferences ordering than its "true" one, it may get a better result, according to its true ordering.

Definition 2

1. Given a SCF F , player i has an opportunity to play strategically (or to manipulate F ) at some profile Π in which its ordering is P i , if there exists a different ordering P i such that, in the profile Π obtained by replacing P i by P i in Π, one has F (Π ) i F (Π) (in the ordering P i ).

2. A SCF F is called strategy-proof if no player ever has an opportunity to play strategically whatever the profile:

∀i ∈ V , ∀Π = (P i , P -i ) ∈ P , ∀P i ∈ P s , F (Π) i F (P i , P -i ) .
The impossibility result is sated in a way similar to Proposition 1:

Theorem 2 ( [Gibbard, 1973, Satterthwaite, 1975]) If a SCF is strategy-proof and an onto function, it is dictatorial.

Analysis

All proofs of theorem 2 are significantly longer than that of Arrow's theorem above.

There are parallel proofs of both (see, e.g. [Reny, 2001]) and even at least one proof of a more general result encompassing both (see [Eliaz, 2004]), but all too long to be detailed here. Here we only want to stress the parallel between the underlying hypotheses in the two theorems, as they do not look similar at first reading.

We need to introduce a rather specific notion of monotonicity.

Definition 3 A SCF is called monotonic if, the following holds: let Π and Π be two profiles. If for all i ∈ V, the subsets of A: {a | F (Π) i a} as defined by Π contain the subsets {a | F (Π) i a} as defined by Π, then F (Π ) = F (Π).

The point we wish to stress in that definition is that it contains a hidden notion of IIA. Indeed, the result should hold irrespective of the relative orders of the alternatives in the subsets quoted and in their complementary subsets in A. Only the rankings with respect to the social choice are relevant. We may also extend here the unanimity rule or Pareto efficiency:

Definition 4 A SCF F satisfies the unanimity rule (is Pareto efficient) if, whenever the same alternative is the preferred alternative of all players, it becomes the social choice:

[∃a ∈ A : ∀i ∈ V , ∀b = a ∈ A , a i b] ⇒ F (Π) = a .
The similarity between the hypotheses of the two theorems follows from the following two propositions:

Proposition 3 ( [START_REF] Muller | The equivalence of strong positive association and strategy-proofness[END_REF]) If a SCF is strategy-proof and onto, it is monotonic and Pareto efficient.

Proposition 4 If a SCF is monotonic and Pareto efficient, it is dictatorial.

The second proposition above is the difficult one to prove. The similarity resides in the first one [START_REF] Muller | The equivalence of strong positive association and strategy-proofness[END_REF] whose proof is easy:

Proof of Proposition 3. Let Π and Π be related as in Definition 3, and suppose that F (Π) = a = F (Π ). Make the change from Π to Π one player at a time in numeric order. Denote by Π k the profile obtained after changing P k to P k . (And Π 0 = Π). At some point, we have that F (Π i-1 ) = a = b = F (Π i ). By strategyproofness, it follows that a i b in P i while b i a in P i . But by hypothesis, if a i b in P i it is a fortiori true in Π . A contradiction. Therefore the SCF is monotonic.

Because F is assumed to be an onto function, for any given a ∈ A, there exists a profile Π such that F (Π) = a. Build Π by moving a at the top of the preferences of all players. By monotonicity, it still holds that F (Π ) = a. Now, get Π by shuffling at will the preferences of all players below a, leaving a at their top position. Because of the definition of monotonicity, and specifically its IIA-like character, it still holds that F (Π ) = a. Therefore the SCF is Pareto efficient.

Cross References

  a ≥ Π b (resp. a > Π b). Let (a , b ) = (a, b) and Π be such that the relative preferences of a and b in Π be the same as those of a and b in Π. (Π and Π may coincide.) Assume first that a = b, b = a. (But we allow either a = a or b = b.) Define another profile Π by modifying Π as follows: if a = a, place alternative a immediately above a and, if b = b, place b immediately below b, This does not modify the relative preferences of a and b, and in this profile Π , a and b have the same relative preferences as have a and b in Π and therefore the same as they have in Π . By unanimity a ≥ Π a and b ≥ Π b . By IIA, it still holds that a ≥ Π b (resp. a > Π b). By transitivity, it follows from these three preferences that a ≥ Π b (resp. a > Π b) and by IIA again, a ≥ Π b (resp. a > Π b ). Finally, if a = b or b = a or both, use a third alternative c and the same reasoning for the sequence of pairs (a, b) → (c, b) → (c, a) → (b, a) → (a , b ) to reach the same conclusion. Definition 1 A coalition S ⊂ A is called (a, b)-decisive in a given SWF f if, when in a profile Π its members agree to set a b while all other players choose b a it results that a > b in the preferences f (Π). It is strongly decisive if the conclusion a > b holds irrespective of the preferences of the players outside the coalition. Proposition 2 It follows from the neutrality lemma that for a SWF satisfying Pareto optimality and IIA, if a coalition is (a, b)-decisive, it is decisive for all pairs of alternatives.

Figure 1 :

 1 Figure 1: Proof of the neutrality lemma. Π and Π are given. In Π , a > a and b > b are by unanimity, a ≥ b by IIA, a ≥ b by transitivity, conclusion in Π by IIA.

Because

  S is decisive, it holds that b > Π c. If a > Π c, the sub-coalition S 1 is decisive, at least in profile Π. If, to the contrary, c ≥ Π a, by transitivity, it follows

Figure 2 :

 2 Figure 2: Proof of Blau's lemma. The social preference shown in Π is by decisiveness of S. In Π , a > c is by decisiveness, c > b by unanimity, the conclusion a > b by transitivity. By IIA, it is not changed if c is moved around.