Vincent Monchiet
email: vincent.monchiet@univ-paris-est.fr

Hai-Bang Ly
email: hai-bang.ly@univ-mlv.fr

Daniel Grande
email: grande@icmpe.cnrs.fr

Macroscopic permeability of doubly porous materials with cylindrical and spherical macropores

Keywords: Permeability, Homogenization, Cylindrical and Spherical Voids, Stokes, Darcy

We derivate analytic expressions for the effective permeability of a doubly porous medium which contains cylindrical or spherical macropores. The homogenization procedure is based on the consideration of a unit cell with a prescribed macroscopic pressure gradient or macroscopic velocity on its boundary. Within the unit cell, we have to solve the coupled Darcy-Stokes equations with appropriate interface models. The effective permeability of the bi-porous solid is determined by considering the composite cylinder and sphere assemblages. The solutions with prescribed macroscopic pressure gradient and velocity are provided and the influence in the choice of the interface model on the effective permeability of the doubly porous solid is studied. Finally, the results are compared with existing expressions provided in the literature and the estimate for the case of a cylindrical macropore is compared with finite element solutions.

Introduction

Geomaterials such as concrete or rocks are naturally porous media and their transport properties are classically described by the Darcy law and the permeability coefficient. The derivation of the permeability in the framework of homogenization procedures has been largely documented in the literature using asymptotic approaches [START_REF] Auriault | Study of macroscopic behavior of a deformable porous medium[END_REF][START_REF] Sanchez-Palencia | Nonhomogeneous Media and Vibration Theory[END_REF][START_REF] Levy | Fluid flow through an array of fixed particles[END_REF] or volume average methods [START_REF] Whitaker | Flow in porous media I: A theoretical derivation of Darcy's law[END_REF][START_REF] Allaire | Homogenization of the Stokes flow in a connected porous medium[END_REF]. In the present paper, we focus on the permeability of a doubly porous material constituted of an initially porous structures containing non interconnected macropores. The apparition of the macropores is due to mechanical loading or chemical reactions and the determination of their role on the mass transfer properties is of particular interest for various applications such as fluid/gaz storage. Obviously, the geometry of fractures in geomaterials must be depicted by flat ellipsoids (see for instance Budiansky and O'Connell [START_REF] Budiansky | Elastic moduli of a cracked solid[END_REF]), however, in this paper, we consider the case of cylindrical and spherical macropores by simplicity. The case of ellipsoidal shapes would be investigated in a next study. Note however that the results in the case of the spherical shape can be used to estimate the permeability of some bi-porous polymers (see for instance [START_REF] Ly | Designing and modeling doubly porous polymeric materials[END_REF][START_REF] Ly | Facile fabrication of doubly porous polymeric materials with controlled nano-and macro-porosity[END_REF]). The problem to study has three scales: the first is that of an interconnected network of submicron channels, the second scale is associated with the random distribution of non interconnected macropores, the last scale is the macroscopic one. The macroscopic permeability can be determined by a double homogenization approach along the lines of a method which has been derived for instance, by the asymptotic expansion method (see Auriault and Boutin [START_REF] Auriault | Deformable porous media with double porosity. Quasi-statics. I: Coupling effects[END_REF][START_REF] Auriault | Deformable porous media with double porosity[END_REF][START_REF] Auriault | Deformable porous media with double porosity III: Acoustics[END_REF], Royer et al. [START_REF] Royer | Macroscopic modeling of double-porosity reservoirs[END_REF], Boutin et al. [START_REF] Boutin | Acoustic absorption of porous surfacing with dual porosity[END_REF], Olny and Boutin [START_REF] Olny | Acoustic wave propagation in double porosity media[END_REF]). At the intermediate scale, a coupled Darcy/Stokes problem has to be solved. However, the asymptotic method is not able to deliver the conditions et the interface between the Darcy region and the Stokes one. Two different approaches are generally considered in the literature: (i) the first uses the interface model of Beavers-Joseph-Saffman (BJS) [START_REF] Beavers | Boundary condition at a naturally permeable wall[END_REF][START_REF] Saffman | On the boundary condition at the interface of a porous medium[END_REF] (see also [START_REF] Nied | The Beavers-Joseph Booundary condition and related Matters: A historical and critical note[END_REF] for a historical background), (ii) the second one uses the Brinkman equation [START_REF] Brinkman | A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles[END_REF][START_REF] Brinkman | On the permeability of media consisting of closely packed porous particles[END_REF]. Some comparisons of these two approaches has been also provided on the literature for 2d geometric configurations (see for instance [START_REF] Chen | Asymptotic analysis of the differences between the Stokes-Darcu system with different interface conditions and the Stokes-Brinkman system[END_REF]). Analytic models has been first provided by Markov et al. [START_REF] Markov | Permeability of the Fluid-Filled Inclusions in Porous Media[END_REF] introducing the concept of an equivalent permeability for the macropores. The latter is determined by solving the coupled Darcy/Stokes equations with the BJS interface model for an isolated cavity embedded in an infinite porous matrix. The results are provided in the cases of a cylindrical or spherical pore. The results have been latter extended to the case of a spheroidal cavity by Rasoulzadeh et al. [START_REF] Rasoulzadeh | Effective Permeability of a Porous Medium with Spherical and Spheroidal Vug and Fracture Inclusions[END_REF] which can mimic fractures by taking an oblate spheroid with a small value of the aspect ratio. Another analytic solutions has been derived by Silva and Ginzburg [START_REF] Silva | Stokes ŰBrinkman ŰDarcy solutions of bimodal porous flow across periodic array of permeable cylindrical inclusions: cell model, lubrication theory and LBM/FEM numerical simulations[END_REF] for a composite cylinder, i.e. two concentric cylinders in which the flow obeys to the Brinkman equation but with different coefficients in the core and in the coating.

In the present paper, we derive the effective permeability of bi-porous materials with cylindrical and spherical macropores by using the approach based on Darcy-Stokes equations. The influence in the choice of the interface model on the effective permeability is then analyzed. The effective permeability is determined by considering the composite cylinder and sphere models which have been often considered by the past by various authors in the context of porous media [START_REF] Happel | Viscous flow relative to arrays of cylinders[END_REF][START_REF] Kuwabara | The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds Numbers[END_REF][START_REF] Stechkina | Drag of porous cylinders in a viscous fluid at low Reynolds numbers[END_REF][START_REF] Boutin | Study of permeability by periodic and self consistent homogenization[END_REF]. The analytic solutions with prescribed macroscopic pressure gradient or macroscopic velocity at the boundary of the representative volume element are provided and the close form expression of the effective permeability is given. The solutions are compared with finite element ones in the case of a cylindrical shape. We also determine the equivalent permeability of the Stokes region and it is compared with the results of Markov et al. [START_REF] Markov | Permeability of the Fluid-Filled Inclusions in Porous Media[END_REF].

2 Homogenization problem

Local equations and interface conditions

Consider a Representative Volume Element (RVE) of a doubly porous solid. Both the porous solid and the macropores are fulfilled by a Newtonian viscous fluid with the dynamic viscosity µ. We denote by Ω s the domain occupied by the porous solid and by Ω f the domain of the pores. Obviously, the permeability of the solid phase is associated with an interconnected porous microstructure at a lower scale (see figure 1). The permeability of the matrix is assumed to be known (obtained from experiments) and the macropores (the first porosity) are assumed to be non interconnected. Denoting by h the characteristic length of the channels of the initial interconnected microporous solid (the second porosity), we assume that h is sufficiently small compared to cavities radius a such that there is a scale separation between the two porosities. The macropores are then embedded in a homogenous porous material having the permeability k.

µ∆v -∇p = 0 ∀x ∈ Ω f (1) div v = 0 x ∈ ∀Ω f (2)
Within the solid, the fluid flow obeys to the Darcy law with the incompressibility condition:

v = - k µ ∇p ∀x ∈ Ω s (3) div v = 0 ∀x ∈ Ω s (4)
At the interface between the fluid and the solid, the Beaver-Joseph (BJ) [START_REF] Beavers | Boundary condition at a naturally permeable wall[END_REF] model is often used:

v (s) .ν = v (f ) .ν ∀x ∈ Γ (5) 2µν.d(v (f ) ).τ = -α[v (f ) -v (s) ].τ ∀x ∈ Γ (6) 2µν.d(v (f ) ).ν = p (f ) -p (s) ∀x ∈ Γ (7)
in which p (f ) , v (f ) and p (s) , v (s) represents the pressure and velocity fields taken at each side of the interface Γ, exponent (s) makes reference to the porous solid and (f ) to the fluid phase (i.e. the macropores). Also, ν and τ represent the normal and the tangential unit vectors acting on Γ. In equation ( 6), α is a coefficient of the BJ model, d(v (f ) ) represents the strain rate tensor computed in the fluid region. These interface conditions represent: the continuity of the mass flux [START_REF] Auriault | Study of macroscopic behavior of a deformable porous medium[END_REF], the discontinuity for the tangential velocity (6) and the continuity for the normal traction [START_REF] Auriault | Deformable porous media with double porosity[END_REF]. At the interface Γ, the normal component of the velocity field is then continuous but the tangential component and the pressure are discontinuous.

Based on their experimental observations, Beaver and Joseph [START_REF] Beavers | Boundary condition at a naturally permeable wall[END_REF] proposed the following expressions for the coefficient α:

α = µ √ k δ (8)
in which δ is a dimensionless coefficient called slip coefficient and is characteristic of the geometry of the porous solid. This coefficient has been determined experimentally by Beavers and Joseph [START_REF] Beavers | Boundary condition at a naturally permeable wall[END_REF] for Nickel foametal and found 0.78, 1.45 and 4.0 for porous microstructures having average pore size of 0.016, 0.034 and 0.045 inches respectively. For Aloxite they found a value of 0.1 with the average pore size of 0.013 and 0.027 inches. Numerical studies have been also provided by Sahraoui and Kaviany [START_REF] Sahraoui | Slip and no slip velocity boundary conditions at interface of porous plain media[END_REF] for 2d periodic structure made of cylindrical particles and found a value of the slip coefficient closed to 2. Some other theoretical support of the BJ model has been provided in the literature. For instance, Saffman [START_REF] Saffman | On the boundary condition at the interface of a porous medium[END_REF] uses a statical treatment and found that the tangential velocity component in the porous solid could be neglected and suggests to use the following condition for the tangential velocity:

2µν.d(v (f ) ).τ = -αv (f ) .τ x ∈ Γ (9)
This form has been considered in numerous works (see for instance [START_REF] Layton | Coupling fluid flow with porous media flow[END_REF][START_REF] Arbogast | A computatonal method for approximating a Darcy-Stokes system governing a vuggy porous medium[END_REF][START_REF] Urquiza | Coupling Stokes and Darcy equations[END_REF]).

A tentative to demonstrate the BJS interface model has been provided by Jager and Mikelic [START_REF] Jager | On the interface boundary condition of Bevears, Joseph and Saffman[END_REF] using the asymptotic series expansion method and a scale separation. Moreover, the authors show that the slip coefficient could be determined through an auxiliary-layer type problem.

Auriault [START_REF] Auriault | About the Beavers and Joseph boundary condition[END_REF] analyses the interface condition at the frontier between a fluid and a porous solid by the method of asymptotic series expansion and conclude that the adherence condition could be used at the frontier between the Darcy and the Stokes region:

v (f ) .τ = 0 x ∈ Γ ( 10
)
The adherence condition is a particular case of the BJS model corresponding to α = +∞. In order to avoid in confusion, the equations ( 5), ( 6) and ( 7) are called Beavers and Joseph (BJ) conditions. We call Beavers-Joseph-Saffman (BJS) the interface model which use conditions (5), ( 9) and [START_REF] Auriault | Deformable porous media with double porosity[END_REF]. The adherence model corresponds to conditions (5), [START_REF] Beavers | Boundary condition at a naturally permeable wall[END_REF] and [START_REF] Auriault | Deformable porous media with double porosity[END_REF]. For all the calculations provided in the next sections, it is preferable to use the following unique equation for the jump of the tangential velocity:

2µν.d(v (f ) ).τ = -α[v (f ) -ǫv (s) ].τ x ∈ Γ (11)
where ǫ = 1 and α is given by ( 8) for the BJ model, ǫ = 0 and α is given by ( 8) for the BJS model while α = +∞ and ǫ = 0 for the adherence condition. Note that other interfacial conditions has been proposed in the literature (see for instance [START_REF] Ochoa-Tapia | Momentum transfer at the boundary between a porous medium and a homogeneous fluid. I. theorethical development[END_REF][START_REF] Ochoa-Tapia | Momentum transfer at the boundary between a porous medium and a homogeneous fluid. II. comparison with experiment[END_REF][START_REF] Valdes-Parada | Jump momentum boundary condition at a fluid-porous dividing surface: derivation of the closure problem[END_REF]) but are not considered in the present study. Note that the existence and uniqueness of the solution of Darcy/Stokes coupled problem with the BJS interface condition has been studied for instance by Arbogast and Lehr [START_REF] Arbogast | Homogebnization of a Darcy-Stokes systen modeling vuggy porous media[END_REF]. Note also that the velocity field is divergence free in the porous solid and in the micro-pores. The pressure is then an harmonic function everywhere in the RVE:

∆p = 0 ∀x ∈ Ω s ∪ Ω f (12)

Boundary conditions and macroscopic permeability

In order to determine the macroscopic permeability of the porous material, let Ω be subjected to the following prescribed pressure on its external surface ∂Ω:

p = G.x, ∀x ∈ ∂Ω (13)
or the prescribed normal velocity:

v.n = V .n, ∀x ∈ ∂Ω ( 14
)
where G represents the macroscopic pressure gradient, V represents the macroscopic velocity and n the normal unit vector taken on the boundary ∂Ω.

A velocity field is said admissible if it is divergence free in Ω f and in Ω s , if the normal velocity v.ν is continuous across Γ and if v satisfies to the boundary condition [START_REF] Brinkman | A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles[END_REF]. For such velocity fields, we have the compatibility condition:

V =< v > Ω (15)
that is easily proved by using the divergence theorem. A pressure field is said admissible if it is continuous and continuously differentiable in each phase and satisfy to the boundary condition [START_REF] Brezzi | Mixed and Hybrid Finite Element Methods[END_REF]. For such pressure field, we have the following compatibility condition:

G =< ∇p > Ω + 1 |Ω| Γ [p]νdx (16)
where [p] = p (s) -p (f ) represents the jump of p across Γ. The above condition extends the classic average relation in the case of non continuous fields. For instance, equivalent relation also appears for the average of stress in the case of elastic composites with imperfect interfaces (see for instance [START_REF] Lequang | Variational principles and bounds for elastic inhomogeneous materials with coherent imperfect interfaces[END_REF]).

If the boundary condition ( 13) is applied to the RVE, due to the linearity of the Darcy-Stokes equations, the local velocity and pressure fields linearly depends on the applied macroscopic pressure gradient G. This mean that there exist a localization tensor A(x) such that:

v = A(x).G (17)
Introducing this expression in the average relation [START_REF] Brinkman | On the permeability of media consisting of closely packed porous particles[END_REF], it leads to:

V = - 1 µ K hom .G (18)
where we have introduced the macroscopic permeability:

K hom = -µ < A(x) > Ω (19)
Alternatively, if the boundary condition ( 14) is used, the local pressure gradient can be expressed as function of V by introducing the localization tensor B(x):

p = B(x).V (20)
Introducing this expression in [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[END_REF], it leads to:

[K hom ] -1 = - 1 µ < ∇B(x) > Ω + 1 |Ω| Γ ν s ⊗ [B(x)]dx (21)
The value of K hom is different from the local permeability k due to the presence of the macropores. Obviously, when the volume of the macropores is null, the RVE is only constituted of the porous medium with the isotropic permeability k and the macroscopic permeability is K hom = kI. The main objective of this work is to determine the homogenized permeability tensor K hom as function of the local permeability k, the macro-pores radius a and of the volume fraction f . Also the choice of the interface model on the effective permeability is investigated considering the BJ, BJS or the adherence interface model. Regarding the boundary condition on the porous cylinder, it reads in the cylindrical frame (r, θ, z):

p = G 1 b cos(θ) + G 2 b sin(θ) (22)
The solution with a prescribed normal velocity at the boundary of the RVE is provided in the appendix A.1. It leads to the same expression for the macroscopic permeability. We assume that only the component G 1 is non null. In order to satisfy the boundary condition [START_REF] Discacciati | Robin-Robin domain decomposition methods for the Stokes-Darcy coupling[END_REF], we choose the following expression for the local pressure: p = F (r) cos(θ) [START_REF] Happel | Viscous flow relative to arrays of cylinders[END_REF] in the fluid and in the solid.

Obviously, the function F (r) takes different expressions in the solid and in the fluid regions. The pressure being a harmonic function, it leads to:

∆p = ∂ 2 p ∂r 2 + 1 r ∂p ∂r + 1 r 2 ∂ 2 p ∂θ 2 = F ′′ (r) + 1 r F ′ (r) - 1 r 2 F (r) cos(θ) = 0 (24)
which is satisfied for:

F (r) =          A 1 r + B 1 r in the solid A 2 r + B 2 r in the fluid (25)
where A 1 , B 1 , A 2 and B 2 are four constants. The term B 2 /r is eliminated since it diverges at r = 0. The velocity field in the solid is given by the Darcy law:

v r = - k µ F ′ (r) cos(θ) = - k µ A 1 - B 1 r 2 cos(θ) (26)
v θ = k µ F (r) r sin(θ) = k µ A 1 + B 1 r 2 sin(θ) (27)
In the macropore, the velocity field must comply with the Stokes equations, which reads, in the cylindrical frame:

µ∆v r - µ r 2 2 ∂v θ ∂θ + v r -F ′ (r) cos(θ) = 0 (28) µ∆v θ + µ r 2 2 ∂v r ∂θ -v θ + F (r) r sin(θ) = 0 ( 29
)
This suggests to use the following expressions for the components of the velocity field:

v r = G(r) cos(θ), v θ = H(r) sin(θ) (30)
The introduction of these expressions in equations ( 28) and ( 29) leads to a linear system of differential equations for G(r) and H(r) which have the following general solution:

G(r) = A 2 8µ r 2 + C 2 + D 2 r 2 , H(r) = - 3A 2 8µ r 2 -C 2 + D 2 r 2 (31)
where C 2 and D 2 are two additional constants. Again, the term D 2 /r 2 is eliminated since it diverges at r = 0. The interface conditions must now be verified. The continuity of the flux reads:

v (f ) r (r = a) = v (s) r (r = a) (32)
The interface condition for the tangential velocity is:

µ   1 r ∂v (f ) r ∂θ - v (f ) θ r + ∂v (f ) θ ∂r   (r=a) = -α v (f ) θ -ǫv (s) θ (r=a) (33)
The continuity of the normal traction is:

2µ ∂v (f ) r ∂r (r=a) = p (f ) -p (s) (r=a) (34)
and finally the boundary condition for the pressure is satisfied if:

A 1 b + B 1 b = G 1 b (35)
Introducing expressions ( 26), ( 27) and ( 30) with [START_REF] Ly | Designing and modeling doubly porous polymeric materials[END_REF] in Eqs. [START_REF] Ly | Facile fabrication of doubly porous polymeric materials with controlled nano-and macro-porosity[END_REF] to (34) leads to a linear set of algebraic equations for the unknown coefficients. The solution of this linear system is:

A 1 = G 1 Q [2aµ + αa 2 + 2α(1 + ǫ)k] B 1 = - G 1 Q (2µa + αa 2 -2αk(1 -ǫ))a 2 A 2 = 8kαG 1 Q C 2 = - kG 1 µQ (4αǫk + 4aµ + 3αa 2 )                                (36)
in which Q is given by: In this section, we consider the case of a spherical cell of radius b containing a macropore of radius a. The permeability of the porous matrix is still denoted by k and the same methodology follows in the case of a cylindrical macropore is still considered. The boundary condition [START_REF] Brezzi | Mixed and Hybrid Finite Element Methods[END_REF] reads in the spherical frame (r, θ, ϕ):

Q = [αa 2 + 2aµ + 2αǫk](1 -f ) + 2αk(1 + f ) (37)
p = G 1 r cos(θ) sin(ϕ) + G 2 r sin θ) sin(ϕ) + G 3 r cos(ϕ) (38)
The solution with a prescribed normal velocity at the boundary of the RVE is provided in the appendix A.2. Again, the same expression for the macroscopic permeability is obtained with this kind of boundary conditions. Due to the symmetry of the problem, only the case G 1 = 0 and G 2 = G 3 = 0 is considered. The verification of the boundary condition suggests to use the following expression for the pressure:

p = F (r) cos(θ) sin(ϕ) (39)
The pressure being harmonic, we deduce that the function F (r) has the following expression:

F (r) =          A 1 r + B 1 r 2 in the solid A 2 r in the fluid (40)
In the solid matrix, the velocity is given by the Darcy law:

v r = - k µ A 1 - 2B 1 r 3 cos(θ) sin(ϕ) (41)
v θ = - k µ A 1 + B 1 r 3 sin(θ) (42)
v ϕ = - k µ A 1 + B 1 r 3 cos(θ) cos(ϕ) (43)
In the fluid phase, we use the following expressions for the velocity field:

v r = G 1 (r) cos(θ) sin(ϕ) (44) v θ = G 2 (r) sin(θ) (45) v ϕ = G 3 (r) cos(θ) cos(ϕ) (46)
The verification of the Stokes equations leads to:

G 1 (r) = C 1 + C 2 r 2 + C 3 r + C 4 r 3 (47)
G 2 (r) = G 3 (r) = A 2 4µ r 2 + C 1 - C 2 2 r 2 + C 3 r - C 4 2r 3 (48)
This field is divergence free for C 2 = A 2 /(10µ).

Obviously, the term in "1/r" and "1/r 3 " are discarded since they diverge à r = 0. Finally, the constants are determined with the boundary condition at r = b and the interface conditions:

v (f ) r (r = a) = v (s) r (r = a), (49) µ
  1 r sin(ϕ) ∂v (f ) r ∂θ - v (f ) θ r + ∂v (f ) θ ∂r   (r=a) = -α v (f ) θ -ǫv (s) θ (r=a) , (50)
µ 1 r ∂v (f ) r ∂ϕ - v (f ) ϕ r + ∂v (f ) ϕ ∂r (r=a) = -α v (f ) ϕ -ǫv (s) ϕ (r=a) , (51)
2µ ∂v (f ) r ∂r (r=a) = p (f ) -p (s) (r=a) (52)
Equations ( 50) and (51) correspond to the discontinuity of the tangential velocity components, along direction e θ and e ϕ respectively. These conditions have found to be equivalent since G 2 (r) = G 3 (r). Additionally, the boundary condition for the pressure is satisfied for:

A 1 b + B 1 b 2 = G 1 b (53)
The identification of the constants leads to:

A 1 = G 1 (a 2 α + 6αk(2 + ǫ) + 3aµ) Q , (54)
B 1 = - G 1 a 3 Q (a 2 α + 3aµ -6α(1 -ǫ)k), (55)
C 1 = - 3G 1 k µQ (3aµ + 2a 2 α + 6αǫk), (56)
A 2 = 30G 1 αk Q (57)
with:

Q = [αa 2 + 3aµ + 6kαǫ](1 -f ) + 6αk(2 + f ) (58)
5 Effective permeability

Analytic solution for cylindrical/spherical macropores

The macroscopic velocity is computed from the average rule [START_REF] Brinkman | On the permeability of media consisting of closely packed porous particles[END_REF]. The in-plane permeability is K hom = K hom I 2 where I 2 is the identity in 2d dimension. Expressions of K hom are:

K hom = k 1 + 2f δ + 2 √ κ [δ + 2 √ κ](1 -f ) + 4δκ BJ model K hom = k 1 + 2f δ(1 -2κ) + 2 √ κ [δ + 2 √ κ](1 -f ) + 2δκ(1 + f ) BJS model K hom = k 1 + 2f 1 -2κ 1 -f + 2κ(1 + f ) Adherence model (59)
The dimensionless coefficient κ is defined by:

κ = k a 2 (60)
and the dimensionless coefficient δ has been introduced in equation ( 8). Coefficient κ accounts for the macropores size.

In the case of a spherical macropore, the permeability tensor is isotropic, K hom = K hom I where K hom is defined by:

K hom = k 1 + 3f δ + 3 √ κ [δ + 3 √ κ](1 -f ) + 18κδ) BJ model K hom = k 1 + 3f δ(1 -6κ) + 3 √ κ [δ + 3 √ κ](1 -f ) + 6κδ(2 + f )) BJS model K hom = k 1 + 3f 1 -6κ 1 -f + 6κ(2 + f ) Adherence model (61)
Considering coefficient κ, it depends on the permeability k of the microporous matrix and on the radius of the macropores. For geomaterials, the permeability is typically inferior to 10 -12 m 2 one Darcy). Assuming that the macropores radius is superior to 10 -6 m, we deduce that the value of κ is strictly inferior to 1. The value of κ can reach very small values, considering for instance a geomaterial with very low permeability (k = 10 -18 m 2 for example) and macropores of 10µm, it gives κ = 10 -8 . From a homogenization point of view, it is raisonnable to suppose that κ is a small parameter. Indeed, the permeability of the porous matrix is

k = O(h 2 )
where h is the characteristic length of the first (sub-micron) porosity, then κ = O(h 2 /a 2 ). Moreover, h << a since we assumed a separation of scales between the two porosities. Assuming that h/a < 0.1, that is the limit for having the scale separation, we deduce that κ ≤ 10 -2 . Consequently, in all the applications proposed in the paper, we limit the value of κ by 10 -2 . Regarding expression (59), the macroscopic permeability decreases with f if:

δ(1 -2κ) + 2 √ κ < 0 (62)
and regarding expression (59), if:

κ > 1 2 (63)
This obviously leading to non physical results since it is expected that the presence of the macropores increases the effective permeability. Similar conditions could be derived for the spherical macropore. As mentioned above, it is reasonable to assume that κ is small and then the above conditions are never attained. It must be mentioned that the effective permeabilities derived with the BJ interface model are monotonously increasing functions with the porosity f whatever the value of δ and κ.

On figure 4, we represent the variations of the dimensionless permeability K hom /k for a cylindrical macropore as function of the porosity f and for different values of the slip coefficient δ and parameter κ. figure 5, we display the same results but for the spherical macropore.

It is observed, for the cylindrical macropore, that the effective permeability obtained with the three interface models leads to the same estimation. There is slight differences when considering the adherence model instead of the BJ or BJS models when δ = 0.1 and κ = 10 -2 . In the case of the spherical macropore, the three interface models leads to the same estimate when κ = 10 -4 .

However, when κ = 10 -2 , the estimate obtained with the adherence model differs from two others. As a first conclusion, we observe that the BJ and BJS interface models leads to the same estimates for the effective permeability for any values of κ and δ. Consequently, the hypothesis formulated by Saffman [START_REF] Saffman | On the boundary condition at the interface of a porous medium[END_REF], which supposes that the tangential components of the velocity field in the Darcy region is negligible compared to its counterpart in the Stokes region, is verified for microstructures with isolated spherical and cylindrical macropores. The variations of the effective permeability with the coefficient κ is now provided in Fig. 6, 7 and 8 for the slip coefficient values δ = 0.01, δ = 1 and δ = 100 respectively. The porosity if f = 0.5. On each figure, it can be observed that the three models (BJ, BJS and adherence) are equivalent at small values of κ (typically inferior to 10 -4 ). Moreover the estimate reach a plateau for very small values of κ. When the slip coefficient is small (see for instance the case δ = 0.01 on Fig. 6), the estimates obtained with the BJ and BJS interface conditions are equivalent but are very different to that derived with the adherence condition. The latter provides a lower estimate for the effective permeability in the range 10 -3 ≤ κ ≤ 1. Moreover the dependence of K hom /k with κ is important for the adherence but less significant with the BJ and BJS interface conditions. At an intermediate value of δ (see for instance the case δ = 1 on Fig. 7). The effect of κ on the effective permeability is more prominent for the BJ and BJS models than in the case δ = 0.01. Also, the three estimates are different in the range 10 -3 ≤ κ ≤ 1. This suggests that the tangential velocity in the microporous solid (that is neglected in the BJS model) is no longer negligible compared to the tangential velocity in the fluid region. For large values of the slip coefficient (see for instance the case δ = 100 on Fig. 8), the permeability with the BJS and adherence conditions are equivalent but differs from that which uses the BJ interface condition. Comparisons with finite element solutions are now provided to evaluate the accuracy of the analytic solutions. The calculations are performed on a squared unit cell containing a circular macropore. The periodic conditions are applied for the velocity and the pressure field on the boundary of the cell. The dimension of the unit cell is L = 10µm. The calculations are performed by considering two values of the permeability of the microporous matrix: k = 10 -15 m 2 and k = 10 -13 m 2 . We vary the radius of the macropore a, by doing so, we modify simultaneously the porosity f = πa 2 /L 2 and coefficient κ = k/a 2 . Incompressible fluid flow problems generally contain velocity and pressure as the unknown variables and fall in the category of mixed formulations [START_REF] Zienkiewicz | The Finite Element Method[END_REF]. It was recognized that the solutions strongly depend upon the particular pair of velocity and pressure interpolations employed. The spaces of discretization must satisfy the inf-sup condition or the Ladyhenskaya-Babuska-Brezzi condition (see Brezzi [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[END_REF]). Following [START_REF] Layton | Coupling fluid flow with porous media flow[END_REF], we use a mixed variational procedure with the Raviart-Thomas elements for Darcy region and MINI element for the Stokes region and the periodic boundary conditions are accounted with Lagrange multipliers. More details can be also found in [START_REF] Ly | Functionalized doubly porous polymeric materials : design and modeling[END_REF]. The variation of the effective permeability with the radius of the macropore represented on figs 9 and 10 for k = 10 -15 and k = 10 -13 respectively. The finite elements (FE) solutions are compared with the analytic expressions in (61). The slip coefficient is δ = 1. A good agreement between the FE solutions and the analytic estimates of the effective permeability is observed. Differences are however observed for the higher values of the macropore radius that is probably attribuable to the shape of the unit cell: squared for the FE solution and circular for the analytic solutions. Also the differences could be attributed to the kind of boundary conditions, periodic for the squared unit cell and Dirichlet for the hollow cylinder.

K hom /k K hom /k K hom /k K hom /k

Approximation with an equivalent Darcy medium

A simplification of the homogenization problem consists to replace the Stokes region by an equivalent Darcy medium with the permeability k ′ . This simplification has been used in various works to determine the effective permeability of doubly porous solids with applications to fractured petroleum reservoir [START_REF] Kazemi | Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution[END_REF][START_REF] De Swaan | Analytic solutions for determining naturally fractured reservoir properties by well testing[END_REF][START_REF] Arbogast | Derivation of the double porosity model of single phase flow via homogenization theory[END_REF]. By combining two Darcy media, the boundary value problem use the same equations for both regions and the problem fall into classic homogenization boundary value problems. Indeed, the Darcy problem is equivalent to the thermal conduction and various results coming from the homogenization theory can then be used. From a numerical point of view, the problem is strongly simplified by considering two Darcy media. The numerical implementation of the coupled Darcy-Stokes problem involves mixed variational formulations and specific interpolation functions which satisfy to the inf-sup BBL condition. This has been already discussed in the previous section. Now, the use of an equivalent Darcy medium also raise the question of the choice of k ′ . It can be identified from the exact solutions derived in this paper. Consider a composite cylinder or sphere with two Darcy media. The core has the permeability k ′ and the surrounded matrix has the permeability k. We still consider the boundary conditions given by 13. The solutions can be found in the book of Milton (established for the thermal conduction) and the resulting macroscopic permeability is:

K hom = k 1 + 2f (c -1) (1 -f )c + 1 + f (64)
for the cylindrical shape and:

K hom = k 1 + 3f (c -1) (1 -f )c + 2 + f (65)
for the spherical shape and where the coefficient c is given by:

c = k ′ k ( 66
)
where c represents the contrast between the two permeabilities (that of the macropore and that of the bulk microporous solid). Regarding first the case of a cylindrical macropore, the equivalent permeability is determined by comparing Eq. (64) to Eqs (59), ( 59) and (59). This leads to:

k ′ = a 2 2 1 + 2κ + 2 √ κ δ BJ model k ′ = a 2 2 1 + 2 √ κ δ BJS model k ′ = a 2 2 Adherence model (67)
For the spherical macropore, the equivalent permeability is:

k ′ = a 2 6 1 + 6κ + 3 √ κ δ BJ model k ′ = a 2 6 1 + 3 √ κ δ BJS model k ′ = a 2 6 Adherence model (68)
The equivalent permeability is proportional to the square of the radius of the macropore. For the BJ and BJS models, the equivalent permeability also depends on the permeability of the microporous matrix and of the slip coefficient δ. We observe that the equivalent permeability k ′ is different for a cylindrical and a spherical macropore. This suggests that the equivalent permeability depends on the shape of the macropore. The present results differs from that of Markov et al. [START_REF] Markov | Permeability of the Fluid-Filled Inclusions in Porous Media[END_REF] who obtained (for the BJS interface model), the following expressions for the equivalent permeability:

k ′ = a 2 2 1 - 3 √ κ δ cylinder k ′ = a 2 6 1 - 4 √ κ δ sphere (69)
These results has been derived by considering an isolated macropore embedded in an infinie microporous matrix. These expressions are different to that obtained in the present paper. Since the equivalent permeability is independent of the first porosity f , it would be expected to obtain the same results for k ′ . Moreover the equivalent permeability obtained by Markov et al. [START_REF] Markov | Permeability of the Fluid-Filled Inclusions in Porous Media[END_REF] is negative if δ > 3 √ κ for the cylinder and if δ > 4 √ κ for the sphere. This can lead to some non physical results that are illustrated on Fig. 11. The effective permeability corresponding to spherical macropores is represented as function of the slip coefficient δ for κ = 10 -2 and for two values of the porosity, namely f = 0.1 (on the figure at the left) and f = 0.5 (on the right). On these figures, we compare the estimate obtained in the present paper with the BJS model and the estimate derived by Markov et al. [START_REF] Markov | Permeability of the Fluid-Filled Inclusions in Porous Media[END_REF]. The results of the present paper show a very low influence of the slip coefficient on the effective permeability in the range [0, 1] while, the estimate of Markov et al. [START_REF] Markov | Permeability of the Fluid-Filled Inclusions in Porous Media[END_REF] has a singularity in that range. This singularity is observed when the denominator in Eq. ( 65) is null, that corresponding to c = -(2 + f )/(1 -f ) that is possible only if the equivalent permeability is negative. When the equivalent permeability is positive, c > 0, no singularity could be noted for the effective permeability.

Conclusion

The permeability of doubly porous solids with cylindrical and spherical macropores has been derived in a homogenization framework. The approach consists to solve a RVE problem with prescribed macroscopic pressure gradient at its boundary. In the RVE, we solve the coupled Darcy-Stokes problem and we use different interface conditions between the darcy and the stokes region (Beavers-Joseph, Beavers-Joseph-Saffman and adherence conditions). Close form solutions are derived by considering a composite cylinder or sphere models. The results show that the interface model has no influence on the macroscopic permeability when there is a strong scale separation between the two porosities. At smaller scale separation, the choice of the interface model, and the value of the slip coefficient has a significant influence on the macroscopic permeability. The analytic estimates have been compared with FE solutions and show a good accuracy. Finally, an equivalent permeability for the Stokes region is identified. The results showed that this equivalent permeability depends on the shape of the macropore (cylindrical or spherical) and on the choice of the interface model. Such results are however not applicable to fractured porous solids, the fractures would depicted by considering ellipsoidal shapes.

pFig. 1 .

 1 Fig. 1. Schematic representation of the RVE

3Fig. 2 .

 2 Fig. 2. Hollow cylinder with external radius b containing a cylindrical void of radius a.

Fig. 3 .

 3 Fig. 3. Hollow sphere with external radius b containing a spherical void of radius a.

Fig. 4 .

 4 Fig. 4. Case of a cylindrical macropore. Variations of the dimensionless permeability K hom with the porosity f for δ = 1 and κ = 1e -4.

Fig. 5 .

 5 Fig. 5. Case of a spherical macropore. Variations of the dimensionless permeability K hom with the porosity f for δ = 1 and κ = 1e -2.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig.6. Case of a spherical macropore. Variations of the dimensionless permeability K hom /k with κ for δ = 0.01 and f = 0.5.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Case of a cylindrical macropore. Variations of the effective permeability with the porosity for k = 10 -15 and δ = 1. Comparison between FE solutions and the analytic expressions.

Fig. 11 .

 11 Fig. 11. Case of a spherical macropore. Variations of K hom /k as function of the slip coefficient δ for κ = 10 -2 and f = 0.1. Comparison between the Markov et al. (2010) estimate and that provided in the present paper with the BJS model.

Acknowledgment

This work has benefited from a French government grant by ANR within the frame of the national program Investments for the Future ANR-11-LABX-022-01 (LabEx MMCD project).

The authors declare that they have no conflict of interest.

A Solution with prescribed velocity on the boundary A. [START_REF] Allaire | Homogenization of the Stokes flow in a connected porous medium[END_REF] The cylindric macropore Consider first the case of the in plane solution. The boundary condition [START_REF] Brinkman | A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles[END_REF] reads in the cylindrical frame:

Only the case V 1 = 0 and V 2 = 0 is considered. The general expressions for the velocity and pressure fields with prescribed macroscopic velocity remains the same than that determined with the applied macroscopic pressure gradient G 1 . These expressions are given in section [START_REF] Arbogast | Homogebnization of a Darcy-Stokes systen modeling vuggy porous media[END_REF] in which the expressions of the constants

Only Eq. ( 35) in section [START_REF] Arbogast | Homogebnization of a Darcy-Stokes systen modeling vuggy porous media[END_REF], that corresponds to the applied pressure gradient, must be changed. Considering Eq. (A.1) with the expression of the velocity given by ( 30), we deduce that the boundary condition is G(b) = V 1 which also read:

the expressions of the constants are:

in which Q is given by:

A.2 The spherical macropore

In the case of a spherical macropore, the condition (53) must be replaced by:

The identification of the constants leads to:


domain: shs sdv sdv.aen

text:
Emmanuelle Kesse-Guyot

Pauline Rebouillat

Joséphine Brunin

Brigitte Langevin

Benjamin Allès

Mathilde Touvier

Serge Hercberg

Hélène Fouillet

Jean-François Huneau

François Mariotti

Denis Lairon

Philippe Pointereau

Julia Baudry

PhD Emmanuelle Kesse
email: e.kesse@eren.smbh.univ-paris13.fr

Environmental and nutritional analysis of the EAT-Lancet diet at the individual level: insights from the NutriNet-Santé study

Keywords: environmental impacts, dietary patterns, cohort study, EAT-Lancet diet

The EAT-Lancet Commission has recently proposed a "universal" healthy reference diet. However, no study has specifically investigated its possible environmental benefits at the individual level based on observed data. Our objective was therefore to characterize the environmental pressures and impacts related to the level of adherence to the EAT-Lancet diet among French adults. Dietary data from a 264item FFQ in 29,210 NutriNet-Santé participants (75% women, mean age=53.5y (SD=14.0)), were used to estimate (i) the level of adherence to the EAT-Lancet diet through the EAT-Lancet diet index (ELD-I), (ii) the food production-related environmental impacts using 3 individual environmental indicators (greenhouse gas emissions, cumulative energy demand and land occupation) and (iii) the overall environmental impact using a validated aggregated partial score (pReCiPe). For clarity purpose, results are presented by quintile (Q) of ELD-I. High ELD-I (Q5), compared to low (Q1), was associated with lower greenhouse gas emissions (-56%), cumulative energy demand (-31%) and land occupation, (-54%). The pRECIPE was 62% lower in high ELD-I than in low ELD-I but the range of pReCiPe in Q1 was large. In this large scale-study of French adults, adherence to the EAT-Lancet recommendations led to lower environmental impacts. Nonetheless, some low-EAT diets (reflecting unhealthy diets), may exhibit low environmental impacts.

Highlights

-the EAT-Lancet diet index was developed to assess the EAT-Lancet diet adherence while accounting for farming practices (organic or conventional).

-The diets highly correlated with the EAT-Lancet diet (compared to low adherence) led to lower impacts: -56% greenhouse gas emissions, -31% cumulative energy demand and -54% land occupation.

-A high variability in environmental impacts was observed among individuals with low adherence to the EAT-Lancet diet, reflecting a discrepancy between human and planetary health.

Introduction

It is now well established that modern eating habits, rich in fat, salt and sugar, largely contribute to the development of chronic diseases (GBD 2017Diet Collaborators, 2019). In 2017, the Global Burden of Diseases estimated that 11 million (95% uncertainty interval 10-12) deaths were attributable to diet-related risk factors (GBD 2017Diet Collaborators, 2019). Besides, diet production cause serious damages and longterm adverse effects on the environment. Indeed, the climate crisis, the depletion of natural resources and the pollution of water and soil require a radical and urgent change at multiple levels of the global food system, from field to plate [START_REF] Clark | The Diet, Health, and Environment Trilemma[END_REF]. In case of no drastic change in the food system by 2050, greenhouse gas emissions (GHGe), land use, water use, as well as nitrogen and phosphorus application would drive natural processes beyond planetary limits [START_REF] Springmann | Options for keeping the food system within environmental limits[END_REF].

A growing body of evidence documents that diets largely based on plant foods with limited amount of animal products could bring benefits to the environment (Lukasz Aleksandrowicz et al., 2016;[START_REF] Auestad | What current literature tells us about sustainable diets: emerging research linking dietary patterns, environmental sustainability, and economics[END_REF][START_REF] Chai | Which Diet Has the Least Environmental Impact on Our Planet? A Systematic Review of Vegan, Vegetarian and Omnivorous Diets[END_REF]. Specifically, vegetarian and vegan diets have been consistently associated with lower environmental impacts compared to meat-based diets [START_REF] Chai | Which Diet Has the Least Environmental Impact on Our Planet? A Systematic Review of Vegan, Vegetarian and Omnivorous Diets[END_REF]. This is in line with findings from modeling studies aiming at determining environmental-friendly diets [START_REF] Gazan | Mathematical Optimization to Explore Tomorrow's Sustainable Diets: A Narrative Review[END_REF][START_REF] Van Dooren | A Review of the Use of Linear Programming to Optimize Diets, Nutritiously, Economically and Environmentally[END_REF]. Besides, other studies have evaluated different types of diet such as the Mediterranean diet, the Nordic diet or adherence to several dietary guidelines [START_REF] Aleksandrowicz | The Impacts of Dietary Change on Greenhouse Gas Emissions, Land Use, Water Use, and Health: A Systematic Review[END_REF][START_REF] Auestad | What current literature tells us about sustainable diets: emerging research linking dietary patterns, environmental sustainability, and economics[END_REF][START_REF] Chai | Which Diet Has the Least Environmental Impact on Our Planet? A Systematic Review of Vegan, Vegetarian and Omnivorous Diets[END_REF][START_REF] Ridoutt | Dietary Strategies to Reduce Environmental Impact: A Critical Review of the Evidence Base[END_REF]. All these findings consistently documented environmental impacts of cropland and livestock [START_REF] Clark | Multiple health and environmental impacts of foods[END_REF]. Livestock, in particular beef, is responsible for a large part of the dietary-related GHGe and also leads to deforestation and loss of biodiversity [START_REF] Gerber | Tackling Climate Change through Livestock -A global assessment of emissions and mitigation opportunities. Food and Agriculture Organization of the United Nations (FAO)[END_REF]. With regard to health aspects, other protein sources need to be favored, while meat -in particular red and processed meat -consumption should be reduced, given the positive link between meat consumption and numerous chronic diseases (cancer, cardiovascular diseases, type 2 diabetes as well as overall mortality) (GBD 2017Diet Collaborators, 2019;[START_REF] Mariotti | Animal and Plant Protein Sources and Cardiometabolic Health[END_REF].

In that context, in 2019, the EAT-Lancet commission proposed a universal healthy diet. The evidence-based EAT-Lancet diet, aligning nutrition with planet preservation, is the first global reference diet that could allow a sustainable trajectory within the planetary boundaries. It provides an anchor point for future national food policies across culturally diverse countries by enabling them to incorporate environmental preservation into their national food-based dietary guidelines (FBDGs). It is also designed to serve as a reference for "estimating health and environmental effects of adopting an alternative diet to standard current diets [START_REF] Willett | Food in the Anthropocene: the EAT-Lancet Commission on healthy diets from sustainable food systems[END_REF]. This diet is based on the available scientific literature on the relationships between food intake and health. The EAT Lancet Commission concluded that "a dietary change towards increased adoption of plant-based diets has high mitigation potential, which is probably needed to limit global warming to a less than 2°C increase" [START_REF] Willett | Food in the Anthropocene: the EAT-Lancet Commission on healthy diets from sustainable food systems[END_REF]. The EAT-Lancet diet is a 2500 kcal daily diet which promotes plant food consumption such as whole grains, fruit and vegetables, legumes and drastically limits the intake of added fat and sugars as well as animal food such as beef, lamb, pork -and to a lesser extent fish, eggs and chicken. A recent study documented nutritional comparison between the EAT-Lancet diet and Dietary Guidelines for American and identified some discrepancies for some plant-based food groups that were more encouraged in the EAT-Lancet such as soy-based food, nuts and seeds and whole grain starch [START_REF] Blackstone | Comparing the Recommended Eating Patterns of the EAT-Lancet Commission and Dietary Guidelines for Americans: Implications for Sustainable Nutrition[END_REF]. Besides, the EAT-Lancet report has generated controversies with studies documenting null (Zagmutt et al., 2020) or protective association as regards the risk of ischemic heart diseases, diabetes and mortality [START_REF] Knuppel | EAT-Lancet score and major health outcomes: the EPIC-Oxford study[END_REF], while another study put into question its affordability for the world's poor [START_REF] Hirvonen | Affordability of the EAT-Lancet reference diet: a global analysis[END_REF]. A recent modelling study however indicates that the EAT-Lancet recommendations were more in line with the World agenda on health and sustainability than most national FBDG [START_REF] Springmann | The healthiness and sustainability of national and global food based dietary guidelines: modelling study[END_REF]. However, no study has explored environmental impact of adherence to the EAT-Lancet diet. Thus, the translation of the EAT-Lancet recommendations at the individual level and in various cultural settings is necessary to thoroughly characterize the sustainability of the EAT-Lancet diet and in particular environmental dimension.

Most studies exploring the associations between environmental pressures and dietary patterns have used a small range of indicators, mainly GHGe and land use [START_REF] Jones | A Systematic Review of the Measurement of Sustainable Diets[END_REF]. Besides, few studies have considered overall impact indicators whereas this would allow to include several pressure indicators, and thus to address trade-offs between the different environmental footprints [START_REF] Kramer | Decreasing the overall environmental impact of the Dutch diet: how to find healthy and sustainable diets with limited changes[END_REF]. In addition, studies have mainly considered the dominant intensive system in their assessment while agroecological systems might differ on some indicators [START_REF] Gomiero | Environmental Impact of Different Agricultural Management Practices: Conventional vs. Organic Agriculture[END_REF]. The use of the organic farming system as an alternative model is therefore of interest, especially since it has been shown that occidental individuals who eat a lot of plant foods also eat more organically-grown products [START_REF] Lacour | Environmental Impacts of Plant-Based Diets: How Does Organic Food Consumption Contribute to Environmental Sustainability?[END_REF].

In the present work, we therefore aimed to explore the link between adherence to the EAT-Lancet diet and associated environmental impacts considering 3 indicator of pressure and 1 impact while considering farming practices for food production. A first dietary index based on binary components has been developed to reflect the adherence to the EAT-Lancet diet in the EPIC-Oxford study [START_REF] Knuppel | EAT-Lancet score and major health outcomes: the EPIC-Oxford study[END_REF]. However, this index does not account for variability in consumption, we thus, chose to develop a new continuous index to better apprehend variability in consumption.

Material and methods

Population

These analyses were based on the NutriNet-Santé study (registered at clinicaltrials.gov as NCT03335644).

NutriNet-Santé is a prospective cohort study implemented in 2009 to investigate the links between food, its determinants and health [START_REF] Hercberg | The Nutrinet-Sante Study: a web-based prospective study on the relationship between nutrition and health and determinants of dietary patterns and nutritional status[END_REF]. Participants are volunteers recruited through a media campaign and who complete regular online questionnaires. At inclusion and each year thereafter, they provided data on their health, their practices (diet, lifestyles) and their socio-demographic characteristics.

This study was conducted in accordance with the Declaration of Helsinki, and all procedures were approved by the Institutional Review Board of the French Institute for Health and Medical Research (IRB Inserm 0000388FWA00005831) and the National Commission on Informatics and Liberty (Commission Nationale de l'Informatique et des Libertés, CNIL 908450 and 909216). Electronic informed consent was obtained from all participants. Data used in this work have been described elsewhere [START_REF] Kesse-Guyot | Sustainability analysis of French dietary guidelines using multiple criteria[END_REF].

Dietary data

In 2014, a web-based semi-quantitative food frequency questionnaire (FFQ) [START_REF] Kesse-Guyot | Relative validity and reproducibility of a food frequency questionnaire designed for French adults[END_REF] was proposed to the volunteers. They were asked to report their frequency of consumption of 264 food items, as well as the portion consumed with photographs helping for the identification of portion size. In addition in this modified version of the FFQ [START_REF] Baudry | Contribution of Organic Food to the Diet in a Large Sample of French Adults (the NutriNet-Santé Cohort Study)[END_REF], for each food item, participants declared the frequency with which the item was organic through the following modalities never, rarely, half of the time, most of the time, always. Weights (0, 0.25, 0.5, 0.75, 1) were allocated to the consumption to split the total consumption of each item into organic and conventional sources. Further details and sensitivity analyses related to computation have been published elsewhere [START_REF] Baudry | Contribution of Organic Food to the Diet in a Large Sample of French Adults (the NutriNet-Santé Cohort Study)[END_REF]. Nutrient intakes were computed using a published composition table and under/over-reporters were excluded as previously described using percentiles of the ratio between energy intake and energy requirement [START_REF] Baudry | Contribution of Organic Food to the Diet in a Large Sample of French Adults (the NutriNet-Santé Cohort Study)[END_REF]. Bioavailable zinc and iron were computed using published equation [START_REF] Armah | A complete diet-based algorithm for predicting nonheme iron absorption in adults[END_REF][START_REF] Miller | A mathematical model of zinc absorption in humans as a function of dietary zinc and phytate[END_REF].

EAT-Lancet diet index (ELD-I)

Based on the definition of the universal healthy diet [START_REF] Willett | Food in the Anthropocene: the EAT-Lancet Commission on healthy diets from sustainable food systems[END_REF], component and cut-off of the EAT-Lancet diet have been already proposed in a previous work [START_REF] Knuppel | EAT-Lancet score and major health outcomes: the EPIC-Oxford study[END_REF], regarding the following 14 food groups: whole grains, tubers and starchy vegetables, vegetables, fruits, dairy foods, beef/lamb/pork, chicken and other poultry, eggs, fish, legumes, nuts, saturated oil, unsaturated oils and sweeteners. Cutoffs for each component are presented in Table 1. For sweeteners component, intake of added sugars was used. To improve the power of discrimination of the dietary index reflecting the adherence to the EAT-Lancet diet, compared to the previously developed score [START_REF] Knuppel | EAT-Lancet score and major health outcomes: the EPIC-Oxford study[END_REF], we accounted for deviation from the cut-off value. The EAT-Lancet diet index (ELD-I) for an individual j with intake for each i was computed as follows, equation (1):

ELD-I j = 100 × {∑ 𝑎 𝑖 × (𝑐𝑢𝑡-𝑜𝑓𝑓 𝑖 - 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑖𝑗 × 2500 𝐸𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑡𝑎𝑘𝑒 𝑗 ) 𝑐𝑢𝑡-𝑜𝑓𝑓 𝑖 14 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑖=1 } 14 (1)
Where i referred to on the 14 food groups and j is the individual. 𝑎 𝑖 = 1 for component to limit and 𝑎 𝑖 = -1 for component to promote.

Environmental data

Three environmental indicators were computed, namely greenhouse gas emissions (GHGe), cumulative energy demand (CED), and land occupation (LO) whose computation has been extensively described elsewhere [START_REF] Baudry | Improvement of diet sustainability with increased level of organic food in the diet: findings from the BioNutriNet cohort[END_REF]. These impacts were considered at the farm level, i.e. without considering post-harvest storage, conditioning and transport since these data were not available for organic farming. It should be however noted that most of the environmental impacts take place during the agricultural production phase [START_REF] Clune | Systematic review of greenhouse gas emissions for different fresh food categories[END_REF]. Data were derived from the DIALECTE tool [START_REF] Pointereau | DIALECTE, a comprehensive and quick tool to assess the agro-environmental performance of farms[END_REF] developed by Solagro (Toulouse, France) aiming to measure the environmental performance of conventional and organic farms. Data were completed using other data sources [START_REF] Baudry | Improvement of diet sustainability with increased level of organic food in the diet: findings from the BioNutriNet cohort[END_REF].

Environmental impacts of 92 conventional and organic raw products were assessed and converted into food items using economic, cooking and edibility coefficients [START_REF] Baudry | Improvement of diet sustainability with increased level of organic food in the diet: findings from the BioNutriNet cohort[END_REF]. Environmental data related to raw products for both organic and conventional farming have been previously disclosed [START_REF] Baudry | Improvement of diet sustainability with increased level of organic food in the diet: findings from the BioNutriNet cohort[END_REF]. Procedures for environmental estimation is summarized on Figure 1.

Intake and related environmental impacts were compiled for each food and then summed up considering organic and conventional systems to compute daily environmental indicator at the diet level.

To provide a synthetic index of available indicators in our study, we used the partial ReCiPe index, based on the ReCiPe method, a synthetic environmental impact indicator [START_REF] Kramer | Decreasing the overall environmental impact of the Dutch diet: how to find healthy and sustainable diets with limited changes[END_REF].

The pReCiPe was calculated for each individual as follows, equation ( 2):

𝑝𝑅𝑒𝐶𝑖𝑃𝑒 = 0.0459 × 𝐺𝐻𝐺𝑒 + 0.0025 × 𝐶𝐸𝐷 + 0.0439 × 𝐿𝑂 (2)
with GHGe, in kg of CO2eq/d, CED, in MJ/d and LO, in m 2 /d. The highest the pReCiPe, the highest the environmental impact.

Other data

As complementary approaches to the planetary health diet, we computed two scores: the PANDiet and the health gain score.

The PANDiet aims at estimating the probability of adequacy of nutrient intakes [START_REF] Gavelle | Patterns of Protein Food Intake Are Associated with Nutrient Adequacy in the General French Adult Population[END_REF]. This score includes an adequacy sub-score (averaging the probabilities of adequacy for 27 nutrients) and a BMI was calculated as the ratio of weight to squared height (kg/m 2 ). The data closest to the FFQ were retained.

Place of food purchase and prices were combined with consumption to estimate the daily cost of the diet.

Briefly, a database of prices by place of purchase was developed using the Kantar database (Kantar Worldpanel, n.d.), involving 20,000 French households, and further completed with collected prices for short supply chains which are not available in Kantar. This procedure has been extensively described elsewhere [START_REF] Baudry | Improvement of diet sustainability with increased level of organic food in the diet: findings from the BioNutriNet cohort[END_REF][START_REF] Seconda | Comparing nutritional, economic, and environmental performances of diets according to their levels of greenhouse gas emissions[END_REF].

Statistical analysis

For the present study, we selected the NutriNet-Santé participants who had completed the Org-FFQ in 2014 (N=37,685), with no missing covariates for weight and height (N=37,305), not detected as under-or overenergy reporter (N=35,196). Next, we selected participants with available data for computation of the cost of the diet (N=29,210).

Participants were ranked and categorized into quintiles (Q) of ELD-I reflecting the level of adherence.

Associations between sociodemographics, cost of the diet, food group consumption, nutritional and environmental indicators and quintiles of the ELD-I were assessed with ANCOVA using observed margins.

Differences across quintiles were estimated providing the means and the confidence intervals of the mean.

For environmental indicators, additional models adjusted on energy intake were performed. For statistical tests, the type I error was set at 5%. Data management and statistical analyses were conducted using SAS ® 9.4 (SAS Institute Inc.) and graphics were performed using R ® (version 3.4.2).

Results

Relationship between adherence to the EAT-Lancet diet and socio-economic characteristics of participants

The characteristics of the sample according to the quintiles of the ELD-I are presented in Table 2. Compared to participants with lower ELD-I, participants with higher ELD-I were less frequently men (Q5 vs. Q1, 34.51% vs. 62.20%), older (mean age about +6y), less often postgraduate (Q5 vs. Q1, 19.14% vs. 25.26%), less often employee or manual worker (Q5 vs. Q1, 26.36% vs. 41.03%), and more often physically active (Q5 vs. Q1, 44.10% vs. 27.15%) or with high income and had lower BMI (mean BMI about -2kg/m²). In addition, Participants with high adherence to the EAT-Lancet had the highest diet cost. However, the association appeared to be J-shaped.

Relationship between adherence to the EAT-Lancet diet and the nutritional characteristics of the diet

Nutritional characteristics according to ELD-I quintiles are presented in Table 3. High ELD-I was negatively associated with energy intake (Q5 vs. Q1, 1935 kcal/d vs. 2099 kcal/d), and positively with the health gain score as well as the PANDiet score (mean PANDiet about +11), proportion of organic food in the diet (Q5 vs. Q1, 0.45 vs. 0.17), of polyunsaturated fatty acids, plant proteins, and intake of fibers, vitamin C, vitamin B9, vitamin E, total iron and bioavailable zinc. Participants with higher ELD-I also exhibited lower % of energy intake from saturated fatty acids and total proteins and lower intakes of sodium, zinc and heme-iron.

Relationship between adherence to the EAT-Lancet diet and environmental impact indicators

The three studied environmental indicators (CED, GHGe and LO) as well as the aggregated score (pReCiPe) across quintiles of ELD-I are presented in Table 4. Negative associations were observed between the ELD-I (modeled as quintiles) and each environmental indicator, which were stronger for GHGe and LO than for CED but showed in each occasion a dropout in the 5 th quintile. High EAT-LS (Q5) compared to low (Q1)

was associated with lower greenhouse gas emissions (-56%), cumulative energy demand (-31%) and land occupation (-54%). As regards the pReCiPe index, a reduction of 63% was observed when comparing Q5 vs Q1. All associations were linear (p<0.0001).

Further adjustment for energy intake did not strongly modify the association (for pReCiPe index: -62%).

The distributions of the pReCiPe across ELD-I quintiles are shown in Figure 3. were almost zero, unsaturated oil consumption was not represented. Gradients across quintiles were expected by construction. However, the most stringent differences between quintiles were for fruits and vegetables and legumes. We can note that fish consumption appeared to be relatively similar across quintiles but a little weaker in the 5 th quintile.

Discussion

In the present analysis, conducted in a large cohort of French adults, we explored, using detailed environmental and food data, the potential environmental benefits associated with adherence to the "Universal" healthy plant-based diet as defined by the EAT-Lancet Commission [START_REF] Willett | Food in the Anthropocene: the EAT-Lancet Commission on healthy diets from sustainable food systems[END_REF]. We showed that a higher adherence to the EAT-Lancet diet, using an individual dietary score accounting for distance to cut-offs value for each component, was markedly related to a better nutritional diet quality (higher PANDiet score). It is noteworthy that a high ELD-I was associated with a slightly lower bioavailable iron intake. However, the mean value for individuals in the 5 th quintile remained higher than the estimated average losses [START_REF] Hunt | Body iron excretion by healthy men and women[END_REF]. In addition, micronutrients found in meat (such as zinc and vitamin B12)

were not particularly low. As expected, fiber intake, which has been associated with lower risk of cancer, cardiovascular diseases, type 2 diabetes and mortality [START_REF] Veronese | Dietary fiber and health outcomes: an umbrella review of systematic reviews and meta-analyses[END_REF], was high among participants with high ELD-I. Besides, plant proteins were high but ratio between plant to total protein reached 48%

only. This may be directly due to the developed methodology as the score was based on probability rather than a threshold to be reached. The overall health gain score (related to nutrient intakes) was 55% higher in participants with high ELD-I compared to those with low ELD-I, arguing for a latent alignment between environmental and nutritional dimensions of the "planetary health diet".

Dietary patterns of participants in Q5 were close to vegetarian dietary patterns, though animal products, especially meat, were not totally excluded from Q5 participants' diets. This was illustrated by an increase in the PANDiet score across quintiles despite a reduced animal-food consumption. Dietary pattern in Q5 of the ELD-I directly compared to the Q5 of PNNS-GS2 reflecting French food-based dietary guidelines (FBDG), which conceptually considered sustainability, after standardization for a 2500 kcal diet, exhibited lower consumption in all food groups specifically, drastic lower consumption of meat (-36%), eggs (-34%), dairy product (-33%), fat and sweet products (-33%), as well as whole-grain (-34%) (data not shown). This stemmed from the fact that participants with high adherence to French FBDG exhibited low energy intake.

In turn, translation for 2500 kcal led to higher levels of consumption [START_REF] Kesse-Guyot | Sustainability analysis of French dietary guidelines using multiple criteria[END_REF]. Another study, conducted in the US compared dietary guidelines for American (DAG) to EAT-Lancet diet [START_REF] Blackstone | Comparing the Recommended Eating Patterns of the EAT-Lancet Commission and Dietary Guidelines for Americans: Implications for Sustainable Nutrition[END_REF]. The authors reported some consistencies between both diets but also divergences in particular concerning whole-grain starch, beans and peas and nuts, seeds and soy-based food leading to high increase in total proteins. This is in line with a global study reporting that most of national FBDG were not compatible with environmental objectives [START_REF] Springmann | The healthiness and sustainability of national and global food based dietary guidelines: modelling study[END_REF].

Adherence to the EAT-Lancet diet was strongly and negatively related to the three studied environmental impacts, namely GHGe, CED and LO, as well as to the synthetic environmental index (pReCiPe), allowing to consider potential trade-offs. A reduction of 62% of the pReCiPe index between participants with high vs low adherence to this diet was observed. This is consistent with the findings of the recent modelling study by Springmann et al. documenting a reduction in the demand for environmental resources associated with compliance with the EAT-Lancet diet [START_REF] Springmann | The healthiness and sustainability of national and global food based dietary guidelines: modelling study[END_REF]. We also showed that participants with high adherence to the EAT-Lancet diet had a significantly higher consumption of organic food in their diet. As previously shown, organic food consumption is positively associated with plant-based diet [START_REF] Baudry | Improvement of diet sustainability with increased level of organic food in the diet: findings from the BioNutriNet cohort[END_REF][START_REF] Lacour | Environmental Impacts of Plant-Based Diets: How Does Organic Food Consumption Contribute to Environmental Sustainability?[END_REF]. Organic farming is an acknowledged agroecological production method that has been shown to be a good proxy of diet-related biodiversity due to non-use of chemical pesticides [START_REF] Tuomisto | Does organic farming reduce environmental impacts?--a meta-analysis of European research[END_REF] and is associated with reduced pesticides exposure from diet in our population [START_REF] Baudry | Urinary pesticide concentrations in French adults with low and high organic food consumption: results from the general population-based NutriNet-Santé[END_REF] and others [START_REF] Mie | Human health implications of organic food and organic agriculture[END_REF].

Our results cannot be directly compared to scientific literature since, to the best of our knowledge, no study has reported environmental values related to the EAT-Lancet diet at the individual level. Nevertheless, our findings can be interpreted in light of findings about observed diets and dietary guidelines.

In France, diet is rather rich in animal products as it has been showed that comparing 4 other European countries, French diet, based on the representative national study INCA-2 Study (2006[START_REF] Miller | A mathematical model of zinc absorption in humans as a function of dietary zinc and phytate[END_REF], exhibited the highest values for land occupation and GHGe [START_REF] Mertens | Dietary choices and environmental impact in four European countries[END_REF]. Besides, our findings are consistent with the scientific literature which have consistently documented that plant-based diets are associated with lower pressures on resources and environment in several observational or modelling studies [START_REF] Springmann | Options for keeping the food system within environmental limits[END_REF][START_REF] Chai | Which Diet Has the Least Environmental Impact on Our Planet? A Systematic Review of Vegan, Vegetarian and Omnivorous Diets[END_REF][START_REF] Aleksandrowicz | The Impacts of Dietary Change on Greenhouse Gas Emissions, Land Use, Water Use, and Health: A Systematic Review[END_REF][START_REF] Clark | Multiple health and environmental impacts of foods[END_REF][START_REF] Hallström | Environmental impact of dietary change: a systematic review[END_REF], in line with the EAT Lancet Commission objectives. For instance, Hallström reported that changes in dietary patterns (with more or less exclusion of animal products) may lead to an up to 50% reduction in GHG emissions and land use demand [START_REF] Hallström | Environmental impact of dietary change: a systematic review[END_REF]. A recent study estimating the impacts of six US consumer diets considered the EAT-Lancet diet among various dietary patterns [START_REF] Laroche | Telecoupled environmental impacts of current and alternative Western diets[END_REF]. In this work, the EAT-Lancet diet showed an intermediate land use, between the lacto-ovo-vegetarian and the low-meat diets. Apart from an expected lower demand for land occupation, this study revealed that lowmeat diets rely more heavily on the abundance and diversity of pollinators and may increase impacts on water resources at least in some countries. In addition, the Mediterranean Diet, permitting some animal food (such as diary) consumption is often advanced as a sustainable diet ( 41). Indeed, one study documented a higher GHGe (+15%) and water use (+9%) for a diet based on the Spanish dietary guidelines compared to the Mediterranean diet [START_REF] González-García | Dietary recommendations in Spain -affordability and environmental sustainability?[END_REF]. However, it has also been reported that Mediterranean Diet had a global warming potential about twice that of the vegan diet [START_REF] Castañé | Assessment of the nutritional quality and environmental impact of two food diets: A Mediterranean and a vegan diet[END_REF].

Also, a recent study was conducted to evaluate sustainability (using a synthetic score summarizing 4 environmental indicators) and to compare environmental performances of different dietary guidelines. The authors showed that environmental values differed according to the diet, Nordic Diet being the most efficient [START_REF] Grosso | Environmental Impact of Dietary Choices: Role of the Mediterranean and Other Dietary Patterns in an Italian Cohort[END_REF]. Within the same large cohort as herein, we compared various dietary patterns and their relationships with various sustainable indicators. Using a validated dietary index reflecting adherence to the new French food-based guidelines [START_REF] Chaltiel | Programme National Nutrition Santéguidelines score 2 (PNNS-GS2): development and validation of a diet quality score reflecting the 2017 French dietary guidelines[END_REF], we recently showed that high adherence was associated with markedly lower environmental impacts compared to low-adherence [START_REF] Kesse-Guyot | Sustainability analysis of French dietary guidelines using multiple criteria[END_REF], with a 50% reduction in pReCiPe and a 46% reduction in GHGE. A lower overall environmental performance for the studied indicators than the EAT-Lancet diet was, though, observed (for a 2500kcal/d diet, data not shown). This seems somehow in accordance with a recent work by Springmann et al, indicating a greater reduction in GHGEs with the global FBGDs than the national FBGDs [START_REF] Springmann | The healthiness and sustainability of national and global food based dietary guidelines: modelling study[END_REF].

In contrast, a recent review focusing on analysis of sustainability of dietary guidelines for Americans reported that healthy US dietary patterns may be responsible for similar or higher GHGe, energy and water use compared to current US diets [START_REF] Reinhardt | Systematic Review of Dietary Patterns and Sustainability in the United States[END_REF]. The review study also underlined, as expected, that plant-based dietary guidelines may only contribute to limit environmental pressures related to food systems. Consistently, the modelling study of Springmann et al., evaluating environmental and health values of food-based dietary guidelines, compared to WHO and Eat-Lancet diet, showed that most of the national food-based dietary guidelines do not allow achieving health and environmental goals [START_REF] Springmann | The healthiness and sustainability of national and global food based dietary guidelines: modelling study[END_REF].

It is noteworthy that the pReCiPe distribution across ELD-I quintiles revealed that low environmental impact was not systematically associated with healthy dietary patterns. Thus, while nutritionally healthy diet and environment preservation are overall in alignment, these dimensions however may not always go hand in hand. This observation seems in line with previous works showing that unhealthy foods, such as sweet and fat food, may exhibit low environmental impacts [START_REF] Clark | Multiple health and environmental impacts of foods[END_REF][START_REF] Perignon | Improving diet sustainability through evolution of food choices: review of epidemiological studies on the environmental impact of diets[END_REF] and observational studies showing that the lowest emitting diets are not systematically the most sustainable as regards their nutritional values [START_REF] Vieux | More sustainable European diets based on selfselection do not require exclusion of entire categories of food[END_REF]. This is of great importance from a public health point of view as it highlights the need of fostering both environment-friendly and healthy diets. It is also notable that unsustainable diets (low ELD-I) were those the least expensive, raising the issue related to affordability of sustainable and healthy diets for vulnerable population.

A global score such as the ELD-I score allows to consider recommendations as a whole as it integrates interrelations between food groups, [START_REF] Burggraf | Review of a priori dietary quality indices in relation to their construction criteria[END_REF]. Based on predefined cut-off values [START_REF] Knuppel | EAT-Lancet score and major health outcomes: the EPIC-Oxford study[END_REF],we developed a continuous index based on the overall consumption distributions to improve power of discrimination of the score and smooth distribution. It has indeed been shown that dietary scores based on binary scoring such as some Mediterranean diet scores led to little consideration of the variability in food consumption [START_REF] Burggraf | Review of a priori dietary quality indices in relation to their construction criteria[END_REF].

Some limitations of the present work should be noted. First, our sample was not representative of the general population as the NutriNet-Santé study included voluntary participants, limiting the external validation of the results. Second, our life cycle assessment did not cover the stages posterior to food production. However, production is one of the major drivers of environmental pressures within the food system, although food loss and waste along the supply chain are substantial [START_REF] Morone | Food waste: Challenges and opportunities for enhancing the emerging bio-economy[END_REF]. Few accurate information covering all the food chain exists especially for alternative production systems. Finally, we did not compute indicators related to water use, biodiversity, excess nitrogen or soil quality as data were not available.

Our study also presents major strengths that include the consideration of two different farming practices (organic and conventional systems) allowing a more accurate environmental analysis. In addition, the three computed environmental impacts reflect major environmental issues [START_REF] Kramer | Decreasing the overall environmental impact of the Dutch diet: how to find healthy and sustainable diets with limited changes[END_REF]. Furthermore, the large sample size permitted the access to a large variety of dietary patterns.

This study documented that the planetary health diet as defined by the EAT-Lancet Commission (i.e. based on a comprehensive review of the scientific literature) was associated with a higher nutritional quality and lower environmental impacts in accordance with the objectives of the EAT-Lancet report. This definition of a universal diet is crucial for aligning the health and environmental dimensions and allows comparable assessments in different contexts. It should however be noted that in some cases, unhealthy diets may also be linked to low environmental impacts arguing for dissemination of guidelines integrating both dimensions.

However, further work is needed to better test for the validation of the EAT-Lancet diet. Future optimization models works to formulate adequate nutritional intakes with reduced environmental footprints and cultural adaptation of the EAT-Lancet diet, as conducted only in Denmark, is also warranted [START_REF] Lassen | Development of a Danish Adapted Healthy Plant-Based Diet Based on the EAT[END_REF].

Relationships between the EAT-Lancet reference diet and various health outcomes should be evaluated in different settings. In addition, as regards environmental pressures, the consideration of a larger number of indicators (e.g. water use) while differentiating the type of farming system may be valuable. Finally, prospective scenarios to assess feasibility would be informative.

Acknowledgements

The authors warmly thank all the volunteers of the NutriNet-Santé cohort.

We also thank Cédric Agaesse (dietitian); Younes Esseddik (IT manager), Thi Hong Van Duong, Régis Gatibelza, Djamal Lamri, Jagatjit Mohinder and Aladi Timera (computer scientists); Julien Allegre, Nathalie Arnault, Laurent Bourhis and Fabien Szabo de Edelenyi, PhD (supervisor) (datamanager/statisticians) for their technical contribution to the NutriNet-Santé study.

Clinical Trial Registry:

The study was registered at ClinicalTrials.gov (NCT03335644).

Financial Support

Zagmutt, F.J., Pouzou, J.G., Costard, S., 2020 F (ranged from 0 to 1, where 1 represents a 100% probability that the usual intake was adequate):

'Probnorm' function in SAS, y is the mean intake, SDy the day-to-day variability of intake, n the number of dietary record days, r the nutrient reference value, SDr the interindividual variability.

2 .

 2 8,257 words moderation sub-score including 6 nutrients and 12 penalty values referring to the probabilities of exceeding upper limits of intakes as showed on the FigureThe health gain score (HS) proposed by[START_REF] Van Dooren | Exploring dietary guidelines based on ecological and nutritional values: A comparison of six dietary patterns[END_REF] is a synthetic score designed to measure health benefits of diets. It includes 10 components, as trans fatty acids were not available in our database, we therefore adapted it as follows, equation (3):where consumptions are in grams, EI denotes energy intake in kcal, and SFA denotes saturated fatty acids.Sociodemographics (gender, age, education, professional categories, monthly incomes, household composition), lifestyle data (smoking status, physical activity), anthropometrics (height and weight) were collected using follow-up questionnaires[START_REF] Mie | Human health implications of organic food and organic agriculture[END_REF]. Monthly household income was calculated per consumption unit using the following weighting ("Définition -Unité de consommation | Insee," n.d.): 1 consumption unit (CU) is attributed for the first adult in the household, 0.5 CU for other persons aged ≥14 y, and 0.3 CU for children aged <14 y. Physical activity was assessed using a short form of the French version of the International Physical Activity Questionnaire[START_REF] Hagstromer | The International Physical Activity Questionnaire (IPAQ): a study of concurrent and construct validity[END_REF] and classified as low physical activity (<30 min of physical activity equivalent to brisk walking/d), moderate physical activity (≥30 and <60 min), or high physical activity (≥60 min), according to the French guidelines.

  Figure 4 showed that despite a statistically significant lowering values across quintiles, a great variability occurs, especially in the 1 st quintile. Similar findings were showed for individual environmental indicators constituting the pReCiPe (data not shown). Pearson correlation coefficient between the ELD-I and the pReCiPe was -0.59. Food consumption (standardized for 2500 Kcal) by quintiles are presented in Figure 4. As consumption

  Abbreviations: MUFA: monounsaturated fatty acids, PUFA; polyunsaturated fatty acids, Q: quintiles;

Figure 1 :

 1 Figure 1: Steps for calculating the environmental indicators

Figure 2 :

 2 Figure 2: Computation of the PANDiet score

Figure 3 :

 3 Figure 3: pRecipe distribution across ELD-I quintiles 1

Figure 4 :

 4 Figure 4: Food consumption (intake in g/d) across ELD-I quintiles 1

  . The EAT-Lancet Commission's Dietary Composition May 536 Not Prevent Noncommunicable Disease Mortality. J. Nutr. https://doi.org/10.1093/jn/nxaa020

Table 3 : Nutritional characteristics of the sample across EAT-Lancet Diet Index quintiles, n= 29,210, NutriNet-Santé 1

 3

		Q1	Q2	Q3	Q4	Q5	P 2
	Energy intake (kcal/d)	2099 (694)	2030 (624)	1994 (606)	1947 (589)	1935 (612)	<.0001
	PANDiet (/100)	59.47 (6.20)	62.54 (6.39)	64.81 (7.01)	67.50 (7.51)	70.55 (7.15)	<.0001
	Health score (/9)	0.91 (0.22)	0.98 (0.21)	1.06 (0.22)	1.14 (0.22)	1.41 (0.38)	<.0001
	Alcohol (g/d)	9.06 (13.23)	9.55 (13.06)	9.47 (13.44)	8.27 (11.45)	6.23 (10.90)	<.0001
	Proportion of organic food 3	0.17 (0.19)	0.23 (0.22)	0.28 (0.25)	0.35 (0.28)	0.45 (0.32)	<.0001
	% Carbohydrates	35.37 (6.98)	38.39 (6.46)	39.65 (6.54)	40.80 (6.83)	43.89 (8.00)	<.0001
	% Lipids 4	42.96 (6.19)	41.89 (6.25)	41.30 (6.63)	40.91 (7.18)	39.05 (8.50)	<.0001
	% SFA 4	17.15 (3.13)	16.16 (3.12)	15.31 (3.15)	14.13 (3.09)	11.87 (3.22)	<.0001
	% MUFA 4	16.48 (3.31)	16.30 (3.52)	16.29 (3.84)	16.59 (4.31)	16.32 (5.17)	<.0001
	% PUFA 4	6.10 (1.80)	6.33 (1.99)	6.64 (2.23)	7.19 (2.61)	7.94 (3.28)	<.0001
	% Proteins 4	21.35 (3.72)	19.37 (3.02)	18.67 (3.09)	17.88 (3.23)	16.57 (3.53)	<.0001
	Proportion of plant proteins	0.22 (0.07)	0.28 (0.07)	0.32 (0.09)	0.37 (0.12)	0.48 (0.19)	<.0001
	Fibers (g/d)	16.03 (5.17)	19.38 (4.47)	22.25 (4.84)	25.74 (5.36)	33.36 (9.40)	<.0001
	Sodium (mg/d)	2717 (574)	2621(473)	2563 (483)	2430 (478)	2181 (558)	<.0001
	Vitamin B12 (µg/d)	8.17 (9.28)	6.84 (3.36)	6.40 (3.02)	5.97 (3.04)	5.15 (3.07)	<.0001
	Vitamin C (mg/d)	108.00 (58.10) 127.08 (58.28) 142.37 (62.11) 161.19 (63.51) 223.82 (105.87) <.0001
	Vitamin B9 (µg/d)	333.10 (108.71) 366.88 (86.55) 400.76 (92.15) 443.67 (99.22) 552.95 (180.94) <.0001
	Vitamin E (mg/d)	11.80 (4.58)	13.06 (4.54)	14.15 (4.62)	15.53 (4.86)	18.34 (5.67)	<.0001
	Calcium (mg/d)	1069 (323)	1119 (316)	1142 (319)	1138 (327)	1107 (331)	<.0001
	Iron	14.33 (3.79)	14.52 (3.15)	15.26 (3.47)	16.05 (3.68)	17.26 (4.10)	<.0001
	Heme-iron	2.17 (1.70)	1.49 (0.66)	1.23 (0.59)	1.02 (0.60)	0.74 (0.60)	<.0001
	Zinc	14.25 (3.25)	12.89 (2.24)	12.58 (2.21)	12.30 (2.26)	11.78 (2.32)	<.0001

Cut-offs for a

kcal diet based on Knuppel et. al[START_REF] Knuppel | EAT-Lancet score and major health outcomes: the EPIC-Oxford study[END_REF].

The NutriNet-Santé study was supported by the following public institutions : Ministère de la Santé, Santé Publique France, Institut National de la Santé et de la Recherche Médicale (INSERM), Institut national de recherche pour l'agriculture, l'alimentation et l'environnement (INRAe), Conservatoire National des Arts et Métiers (CNAM) and Université Sorbonne Paris Nord. Researchers were independent from funders.

Funders had no role in the study design, the collection, analysis, and interpretation of data, the writing of the report, and the decision to submit the article for publication.

Data sharing statement

Data of the study are protected under the protection of health data regulation set by the French National Commission for Information Technology and Liberties (Commission Nationale de l'Informatique et des Libertés, CNIL). The data are available upon request to the study's operational manager, Nathalie Pecollo (n.pecollo@eren.smbh.univ-paris13.fr), for review by the steering committee of the NutriNet-Santé study.

Tables

 Table 1: