Carlos Conca
email: cconca@dim.uchile.cl

AND Antoine Laurain
email: laurain@math.hu-berlin.de

Rajesh Mahadevan
email: rmahadevan@udec.cl

MINIMIZATION OF THE GROUND STATE FOR TWO PHASE CONDUCTORS IN LOW CONTRAST REGIME

In this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain, in order to minimize the first eigenvalue (the ground state) of a Dirichlet operator. It is known, when the domain is a ball, that the solution is radial, and it was conjectured that the optimal distribution of the materials consists in putting the material with the highest conductivity in a ball around the center. We show that this conjecture is not true in general. For this, we consider the particular case where the two conductivities are close to each other (low contrast regime) and we perform an asymptotic expansion with respect to the difference of conductivities. We find that the optimal solution is the union of a ball and an outer ring when the amount of the material with the higher density is large enough.

INTRODUCTION

Let Ω be a bounded domain in R d which is to be called the design region. Let m be a given positive number, 0 < m < |Ω|, where |Ω| is the Lebesgue measure of the design region Ω. Two materials with conductivities α and β (0 < α < β) are distributed in arbitrary disjoint measurable subsets A and B, respectively, of Ω so that A ∪ B = Ω and |B| = m. Consider the two-phases eigenvalue problem:

-div(σ∇u) =λu in Ω, [START_REF] Alvino | On optimization problems with prescribed rearrangements[END_REF] u =0 on ∂Ω, [START_REF] Conca | An extremal eigenvalue problem for a two-phase conductor in a ball[END_REF] with the conductivity σ = αχ A + βχ B . Let λ be the first eigenvalue of (1)-( 2) and u the associated eigenvector. The variational formulation for λ is [START_REF] Conca | Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball[END_REF] λ = min

u∈H 1 0 (Ω) Ω σ|∇u| 2 Ω u 2 = min u∈H 1 0 (Ω), u 2=1 Ω σ|∇u| 2 ,
where u 2 denotes the L 2 -norm of u. In this paper the set Ω is fixed and we are interested in the dependence of λ on A and B. Since A = Ω \ B, λ may be described as a function of B and we write λ = λ(B). We consider the problem of minimizing λ(B) with the constraint that the two phases are to be distributed in fixed proportions: minimize λ(B) [START_REF] Cox | Extremal eigenvalue problems for two-phase conductors[END_REF] subject to B ∈ B [START_REF] Dambrine | On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems[END_REF] where [START_REF] Gilbarg | Elliptic Partial Equations of Second Order[END_REF] B = {B ⊂ Ω, B measurable, |B| = m}

The existence of a solution to the problem (4)-( 6) remains an open question. In general, one may evidence microstructural patterns in relation to minimizing sequences and the original problem may have to be relaxed to include microstructural designs. Existence of a solution and optimality conditions in the class of relaxed designs has been discussed in Cox and Lipton [START_REF] Cox | Extremal eigenvalue problems for two-phase conductors[END_REF]. However, the original problem ( 4)-( 6) may still have a solution for particular geometries as is the case when Ω is a ball. When Ω = B(0, R) is a ball, the existence of a radially symmetric optimal set has been proved in [START_REF] Alvino | On optimization problems with prescribed rearrangements[END_REF], using rearrangement techniques and a comparison result for Hamilton-Jacobi equations and later, only using rearrangement techniques in [START_REF] Conca | An extremal eigenvalue problem for a two-phase conductor in a ball[END_REF]. Even, in this case an explicit solution to the problem is not known. It was conjectured in [START_REF] Conca | An extremal eigenvalue problem for a two-phase conductor in a ball[END_REF][START_REF] Conca | Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball[END_REF], for higher dimensions, that the solution B * to this problem is a ball B(0, R * ) like in the one-dimensional case [START_REF] Krein | On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability[END_REF], a result known since the 1950's. This conjecture has been recently reinforced by numerical tests in [START_REF] Conca | Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball[END_REF] and by the result in [START_REF] Dambrine | On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems[END_REF], where it is shown, using second order shape derivative calculus, that such a configuration is a local minimum for the problem when the volume constraint m is small enough.

In spite of these results, we prove in this paper that the conjecture is not true in general. Indeed, the optimal domain B * cannot be a ball when α and β are close to each other and m is sufficiently large (cf. Theorem 3). The theoretical base for this result is provided by an asymptotic expansion of the eigenvalue with respect to β -α as β → α, which allows us to approximate ( 4)-( 6) by a simpler optimization problem (cf. Theorem 2). This is done in Section 2. Through this asymptotic formulation we are not only able to show that the previous conjecture is false but also we are able to compute numerical approximations of the solution in design domains other than balls. The numerical results are presented in Section 4. Another main feature of the paper is the proposal, in Section 3, of a descent algorithm to solve the problem in general. This also permits to establish some necessary optimality conditions and allows us to deduce certain features of the optimal solution.

OPTIMAL SETS FOR SMALL CONDUCTIVITY GAP

2.1. Asymptotic expansion. In this section we shall look at the problem of minimization of the first eigenvalue in the special case where the conductivities of the two materials, α and β, are close to each other (i.e. are in low contrast regime). Thus, we assume that β = β ε := α+ε with ε > 0 converging eventually to zero. If the material with conductivity β ε occupies the sub-domain B of Ω, the conductivity coefficient is, in this case, [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] σ = σ ε (B) :

= αχ A + β ε χ B = α + εχ B .
Let λ ε (B) be the first eigenvalue in the problem

-div(σ ε (B)∇u ε ) =λ ε (B) u ε in Ω, (8) u ε =0 on ∂Ω (9)
for the conductivity σ ε (B). It is well-known, from the Kreȋn-Rutman theorem [START_REF] Krein | Linear operators leaving invariant a cone in a banach space[END_REF], that the first eigenvalue of a linear elliptic operator is simple and the corresponding eigenfunction is of constant sign (and is the only eigenvalue whose eigenfunction does not change sign). So, we can choose the eigenfunction u ε = u ε (B) corresponding to λ ε (B) to be positive and normalize it using the condition ( 10)

Ω (u ε ) 2 = 1,
In this way, u ε is uniquely defined. We affirm that, for fixed B, both λ ε (B) and u ε (B) depend analytically on the parameter ε. This result is classical in the perturbation theory of eigenvalues and follows readily, for instance, from Theorem 3, Chapter 2.5 of Rellich [START_REF] Rellich | Perturbation Theory of Eigenvalue Problems[END_REF]. This justifies the ansätze

λ ε (B) = λ 0 (B) + ελ 1 (B) + . . . (11) u ε (B) = v 0 (B) + εv 1 (B) + . . . ( 12
)
The convergence of the series in [START_REF] Krein | On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability[END_REF] holds in the space H 1 0 (Ω). We first make some useful observations about the the terms in ansätze ( 11)- [START_REF] Krein | On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability[END_REF]. Proposition 1. In ansätze [START_REF] Kondratiev | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF] and [START_REF] Krein | On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability[END_REF], the terms λ 0 (B) and v 0 (B) are independent of B. In fact, λ 0 (B) = λ 0 is the first eigenvalue in the problem

-α∆v 0 =λ 0 v 0 in Ω, (13) v 0 =0 on ∂Ω . ( 14
)
The function v 0 is the positive eigenfunction corresponding to λ 0 and satisfies the normalization condition Ω v 2 0 = 1. Proof. In view of the analytic dependence of λ ε (B) and u ε (B) on ε, it follows that λ 0 (B) is the limit, as ε → 0, of λ ε (B) and that v 0 (B) is the limit of u ε (B) in H 1 0 (Ω) as ε → 0. Recalling that the eigenfunctions u ε (B) are positive it follows that v 0 (B) is non-negative. Passing to the limit in ( 8)- [START_REF] Kao | Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains[END_REF], as ε → 0, we obtain that λ 0 (B) and v 0 (B) solve the eigenvalue problem ( 13)- [START_REF] Lou | On singular sets of local solutions to p-laplace equations[END_REF] and Ω (v 0 (B))

2 = 1. As v 0 (B) is a positive eigenfunction,
it follows by the Kreȋn-Rutman theorem that λ 0 is necessarily the first eigenvalue of the eigenvalue problem ( 13)-( 14). Thus, λ 0 (B) and v 0 (B) are independent of B and shall be denoted by λ 0 and v 0 , respectively.

Proposition 2. In the ansatz (11), λ 1 (B) is given explicitly in terms of v 0 as follows

(15) λ 1 (B) = B |∇v 0 | 2 dx .
The following orthogonality relations hold true

Ω v 0 v 1 (B) dx = 0 = Ω ∇v 0 • ∇v 1 (B) dx . (16)
Proof. The term λ 1 (B) in the ansatz [START_REF] Kondratiev | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF] is the derivative of λ ε (B) with respect to ε at ε = 0, whereas the term v 1 (B) in the ansatz [START_REF] Krein | On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability[END_REF] is the derivative of u ε (B) with respect to ε at ε = 0. Differentiating the equations ( 8)- [START_REF] Kao | Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains[END_REF] with respect to ε at ε = 0, we obtain the equations

-div(α∇v 1 (B)) -λ 0 v 1 (B) = div(χ B ∇v 0 ) + λ 1 (B) v 0 in Ω, (17) v 1 (B) =0 on ∂Ω (18)
and the first of the orthogonality relations in [START_REF] Rellich | Perturbation Theory of Eigenvalue Problems[END_REF]. We have seen in Proposition 1 that λ 0 is the first eigenvalue of the problem (13)- [START_REF] Lou | On singular sets of local solutions to p-laplace equations[END_REF] and is simple, the eigenspace being generated by the eigenfunction v 0 . Taking v 1 (B) as a test function in ( 13)-( 14) and using the first orthogonality relation in [START_REF] Rellich | Perturbation Theory of Eigenvalue Problems[END_REF], we obtain the second orthogonality relation in [START_REF] Rellich | Perturbation Theory of Eigenvalue Problems[END_REF]. Finally, the system (17)-(18) admits a solution, by the Fredholm alternative, if and only if the right hand side is orthogonal to the eigenfunction v 0 . This condition leads to the relation

Ω div(χ B ∇v 0 )v 0 dx + λ 1 (B) Ω v 2 0 dx = 0.
As Ω v 2 0 = 1, we obtain

λ 1 (B) = - Ω div(χ B ∇v 0 )v 0 dx = - ∂Ω χ B v 0 ∇v 0 • n dS + B |∇v 0 | 2 dx = B |∇v 0 | 2 dx .
Let us denote by

λε (B) = λ ε (B) -λ 0 -ελ 1 (B)
the remainder in the ansatz [START_REF] Kondratiev | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF]. Although λε (B) is of order ε 2 for fixed B, we need estimates for λε (B) which are uniform with respect to B. This is given by the following theorem.

Theorem 1. For ε > 0 sufficiently small, there exists a constant C independent of ε and B such that

(19) | λε (B)| ≤ C ε 3 2 ∀B ∈ B .
Proof. For the sake of clarity we divide the proof into several steps. Let ε > 0 be a constant which is small compared to 1. In what follows, we use C to denote a generic constant independent of ε and B.

STEP 1: We first show that (20)

λ ε (B) ≤ C , u ε (B) H 1 0 (Ω) ≤ C .
The first inequality in (20) follows readily from the variational characterization of λ ε (B). Indeed, choosing a function ϕ ∈ H 1 0 (Ω) with Ω ϕ 2 = 1, we have

λ ε (B) = inf Ω σ ε (B)∇u • ∇u dx : u ∈ H 1 0 (Ω), Ω u 2 = 1 (21) ≤ Ω σ ε (B)∇ϕ • ∇ϕ dx ≤ (α + 1) Ω ∇ϕ • ∇ϕ dx
which proves the first estimate in (20). Now, using the uniform bound for λ ε (B) and using the fact that the coefficients σ ε (B) are uniformly elliptic, we have

α u ε (B) 2 H 1 0 (Ω) ≤ Ω σ ε (B)∇u ε (B) • ∇u ε (B) dx = λ ε (B) ≤ C (22)
which proves the second estimate in (20). As σ ε (B) ≥ α for all ε > 0, it follows from the variational characterization (21) of λ ε (B) that (24) λ 0 ≤ λ ε (B) for all ε > 0 and for all measurable B ⊂ Ω .

On the other hand, using the variational characterization (21) and the fact that

λ 0 = Ω α∇v 0 • ∇v 0 dx,
we get the following estimate

λ ε (B) -λ 0 ≤ Ω σ ε (B)∇v 0 • ∇v 0 dx - Ω α∇v 0 • ∇v 0 dx = ε Ω χ B ∇v 0 • ∇v 0 dx . ( 25
)
The claim (23) follows from (24) and (25). STEP 3: Now, we use the above estimate to show that

(26) u ε (B) -v 0 H 1 0 (Ω) ≤ C √ ε .
To begin with, we have

λ ε (B) -λ 0 = Ω σ ε (B)∇u ε (B) • ∇u ε (B) dx - Ω α∇v 0 • ∇v 0 dx = ε Ω χ B ∇u ε (B) • ∇u ε (B) dx +α Ω ∇u ε (B) • ∇u ε (B) dx - Ω ∇v 0 • ∇v 0 dx = ε Ω χ B ∇u ε (B) • ∇u ε (B) dx + 2α Ω ∇v 0 • ∇(u ε (B) -v 0 ) dx + α Ω ∇(u ε (B) -v 0 ) • ∇(u ε (B) -v 0 ) dx .
Rewriting the previous equality we get

α Ω |∇(u ε (B) -v 0 )| 2 dx + 2α Ω ∇v 0 • ∇(u ε (B) -v 0 ) dx = λ ε (B) -λ 0 -ε Ω χ B ∇u ε (B) • ∇u ε (B) dx . (27)
Finally we obtain

α Ω |∇(u ε (B) -v 0 )| 2 dx + 2α Ω ∇v 0 • ∇(u ε (B) -v 0 ) dx ≤ |λ ε (B) -λ 0 | + ε Ω χ B ∇u ε (B) • ∇u ε (B) dx ≤ Cε (28)
where the last inequality is a consequence of ( 23) and (20). Dividing by ε and passing to the limit in (28), we have

(29) lim ε→0 α Ω |∇(u ε (B) -v 0 )| 2 dx ε + 2α Ω ∇v 0 • ∇v 1 (B) dx ≤ C .
In view of the second orthogonality relation in [START_REF] Rellich | Perturbation Theory of Eigenvalue Problems[END_REF] we conclude that (26) holds.

STEP 4: Finally, we show the estimate for the remainder λε (B)

(30) | λε (B)| = |λ ε (B) -λ 0 -ελ 1 (B)| ≤ C ε 3 2 .
On the one hand, we observe that

λ 0 + ελ 1 (B) -λ ε (B) ≥ Ω α∇v 0 • ∇v 0 dx + ε Ω χ B ∇v 0 • ∇v 0 dx - Ω σ ε (B)∇v 0 • ∇v 0 dx = 0 . (31)
On the other hand, we have

λ 0 + ελ 1 (B) -λ ε (B) ≤ Ω α∇u ε (B) • ∇u ε (B) dx + ε Ω χ B ∇v 0 • ∇v 0 dx - Ω σ ε ∇u ε (B) • ∇u ε (B) dx (32) ≤ ε Ω χ B ∇v 0 • ∇v 0 dx -ε Ω χ B ∇u ε (B) • ∇u ε (B) dx = ε Ω χ B ∇(v 0 -u ε (B)) • ∇(v 0 -u ε (B)) dx + 2ε Ω χ B ∇u ε (B) • ∇(v 0 -u ε (B)) dx .v (33)
So, by the result of Step 3, it follows from (31) and (33) that

|λ 0 + ελ 1 (B) -λ ε (B)| ≤ ε v 0 -u ε (B) 2 H 1 0 (Ω) + 2Cε v 0 -u ε (B) H 1 0 (Ω) ≤ Cε 2 + Cε 3 2 ≤ Cε 3 2 .
This completes the proof of the estimate (19).

Corollary 1. Let λ ε (B) be the first eigenvalue in (8)-( 9), λ 0 the first eigenvalue of problem (13)-( 14) and let λ 1 (B) be as in Proposition 2. Then, we have

(34) inf B∈B λ ε (B) -λ 0 -ε inf B∈B λ 1 (B) ≤ Cε 3 2 ,
where C is a constant independent of ε and B ∈ B

Proof. The proof is immediate from the estimate (19).

Corollary 2. Let C be the constant independent of ε and B ∈ B appearing in (19) of Theorem 1. If B ε ∈ B is a minimizer of λ ε (•) then we have the following estimate:

(35) λ 1 (B ε ) -inf B∈B λ 1 (B) ≤ 2 Cε 1 2 .
Proof. Let B ε ∈ B be a minimizer for λ ε (B). Then, we have

ε λ 1 (B ε ) -inf B∈B λ 1 (B) = (λ ε (B ε ) -λ 0 -ελ 1 (B ε )) -(λ ε (B ε ) -λ 0 -ε inf B∈B λ 1 (B)) ≤ |λ ε (B ε ) -λ 0 -ελ 1 (B ε )| + inf B∈B λ ε (B) -λ 0 -ε inf B∈B λ 1 (B) ≤ 2 Cε 3 2 (36)
where the inequalities follow from (30) and (34). Thus, we have obtained (35).

Remark 1. The Corollary 1 tells us that, in low contrast regimes, asymptotically the minimum value of λ ε (•) can be calculated approximately by calculating the minimum of λ 1 (•) which is easily achieved using Theorem 2. In addition, the previous corollary gives us to understand that a minimizer for λ ε (•) is approximately a minimizer for λ 1 (•) when ε is small. It can be seen, by similar arguments, that a minimizer for λ 1 (•) is approximately a minimizer for λ ε (•) when ε is small. This remark will be used later in Section 4 to determine numerical approximations of the set B minimizing λ(B) in low contrast regime.

We prove the following theorem which provides a characterization of the minimizer of λ 1 (•), in terms of the level sets of the gradient of v 0 .

lim k→∞ f (c k ) = lim k→∞ |{x ∈ Ω : |∇v 0 (x)| ≤ c k }| = | ∩ k∈N {x ∈ Ω : |∇v 0 (x)| ≤ c k }| = |{x ∈ Ω : |∇v 0 (x)| ≤ c * }| = f (c * ), so that f (c * ) ≥ m.
In a similar way we have |{x :

|∇v 0 (x)| < c * }| ≤ m, indeed let c k < c * be an increasing sequence with c k → c * . On one hand f (c k ) < m and lim k→∞ f (c k ) ≤ m.
On the other hand

lim k→∞ f (c k ) = lim k→∞ |{x ∈ Ω : |∇v 0 (x)| ≤ c k }| = | ∪ k∈N {x ∈ Ω : |∇v 0 (x)| ≤ c k }| = |{x ∈ Ω : |∇v 0 (x)| < c * }|, so that |{x ∈ Ω : |∇v 0 (x)| < c * }| ≤ m. If B is a measurable set such that {x : |∇v 0 (x)| < c * } ⊂ B ⊂ {x : |∇v 0 (x)| ≤ c * }
and |B| = m then B is an optimal solution for the problem of minimization of λ 1 (•) over B. Indeed if D is any measurable subset of Ω such that |D| = m, we shall have

|B ∩ D c | = |B c ∩ D|. Therefore, D |∇v 0 | 2 dx = D∩B |∇v 0 | 2 dx + D∩B c |∇v 0 | 2 dx ≥ D∩B |∇v 0 | 2 dx + D c ∩B |∇v 0 | 2 dx = B |∇v 0 | 2 dx as |∇v 0 | ≥ c * on D ∩ B c whereas |∇v 0 | ≤ c * on D c ∩ B.
Remark 2. If {x : |∇v 0 (x)| = c * } is of measure zero, then the unique solution (up to a set of measure zero) is the set {x : |∇v 0 (x)| < c * } which shall also be open if Ω is a sufficiently smooth domain (as by Theorem 8.14 [START_REF] Gilbarg | Elliptic Partial Equations of Second Order[END_REF] it can be concluded that ∇v 0 is of class C 1 ). In view of Proposition 3 proved below, this will be the case when Ω is a disk.

2.2.

Disproving the disk conjecture. In this section we show that, for ε sufficiently small, the distribution of materials which corresponds to placing material β in a ball around the centre of Ω when Ω is a ball is not optimal for the problem ( 4)-( 6).

In particular, let Ω = B(0, 1) ⊂ R d be the ball of center 0 and radius 1. Then, the solution v 0 of ( 13)-( 14) is radial and smooth. By setting w 0 (|x|) := v 0 (x), equation ( 13)-( 14) becomes, using the Laplacian in polar (r, θ) or spherical (r, θ, ϕ) coordinates, for d = 2, 3 r 2 w 0 (r) + (d -1)rw 0 (r) + r 2 λ 0 α w 0 (r) = 0, (37)

w 0 (0) = 0, w 0 (1) = 0. ( 38
)
where the boundary conditions (38) correspond to the continuity of the gradient at the origin and the Dirichlet condition on the boundary, respectively. The solution of this equation is

w 0 (r) = J 0 (η d r) if d = 2, ( 39
)
w 0 (r) = j 0 (η d r) if d = 3, (40)
where J 0 and j 0 denote Bessel functions of the first and second kind, respectively and η d (d=2,3) are their respective zeros. We introduce the notation w 1 (r) := -w 0 (r). The function w 0 is decreasing and so w 1 is positive. The behaviour of the functions w 0 and w 1 is depicted in Figure 1. In the case m > ω d (r 0 d ) d , the equation w 1 (r) = c * has two solutions ξ 0 and ξ 1 in (0, 1). Then the solution set is

B * = B(0, ξ 0 ) ∪ B(0, 1) \ B(0, ξ 1 )
which completes our proof.

In [START_REF] Conca | Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball[END_REF] it was conjectured, based on numerical tests, that the infimum of the first eigenvalue of ( 4)-( 5) is attained when the material with the highest conductivity is placed in a concentric disk in the center of the domain. We prove that this conjecture is false, at least in two-or three-dimensional space, for β close to α and when the proportion of the material β exceeds a certain quantity. Theorem 3. When Ω = B(0, 1), for β = α + ε sufficiently close to α and given an m > ω d (r 0 d ) d , the distribution of the materials wherein the material with the higher conductivity β is placed in a concentric disk in the center of the domain is not optimal for the problem (4)- [START_REF] Dambrine | On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems[END_REF].

Proof. Our proof is based on Corollary 2 and Proposition 4. We shall assume that m > ω d (r 0 d ) d . Let B be the minimizer of λ 1 (•) over B ∈ B, which according to Proposition 4, is the union of a ball B(0, ξ 0 ) and a ring or shell B(0, 1) \ B(0, ξ 1 ). Let B m be the concentric ball in B(0, 1) having the volume m. We show that B m cannot be the minimizer of λ ε (•) in the problem (4)-( 5) whenever β = α + ε is less than a certain quantity, to be precise, if 2Cε

1 2 < |λ 1 (B m ) -λ 1 (B * )| .
For such an ε > 0, if we assume that B m is a minimizer for λ ε (•) in the problem (4)-( 5), then, by (35) of Corollary 2, we obtain

|λ 1 (B m ) -λ 1 (B * )| ≤ 2Cε 1 2
. This is clearly a contradiction. So, for ε > 0 as above, B m cannot be a minimizer of λ ε (•) in the problem (4)-(5).

DESCENT ALGORITHM AND NECESSARY OPTIMALITY CONDITION

In this section we consider domains Ω with any shape, and not only the case of a disk as in the previous section. We describe an algorithm which, starting from a given initial measurable set B 0 , allows to find a new measurable set B 1 such that the first eigenvalue λ 1 is decreased, i.e.

λ(B 1 ) ≤ λ(B 0 ).

Algorithms based on a similar principle have been used successfully to minimize eigenvalues in problems with indefinite weight [START_REF] Hintermüller | Principal eigenvalue minimization for an elliptic problem with indefinite weight and Robin boundary conditions[END_REF][START_REF] Kao | Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains[END_REF]. It may be used, as we shall see in Subsection 4.2, to obtain numerical results. This algorithm allows to derive necessary optimality conditions; see Corollary 3, which in turn may be used to derive properties of the optimal set B for particular geometries such as symmetric domains or polygonal domains. More details may be found at the end of this section. Given an initial measurable set B 0 , let u B0 and λ(B 0 ) denote the first eigenvector and eigenvalue, respectively, for problem ( 1 . Also, M (B 0 , c) → 0 as c → 0 as it is known that the set of critical points for u B0 is of measure zero [START_REF] Lou | On singular sets of local solutions to p-laplace equations[END_REF]. The right continuity may be seen from the following

M (B 0 , c+) -M (B 0 , c) = lim cn c M (B 0 , c n ) -M (B 0 , c) = lim cn c |{x : c < |∇u B0 (x)| ≤ c n }| = | ∩ n {x : c < |∇u B0 (x)| ≤ c n }| = 0
as the latter intersection is empty.

On the other hand,

M (B 0 , c) -M (B 0 , c-) = lim cn c M (B 0 , c) -M (B 0 , c n ) = lim cn c |{x : c n < |∇u B0 (x)| ≤ c}| = | ∩ n {x : c n < |∇u B0 (x)| ≤ c}| = |{x : |∇u B0 (x)| = c}|,
which yields the assertion on left continuity.

Now define (42)

c 0 := inf{c : M (B 0 , c) ≥ m}.
As in the proof of Theorem 2, it may be shown that |{x :

|∇u B0 (x)| ≤ c 0 }| ≥ m and |{x : |∇u B0 (x)| < c 0 }| ≤ m. Let B 1 be any measurable subset of Ω which satisfies {x : |∇u B0 (x)| < c 0 } ⊂ B 1 ⊂ {x : |∇u B0 (x)| ≤ c 0 } and |B 1 | = m.
We have the following result Theorem 4. Given an initial measurable set B 0 ⊂ Ω with |B 0 | = m, let c 0 be as defined in (42) and let B 1 be a measurable set as above having measure m. Then we have λ(B 1 ) ≤ λ(B 0 ). Furthermore, if M (B 0 , c) is continuous, then equality holds if and only if B 1 = B 0 almost everywhere.

Proof. Consider the decompositions

B 0 = (B 0 ∩ B 1 ) ∪ (B 0 ∩ B c 1 ), B 1 = (B 0 ∩ B 1 ) ∪ (B c 0 ∩ B 1 ). Since |B 0 | = |B 1 | = m we have with the above decompositions |B 0 ∩ B c 1 | = |B c 0 ∩ B 1 |. Noticing that |∇u B0 | ≥ c 0 on B c 1 and |∇u B0 | ≤ c 0 on B 1 , we may write B0 |∇u B0 | 2 = B0∩B1 |∇u B0 | 2 + B0∩B c 1 |∇u B0 | 2 ≥ B0∩B1 |∇u B0 | 2 + c 2 0 |B 0 ∩ B c 1 | (43) = B0∩B1 |∇u B0 | 2 + c 2 0 |B c 0 ∩ B 1 | ≥ B0∩B1 |∇u B0 | 2 + B c 0 ∩B1 |∇u B0 | 2 = B1 |∇u B0 | 2 , (44) Therefore λ(B 0 ) = α Ω |∇u B0 | 2 + (β -α) B0 |∇u B0 | 2 ≥ α Ω |∇u B0 | 2 + (β -α) B1 |∇u B0 | 2 (45) ≥ min u∈H 1 0 (Ω),||u||2=1 α Ω |∇u| 2 + (β -α) B1 |∇u| 2 = λ(B 1 ) , ( 46
)
and the inequality is proved. Now, let us assume that M (B 0 , c) is continuous. In view of Lemma 1, the continuity of M (B 0 , c) implies that the set {x : |∇u B0 (x)| = c 0 } has zero measure, so that B 1 can be taken to be {x : |∇u B0 (x)| ≤ c 0 }. The equality λ(B 1 ) = λ(B 0 ) holds if and only if equality holds in (45), (46). In particular, equality in (45) holds only if equality holds in (43) and (44). Thus, as |∇u B0 (x)| > c 0 on B c 1 , for equality to hold in (43) we need that |B 0 ∩ B c 1 | = 0. Consequently, we also have |B c 0 ∩ B 1 | = 0. Thus, we have shown that B 1 = B 0 except for a set of measure zero. Remark 3. Using Theorem 4, we may obtain a sequence of domains B n such that

λ(B n ) ≤ λ(B n-1 ).
However, it is not a priori guaranteed that B n converges to an admissible set B * in some topology for which the map B → λ(B) is lower semicontinuous. Even by supposing this, we cannot say that B * is a global optimum for the problem.

The following corollary derives immediately from Theorem 4.

Corollary 3 (Necessary optimality condition). If a measurable set B * is optimal for the problem (4) -( 5) and if M (B * , c) is continuous at c * for c * defined analogously as in (42) with B * replacing B 0 , then up to a set of measure zero, B * is equal to the level set {x :

|∇u B * (x)| ≤ c * }.
Proof. It is enough to apply the previous theorem taking B * in place of B 0 , c * instead of c 0 and then to take

B 1 = {x : |∇u B * (x)| ≤ c * }, which is allowed due to the continuity of M (B * , •) at c * . Then, λ(B 1 ) ≤ λ(B *
) and B * is optimal, so we have the equality λ(B 1 ) = λ(B * ). The conclusion follows, as by Theorem 4, equality holds only if B 1 is almost everywhere equal to B * .

Assuming that the hypotheses of Corollary 3 are satisfied we obtain the following results for certain geometries.

The disk case. In the case Ω = B(0, R) the optimal set B * should include the origin. Indeed, in [START_REF] Conca | An extremal eigenvalue problem for a two-phase conductor in a ball[END_REF], it is shown that the optimal domain is radially symmetric. The regularity and the radial symmetry of the solution imply that the gradient of u vanishes at the origin 0 and therefore, by Corollary 3, it follows that 0 ∈ B * .

The ring or torus case. If one is able to show the radial symmetry of the solution as in the disk case, then due to the Dirichlet condition and the positivity of the solution, it is clear that the gradient of u vanishes at one point along a radius of the domain and by radial symmetry, the gradient of u vanishes on a whole circle whose center is the center of the ring or torus. Using Corollary 3 we obtain that this circle is in B * . This property may be observed for instance in Figure 5 for different ratios m/|Ω|. Domains with corners in two dimensions. In this case the optimal set B * contains a neighbourhood of the corners with angle smaller than π while its complement A * = Ω\B * contains a neighbourhood of the corners with angle greater than π. Indeed, let P ∈ ∂Ω be a conical (corner) point of Ω and denote ϑ the associated angle at this corner. The classical theory of solution of elliptic partial differential equations in non-smooth domains [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF][START_REF] Grisvard | Singularities in boundary value problems[END_REF][START_REF] Kondratiev | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF] establishes that in view of the Dirichlet boundary conditions, u B * may be written as

u B * (x) = cr π/ϑ sin πθ ϑ + u s (x),
where c is the coefficient of the singularity which depends on the geometry of the domain, θ corresponds to polar coordinates with center P and θ = 0 or θ = ϑ on the tangents to ∂Ω at P , and u s ∈ H 2 (Ω) satisfies u s (x) = O(r π/ϑ+1 ) as x → P . Therefore Symmetrical domains. If the domain Ω has symmetries, then it is probably possible to show that the optimal domain has the same symmetries, as in [START_REF] Conca | An extremal eigenvalue problem for a two-phase conductor in a ball[END_REF]. Thus, if Ω ⊂ R d has d independent hyperplanes of symmetry, it is expected that the solution set B * includes the point which is the intersection of these hyperplanes. If Ω is a square for instance, as in Figure 2, the center of this square belongs to B * .

|∇u B * (x)| = O(r π/ϑ-1 ) as x → P and |∇u B * (x)| → 0 if ϑ < π, |∇u B * (x)| → ∞ if ϑ > π as x → P.
These properties are corroborated by numerical results shown in Section 4.1 for such domains in the case when α and β are close enough.

NUMERICAL RESULTS

The aims of this section are twofold. On the one hand we would like to obtain approximate numerical solutions of problem ( 4)-( 5) by applying Remark 1 and Theorem 2 in the case of nearly equal conductivities. On the other hand, in the general case, we would like to explore the numerical utility of the algorithm described in Section 3.

4.1.

In low contrast regime. In this section we compute numerical approximations of solutions of the optimization problem (4)-( 5) for general geometries Ω, under the assumption of low contrast regime, i.e. β = β ε = α + ε for small ε. Following Remark 1, the solution B * of the auxiliary problem corresponding to the minimization of λ 1 (•) (see Theorem 2) is then an approximate solution for (4)-( 6) for small ε. For the computation of B * , we compute the function f (c) defined in the proof of Theorem 2 and look for c * such that f (c * ) = m. Numerically, this may be achieved by simple dichotomy for instance.

In Figures 2 to 6, the nearly optimal distribution B * for various geometries Ω and values m are plotted. We also plot the isolines of |∇v 0 | 2 , where v 0 is the solution of ( 13)-( 14), i.e. v 0 is the first eigenvector with constant conductivity α. In all these examples, the features mentioned at the end of Section 3 are seen to hold. For domains with salient angles, such as in Figures 2 to 4, the set B * contains a neighbourhood of these corners. On the contrary, reentrant corners such as in Figure 4 are always in A * since the gradient is unbounded at these points. We also observe that the set B * always contains a point (or more in the case of radial symmetry as in Figure 5) inside the domain. This interior point corresponds to the place where the gradient of v 0 vanishes and thus is expected to be in B * according to Corollary 3. For domains without the optimal set may or may not touch the boundary as show Figures 5 and6.

In general the optimal set B * seem to have a complex structure and a certain regularity, except for some particular values of m as in Figure 2(c). 4.2. The disk case for general β. We shall apply the algorithm of Section 3 to get an idea of the optimal solution in the case of a disk for any value of β. As was commented in Remark 3, the convergence of the sequence B n is not guaranteed and even if the domains B n do converge to a domain B * , this domain is not necessarily optimal for problem (4)- [START_REF] Dambrine | On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems[END_REF]. So as to make the algorithm more effective, we include a second step, wherein we make local perturbations to the B * in some descent directions with the aid of the shape derivative obtained in [START_REF] Conca | Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball[END_REF] (cf. the same for more details). By repeating these two steps, successively, it is plausible that these iterations lead to a global optimum.

One of the main difficulty when solving the discrete version of ( 4)-( 5) comes from the volume constraint |B| = m. Indeed from a numerical point of view it is not possible to satisfy the volume constraint exactly due to the discretization. Given an initial set B 0 , the new set B 1 is numerically determined by dichotomy using the gradient of the solution u B0 according to (42). The new set B 1 does not satisfy exactly the constraint |B 1 | = m, but we may compute an intermediate point I lying between grid points, and such that the constraint |B 1 | = m is satisfied. The numerical scheme is modified accordingly using interpolation.

We present results in dimension d = 2. For the numerical tests we take α = 1, Ω = B(0, 1). The initial domain is B 0 = B(0, 0.75) (cf. Figure 7) and we test different values of β > α.

In Figure 8, the optimal sets B * with conductivity β are depicted in black. These results show that the algorithm is able to perform topological changes, since the initial set B 0 has only one connected component, while the optimal domain B * , 1 = α < 1.01 < β ≤ 1.15, has two connected components (Type II). For β = 1.16 and higher initial values β, we observe an optimal set B * of Type I mentioned in Proposition 4. This leads us to think that, for β ≥ 1.16, the proportion of the material β may need to be higher in order to produce solutions of Type II.

STEP 2 :

 2 Next, we show that (23) |λ ε (B) -λ 0 | ≤ C ε .

Theorem 2 .

 2 There exists c * ≥ 0 such that whenever B is a measurable subset of Ω satisfying {x : |∇v 0 (x)| < c * } ⊂ B ⊂ {x : |∇v 0 (x)| ≤ c * } and |B| = m, then B is an optimal solution for the problem of minimizing λ 1 (B) over B ∈ B. Proof. Let f (c) := |{x ∈ Ω : |∇v 0 (x)| ≤ c}|, f is clearly an increasing function with 0 ≤ f (c) ≤ |Ω|. Let c * := inf{c : f (c) ≥ m}. We have f (c * ) ≥ m, indeed, let c k > c * be a decreasing sequence such that c k → c * . On one hand f (c k ) ≥ m and lim k→∞ f (c k ) ≥ m. On the other hand

FIGURE 1 .

 1 FIGURE 1. Functions w 0 (r) (plain), and w 1 (r) = -w 0 (r) (dashed) in dimensions d = 2 (left) and d = 3 (right). r 1 d is such that w 1 is increasing on [0, r 1 d ] and decreasing on [r 1 d , 1], and r 0 d is such that w 1 (r 0 d ) = w 1 (1).

Proposition 3 .

 3 For Ω = B(0, 1), the level set {|∇v 0 | = c} has zero measure for each c ≥ 0.Proof. Since v 0 (x) = w 0 (|x|), we have ∇v 0 (x) = w 0 (|x|)x |x| and consequently we have|∇v 0 |(x) = -w 0 (|x|) .Thus, the measure of the set {|∇v 0 |(x) = c} is positive if and only if the measure of one of the sets {x : w 0 (|x|) = c} or {x : w 0 (|x|) = -c} is positive. Thus, if {x : w 0 (|x|) = -c} has positive measure, this is the same as saying S := {x : ∇v 0 (x) = -c x |x| } has positive measure. That is, the set {x : ∇ (v 0 (x) + c|x|) = 0} has positive measure. By a classical result due to Morrey[START_REF] Morrey | Existence and differentiability theorems for the solutions of variational problems for multiple integrals[END_REF], if ψ ∈ W 1,1 loc , then ∇ψ = 0 almost everywhere on the set {ψ = 0}. Applying this result to the function ∇ (v 0 + c|x|), we get∇ 2 (v 0 + c|x|) = 0almost everywhere on the set of positive measure S defined above. So, now taking the trace we obtain-∆v 0 (x) = c(d -1) |x|on a set of positive measure. Then, by equation[START_REF] Krein | Linear operators leaving invariant a cone in a banach space[END_REF], this means that we havew 0 (r) = αc(d -1) λ 0 ron a set of positive measure which is clearly in violation of the behaviour of the Bessel function J 0 or j 0 . This concludes our proof. Let ω d denote the volume of the unit ball, i.e. we have ω d = π for d = 2 and ω d = 4π/3 for d = 3 and let r 0 d and r 1 d be the constants mentioned in the caption of Figure 1.

Proposition 4 .

 4 When Ω is a ball, in two or three dimensional space, the unique symmetrical optimal domain B * which is solution of the minimization problem for λ 1 (B) over B ∈ B is of two possible types• Type I: If m ≤ ω d (r 0 d ) d then B * = B(0, (m/ω d ) 1/d ) or, • Type II: If m > ω d (r 0 d ) d then there exists ξ 0 and ξ 1 with (m/ω d ) 1/d < ξ 0 < ξ 1 < 1 and B * = B(0, ξ 0 ) ∪ B(0, 1) \ B(0, ξ 1 ) .Proof.Inview of Theorem 2, Proposition 3 and Remark 2, a solution to the problem of minimization of λ 1 (•) over B is the set B * = {x ∈ Ω : |∇v 0 (x)| ≤ c * } where c * is as in Theorem 2. Moreover, it is the unique solution (up to a set of measure zero). We may also write it as B * = {x ∈ Ω : w 1 (|x|) ≤ c * }. Whenever m ≤ ω d (r 0 d ) d , the equation w 1 (r) = c * has exactly one solution ξ in (0, 1). In this case the solution to our minimization problem is B * = B(0, ξ) consisting of a single component in the center. The constraint |B * | = m implies ξ = (m/ω d ) 1/d .

1 .

 1 )-(2). Introduce the quantity (41) M (B 0 , c) = |{x : |∇u B0 (x)| ≤ c}| Lemma The function M (B 0 , c) is non-decreasing with respect to c and is such that M (B 0 , c) → 0 as c → 0 and M (B 0 , c) → |Ω| as c → ∞. Furthermore, it is a rightcontinuous function. It is also left continuous at any c if and only if the Lebesgue measure of {x : |∇u B0 (x)| = c} is zero. Proof. The function M (B 0 , c) is monotone non-decreasing due to the set inclusion {x : |∇u B0 (x)| ≤ c} ⊂ {x : |∇u B0 (x)| ≤ c } whenever c ≤ c . Due to the integrability of |∇u B0 |, we have |{x : |∇u B0 (x)| > c}| → 0 as c → ∞ and therefore, we can conclude that M (B 0 , c) → |Ω| as c → ∞

Therefore, in view of Corollary 3 ,

 3 when ϑ < π, B * contains a neighbourhood of P since |∇u B * (x)| → 0 as x → P and we may find a small enough neighbourhood of P that will be included in {x : |∇u B * (x)| ≤ c * }. When ϑ > π then A * = Ω \ B * contains a neighbourhood of P since |∇u B * (x)| → ∞ as x → P . These properties may be observed in theFigures 2, 3 and 4.

2 FIGURE 2 . 2 FIGURE 3 .

 2223 FIGURE 2. Nearly optimal distribution B in the square case. The dark region corresponds to B and the material β, the white region to A and material α. As predicted by Corollary 3, the optimal domain B contains the corners of the square, as well as the center since the gradient of the eigenfunction vanishes at that point.

  (a) m/|Ω| = 0.3370 (b) m/|Ω| = 0.5673 (c) m/|Ω| = 0.7546 (d) m/|Ω| = 0.8424 (e) isolines of |∇v 0 | 2

FIGURE 4 . 2 FIGURE 5 .

 425 FIGURE 4. Nearly optimal distributions B in the polygon case for different values of m. The dark region corresponds to B and the material β, the white region to A and material α. As predicted by Corollary 3, the approximately optimal domain B contains the corners of the polygon with angle less than π and a point in the center where the gradient of the eigenfunction vanishes. Note that A * = Ω \ B * contains a neighbourhood of the point with the reentrant corner, i.e. the corner with angle greater than π. Indeed, in this corner the gradient is unbounded.

2 FIGURE 6 .FIGURE 7 .

 267 FIGURE 6. Nearly optimal distributions B in the case of an ellipse with two holes for different values of m. The dark region corresponds to B and the material β, the bright region to A and material α. In this case we observe that the approximately optimal set B * has several connected components, except maybe for large m, and touches the boundary. The set B * always has a connected component in the center of the domain. This corresponds to a point where the gradient of the eigenfunction vanishes.

FIGURE 8 .

 8 FIGURE 8. Optimal sets A * and B * for different values of β close to α. The set B * is represented in black, the set A * in white. The optimal region B * is of type II for 1 = α < 1.01 < β ≤ 1.15 and of type I for β = 1.16.

Acknowledgements. This research project was initiated during the workshop on partial differential equations and optimization at the Centro de Ciencias de Benasque Pedro Pascual, Spain. The authors would like to thanks the organizers for the fruitful conference. The second author acknowledges financial support by the Austrian Ministry of Science and Education and the Austrian Science Foundation FWF under START-grant Y305 "Interfaces and Free Boundaries" and the subproject "Freelevel" of the SFB F32 "Mathematical Optimization and Applications in Biomedical Sciences". The third author acknowledges financial support by CONICYT-FONDECYT project N o 1090305. The first author thanks the MICDB for partial support through Grant ICM-P05-001-F, Fondap-Basal-Conicyt.


L Gourcy
email: l.gourcy@brgm.fr

A Brenot

Multiple environmental tracers for a better understanding of water flux 1 in a wetland area (La Bassée, France) 2

Keywords: δ 18 O, δ 2 H, 87

Understanding water exchange between groundwater and streams, or groundwater/surface-12 water relationships, is of primary importance for solving conflicts related to water use and for 13 restoring water ecosystems. To this end, we tested a combination of classic geochemical 14 tools and isotopic tools on the Bassée study site, located in the alluvial plain of the River 15 Seine, to see whether they are relevant for tracing multiple and complex 16 groundwater/surface-water relationships. The Ca/Sr ratio associated with Sr isotopes shows 17 contrasted values and suggests that infiltration of surface water to groundwater increases 18 when approaching the Seine. Furthermore, stable isotopes of the water molecule indicate 19 that water from gravel-pit lakes may infiltrate into groundwater. Tritium and CFC tools 20 confirmed surface-water influence on the Alluvial and Chalk aquifers. This geochemical 21 approach, tested on the Bassée site, clearly demonstrates the need of using various 22 geochemical tools for describing groundwater/surface-water relationships, and can be 23 conclusively addressed to other case studies for helping decision makers in their 24 management of natural water resources.

INTRODUCTION 31 32

In studies of hydrological systems over the past few years, new environmental tracers have 33 been successfully applied, such as strontium, boron and sulphate isotopes, as well as CFCs 34 and SF 6 (e.g. Vengosh et al., 2002;Négrel and Pauwels, 2004;Négrel and Petelet, 2005; 35 [START_REF] Gooddy | Using chlorofluorocarbons 550 (CFCs) and sulphur hexafluoride (SF6) to characterise groundwater movement and 551 residence time in a lowland Chalk catchment[END_REF]. New-generation inductively-coupled plasma mass spectrometers (ICP-36 MS) have allowed the development of much more precise isotopic measurements for various 37 elements and a wider use of these tools (e.g. Luais et al., 1997;Marechal et al., 1999;Halicz 38 et al., 1999;Rehkamper and Halliday, 1999;[START_REF] Galy | High-precision measurement of 544 magnesium isotopes by multiple-collector inductively coupled plasma mass spectrometry. 545[END_REF]Rouxel et al., 2002;Beard et and Piketti, 2001). The intensive sand and gravel extraction throughout the alluvial plain 77 since 40 years has created large borrow-pits that, at the end of the exploitation period, are 78 filled by ground-and rain-water. The Bassée site is also the major wetland area of the Ile-de-

79

France region, labelled "Natura 2000", and part of a Europe-wide network of sites tasked with 80 the preservation of our natural heritage. It is one of the 87 wetlands of major importance in

81

France and governmental action plans were established in order to protect the surface water 82 covering 20% of the plain (http://natura2000.environnement.gouv.fr /sites/FR1112002.html).

83

The groundwater here is of great importance as a drinking-water supply and is considered as 84 a reservoir to be protected for the Paris metropolitan-area water supply (SDAGE,

85

Management Plan of Seine-Normandy basin). The study area covers 23 km 2 of the 86 downstream part of the alluvial plain. Previous hydrogeological studies (Mégnien, 1965;87 Sogreah, 2004;Armines, 2005) using classical hydrogeological investigation methods

88

(water-level monitoring, geology, geophysics, geotechnics, and modelling) left uncertainties 89 on the magnitude of the relations between deep and alluvial aquifers, on the possible 90 hydraulic connections between deep aquifers and the river, and on the extent of the 91 relationships between the River Seine and shallow aquifers.

92

The scope of this study was to improve the hydrological understanding of the system through 93 the use of multiple geochemical tools, in order to arrive at a better protection of this floodplain 94 from qualitative and quantitative points of view. To this end, three different types of water 95 relationships had to be evaluated in the alluvial plain: Chalk and Alluvial groundwater 96 connections, groundwater (Chalk and Alluvial) and river water relationships, and the relation 97 between other surface waters (oxbow lakes, ponds, gravel pit lakes) and groundwater. The 98 alluvial plain in the past having been subject to periodic flooding by the Seine and today 99 being intensively exploited for sand and gravel extraction, the existing surface water is of 100 particular ecological and hydrological importance in the studied area. 101 102

CHARACTERISTICS OF THE BASSÉE SITE 103

The Bassée site, south-east of Paris, is of major importance for the drinking-water supply of 104 Paris and surroundings. The site lies in the large Seine floodplain, delimited upstream by 105 Bray-sur-Seine and to the west by the Montereau-Fault-Yonne at the junction between the 106 rivers Seine and Yonne (Fig. 1). In view of its numerous ponds, canals and river arms, the 107 Bassée plain was included in the "Natura 2000" program.

108

Sand and gravel exploitation and agriculture are the main economic activities in the area.

109

Three aquifers are interconnected: i) the Alluvial aquifer of the River Seine located within the 110 whole studied area, ii) the mainly unconfined Chalk aquifer below the alluvial one, and iii) the 111 Tertiary calcareous formations in the northern part of the area (Fig. 2). The Alluvial and Chalk 112 aquifers discharge into the Seine (Weng et al., 1999;Vernoux et al., 2004). Groundwater -5 -and SF 6 .

149

The boreholes were sampled after pumping at least three purge volumes and stabilizing the

162

Sr concentrations were measured by inductively coupled plasma mass spectrometry (ICP-

163

MS) (uncertainty 5%). Alkalinity was determined by HCl titration and Gran"s method.

164

Strontium for isotopic analysis was separated using an ion-exchange column (Sr-Spec resin).

165

Procedural blank levels were lower than 0.5 ng.

174

Samples for tritium activity measurements were collected in one litre polyethylene bottles.

175

Analyses were done by direct liquid scintillation counting. The detection limit was 10 TU

176

(Tritium Unit with 1 TU equal to 1 tritium atom in 10 18 hydrogen atoms). All isotopic analyses

177

were carried out by the BRGM Geochemistry Laboratory.

178

Samples for CFC and SF 6 were collected in 500 mL glass bottles. measured in this study (Fig. 8). The Ca/Sr ratio was demonstrated elsewhere to be a good 251 tool for tracing contrasting lithological origins [START_REF] Brenot | Interaction between different water 525 bodies in a small catchment area in the Paris basin (France): Sr isotopic tracing of multiple 526 Sr sources[END_REF]. As there is no correlation 252 between 87 Sr/ 86 Sr (or for the Ca/Sr ratio) and the NO 3 concentration (index of anthropogenic 253 origin), End-member 2 most probably has a lithological origin. Nevertheless, the 87 Sr/ 86 Sr and 254 Ca/Sr ratios measured for the Sc22 and Sa37 sampling points (defining End-member 2) are 255 much higher than expected values for local carbonate rock (as discussed above). A similar 256 deviation towards a radiogenic composition (higher 87 Sr/ 86 Sr values) was earlier observed that radiogenic Sr might derive in part by thin marly-clayey beds in carbonate rock through 260 ion exchange or desorption of Sr, or by dissolution of the potential dolomitic part of carbonate 261 rock with a more radiogenic isotopic signature. End-member 3 is close to the composition of 262 Seine water, water from gravel-pit lakes and the Sa48 and Sc13 sampling points.

263

Furthermore, the Sr-isotope composition of Seine water (Roy, 1996) falls between 0.70777 264 and 0.70793, close to the composition expected for End-member 3. Thus, End-member 3 265 could potentially reflect a contribution of dissolved elements in surface water. Water in the

266

Voulzie River has a different Sr isotopic signature compared to that of the Seine (Fig. 7) and

267

water from gravel-pit lakes, which could be explained by significant differences in water-rock -9 -probably caused a degradation of CFCs [START_REF] Horneman | Degradation rates of CFC-11, CFC-12 and CFC-557 113 in anoxic shallow aquifers of Araihazar, Bangladesh[END_REF], in which case SF 6 was 296 then used for dating the water, this element being less sensitive to microbial degradation.

297

All water samples could be dated at least once. Determination of apparent water ages using 298 CFC and SF 6 (Plummer and Busenberg, 2000) was done by applying a binary mixing model 299 considering old water before 1950 (CFC-free) and young water with an age between 1985 300 and Present. This model was given the best results in "age-dating" (example for CFC-11 and 301 CFC-113 given in Fig. 8). Young-water mixing fractions ranged from 0.20 to 0.99. Results

302

from the two sampling campaigns are quite similar. The uncertainty in recharge temperature 303 estimation (about 1°C what is the annual variability of groundwater temperature) is leading to 304 an uncertainty in mixing ratio of a maximum of 5%. Anthropogenic contamination was 305 observed in few samples, mainly for CFC-12 and SF 6 possibly due to the proximity of large 306 urban areas (Oster et al., 1996;[START_REF] Horneman | Degradation rates of CFC-11, CFC-12 and CFC-557 113 in anoxic shallow aquifers of Araihazar, Bangladesh[END_REF].

307

The dissolved gases and age-dating interpretation show a significant differentiation between

345

and therefore with different isotope contents. Variation of the Seine isotopic composition is 346 about 1 ‰ (Conrad et al., 1978).

356

floodplain; here, water renewal is limited and therefore evaporation is expected to be high.

357

Though to a smaller extent, river water is also evaporated and falls on the same evaporation 358 line. This clearly indicates that some sampling points in groundwater integrate evaporated 359 water that could be provided either by gravel-pit lakes or by river water.

Relationship between groundwater and gravel-pit lakes 401

Numerous gravel-pit lakes or natural ponds, swamps, oxbow lakes, and other backwaters 402 exist in the flood plain. Chemical data and 87 Sr/ 86 Sr ratios do not show significant differences

403

-12 -between water from the aquifers and such surface waters. This was expected, as such 404 waters mostly consist of groundwater.

405

In these water bodies, water renewal is limited and evaporation is quite high. Evaporation of 406 water before infiltration was highlighted for some wells using the δ 2 H and δ 18 O isotope

416

The Sc13 water may result from a mixture of surface water with a more fractionated isotopic 417 signature (higher H and O isotopic signatures) than the two gravel-pit lakes sampled in this 418 study or the groundwater. The infiltration of surface water to the aquifer is also clearly shown 419 at sampling points Sa26, Sa38, Sc13 and Sc16 (Fig. 9).

420

Dissolved gases SF 6 ) and age-dating interpretation give quite 421 heterogeneous results. Piezometers Sc13 and Sc09 in the Chalk aquifer mainly contain 422 young water, and a direct influence of surface water on these points is thus expected. Being 423 far from the Seine, the influence of ponds or gravel-pit lakes is most likely for these two wells.

424

However, the recharge functioning is quite complex for Sc09 located in the eastern part of 425 the study area, far from the Seine and any flooded area. No explanation could be found to 426 explain the geochemical parameters of groundwater at this point. 427

Impact of the water dynamics on groundwater quality 428

An understanding of the hydraulic connections in the Seine floodplain may help in 429 understanding groundwater contamination. Borrow pits were initiated in the area in the 1970s 430 and rapidly increase in number. This extraction activity does not seem to have a direct 431 qualitative impact on the groundwater. Agriculture today occupies 40% of the Bassée plain 432 as against 46% in 1976 [START_REF] Fustec | Fonctionnement du 542 territoire 'Bassée[END_REF]. The dominant crops are cereals (wheat, maize, 433 barley, rape, sunflower) and sugar beets, and diffuse pollution is expected at the study area.

434

Groundwater contamination can be quite high in the left bank of the Seine. Nitrate

444

The percentage of young groundwater was compared to nitrate concentrations (Fig. 10). A 445 positive correlation was observed between the percentage of young water and nitrate 446 concentrations for water in aerobic conditions, circulating in the Alluvial and Chalk aquifers.

447

For most sampling points, the more important is the influence of the young groundwater, the 448 higher are the nitrate concentrations (Fig. 11), whereby the "youngest" groundwater dates 449 from 1985 up to 2007. In captive parts of the aquifer or around wetlands, the nitrate 450 concentrations are below the detection limit (0.5 mg/L), most probably due to denitrification 451 processes (in grey on Fig. 11). Seine water upstream and downstream from the study area 452 has an average nitrate concentration lower than groundwater (<0.5 mg/L to 35.4 mg/L).

453

For Sa61, Sa41, Sa48 relatively low nitrate concentrations compared to the apparent CFC 454 ages may be explained by surface water influence (Fig. 11). In fact, the very high tritium 455 content in groundwater from these three wells indicates a stronger influence of the Seine on 456 groundwater, an influence that was also felt in Sc13 and Sc09. At the latter two points, a low 457 redox potential suggests possible denitrification processes. In Sc09, the nitrate concentration

463

The two sampled gravel-pit lakes have nitrate concentrations of 2.8 and <0.5 mg/L. Low 464 nitrate concentrations can be explained by dilution (from rainfall and/or rivers), adsorption by 465 plants and denitrification processes, as was earlier demonstrated in riparian wetlands 466 upstream from the present study area [START_REF] Bendjoudi | The riparian wetland of the middle 518 reach of the Seine River: Historical development, reconnaissance and present hydrologic 519 functioning, a case study[END_REF]. Concentrations of other

485

 Age-dating highlighted the presence of young, i.e. less than 10 years old, 486 groundwater in the wells Sc09 and Sc13 tapping the Chalk aquifer.

487

 The influence of Seine water, direct or indirect through the alluvial deposits, was 488 shown using tritium and Sr isotope data for Sa41, Sa48 and Sa61.

489

 The impact on the aquifer of water stored in the gravel-pit lakes or other natural 490 shallow lakes, ponds or swamps, was highlighted by using stable-isotope tools. The 491 evaporation effect was clear on groundwater collected in wells Sc13, Sc16, Sa26 and 492 Sa38. For Sc13, both a surface-water influence by the Seine through alluvial deposits

493

and by infiltration from gravel-pit lakes is possible.

494

 δ 2 H and δ 18 O, tracers of water-rock interaction processes ( 87 Sr/ 86 Sr, Ca/Sr) and of

499

 The geochemical approach tested on this site clearly demonstrates the efficiency of 500 geochemical tools for describing the relationships between groundwater, river water 501 and lake water. In addition, it underlined the need of using multiple tools for 502 understanding a complex water system.

503

 The approach described in this paper can be usefully applied to other case studies, 504 and will help decision makers in the context of natural water management.

506

Figure and table captions 649

239

  concentrations and nitrate (Fig.6b) or chloride concentrations (not shown here). On a 240 87 Sr/ 86 Sr vs. 1/Sr (Fig. 7a) diagram, samples plot globally in a triangle. In this classic diagram, 241 such a distribution suggests that, at least, three sources of dissolved Sr must explain the 242 ground-and surface-water composition. One end-member should display low 87 Sr/ 86 Sr 243 values and high Sr concentrations (End-member 1). Another end-member is expected to 244 have high 87 Sr/ 86 Sr values and intermediate Sr concentrations (End-member 2). A third end-245 member would display low 87 Sr/ 86 Sr values and low Sr concentrations (End-member 3). End-246 member 1 displays a Sr concentration and Sr-isotope composition close to the composition 247 of the Sa26 sampling point ( 87 Sr/ 86 Sr = 0.70792 and 0.70796; 7.51 and 7.19 μmol/L) and 248 compatible with a Sr contribution from carbonate dissolution. End-member 2 is close to the 249 Sc22 and Sa37 sampling points, both of which display the highest 87 Sr/ 86 Sr and Ca/Sr ratios 250

  268interaction. Indeed, the Voulzie River flows through Early Tertiary layers composed of clay 269 and organic matter, whereas the Seine above the Bassée floodplain mainly flows over 270 carbonate rock(Roy et al., 1999). The Chalk aquifer sampled upstream also has a quite 271 different Sr-isotope signature compared to the Chalk aquifer sampled in the Bassée 272 floodplain (Fig.7).values and the Ca/Sr ratio for all sampling points. The lowest Sr-isotope composition and275Ca/Sr ratio in this study are in the Seine ( 87 Sr/ 86 Sr=0.70793 to 0.70778; Ca/Sr=424 to 964),276whereas the highest Sr-isotope compositions and Ca/Sr ratios occur in groundwater at the Based on only 87 Sr/ 86 Sr values, and 1/Sr and Ca/Sr ratios (Fig.7), one cannot discriminate 279 the groundwaters of the Alluvial and Chalk aquifers sampled in the Bassée floodplain.

  and CFC data 281The trends of CFCs and SF 6 used as input functions in air correspond to those of the Mace 282 Head (Ireland) station from the ALE/GAGE/AGAGE program (available from 283 cdiac.esd.ornl.gov/ftp/ale_gage_Agage/; Prinn et al., 2000), the closest survey station to the 284 Bassée site. Apparent groundwater ages were evaluated by using Excel® programs for CFC 285 developed by the International Atomic Energy Agency (IAEA, 2006). The recharge date was 286 determined by comparing the calculated partial pressure of CFCs and SF 6 in solubility 287 equilibrium with the water sample, with historical CFC concentrations in local air. The288 equilibrium concentration of CFCs in groundwater depends on the air pressure (i.e. altitude) 289 and temperature in the saturated zone. Recharge temperatures used are the temperature of 290 groundwater at high-water-stage sampling time. The mean altitude of recharge is close to 291 55 m, the average altitude of the alluvial plain. The uncertainty in recharge temperatures and 292 recharge elevations is respectively maximum 1°C and 50 meters.

  308 the Chalk and Alluvial aquifers. Only two wells (<20%) located in the Alluvial aquifer (Sa5, 309 Sa20) have a percentage of young water below or equal to 50%, whereas five deep wells 310 (>71%) tapping the Chalk aquifer (Sc11, Sc16, Sc22, ScH3, ScH4) show a percentage of 311 young water below 50%. Surprisingly, wells Sc13 and Sc09 in the Chalk aquifer mainly 312 contain young water, and a direct influence of surface water on these points is expected. 313 Being far from the River Seine, the influence of ponds or gravel-pit lakes is supposed. The 314 Alluvial groundwater in 75% of the wells is represented by water with a present (1985-2007) 315 component below 80%. Therefore, the older and deeper groundwater component is 316 significant, confirming the initial conceptual hydrogeological model of the Chalk aquifer 317 discharging into the upper aquifer layers.

  318A few kilometres upstream from the Bassée floodplain, the nuclear power plant at Nogent-319 sur-Seine regularly has released tritium into the Seine since its start-up in 1988-1989. Tritium 320 can thus be used as a radionuclide tracer (Sanchez-Cabeza and Pujol, 1999) of the 321 infiltration of the Seine to groundwater since then. In the Seine at Bray-sur-Seine, the tritium 322 concentration measured during the project is quite variable, from 21 TU in July 2008 up to 323 573 TU in April 2008, reflecting the variable release from the power plant (3.1 to 324 11.7 TBq/month for the period August 2006-August 2008, data provided by EDF, the 325 operator). By comparison, the tritium values for Orléans precipitation (closest station at about 326 120 km from the Bassée site) have on average below 6 TU since 1999 (IAEA/WMO, 2009).

327

  Tritium values measured in groundwater are very high (>60 TU) for Sc13, Sa41, Sa48 and 328 Sa61, and high (>15 TU) for Sc09, indicating a more or less important influence of surface 329 water on the Chalk and Alluvial aquifers at these points.

  isotopes significant difference between the Alluvial and Chalk aquifers and the River Seine. The 335 average annual composition of δ 2 H and δ 18 O of the Seine water measured from oct.74 to 336 sep.76 at Poses (upstream Paris, 200 km far from La Bassée; Conrad et al., 1978) is falling 337 very closed to the 1996-2010 weighted mean of precipitation collected at Orléans. 338 The most isotopically depleted groundwater is that sampled in the Chalk aquifer upstream 339 from the Bassée area, which thus can be considered as the Chalk aquifer without influence 340 by surface and/or alluvial water. Another groundwater sample taken in 2009 at 120 km west 341 of the Bassée area in the Chalk aquifer of Paris Basin, has a similar isotopic content as that 342 of the Chalk water sampled at La-Chapelle-sur-Oreuse. River water from the Seine or the343Voulzie shows more enriched values. All other samples may be considered to be a mix 344 between "pure" groundwater and surface (rainfall and/or river) water from different periods,

347

  Some points in the Alluvial (Sa26, Sa38) and Chalk (Sc13, Sc16) aquifers fall on a line with a 348 slope of 4.3 that may correspond to isotopic fractionation caused by evaporation349(evaporation lines have a slope between 3.5 and 6 as function of local atmospheric humidity ; 350[START_REF] Gat | Stable Isotope Hydrology -Deuterium and Oxygen-18 547[END_REF]. To test this hypothesis, we calculated the evaporation line with as starting point 351 the average weighted δ 18 O and δ 2 H values of precipitation in Orléans for the period 1999-352 2009, and which would explain the compositions of samples Sa26, Sa38 (Alluvial aquifer) 353 and Sc13, Sc16 (Chalk aquifer). The calculations show that the two gravel-pit lakes sampled 354 in November 2007 have a stable-isotope composition falling onto the same evaporation line 355 and display the most fractionated isotopic composition from all samples in the Bassée

  Mégnien, 1979; Armines, 2005), consider that the Chalk aquifer supplies -11 -Based on only the major element concentrations, the geochemical composition of 367 groundwater and surface water cannot be discriminated. Other geochemical tools are thus 368 needed to reach a better understanding of the potential relationships between rivers and 369 groundwater.370Sr isotopes are known(Négrel and Petelet-Giraud, 2005) to be an interesting tool for 371 groundwater-surface water relationships. In the Bassée site the Sr ratio contrast observed 372 between groundwater and surface water confirms its potential use in this context.373The spatial distribution of sampling points in the Bassée floodplain allows the construction of 374 two north-south profiles perpendicular to the Seine. Along these profiles, the Ca/Sr ratio and 375 87 Sr/ 86 Sr values (Fig. 10) show a significant decrease, with the lowest values close to the 376 Seine for the high-and the low-water-stage sampling campaigns. Knowing the Ca/Sr ratio 377 and 87 Sr/ 86 Sr values of the Seine water, the concomitant decrease of these two geochemical 378 tracers suggests an increasing contribution of Seine water to the groundwater close to the 379 riverbanks. Furthermore, the Ca/Sr ratio (1139) and the Sr isotopic signature 380 ( 87 Sr/ 86 Sr=0.70812) of Chalk aquifer groundwater collected at "La Chapelle", 10 km away to 381 the north (Fig. 1), are significantly higher than the signatures measured for groundwater 382 collected closed to the Seine (Sa48, Sa41, ScH3, Sa26, Sa21, Sa20 and Sc16). The Ca/Sr 383 ratio and Sr isotopic compositions thus clearly show a recharge of the aquifers by surface 384 water during the high-and low-water stages. Surface water contribution seems to be of the 385 same importance for both Alluvial and Chalk aquifers. River-water infiltration to groundwater 386 is possible close to the river banks, and around ponds and gravel-pit lakes. 387 The sampling points located upstream and far from the Seine (Sc22, Sa37, Sc09, Sa61 and 388 Sa65) have the highest Ca/Sr ratio (930 to 1134) and 87 Sr/ 86 Sr (0.70817 to 0.70835) 389 compared to the Seine water composition. At the opposite, sampling points located 390 downstream of the Bassée alluvial plain and close to the Seine (Sc11, SCH3 and Sa02) have 391 a Ca/Sr (876 to 916) close to the ratio measured at the upstream Seine station (not all points392fall onto the N-S profiles and therefore cannot be seen in Figure10).

393

  Tritium-content data show that very high (>60 TU) values for Sc13, Sa41, Sa48 and Sa61 394 and high (>15) values for Sc09 indicate a direct influence of river water on the Chalk and 395 Alluvial aquifers at these points. For well Sc13, located far from the main stream but close to 396 a large flooded area, many of the ponds and gravel-pit lakes are most probably fed directly 397 by the River Seine and the canal parallele to the Seine. Tritium content then indicates the 398 input of river water into groundwater via a residence period in ponds and gravel-pit lakes. A 399 quantitative estimation is not possible as tritium input varies throughout the year.

  400

  lakes were created by gravel extraction, causing a change in the hydraulic 409 properties of the Alluvial aquifer. It is expected that these artificial holes also facilitate the 410 infiltration of surface water down to the deeper parts of the aquifers. The high permeability of 411 the walls suggests that water will mainly infiltrate horizontally into both the Chalk and Alluvial 412 aquifers. 413 Water sampled at Sc13 in the Chalk aquifer is the most intensively evaporated. Due to the 414 strong variability in H and O isotope signatures of the gravel-pit lakes (only two of them were 415 sampled), it is possible that fractionation is even stronger in other superficial water bodies.

435

  concentrations measured in June and November 2007 were, for 9 wells of the 19 sampled, 436 over 17 mg/L and up to 83.1 mg/L (Sa21), confirming an anthropogenic impact. For 5 wells -13 -in the past showed triazine contamination at least once. Pesticide (mainly atrazine and DEA) 441 and nitrate contamination have also been detected in wells tapping the Chalk aquifer. For the 442 Water Framework Directive, both groundwater bodies (FRHG006 Alluvial aquifer on the 443 Bassée site and FRHG209 Chalk aquifer) were classified as having a poor chemical status.

458in

  November 2007 (74.2 mg/L) and the redox potential (311 mv NHE) are higher than during 459 the high water period (<0.5 mg/L and 345 mV NHE, respectively). As other chemical 460 elements have similar concentrations for the high and low water sampling campaigns, 461 denitrification is the most probable process for explaining major changes for only the NO 3 462 concentration.

467

  chemical elements are similar or lower than in river water. Dilution by rainfall input is 468 probably the most important factor, as Redox conditions may not allow denitrification 469 processes and there are no or very few plants in the gravel-pit lakes. Previous work (Fuster 470 et al., 2001) showed that silt deposits at the bottom of the gravel-pit lakes greatly reduce 471 groundwater exchanges. Evaporation and recharge by precipitation therefore are dominating 472 factors of the hydraulic balance of these specific surface-water bodies. Infiltration of water 473 from gravel-pit lakes has a positive influence on groundwater quality. As for river water, the 474 infiltration of water from gravel pits reduces, at least locally, the nitrate concentration in wells 475 upstream from these artificial lakes. 476 6. CONCLUSIONS regional scale, which considers that Chalk groundwater supplies water to the Alluvial 481 aquifer. However, at the local scale more complex hydraulic connections exist 482 between the different water bodies, as was highlighted by using isotope tools and 483 age-dating. Both the Chalk and the Alluvial aquifers may be influenced by surface 484 water.

495

  water origin ( 3 H) differentiated the role of the River Seine, gravel-pit lakes (and most 496 probably other superficial water bodies) in local groundwater recharge. The 497 geochemical approach provided a spatial characterization of the origin of water and 498 dissolved elements from surface and groundwater bodies.

Figure 1 -Figure 3 -Figure 4 -Figure 5 -Figure 6 -

 13456 Figure 1 -Map showing the studied area and location of sampling points

  Figure

Figure 2 :Figure 3 :Figure

 23 Figure 2 : Synthetic geological profile of the La Bassée area

Figure 4 :

 4 Figure 4 : SO 4 vs. Cl and Na vs. Cl for surface and groundwater sampled at the La Bassée plain in June and November 2007 and groundwater sampled upstream (Chalk aquifer)

Figure 5 :

 5 Figure 5 : NO 3 vs. Cl for surface and groundwater sampled at the La Bassée plain in June and November 2007 and groundwater sampled upstream (chalk Aquifer)

Figure 6 :

 6 Figure 6 : Ca vs. Sr and NO 3 vs. Sr for surface and groundwater sampled at the La Bassée plain in June and November 2007 and groundwater sampled upstream (Chalk aquifer)

Figure 7 :FigureFigure 8 :Figure

 78 Figure 7 : 87 Sr/ 86 Sr vs. 1/Sr (molar) and 87 Sr/ 86 Sr vs. Ca/Sr (molar) for surface and groundwater sampled at La Bassée plain in June and Novembre 2007 and one upstream well

Figure 9 :Figure

 9 Figure 9 : δ 2 H vs δ 18 O for surface and groundwater collected in the La Bassée site in June and November 2007; local meteoric line and mean weighted values for precipitations at Orléans (1996-2010), mean of Seine water at Poses (1974-1976) and calculated evaporation line

Figure 10 :

 10 Figure 10: 87 Sr/ 86 Sr vs. distance and Sr/Ca vs. distance for the groundwater collected in June and November 2007 following the two North-South profiles

Figure

  Figure

  150 chemical groundwater parameters, such as pH and conductivity. Samples for cation and 151 87 Sr/ 86 Sr determinations were kept in polyethylene bottles after acidification with nitric acid.

	152
	153	Raw water samples of 50 ml and 1 litre in HDPE bottles were used for δ 2 H, δ 18 O and 3 H
	154	analyses. Glass sampling bottles of 500 ml were used for CFC and SF 6 analyses (IAEA,
	155	2006).
	156	3.2. Analytical methods
	157	Concentrations were measured on water samples by inductively coupled plasma-atomic
	158	emission spectrometry (ICP-AES) for Ca 2+ , Na + , K + and Mg 2+ , and by ion chromatography for
	159	Cl -, NO 3 -, SO 4 2-and PO 4	2-(uncertainty for both methods 5-10%). Anion analyses were done

by ion chromatography (model DX120, Dionex®), cations by Inductively Coupled Plasma 160 Atomic Emission Spectroscopy (ICP-AES; Ultima-2 model, Jobin Yvon®). The accuracy of 161 both techniques was around 5-10% depending upon the concentration.

Table 1 -

 1 Analytical results for June and Novembre 2007 sampling campaign

	N°type	Date	Cond. Eh(NHE) Temp.	pH	Ca	Mg	Na	K	HCO 3	Cl	NO 3	SO 4	Sr	Rb	87 Sr/ 86 Sr	δ 18 O	δ 2 H
				μS.cm -1	mV	°C		µmol/L µmol/L	µmol/L	µmol/L	µmol/L	µmol/L	µmol/L	µmol/L µmol/L µmol/L		‰	‰
	June-07 (high flow period)
	Sa02	Alluvial aquifer	12/06/2007	734	383	11.9	7.1	3671	86	278	49	4607	775	969		4.01	0.01 0.708001	-6.4	-42.3
	Sa05	Alluvial aquifer	12/06/2007	643	359	12.3	7.1	3150	136	300	49	4869	541	658		3.89	0.00 0.708063	/	/
	Sa16	Alluvial aquifer	12/06/2007	725	250	12.8	7.1	3549	152	513	69	4902	868	161		3.40	0.01 0.708152	-6.4	-42.3
	Sa20	Alluvial aquifer	12/06/2007	579	311	12.0	7.2	2746	115	413	90	4721	515	458		3.38	0.02 0.707981	-6.7	-45.1
	Sa21	Alluvial aquifer	12/06/2007	578	350	12.5	7.4	2618	66	239	74	3508	617	1332		2.67	0.00 0.707940	-6.4	-42.5
	Sa26	Alluvial aquifer	13/06/2007	1135	173	11.7	7.0	6426	222	361	18	5639	606	<DL		7.51	0.00 0.707925	-4.8	-35.7
	Sa37	Alluvial aquifer	13/06/2007	786	358	11.3	7.0	3469	169	983	128	5459	1270	540		3.06	0.01 0.708325	-6.6	-43.2
	Sa38	Alluvial aquifer	13/06/2007	668	154	12.0	7.0	3337	123	465	23	5459	676	94		3.29	0.00 0.708184	-4.8	-35.7
	Sa41	Alluvial aquifer	13/06/2007	541	351	13.2	7.2	2471	152	339	100	4164	361	276		3.34	0.02 0.708042	-6.6	-45.1
	Sa48	Alluvial aquifer	13/06/2007	533	302	15.9	7.3	2349	160	470	72	4066	637	95		3.03	0.01 0.707777	-6.5	-44.1
	Sa61	Alluvial aquifer	12/06/2007	860	287	12.1	6.9	4005	387	670	348	7213	901	202		3.77	0.00 0.708168	-6.8	-44.0
	Sa65	Alluvial aquifer	13/06/2007	763	375	17.3	7.1	3436	140	465	51	5049	989	989		3.56	0.00 0.708246	-6.4	-42.4
	Sc09	Chalk aquifer	13/06/2007	645	345	18.6	7.2	2885	128	474	0	4213	769	<DL		3.07	0.01 0.708168	-7	-46.9
	Sc11	Chalk aquifer	12/06/2007	525	386	12.3	7.2	2731	78	487	28	4557	538	432		3.09	0.00 0.708119	-6.9	-45.9
	Sc13	Chalk aquifer	12/06/2007	414	105	13.4	7.4	1788	111	313	38	3656	437	<DL		2.47	0.00 0.707807	-2.4	-25.2
	Sc16	Chalk aquifer	13/06/2007	682	175	11.5	7.0	3479	189	330	31	5705	462	<DL		3.81	0.01 0.708063	-6	-41.8
	Sc22	Chalk aquifer	12/06/2007	700	321	12.5	7.0	3519	103	452	15	5443	854	<DL		2.81	0.00 0.708582	-6.7	-44.5
	ScH3	Chalk aquifer	12/06/2007	500	282	14.7	7.3	2259	144	270	41	3787	352	626		2.82	0.00 0.707798	-6.8	-46.0
	ScH4	Chalk aquifer	13/06/2007	733	162	11.8	6.9	3451	193	643	33	5721	814	<DL		3.39	0.00 0.708156	-6.8	-45.1
	Seine Amont	River	13/06/2007	536	344	23.5	8.2	2334	165	591	66	3607	732	361		3.25	0.02 0.707728	-6.3	-43.6
	Voulzie	River	13/06/2007	661	364	18.7	8.2	2339	165	535	64	3689	718	368		2.24	0.00 0.708282	-6.6	-44.0
	Seine Aval	River	13/06/2007	538	391	23.1	8.2	3027	210	604	79	4836	941	600		3.14	0.00 0.707749	-6.2	-42.0
	October-07 (low flow period)
	Sa02	Alluvial aquifer	06/11/2007	736	271	12.9	7.1	3641	74	243	49	4770	868	985		4.16	/	0.708019	-6.5	-41.3
	Sa05	Alluvial aquifer	06/11/2007	640	265	13.5	7.3	3137	115	257	51	5016	552	650		3.89	/	0.708108	-6.7	-42.8
	Sa16	Alluvial aquifer	06/11/2007	735	243	13.5	7.2	3456	160	296	28	5574	501	<DL		4.16	/	0.708242	-6.3	-41.5
	Sa20	Alluvial aquifer	06/11/2007	563	378	12.8	7.5	2626	95	343	90	4656	510	571		3.32	/	0.708002	-6.9	-45.7
	Sa21	Alluvial aquifer	06/11/2007	575	301	14.2	7.6	2656	58	200	64	3459	668	1340		2.72	/	0.707974	-6.4	-42.3
	Sa26	Alluvial aquifer	06/11/2007	1095	139	13.2	7.2	6095	189	335	18	5410	631	<DL		7.19	/	0.707964	-4.7	-34.2
	Sa37	Alluvial aquifer	06/11/2007	818	302	13.5	7.4	3651	156	961	125	5705	1693	518		3.31	/	0.708349	-6.6	-42.5
	Sa38	Alluvial aquifer	07/11/2007	661	196	12.9	7.1	3257	119	430	28	5393	735	<DL		3.20	/	0.708231	-4.3	-34.3
	Sa41	Alluvial aquifer	07/11/2007	593	330	14.2	7.3	2698	148	330	113	4803	400	353		3.73	/	0.708066	-6.9	-45.6
	Sa48	Alluvial aquifer	07/11/2007	425	276	15.6	7.5	1900	119	287	51	3492	414	13		2.52	/	0.707802	-6.3	-43.0
	Sa61	Alluvial aquifer	06/11/2007	841	306	13.2	7.1	3868	296	587	315	6951	1090	192		3.55	/	0.708195	-6.8	-44.5
	Sa65	Alluvial aquifer	07/11/2007	678	325	12.8	7.4	3057	119	504	64	4459	1096	1129		3.28	/	0.708244	-6.6	-44.5
	Sc09	Chalk aquifer	07/11/2007	614	311	12.5	7.5	2890	107	422	0	4033	893	1197		3.11	/	0.708177	-7.0	-46.6
	Sc11	Chalk aquifer	06/11/2007	565	282	12.0	7.2	2678	70	387	31	4672	603	484		3.05	/	0.708127	-6.9	-46.2
	Sc13	Chalk aquifer	06/11/2007	418	88	14.9	7.6	1798	103	300	38	3885	462	<DL		2.49	/	0.707820	-2.4	-24.7
	Sc16	Chalk aquifer	06/11/2007	679	146	12.7	7.3	3579	148	413	51	5180	910	519		3.37	/	0.708015	-6.2	-41.5
	Sc22	Chalk aquifer	06/11/2007	645	283	12.2	7.3

The Seine has been channelled since decades and natural flooding is quite reduced in the

Chapelle-sur-Oreuse in the Chalk aquifer upstream from the Bassée alluvial plain were

sampled in November 2007 (Fig.1). A complementary sampling campaign tooks place in

July 2008. Tritium analyses were carried out on water of all groundwater points sampled in

and the Seine at Bray-sur-Seine.

The analyses of major chemical elements and stable water isotopes (δ 2 H, δ 18 O), 3 H,

Sr/ 86 Sr, were completed by the determination of apparent groundwater ages using CFCs

0.01 pmol L -1 and SF 6 concentrations to 0.1 fmol L -1 . The reproducibility was about ±5% for

water samples.

RESULTS

4.1 Chemical composition of groundwater

Physico-chemical parameters (Table1) indicate that groundwater conditions in the northern

waters in wells Sa26, Sc09, Sc13, Sc16, SC22, ScH4, Sa16 and Sa38 (only for the

November campaign). In addition, the gravel-pit lakes have very low nitrate concentrations

pesticides in 116 analysed samples, which were detected at concentrations above the

detection limits. These pesticides and degradation molecules are atrazine (max. 0.091 mg/L

in the Voulzie River), deethylatrazine (max. 0.3 mg/L in Sc09), glyphosate (max. 5.59 mg/L in

Sa37) and aminomethylphosphonic acid, AMPA (max. 2.52 mg/L in Sa21). Atrazine has not

been used in this area since 2003. Glyphosate has both agricultural and non agricultural

uses, which makes its origin difficult to identify.

locally in the Seine Basin(Roy et al., 1999;[START_REF] Brenot | Interaction between different water 525 bodies in a small catchment area in the Paris basin (France): Sr isotopic tracing of multiple 526 Sr sources[END_REF], and for the Chalk aquifer in

On a 87 Sr/ 86 Sr vs. Ca/Sr diagram (Fig.7b), a positive correlation is seen between 87 Sr/ 86 Sr

Sc22 and Sa37 sampling points ( 87 Sr/ 86 Sr=0.70832 to 0.70858; Ca/Sr=1103 to 1253).

 and SF 6 were systematically measured for groundwater dating.

For five sampling points, Sa26, Sa38, Sa48, Sc16 and ScH4, semi-anoxic conditions most

The δ 2 H and δ 18 O isotope composition of groundwater is -47 to -25‰ and -7.1 to -2.4‰,

respectively (Table1). In the δ 2 H vs. δ 18 O diagram (Fig.9), most of the water falls on the

local meteorological line (LML) represented by Orléans station (IAEA/WMO, 2009) with no

water to the Alluvial aquifer, which then discharges into the Seine. More specific studies

[START_REF] Chabart | Étude des caractéristiques 531 hydrochimiques et hydrodynamiques du champ captant de la Grande-Paroisse[END_REF] have shown that, locally and occasionally, the River Seine can

influence the Alluvial aquifer.

(4 in the Chalk and 1 in the Alluvial aquifer), reducing conditions may have led to

denitrification. Existing data (www.eaufrance.ades.fr) underlines the importance of

contamination of the Alluvial aquifer. Actually, 40% of the wells in the Bassée Alluvial aquifer

From the geochemical tools the following conclusions may be drawn:

 No systematic chemical differentiation exists between the Chalk and Alluvial aquifer

waters. This confirms that the conceptual hydrogeological model is correct at a

project, and BRGM for this research project.

de faisabilité. Mine Paris.

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