Antonio Ventosa Cutillas

Carolina Albea

Alexandre Seuret

Francesco Gordillo

Relaxed periodic switching controllers of high-frequency DC-DC converters using the δ-operator formulation

This paper deals with the design of new periodic switching control laws for high frequency DC-DC converters. The contributions are twofolds. On a first hand, the DC-DC converter model are rewritten as a periodic switched affine systems thanks to a δ-operator formulation, which represent an efficient framework for the numerical discretization at high frequencies. On a second hand, three different control laws are provided, the first one being the usual Lyapunov-based control law and the two others being relaxed versions of this first solution. The benefits of these two new control laws over the usual Lyapunovbased one are demonstrated on an simple example. More particularly, it is showed that the selection of sampling period and of the control law strongly influence the size of the region of attraction.

I. Introduction

Nowadays, there is a relevant interest for DC-DC converters due to their numerous applications in the industry, as for example in computer power supply, cell phones, appliances, automotive, aircraft, etc. These systems can be modeled as switched affine systems (SASs), which represent a particular nonlinear class of switched systems. They correspond to a class of hybrid dynamical systems consisting of several operating modes represented by continuous-time subsystems and a rule that selects between these modes [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]. Compared to the linear case, the affine structure of these systems imposes a set of operating points defined for an averaged dynamic, leading to solutions in the generalized sense of Krasovskii.

Many works found in the literature in continuoustime control the SASs by a min-projection strategy [START_REF] Albea | Hybrid dynamic modeling and control of switched affine systems: application to DC-DC converters[END_REF], [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF], [START_REF] Pettersson | Stabilization of hybrid systems using a min-projection strategy[END_REF], even for systems with a general nonlinear form [START_REF] Liu | On the (h0, h)-stabilization of switched nonlinear systems via state-dependent switching rule[END_REF], [START_REF] Lu | A piecewise smooth control-lyapunov function framework for switching stabilization[END_REF]. In these works the provided controllers are good, but may lead to arbitrarily fast switching control. Some solutions to this problem can be found in the literature, as [START_REF] Buisson | On the stabilisation of switching electrical power converters[END_REF], [START_REF] Senesky | Hybrid modelling and control of power electronics[END_REF], [START_REF] Theunisse | Robust global stabilization of the dc-dc boost converter via hybrid control[END_REF], where, the authors aim at ensuring a dwelltime associated with an admissible chattering around the operating point. Nevertheless, [START_REF] Buisson | On the stabilisation of switching electrical power converters[END_REF] does not prove a minimum time associated to the spacial regularization. In [START_REF] Theunisse | Robust global stabilization of the dc-dc boost converter via hybrid control[END_REF], a focus on specific electronic architecture related to boost converters is proposed. In addition, the contributions of [START_REF] Senesky | Hybrid modelling and control of power electronics[END_REF] do not provide a complete stability proof. On the other hand, in [START_REF] Colaneri | Stabilization of continuous-time switched nonlinear systems[END_REF], the authors present an open-loop stabilization strategy based on dwell-time computation, Antonio Ventosa Cutillas and Francisco Gordillo are with University of Seville. Ad. Camino de los Descubrimientos s/n. 41092, Sevilla, Spain. aventosa,gordillo@us.es C. Albea and A. Seuret are with LAAS-CNRS, Univ. de Toulouse, UPS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France. calbea, aseuret@laas.fr [START_REF] Albea | Practical stabilisation of switched an systems with dwell-time guarantees[END_REF] proposes a minimum dwell-time with a space and time regularization, [START_REF] Hauroigne | Switched affine systems using sampled-data controllers: Robust and guaranteed stabilisation[END_REF] guarantees a minimum and maximum dwell-time by solving optimization problems. These solutions presents a common characteristic: systems are controlled by aperiodic switching.

In many occasions, it is necessary to control this class of systems with periodic switching, due to physical constraints. In order to deal with this issue, a solution consisting in the discretization of the continuous-time model with a fixed periodic sampling time was provided in [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF]. The authors of [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF] present a controller based on a Lyapunov function synthesized by solving an optimization problem, whose objective is to minimize the area around the equilibrium, where the solutions converge. On the other hand, in [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF], the authors design a sampleddata switching control with an upper-bound designed to ensure robustness in continuous-time systems. Through Linear Matrix Inequalities (LMI) based conditions, the upper-bound of the length of the inter-sampling interval can be directly related to the size of the asymptotic stability set around the considered equilibrium. Practical stability is obtained using Lyapunov-Krasovskii functional and the Jensen's inequality. Both solutions do not consider a high-frequency sampling, beside the fact that they are conservative because the controller are based on a Lyapunov-function.

In some applications when discretizing these systems, if the sample time is very low, several problems may appear to assess stability, because of numerical issues. Several solutions have been considered in the literature of automatic control. Among them, the δ-operator has been introduced in [START_REF] Middleton | Improved Finite Word Length Characteristics in Digital Control Using Delta Operators[END_REF]. It consists in a discretization method that becomes sufficiently close to the continuoustime model, ensuring continuity of the conditions for high-frequency samplings. In [START_REF] Eidson | Proc. IEEE South East conference[END_REF], a comparison is made between the operator q and the operator δ to perform this discretization. Here, it is possible to observe as the operator δ presents as advantages the natural convergence to a continuous system, while avoiding numerical problems when the sampling period is very short. In [START_REF] Viji | Improved Delta Operator based Discrete Sliding Mode Fuzzy Controller for Buck Converter[END_REF] it is possible to observe that the delta operator is used to perform the sliding mode fuzzy controller of a DC-DC buck converter due to the need for very fast sampling. Therefore, the use of the δ-operator presents a great advantage in the design of controllers with very fast sampling times like the one presented in this paper.

In this paper, we model the DC-DC converters in discrete-time by using the δ-operator and we control the system with a well-known min-projection strategy, based on a Lyapunov function. Then, we propose a periodic-sampling relaxed controller for these systems, allowing to obtain results less conservatives, even for highfrequency systems. Moreover, this approach presents a tradeoff between the sampling period and the size of the chattering effects. Some simulations in Matlab valid our contribution.

The paper is organized as follow: the problem formulation is stated in Section II. Then, a classical controller is presented in Section III. From this, Section IV proposes some relaxed controllers. An optimization of the controllers is given in Section V. Section VI illustrates the potential of this method on a particular DC-DC converter. The paper ends with a conclusion section.

Notation: Throughout the paper N and R denote the set of natural and real numbers, respectively. R n the ndimensional Euclidean space and R n×m the set of all real n×m matrices. The set composed by the first N positive integers, namely {1, 2, ..., N }, is denoted by K N . I is the identity matrix of suited dimension. The Euclidean norm of vector x ∈ R n is denoted by |x|. For any symmetric matrix M of R n×n , the notation M 0 (M ≺ 0) means that the eigenvalues of M are strictly positive (negative).

II. Problem formulation

A. System data

Inspired by the work in [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], we focus on the following class of switched affine systems, which is relevant in the context of DC-DC converters

ż = A σ z + a σ , (1)
where z ∈ R n is the state and it is accessible, A σ and a σ present suited dimensions. The control action is performed through the high frequency switching signal σ ∈ K N := {1, 2, ..., N }, which may be only modified at sampling instants t k , with k ∈ N. In this paper, the length of the sampling interval t k+1 -t k = T is assumed to be constant, known and small enough. This paper focuses on the design problem of a feedback law for the high frequency periodic switching signal σ, in such a way to ensure suitable practical convergence properties of the plant state z to a operating equilibrium z e , which is not necessarily an equilibrium for the continuous-time dynamics in ( 1), but can be obtained as an equilibrium for the switching system with arbitrary switching. A necessary and sufficient condition characterizing this equilibrium is then represented by the following standard assumption (see [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF], [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]).

Assumption 1: There exists λ = [λ 1 , λ 2 , ..., λ N ] satisfying i∈K λ i = 1, such that the following convex combination holds:

i∈K λ i (A i z e + a i ) = A(λ)z e + a(λ) = 0. ( 2
)
Remark 1: It is emphasized that Assumption 1 is both necessary and sufficient for the existence of a suitable switching signal ensuring forward invariance of the point z e (namely inducing an equilibrium at z e ) when understanding solutions in the generalized sense of Krasovskii or Filippov. Indeed, under (2), we can conclude that the error equation of (1):

ẋ = A σ x + B σ , (3)
where the error vector is denoted by x := z-z e and where the matrices B σ are defined by B σ := A σ x e + a σ and verify the following convex combination i∈K λ i B i = 0. The objective is to ensure that the error state x converges to the equilibrium x = 0 in the Filippov sense.

In addition, the following property is assumed. Assumption 2: The matrices A i , for i ∈ K N are nonsingular and A(λ) is Hurwitz.

B. δ-operator for high frequency switching function

In this paper, we will propose a discrete-time model based on the δ-operator [START_REF] Middleton | Improved Finite Word Length Characteristics in Digital Control Using Delta Operators[END_REF], which is suitable for high switching frequencies. The δ-operator has been widely used in the literature to avoid numerical problems in the computation of discrete-time dynamics. This is based on the continuous ones in the situation where the sampling period T is potentially very small. The definition of the δ-operator is as follows. For any function ξ from R + to R n , the vector δξ k , at any sampling instant t k ∈ R + , is defined as follows

δξ k := 1 T (ξ k+1 -ξ k ), ∀k ≥ 0,
where we used the convention ξ k = ξ(t k ) and δξ k = δξ(t k ), for all integer k ≥ 0. Hence the dynamics of system (1) can be rewritten in the framework of the δoperator, which yields the following dynamics

δx k = E σ x k + F σ (4)
where the matrices that defines the system dynamics are given by

E σ = 1 T (e AσT -I), F σ = 1 T T 0 e Aσ(T -s) dsB σ . (5)
The interest of this formulation compared to the usual discrete-time formulation comes from the fact that, when T goes to zero, matrices E σ and F σ converge to A σ and B σ , respectively. Another important issue is that matrices E σ and F σ depend explicitly on the switching period T . Indeed, considering small values of T may lead to several numerical problems when discretizing (3).

Remark 2: Note that if matrix A σ is non singular, then a simple expression of F σ is provided by

F σ = e AσT -I T A -1 σ B σ .
It is worth noting that model (4) does not account for the continuous evolution of (1) during the intersampling time. It is however possible to characterize the continuous solution by integrating the solution over a sampling interval, leading, for all t ∈ [t k , t k + T ] and for all k ∈ N, to

x(t) = e Aσ(t-t k ) x(t k ) + t t k e Aσ(τ -t k ) dτ a σ
Since, the matrices A σ are assumed to be Hurwitz, and since t belongs to the bound interval [t k , t k + T ], the solutions to the system are obviously bounded during the inter sampling time.

C. Control objectives

When considering such switching affine systems, asymptotic stability to zero is in general not possible. Therefore one has to relax the control objectives and to consider attractor sets, which are not necessarily reduced to the equilibrium set. In this paper, we will consider an estimation of the attractive set, which is of the following quadratic form

E := x ∈ R n , x Sx ≤ 1 . (6)
with S being a symmetric positive definite matrix to be optimized. This formulation is quite usual and has been used in other contexts as in [START_REF] Albea | Practical stabilisation of switched an systems with dwell-time guarantees[END_REF], [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF]. This paper focuses on the design problem of a feedback law for the high frequency periodic switching signal σ, in such a way to ensure suitable practical convergence properties of the plant state x to 0, which is not necessarily an equilibrium for the continuous-time dynamics in ( 1), but can be obtained as an equilibrium for the switching system with arbitrary switching. A necessary and sufficient condition characterizing this equilibrium is then represented by the following standard Assumption 2 (see [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF], [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF]). The problem can be summarized as follows Problem 1: For any small sampling period T , the problem is to find a switching control law that selects, at each sampling time, the mode or subsystem among all possibilities that stabilizes system (1) with certain performance guarantees at its equilibrium.

III. Lyapunov-based switching control

Looking at the literature on switched affine systems, one can find the well-known min-projection control law for such a class of systems [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF], [START_REF] Pettersson | Stabilization of hybrid systems using a min-projection strategy[END_REF]. The underlying idea of this control law is to select the mode of the system which minimizes the decrease of a quadratic Lyapunov function given by

V (x) := x P x, ∀x ∈ R n (7)
where P 0 is a positive definite matrix of R n×n . This idea is formalized in the following theorem.

Theorem 1: Consider Assumption 1 and Property 2 and matrices P 0 and S 0 of suited dimension that are solution to the feasibility problem

Γ (1) ij ≺ 0, ∀i, j ∈ K (8)
for any pair (i, j) in K 2 .

Γ

(1) ij

= Ψ i (P ) + µ i S 0 0 -1 + γ i [Ψ j (P ) -Ψ i (P )] , Ψ i (P ) = P E i + E i P + T E i P E i P F i + T E i P F i F i P + T F i P E i T F i P F i , (9)
for some given parameters γ i > 0 and µ i > 0. Then the switching control (C1) law defined by

(C1) σ(x k ) = argmin i∈K N x k 1 Ψ i (P ) x k 1 (10)
guarantees ∆V <0 outside of set E. Proof: Consider the Lyapunov function given in [START_REF] Eidson | Proc. IEEE South East conference[END_REF], where P 0 ∈ R n×n . Let us first compute the expression of δV k as follows

δV (x k ) = 1 T (V (x k+1 ) -V (x k )) = 1 T ((x k + T δx k ) P (x k + T δx k ) -x k P x k ) = 2δx k P x k + T δx k P δx k . (11)
Replacing δx k by its expression given in (4), and using the definition of the matrix Ψ i (P ) provided in (9) yields

δV (x k ) = x k 1 Ψ σ (P ) x k 1 .
Using ( 9), the previous expression can be rewritten as follows, for any j in K

δV (x k ) = x k 1 Γ (1) σj -µ σ S 0 0 -1 -γ σ [Ψ j (P ) -Ψ σ (P )] x k 1
Since the matrix inequalities Γ

i,j ≺ 0 holds for any pair (i, j) in K 2 , we have

δV (x k ) < µ σ (1 -x k Sx k ) -γ σ x k 1 [Ψ j (P ) -Ψ σ (P )] x k 1
Note that the switching control law [START_REF] Hauroigne | Switched affine systems using sampled-data controllers: Robust and guaranteed stabilisation[END_REF] ensures that the last term of the right hand side of the previous inequality is negative, which guarantees that

δV (x k ) < µ σ (1 -x k Sx k )
The previous inequality finally guarantees that for any values of x k outside of E (i.e. 1 -x k Sx k < 0), the quantity δV (x k ) is strictly negative, which concludes the proof.

IV. Relaxed switching controller

In the previous section, a Lyapunov-based switching control was presented. This control was clearly inspired from the existing literature on switched affine systems such as [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF], [START_REF] Hetel | Robust sampled-data control of switched affine systems[END_REF] but adapted to the δ-operator modelling of DC-DC converters. The motivation of this section is to present a relaxed version of the previous control law. This relaxation considered here is related to the fact that the a priori intuition behind this Lyapunov-based control law might be too restrictive in the sense that the selection of the Lyapunov matrix P is done to verify two distinct purposes, namely, the definition of the Lyapunov function and the construction of the switching signal. Based on this comment, a relaxed control law can be provided by decoupling the two problems of selecting a Lyapunov function and of designing the switching control law.

The relaxed control law is based on the simple idea consisting of keeping the same structure of control law presented in [START_REF] Hauroigne | Switched affine systems using sampled-data controllers: Robust and guaranteed stabilisation[END_REF]. However, instead of using the Lyapunov matrix P , a new unconstrained matrix is introduced to define the switching law. This is formalized in the following theorem.

Theorem 2: Consider Assumption 1 and Property 2 and matrices P 0 and S 0 of suited dimension, a new matrix N ∈ R n×n that are solution to the feasibility problem Γ

(2)

ij ≺ 0, ∀i, j ∈ K (12)
where, for any pair (i,

j) in K 2 , Γ (2)
ij = Ψ i (P ) + µ i S 0 0 -1 + γ i [Ψ j (N ) -Ψ i (N )] , (13)
where the matrix Ψ i (P ) and Ψ i (N ) are given in [START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF] and, again, for some given parameters γ i > 0 and µ i > 0. Then the switching control (C2) law defined by

(C2) σ(x k ) = argmin i∈K N x k 1 Ψ i (N ) x k 1 (14)
guarantees ∆V <0 outside of set E. Proof: The proof strictly follows the proof of Theorem 1, except that, now, the switching control law is not characterized by the Lyapunov matrix P but by an arbitrary matrix N , which only has to be a solution to the feasibility problem [START_REF] Liberzon | Basic problems in stability and design of switched systems[END_REF].

Remark 3: Compared to Theorem 1, there are two main advantages. The first one relies on the fact that matrix N is not required to be symmetric nor positive. The second one consists in the fact that the switching law is now completely decoupled from the definition of the Lyapunov function. Moreover, one can see that selection N = P in Theorem 2 leads to the same statement as in Theorem 1. This ensures that the set of feasible solutions of ( 12) is greater than the ones of [START_REF] Grant | Graph implementations for nonsmooth convex programs[END_REF]. It is then expected to derive relaxed solutions that will be presented in the example section where the optimization procedure presented later on in Section V is included.

V. Optimisation procedure

The feasibility problems proposed in Theorems 1 and 2, only ensure that there exists a switching control law that stabilizes the system to a bounded region around the equilibrium. Without an optimization process, the resulting regions might be too large to be relevant from the physical point of view. Indeed, considering a too large set E can possibly increase the chattering effects which are the main phenomena to avoid or limit in the control design of DC-DC converters. This chattering behavior can damage or even break the devices. Therefore, it is necessary to include an optimization procedure in these theorems, whose objective is to minimize the size of the set E. Since this set is fully characterized by the symmetric positive definite matrix S, minimizing the size of E can be achieved by maximizing the determinant of S. Based on the discussion above, the following proposition of stated dealing with the optimization of the solutions to the conditions of Theorems 1 and 2 is stated.

Proposition 1: For any a priori fixed scalar parameters µ j and γ i , the optimisation problem max

S,P det(S), s.t. Γ (c) ij ≺ 0, ∀i, j ∈ K (15)
for any c = {1, 2}, minimizes the size set [START_REF] Deaecto | Discrete-time switched linear systems state feedback design with application to networked control[END_REF] guaranteeing ∆V <0 outside of set E. Remark 4: Note that we do not optimize the chattering region, because it is constrained to an ellipse form and, the controller is also constrained for a given structure. Thus, it is expected that set E will be relaxed with the control given in Theorem 2, with respect to the control law C1..

Remark 5: This optimization process strongly depends on the selection of the scalar parameters γ i and µ j . Hence it is expected that an iterative procedure to selected the best parameters needs to be included.

Remark 6: Other optimization objectives can be considered such as the maximization of the eigenvalues of S, which can be done by maximizing a scalar τ such that τ I ≤ S.

VI. Application to a boost converter

The control laws introduced above are evaluated on a classical boost converter system. This converter switches at high frequency between two modes (N = 2) corresponding to two affine subsystems. The state variable is defined by x = [i L v c ] , where i L denotes the inductor current and v c the capacitor voltage.

We take the parameters given in [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF] for comparison with the switched control algorithm presented therein, which switches with arbitrary aperiodic switching in the steady-state. This type of switching tends to be complicated in physical applications. The considered nominal values are: V in = 100V , R = 2Ω, L = 500µH, C o = 470µF and R o = 50Ω. The switched system state space model ( 1) is defined by the following matrices for i = 1, 2:

A i = -R L (i-1) L (i-1) C0 -1 R0C0 , a i = 1 L 0 .
The chosen simulation parameters are given by z e = 3 120 , λ = [0.22 0.78] for which simple calculations ensure the satisfaction of Assumption 2. The optimization problems given in Proposition 1 is solved using the CVX solver [START_REF] Grant | Graph implementations for nonsmooth convex programs[END_REF], [START_REF] Grant | CVX: Matlab software for disciplined convex programming[END_REF]. The results obtained with this software illustrate our the efficiency of the new control law presented in Theorem 2 with respect to the Lyapunov-based controller employed in the literature, presented in Theorem 1.

As pointed out in Remark 5, the optimization scheme presented in Proposition 1 delivers different results for different values of γ i and µ j . Therefore, a random algorithm has been considered to obtain the best tuple of parameters. The values of these parameters µ 1 , µ γ 2 , obtained for several values of the sampling period T s are provided in Table I.

Figure 1 shows the state trajectories, set E and δV > 0 surface in the state-plane for the different controllers given in Proposition 1, as well as for different sampling periods. Note that the δV > 0 surface is in the interior of the set E as is expected from Theorem 1 and 2. Remark also as E is reduced, as T s decreases. We can see that set E is reduced with control law C2 w.r.t. C1, showing as the control law C2 provide reduced region of attraction with respect to C1. This is consistent with Remark 3. Another important remark concerns the fact that the region where δV > 0 is concentrated in a smaller area in control law C2 with respect to C1.

These simulations demonstrate the advantages of the relaxed control law over the existing Lyapunov-based one. Indeed, our proposed controller C2 allow to control efficiently the DC-DC converters under study even with relatively high sampling period with a notable reduction of the switches of these systems, which ensures an increase of the lifespan and the reduction of the dissipated energy.

VII. Conclusions and future work

In this paper, we have presented two main contributions. The first one deals with the definition of accurate discrete-time model for high frequency sampling DC-DC converters. The second contributions consists in the extension of the usual Lyapunov-based control laws employed in this domain to a less restrictive control law that encompass this first formulation. This new control law deliver notable improvements with respect to the Lyapunov-based control law in terms when comparing the estimate of the set E of the switched affine system.

  (a) C1 with T = 10 -4 . (b) C1 with T = 10 -5 . C1 with T = 10 -6 ; (d) C2 with T = 10 -4 .(e) C2 with T = 10 -5 . C2 with T = 10 -6 .

Fig. 1 :

 1 Fig. 1: Numerical results of Proposition 1. Time trajectories in green, set E in yellow and δV > 0 in red.

  2 , γ 1 and

	Parameters	Ts	µ 1	µ 2	γ 1	γ 2
	Th. 1	10 -4	9.84	9.86	0.948 0.0515
		10 -5	0.16	0.157 0.777	0.223
		10 -6	0.907 0.907 0.782	0.218
	Th. 2	10 -4	3.16	3.16	0.916	0.327
		10 -5	1	1	0.349	0.1
		10 -6	0.313 0.313	10	2.793

TABLE I :

 I Numerical values of µ 1 , µ 2 , γ 1 and γ 2 .


V Renaud

F Lahaie

G Armand

T Verdel

P Bigarré

Conception, numerical prediction and optimization of geomechanical measurements related to a vertical Mine-by-Test at the Meuse/Haute-Marne URL

Keywords: instrumentation, conception, design, under-excavation technique, numerical modelling, field stresses, measurement

Andra is conducting scientific experiments in the Meuse/Haute-Marne Underground Laboratory among which REP experiment is a vertical mine-by-test focusing on short and long term hydromechanical response of the argilite to the main shaft sinking. Displacements, strains, and pore pressures will be monitored while the shaft is passing down. Andra and INERIS intend to back-analyse most recorded geomechanical data based on under-excavation

numerical technique in order to estimate pre-existing field stresses. The under-excavation interpretative technique consists in determining the pre-existing stress tensor related to a quite large volume of rock based on generalized inversion of geomechanical measurements recorded during the disturbance of the host rock (typically the excavation of an underground opening). In the framework a numerical study aiming to test accurately the sensitivity and numerical stability of this interpretative technique, 3D modelling of a step-by-step vertical mine-by-test, based on REP design, has been undertaken. One major step of the numerical

INTRODUCTION 1.1 OBJECT/CONTEXT

This study is part of the research related to REP experiment in the underground laboratory of Meuse/Haute-Marne (located at Bure, France).

Main objective of REP experiment is to study the short and long-term response of argillite to the sinking a vertical shaft (REP). The experiment will allow to record a great number of geomechanical measurements (strains, displacements, tilts.) around the excavated works. It is then possible to estimate, through a generalized numerical inversion, the pre existing stress field in the virgin rock mass.

Background literature and field experimentations show both this technique, known as the "under-excavation" or "undercoring" technique, as very promising.

The under-excavation technique was first described and proposed by Wiles and Kaiser [START_REF] Wiles | In Situ Stress Determination Using the Underexcavation Technique -I[END_REF][START_REF] Wiles | In Situ Stress Determination Using the Underexcavation Technique -II[END_REF], which applied it quite successfully in the granitic context of the underground laboratory of the AECL (Atomic Energy off Canada Ltd, [START_REF] Read | Technical summary of AECL's Mine-by Experiment Phase 1: Excavation response[END_REF][START_REF] Thompson | In situ rock stress determinations in deep boreholes at the Underground Research Laboratory[END_REF][START_REF] Tonon | Stresses in anisotropic rock masses: an engineering perspective building on geological knowledge[END_REF]). This technique was evaluated by Andra and INERIS in a marl formation (potash mining in Alsace: MDPA, France [START_REF] Bigarré | Casamance, Mesures de déformations, interprétation des données[END_REF]) and clays type (Mont Terri, [START_REF] Bigarré | Mt. Terri Project, Phase III, Stress Measurement Experiment[END_REF][START_REF] Martin | Measurement of in-situ stress in weak rocks at Mont Terri Rock Laboratory, Switzerland[END_REF]). These experiments allowed to confirm both all the interest of this technique [START_REF] Galybin | A measuring scheme for determining in situ stresses and moduli at large scale[END_REF] (even if it is not yet largely used [START_REF] Martin | Stress, instability and design of underground excavations[END_REF]) while raising up some issues concerning its sensitivity and stability as regards both uncertainties affecting the important amount of input data handled and moreover the model used.

Research has then been undertaken based on numerical simulation of a standard, synthetic 3D experiment, aiming to a careful, detailed evaluation of this technique.

DESCRIPTION OF THE UNDER-EXCAVATION TECHNIQUE

The under-excavation technique requires several assumptions, whose principal one is the linear elastic behaviour of the rock mass. If several types of instruments are to be set up around the work (CSIRO cells, extensometers, inclinometers, convergence meters, clinometers, etc. figure 1), a linear relationship between strains/displacements and stresses can be expressed in the following matrix form:

  ... M      
(this implies an elastic behaviour).

[M] is the influence matrix.

                                                         N N N N N N N P Q Q a a a
                                                       X Y P P P P P P Z YZ XZ Q Q Q Q Q Q XY R R R R R R b b b b b b b b b b b c c c c c c c c c c c c d d d d d d d d d d d d      (1)
 

M

where:    is the initial stress tensor and the a ij , b ij and c ij are the influence coefficients respectively connecting the measurement variations of the sensors, with the components of the initial stress tensor    on the assumption of a linear elastic behaviour of the rock mass.

Determining the six components of the initial stress tensor amount thus to solve the system of N+P+Q+… linear equations with 6 unknown. The difficulty of the under-excavation method lies in the direct problem: i.e. the determination of the coefficients of the matrix [M] which requires the numerical modelling of the experiment with 6 canonical loading schemes.

For each one of these 6 simulations and with each stage of work excavation, the method consists in recovering, at the location of each virtual sensor, the local value of the stress shift (for CSIRO cells), the new position (for extensometers and inclinometers), the new angle (clinometer), i.e. all induced perturbations due to the mining work progress. These values are then transformed into virtual measurements of strain (CSIRO cells), relative displacement (extensometer) and so on.

GENERAL CONSIDERATIONS

The synthetic experiment run consisted on the "virtual" monitoring of the progressive excavation of a 6.25 m in diameter vertical shaft around which were laid out beforehand different sensors,. Simulations have been carried out starting from the computer code FLAC 3D

(transverse isotropic elastic behaviour + continuous and homogeneous medium).

3D mesh has been designed in order to fit as much as possible location of sensors, made in this case of 3 CSIRO cells, 3 multipoint extensometers and 1 multipoint inclinometer.

An additional calculation was carried out using results of triaxial loading tests in accordance with the one already estimated on the field. This was needed in order to check numerically several supplementary functionalities implemented in SYTGEOmath interpretative tool developed by INERIS.

DESCRIPTION OF THE NUMERICAL MODEL

MODEL GEOMETRY

The model geometry is presented on figure 1. Dimensions of the model are 75 m x 75 m x 95 m: the lower and higher dimensions being respectively -515 m and -420 m. A meshing made up of 114009 elements, that is to say 120870 nodes was generated.

The model is defined in the coordinates system of principal stresses (X, Y, Z), which corresponds to a rotation of 25° of the general East-North (x, y, z) coordinates system. The meshing has been adapted to take into account the theoretical location of some measurement points (figure 1).

MECHANICAL PROPERTIES -EXCAVATION PROCESS

Input mechanical properties data are those presented in chapter VI of Andra report "Geological Referential of the site of East" [START_REF] Collectif | Référentiel Géologique du site de Meuse/Haute-Marne -Tome 4 Le Callovo-Oxfordien[END_REF]. They are summarized for each main geological facies in table 1. The shear modulus G 13 of the transverse isotropic elastic law was calculated with the relation of Lekhnitskii [START_REF] Lekhnitskii | Theory of elasticity of an anisotropic elastic body[END_REF], based on laboratory tests:

  13 13 13 1 3
12

EE G EE    (2)
The boundary conditions correspond to a null normal displacement on the vertical faces and the lower horizontal face. The upper horizontal face is loaded with stress components.

All simulations have been carried out following two principal phases:

 first phase corresponds to the calculation of the initial state of equilibrium before shaft sinking onset ;

 second phase corresponds to the shaft sinking simulation in 31 successive stages.

MODEL VALIDATION

The numerical model was validated based on 3 x 3 x 4 calculated stress profiles compared with the analytical solutions of the stress field in elastic homogeneous medium around an infinite cylindrical shaft. Error acceptance has been set as very low in order to minimize errors due only to numerical modelling artefacts in the overall procedure.

Table 2 recapitulates the maximum absolute and relative error (between numerical and analytical results) made near the various measurement points.

ANALYSIS OF THE MEASUREMENTS OBTAINED ON THE VARIOUS INSTRUMENTS OF THE VIRTUAL EXPERIMENTAL DEVICE

Measurements obtained in the canonical loading simulations

Measurements obtained correspond to the influence coefficients relating these measurements to the corresponding components of the initial stress tensor. Basic analysis of these measurements allows to evaluate amplitude versus time fonction of each sensor (maximum, signal to noise ratio, gradients, etc.). Then analysis of these data (of the 42 graphs like those presented on figure 4.), instrument by instrument, offers a unique mean to identify quantitatively numerous singular conditions as, for example:  redundancy of sensors inside a same instrument (extensometer) can not be justified in terms of quantitative improvement of the overall instrumentation set up;


 a sensor may show a narrow predicted useful data range to be back analysed; this can be anticipated by further considerations on the front face working progress of the opening;

 best numerical conditioning is to be obtained by combining instruments bringing of additional information, for example CSIRO cells (information on all the components of [ 0 ] except  0 ZZ ) and axial extensometer 3 (information only on  0 ZZ ).

Numerical results of the shaft sinking

The graphs of evolution of the numerically simulated measurements obtained on each instrument in the triaxial case (loaded with an estimated stress state representative of the experiment depth) are presented on figure 4. On can note some of the relevant points below:

 the order of amplitude of the measurement variations obtained on CSIRO 1 cell (more than 4000 µm/m in extension) is relatively high taking into account the range of recommended use for CSIRO cells, i.e. 2500-3000 µm/m. This remark thus encourages to recommend the taking of this cell away the shaft side wall;

 the variations of maximum displacement obtained on the sensors of extensometers 1 and 2 lie between 500 and 2400 µm (4000 to 8500 µm for extensometer 3). These variations are at the same time sufficient with respect to the precision of these instruments ( 50 µm)

and remain quite lower than their measurement range (105 µm);

 the variations of displacement obtained on extensometer 3 are at the same time sufficient with respect to their precision ( 50 µm) and lower the tolerance range considered for this instrument (100000 µm);

 the variations of measurements obtained by the inclinometer lie between 100 and 1000 µm. These variations are weak, but remain sufficient for the points closest to the shaft (points n°3 to 5). On the other hand, they become insufficient, with respect to the device precision ( 100 µm/m) for the points furthest away from the shaft (points n°1 and 2);

 the comparison (figure 3) of the displacements obtained at the position of the reference points of extensometers 1, 2 and of the inclinometer with those obtained at the position of their first measurement point (point n°1) shows well that displacements of the reference positions are significant and do not have to be neglected.

Those simulations have been completed with elastoplastic modelling of the shaft sinking in order to finalize the layout of the experiment. As this article is focus on the under excavation method which requires elastic model, the result of elastoplastic are not shown here.

DATA INVERSION STRATEGY

PRESENTATION

Increasing number of measurements of varied types make the data to be selected and to be inversed quite difficult. Because there are a too great number of possible choices which relate at the same time to:

 number of sensors (data sub sets) to be inversed;  temporality of the considered measurement intervals compared to the face advance;

 number and width of measurement intervals;

 relative temporal shift between the considered intervals, etc…

At the same time, as it was mentioned for the under-excavation technique, by Wiles & Kaiser [START_REF] Wiles | In Situ Stress Determination Using the Underexcavation Technique -II[END_REF] who proposed a methodology of selection of the data to be taken into account in the inversion (figure 5). Wiles and Kaiser propose for the stage B three different strategies (figure 6):

 "simple interval" (SI);

 "multiple intervals with shifted origins" (MISO);

 "multiple intervals with common origins" (MICO).

In this study, we explored a large set of possible choices for combination of instruments at stage A (12 combinations of instruments: C1, C2, C3, C1-E3, C2-E3, C3-E3, C1-C2-C3, C1-C2-C3-E3, E1-E2-I1, E1-E2-I1-E3, C1-C2-C3-E1-E2-I1 and C1-C2-C3-E1-E2-I1-E3) and all possible combinations for stages B and C.

Moreover, for stage B, we tested a fourth procedure (called TOT), which consists in taking into account all the possible intervals of measurement in the inversion. The aim of this last approach is to be able to give a more complete answer on the influence of each assumption on the estimated initial stresses. It is thus possible to establish the type of instrument or the intervals of measurement which it is necessary to consider in the inversion to be able to limit to the maximum the influence of the assumptions that one wishes to test.

APPLICATION TO THE TRIAXIAL LOADING SIMULATION

The triaxial tests simulation allows to select the most favourable inversion methods. The application of the methodology of inversion led to 15660 inversions to be run through an automated function implemented in the interpretative tool (1305 by combination of instruments). All possible inversions are studied in order to have a better estimation of stresses and their linked error.

The assumptions used in this method being the same as those supposed to build the matrix

[M], we have to find, after inversion, the initial stress tensor imposed on the model boundaries if the matrix [M] is well conditioned (the matrix conditioning is the ratio of its greater eigenvalue on its smaller eigenvalue). It is allowed that a conditioning ranging between 0 and 10 is "very good", "good" between 10 and 20, "acceptable" between 20 and 30, and that beyond 30, the inversion of measurements presents a significant amplification risk of numerical errors.

Figure 7 shows the cumulative distribution (P<(x), percentage of case where x is lower than a given value) of the conditioning value according to the combination of instruments taken into account in the inversion. The conditioning value of the influence matrix (figure 7) is very variable according to the combination of instruments considered in the inversion. Best conditioning is not obtained by considering all the instruments, but only the CSIRO cells and the extensometer located in the shaft axis (E3). For these two combinations of instruments, almost all the inversion methods lead to an acceptable conditioning (< 30). On the other hand, the inversions only carried out on the extensometers except the one placed axially inside the front face of the shaft and inclinometer (E1-E2-I1) never lead to an acceptable conditioning.

As for conditioning, we note that the difference between the back calculated and prescribed stresses is very variable according to the combination of instruments considered in the inversion. The combinations of instruments giving place to a bad conditioning tend overall to generate a more important error on the estimated stresses. This global correlation between the conditioning (COND) of the influence matrix and the made error DEVMAX on the estimated stresses is shown on figure 9.

However, one can note certain differences between figures 7 and 8. For example, the combination of instruments giving place to best conditioning (C1-C2-C3-E3) is not that which produces less error on the estimated stresses (even if it remains among the best).

A limited number of inversion cases were selected. These favourable inversion cases are those which lead at the same time to best conditioning and the weakest error on the prescribed stresses (COND < 30, DEVMAX < 1%). The number of cases thus selected for the study continuation is 2151, that is to say 13.7 % of the number of initial inversions.

CONCLUSION

Within the framework of REP experiment in Bure (France), Andra and INERIS intend to develop an under-excavation interpretative technique in order to reduce uncertainty on the insitu stress state in the Callovo-Oxfordian argillite formation.

A 3D synthetic numerical study has then been completed in order to assess quantitatively the overall reliability and performance of the under-excavation technique. The methodology has been extended in the way that:

 intermediate results needed as calculated influence coefficients and total predicted measurements of all varied sensors to be implemented could be back analysed in terms of operational recommendations, regarding a given sensor or a subset of sensors;

 inversion numerical strategies are explored based on calculated indicators able to quantify comparatively different instrumentation schemes versus excavation front face overall lay outs, aiming to minimize computational undesired artefacts.

This study is one of the studies necessary to the REP experiment design, nevertheless, the calculations presented in this article allow to improve the experimental device by:  moving away CSIRO 1 from the shaft side wall of approximately 3 m;  relocating the inclinometer sensors by reducing the bars length and by increasing the number of measurement points.

Moreover, this study shows that the reference points of the extensometers out of shaft and the inclinometer will move significantly in the shaft passage (comparison of figure 3a with figure 3b). This problem can be avoided by not inversing measurements out of shaft that starting from the stage -460 m.

3D numerical conception of an under-excavation interpretative experiment, e.g. aiming to back estimate an unique, overall quantitative result as field stresses, appears to be of major interest for optimizing data quality to be recorded, first for a specific sensor considered, secondly and moreover for the quality of the overall instrumentation scheme, or whatever can be called "information wealth" of data set to be recorded. This "optimal 3D design" approach includes complex factors usually difficult to handle at the same time for the rock mechanics engineer as:

1-best instrumentation coverage of the 3D complex geometry of the advancing excavation inside the instrumented volume of rock;

2-geomechanical properties of host rock to be monitored;

3-correct spreading of the varied instruments to be set up.

  no measuring instrument considered individually shows a satisfactory sensitivity to the whole components of [  ]. This result is clearly accentuated by the 2D final geometry of the experiment, once the shaft has been completely passing by the monitored volume of rock;

Figure 8

 8 Figure 8 shows the distribution of the maximum relative difference (noted DEVMAX) between the values of stresses imposed on the model  0 i imp and those estimated by inversion

Figure 10 shows

 10 Figure10shows the distribution of the favourable inversion cases according to the

2 C

 2 o m è tr e I1 E xte n s o m è tr e E X 1 E xte n s o m è tr e E X 2 C ellu le C S IR O 1 C ellu le C S IR O

Figure 2 :

 2 Figure 2 : 3D overview of the measurement device around the shaft

Figure 3 : 3 Figure 4 :Figure 5 :

 3345 Figure 3 : Comparison between displacements of the n°1measurement and reference points of extensometer 2

Figure 6 :Figure 7 :

 67 Figure 6 : Selection procedures of intervals of measurements taken into account in the inversion = stage B (adapted of Wiles & Kaiser [15])

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8 : Distribution of the maximum relative difference (DEVMAX) between the estimated stresses and the prescribed stresses in the triaxial case

Table 1 :

 1 Mechanical characteristics of facies A, B and C

	Parameter		Facies A	Facies B & C
	Wet density			 h = 2420 kg/m 3 
	Young modulus  to the plan of transverse isotropy	E 3 = 5200 MPa	E 3 = 5200 MPa
	Young modulus  to the plan of transverse isotropy	E 1 = 6300 MPa	E 1 = 6300 MPa
	Poisson's ratio			  =   = 0,30
	Shear modulus G 13		G 13 = 2144 MPa	G 13 = 2144 MPa
		 rr 	  	 zz 	 r 
	Relative error	1.58%	1.01%	1.21%	0.62%
	Absolute error (MPa)	6.71E-02	1.00E-03	1.00E-03	2.13E-03

Table 2 :

 2 Absolute and relative maximum errors between numerical and analytical resultsFigure 1 : Detail of the horizontal meshing and localisation of measurement points


Renaud Vincent
email: vincent.renaud@ineris.fr

Tritsch Jean-Jacques
email: jacques.tritsch@ineris.fr

Franck Christian
email: christian.franck@industrie.gouv.fr

MODELING AND ASSESSMENT FOR SUBSIDENCE HAZARD IN INCLINED IRON MINING

Keywords: hazard, subsidence, inclined seams, iron mine, modeling aléa, affaissement, gisement penté, mine de fer, modélisation

The old iron mines of the North-West of France have geometrical and exploitation configurations appreciably similar, with dips varying between 30° and 90°. Within the framework of the establishment of risk maps related to these exploitations, the observation of subsidence in certain basins leds us to try to better know the conditions of occurrence and the consequences on the surface of these phenomena, and in particular the influence of the dip on their relevance. A modeling was thus undertaken, consisting initially of back-analysis of a subsidence trough observed and studied, in order to seek the initiating mechanism within mining work and to appreciate the influence and the degree of reliability of the parameters, and in the second time the parameterised analysis of the zones of potential failure according to the dip, the opening of the mine seam, the extraction ratio and the thickness of the overburden. The contribution of this modeling and the experience feedback of other mining basins allowed to fix the principles of evaluation of the subsidence alea, in terms of intensity and occurrence, of these deposits.

Introduction

In the framework of the assessment and the prevention of mining hazards, the establishment of risk maps related to the movements above the iron deposits exploited in the North-West of France (figure 1) highlighted their relative homogeneity and singularity.

These basins indeed have geological and exploitation characteristics of formation relatively homogeneous. In addition, and mainly because these exploitations have an important dip (between 30° and 90°), it quickly appeared during the information collection and the first observations of disorders on the surface that the risk evaluation of ground movement, and especially of subsidence occurrence, had to take into account the singularity of these deposits and could not be completed with the analyses made for horizontal mining works. This article initially describes the characteristics specific to these exploitations. Then, the objectives are presented, steps and results of the modeling made on the basis of back-analysis of an observed subsidence phenomenon. Finally the transcription of these results and the evaluation of the subsidence risk are discussed.

Characteristics of North Western iron mining exploitations

The risk analysis is developed for studies at the scale of a whole basin of risk, even several risk basins, if they present strong analogies. It is the case of the iron deposits of the synclinals of Soumont, May/Orne, La Ferrière-aux-Etangs (Normandy) and Segré (Pays-de-Loire).

Figure 1. Localisation of the iron-bearing basins of the North-West of France (Varoquaux and Gerard, 1980).

The various basins present much analogies on the geological and exploitation aspects. These deposits fit in the dissymmetrical synclinal whose periods of deposit (Ordovician or Silurian) and of crumpling are near on a geological scale. They are fairly to strongly slopes (figure 2), located at very close depths (between 10 and 600 m) and hold one or two veins of low or average thickness (overall 2 to 4 m, locally more). [START_REF] Maury | Aménagement de la mine de May-sur-Orne en stockage souterrain d'hydrocarbures[END_REF].

The nature and the strength of the iron ore are relatively variable. The ore is mainly constituted of haematite at May-sur-Orne and at a shallow depth under the calcareous overthrust of Soumont. The ore is carbonated in-depth in others basins, like at Segré and La Ferrière-aux-Etangs. The compressive strength of the ore is about 100 MPa at May-sur-Orne and 200 MPa (perpendicular to the bedding plane) at La Ferrière-aux-Etangs.

The mining methods used in those several basins are appreciably similar (figure 3). The oldest mining sites were exploited by short dip faces also called stops, then by dip strike faces. Thereafter, one systematically applied the method of the rise faces or the mechanised strike faces for the mining sites with low slopes (dip lower than 50°) and the shrinkage method for the mining sites slopes to high slopes (dip higher than 50°). These works are connected by level galleries connected to the works of ore extraction, and spaced of 30 m to 75-80 m in altitude, according to the basins and the methods used.

The observed disorders (table 1) in these various basins are similar (primarily some localized sinkholes by crown section rupture, shaft or raise clearings, or collapses of galleries). One notes however the existence of collapses of important districts at the bottom, in production run, in general without repercussions on the surface, except for Soumont and La Ferrière-aux-Etangs. The observed disorders on surface are traditional depressions with spread out board, with opened cracks but without frank breaks of shearing, which can be connected with subsidence troughs.

On the other hand, the documentary analyses do not identify any accident of huge collapse type: the only events known in the western French basins are exclusively the fact of slate exploitations whose common factors are their complex geometry, very different from iron mining works, and the presence of important residual voids [START_REF] Tritsch | Assistance technique à l'élaboration d'un dossier de demande d'abandon, carrières de Misengrain -site de Noyant[END_REF].

Modeling of the inclined deposits of the West of France

Modeling approach

The study of modeling, using code UDEC (2D calculations in discontinuous medium), was organised in two stages:

 the back-analysis of a subsidence trough in Soumont developed in 1966 above a well delimited underground collapse;  parametric analysis of the zones of potential failure (extension, amplitude) according to the dip (30 to 65°), the extraction ratio (70 to 90 %), the layer thickness (1.5 to 5 m), and thickness of the formations of overburden in discordance (0 to 50 m).

Back-analysis of the subsidence phenomenon of Soumont appeared in 1966

Several collapses occurred in the mine of Soumont between 1929 and 1966. They mainly induced subsidence troughs on the surface (figure 4). The latest one is the most documented for underground visits and analyses of the causes were carried out in close connexion with this collapse. Thus this event has been selected for the back-analysis. Collapse occurred between the levels -120 and -250 m, 40 years after the exploitation of this sector of mining works. The dip of the layer is 30° and the extraction ratio is high (80-85 %).The maximum subsidence measured at that time was 65 cm. The back-analysis was based on a modeling on the collapsed district scale. The objective was to specify the conditions which were at the origin of collapse and the most relevant mechanisms. This work was carried out by respecting the three following checking points:

 A subsidence amplitude of 0.65 m was measured on surface;  The absence of collapse in work of depth higher than 220 m;  The stability of the mining works exploited with the same depth by shrinkage, to the west of the collapsed districts.

Description of the mining conditions

The extraction ratio of the lower stages decreases with depth. The section of the way is trapezoidal and the opening of these levels is 4.5 m. The pillar width of the lower stages varies between 4.5 m and 6 m. Above the ore layer, one meets massive and resistant schist beds for a total thickness of 120 m, then sandstone beds over 95 m, then a schist alternation and sandstone of weak thickness (10 m) and finally again sandstone. In lower part of the ore layer, there is a 10 m thick schist bed then a series of sandstones. The whole of these formations is covered by a calcareous slab whose thickness can be evaluated to 30 m at the location of the concerned sector [START_REF] Tincelin | Mine de Soumont -Mesures à entreprendre pour prévoir l'imminence d'un risque d'effondrement survenant à l'aplomb des routes nationales n°158 ou départementales n°43[END_REF].

Geomechanical characteristics

The geomechanical characterisation of various materials of the southern side of the mine of Soumont is not complete. Only the iron ore and its immediate roof and floor have been tested in laboratory. Hence we estimated the data according to various sources:  values obtained in laboratory: bibliographical study carried out at the time of the preliminary phase [START_REF] Delaunay | Phase préliminaire à la réalisation d'une modélisation numérique sur les gisements pentés des bassins ferrifères de Soumont[END_REF];  data resulting from a study on the slate mine of Misengrain [START_REF] Tritsch | Assistance technique à l'élaboration d'un dossier de demande d'abandon, carrières de Misengrain -site de Noyant[END_REF];  data from the database by [START_REF] Fine | Le soutènement des galeries minières[END_REF];  data (for the sandstone) resulting from the synthesis of the mechanical characterisations for the HBL [START_REF] Mery | Synthèse des caractérisations géomécaniques[END_REF];  data from the geological map of Mézidon (BRGM); The values of strengths (tensile and compressive) were then degraded while taking into account:

 the scale effect being estimated at 0.47: according to [START_REF] Bieniawski | The significance of in-situ tests on large rock specimens[END_REF], referring to a curve obtained on unconfined iron ore samples;  the 2D aspect of modeling by preserving the strength/stress ratio for the pillar in 2D and 3D by decreasing the compressive strength of the seam (equation 1):

Let us recall that for square rooms and pillars:  the time influence on the material (by estimating that the coefficient of reduction of strengths is founded on a ratio elastic strength/ peak strength). Table 2 shows a synthesis of all strengths values obtained according to the various effects taken into account. The behaviour law retained takes into account a hardening then softening post-failure behaviour. 1981) showed that  h / v = 0.5. However, many stress measurements carried out in the West of France, within synclinal structures, show that the horizontal stress is always higher or equal to the vertical stress.

For the sites of Grais (May/Orne) and St-Sigismond (Maine-et-Loire), the ratio  h / v varies between 1 and 1.5 [START_REF] Burlet | Détermination du champ de contrainte régional à partir de tests hydrauliques en forages, résultats de neuf expérimentations in-situ réalisées en France[END_REF]. The stress tensor being of doubtful validity, we considered three values for the ratio  h / v : 0.5, 1 and 2.

Results

The solutions being able to explain the collapse of 1966 being plural, some certain data input were regarded sure (extraction ratio, dimensions of the various stages, width of the stage pillars, exploitation thickness, geology, geomechanical characterisation of materials other than the iron ore) and others were regarded as variables of the study.

Each method of calculation of this study was analysed in terms of subsidence on surface, distribution of plasticity, displacements, principal stresses and plastic deformation in five tested pillars (figure 5). To sum up, the different methods of calculation carried out made it possible to study the influence of the:  the length of the plastic zone cannot exceed 200 m. The mechanism highlighted cannot thus be repeated in the lower stages;  the verification of the 2 nd checking point showed the importance of the strength value of the barrier pillars. The pillar strength must be higher (51 MPa) than that of the pillars of the higher stage (34 MPa) for the mechanism to occur. That is compatible with the fact that the iron ore of the lower stages is more carbonated (so more resistant);  for  h / v = 0.5 and 1, we notice that the value of maximum subsidence on the surface is close to that measured in 1966: 65 cm;  the state of the initial stresses has a relatively weak influence on the thresholds of pillar strength of the lower stages;  the characteristics (spacing and friction angle) of the stratification network parallel to the dip are essential parameters in the mechanism which we highlighted: the increase in spacing between the joints inhibits the mechanism of collapse. It is the same for the friction angle;  the values of strengths which we introduced into our models are compatible with the intervals of variation of the in-situ characteristics [instantaneous value ; value in the long term integrating the effect of time].



Parametric analysis

The second part of this study consisted in carrying out a numerical modeling on the scale of the mine in order to develop the back-analysis collapse (of Soumont in 1966) and to evaluate the criteria to specify the risk by carrying out a parametric study (allowing a valorization on the whole inclined deposits of the same type). We thus studied the sensitivity of four parameters by carrying out twenty calculations:

 the dip (between 30 and 65°);  the extraction ratio (between 70 and 90 %);  the exploitation thickness (between 1.5 and 5 m);  the height of overburden (between 0 and 50 m). The analysis of these twenty calculations was focused on the extension of the zones of potential failures (plasticity), on the value of maximum displacement in the pillars and on the value of the maximum subsidence on the surface. This reveals that the mechanism identified at the time of the back-analysis can be reproduced under the geometrical conditions synthesised in table 4. In addition, we noticed that the subvertical faults can inhibit or amplify the mechanism of failure by shearing. Moreover, the reduction of the height/width ratio of the pillars (or thickness reduction) has a very significant positive role on the exploitation stability.

Evaluation of the "subsidence" hazard

The hazard assessment is classically made by combining the awaited intensity of the phenomenon with its probability of occurrence, this being the predisposition of the site with respect to the dreaded phenomenon.

Qualification of the intensity

It is recognised that the characteristics of depression which materialise the most severe damage for the goods located on surface are the horizontal differential strains and movements of ground inclined setting rather than maximum vertical subsidence in itself. Table 3 gives indicative values of the strains and slopes which make it possible to evaluate the phenomenon intensity.

Negligible  < 1  < 0.2 Very low 1 <  < 5 0.2 <  < 1 Low 5 <  < 10 1 <  < 2 Medium 10 <  < 30 2 <  < 6 High  > 30  > 6
The value of these two parameters can be appreciably influenced by different factors studied before.

It appears so that value of maximum subsidence is in the form: A max = 0.3. w., with:

 A max = maximum subsidence;  w = exploited thickness (in the districts exploited by shrinkage);   = extraction ratio (or recovery factor).

It can be easily deduced from them the values from the strains ( max ) and slopes ( max ) starting from the following traditional relations:

 max =  . A max / P  max =  . A max / P
Where:

 P is the average depth of the panel;   and  of the coefficients estimated respectively at 1.5 and 5 in the western iron basin.

The values of the coefficients  and  are deduced from the studies in experience feedback carried out on the Iron Mines of Lorraine and adopted for their drastic security character.

Qualification of the occurrence probability

In the inclined exploitations of the iron deposits of the West of France, it is mainly the stability of the barrier pillars, the slabs or the pillars left in place to ensure the behaviour of the immediate strata which controls the subsidence predisposition. To evaluate the long-term stability of the undermined surface, main factors that have to be take into account are:

 dimensions of the panels;  dip of the layers;  extraction ratio;  opening (height exploited between immediate strata);  strength of pillars.

In a more precise way, the parametric analysis described previously provides fundamental indications on the configurations of layer and exploitation for which the occurrence of a subsidence can be excluded (table 4, below).

> 55° ≤ 90% ≤ 4 m ≤ 85% ≤ 5 m 45° to 55° ≤ 90% ≤ 3 m ≤ 80% ≤ 5 m 30° to 45° ≤ 80% ≤ 3 m ≤ 70% ≤ 5 m
The influence of an increase in the dip appears by a displacement of the zones of failure toward the surface (or of the outcrop): the greater the dip value, the more one affects the grounds close to surface (plastic points).

In addition to these configurations of exploitation, other conditions must be taken into account for a reduction of the hazard level, like:

 condition n° 1: for a subsidence to occur entirely, it is necessary that dimensions of the mining sites (width L) reach or exceed the depth (H) (that is: L ≥ H), which represents, in the context of these exploitations, a width along the dip from 250 to 290 m (depth lower than 220-250 m). In lower part (L < H), subsidence is all the more, the hazard level is lower;  condition n° 2: it is considered that there are no repercussions on surface (non perceptible subsidence) if the mining site has a width L < 0.4 H;  condition n° 3: if the minimum depth of mine working is higher than 250-300 m (according to the geometry of the mining sites), it is considered that the failure zones are not likely to reach surface.

Hazard zoning

The limits materialising on surface the zone influenced by subsidence are established, taking in account an angle called "influence angle", measured from the vertical, which connects the end of the panel, at the bottom, to the points of surface where subsidence, strains or slopes are regarded as unperceivable or null. Although an single influence angle () value of 30° to 35° is retained for flat veins, three angle limit values are defined for inclined layers (exploitations) These are:

 the limiting angle value (), in the direction of drivage which is equal to the limiting angle in flat vein;  the "upstream" angle value, lower than the angle ;  the "downstream" angle value, always greater than the angle ;

Looking at the data obtained in Soumont, it can be noticed (table 6) that the values of failure angles measured upstream and downstream (on average respectively about 7° and 30°), for a dip ranging between 30° and 40°, are very close to the corresponding values of the abacuses of the Lorraine coalfields or the Nord Pas-de-Calais region [START_REF] Proust | Etude sur les affaissements miniers dans le Bassin du Nord et du Pas-de-Calais[END_REF]. Hence, it can be deduced that the influence angles must be also very close and take for the layer of Segré some values of influence angle equal to 30° (upstream side) and 45° (downstream side). Let us specify that the downstream influence angle is taken at the base of the exploited panels, and the upstream influence angle at the higher part of the panels.

Figure 2 .

 2 Figure 2. Example of the mine configuration of May-sur-Orne (according to Maury, 1972).

Figure 3 .

 3 Figure 3. Mining methods of the basin of Soumont (according to Perrotte and Lidou, 1983).

Figure 4 .

 4 Figure 4. Plane view of the seam worked by rooms and pillars at Soumont in the collapse zone of 1966.

  stress field with the ratio  h / v (3 values: 0.5, 1 and 2);  density of the roof stratification;  friction angle of the bedding planes;  effects of faults on the collapse mechanism;  joint behaviour law;  strength of the pillars;  panel width;  opening effect. The various calculations allow to reproduce a mechanism and explain the subsidence observed on surface in 1966. It is due initially to the relative compressions of pillars and then to the deflection of the roof. These two zones are the place of strong shear mechanisms which imply a potential failure by shearing up to surface. The three checking points (collapse of 1966 in the upper stage, stability of the lower stages and shrinkage stability) were checked.

Figure 5 :

 5 Figure 5: Distribution of plasticity: joint spacing of 10 m + variation of subsidence on the surface

Figure 6 :

 6 Figure 6: Diagram showing the dissymmetry of the upstream and downstream influence angles in inclined deposit

Table 1 :

 1 Comparative analysis of various Western iron-

bearing basins Basin MAY-SUR-ORNE SOUMONT LA FERRIERE-AUX- ETANGS SEGRE

	Dates of exploitation	1896 -1968	1907 -1989	1905 -1970	1907 -1984
	Maximum depth	450 m	650 m	400 m	490 m
	Mining methods	stoping, dip faces, shrinkage	Rise faces, shrinkage, strike faces or "stoping"	Stopings, rise faces, retreating workings, shrinkage	shrinkage
	Dip	45° to 90°	30° to 60°	25° to 45°	60° to 90°
	Number of worked seams	1 (very locally 2)	1	1	2 (intercalated bed of 40 to 50 m thickness)
			Haematite under
	Dominant nature of the iron ore	Facies haematite	calcareous overthrust, Carbonated and siliceous	Chlorito-carbonated. Little haematite	Carbonated
			in-depth
	Content of iron	35-50%	36-50%	35-50%	Average 52%
	Compressive			80 MPa parallel to the bedding
	strength of the	100 MPa	115 MPa	plane, 200 MPa perpendicular	???
	ore			to the bedding plane

Table 2 .

 2 Synthesis of the compressive strengths for various materials of the study.

	Iron ore

Table 3 :

 3 Classes of intensity of the risk "subsidence" (purely indicative values)

	Classify intensity	Horizontal differential strains  (in mm/m)	Surface inclination  (in %)

Table 4 :

 4 conditions of exclusion of the process of subsidence (according to[START_REF] Renaud | Contribution à l'analyse des conditions d'effondrement des gisements pentés des bassins ferrifères de Soumont[END_REF]

	Dip	Extraction ratio ( %)	Thickness (W)

Table 5 :

 5 Classes of predisposition of the site for the risk "subsidence"

	Site predisposition	Ratio L/H	Depth (H)
	Very sensitive	L	< 250 m
	sensitive	L/H # 1	< 250 m
	Not very sensitive	0.4 < L/H <1	< 250 m
	negligible	L/H < 0.4 L	< 250 m > 250-300 m

Table 6 :

 6 Values given in the subsidence abacuses of Nord/Pas-de-Calais, Saar and Lorraine basins

	Dip values		0°	15°	25°	30°	40°	50°	60°
	Angles of rupture giving the limits of	Upstream angle	18	14	12	11	9	7	6
	fracturing on the surface	Downstream angle	18	22	25	27	30	33	36
	Angles of influence giving the limits of null subsidence	Upstream angle Downstream angle	35 35	32 38	30 40	30 43	30 45	28 47	27 48


Yannick Wileveau

Vincent Renaud

Jean-Bernard Kazmierczak

RHEOLOGICAL CHARACTERIZATION OF A CLAY FORMATION FROM DRIFTS EXCAVATION : ELASTIC AND ELASTOPLASTIC APPROACH

An extensive scientific programme has been carried out by Andra (French Agency in charge of radioactive waste management) for investigating feasibility of High Level Activity Waste disposal in deep geological formation. An Underground Research Laboratory (URL) is currently being constructed in North-eastern France to assess the adequacy of a hard-clay argillite layer (Callovo-Oxfordian formation) situated between 420 m and 550 m of depth. Geotechnical measurements have been carried out during the shafts and drifts excavation and particularly upon the main level of the laboratory (-490 m). The drifts are "horseshoe section" type with about 17 m² in area mainly supported by metallic ribs and rock bolts. The digging has been performed with classical pneumatic hammer. Measurement sections have been instrumented very close to the front face using convergencemeters and radial extensometers. This paper presents a comparison between in situ measurements and numerical modelling. Elastic calculations are not in agreement with the measured deformations. An elastoplastic constitutive model considering damage and using Hoek & Brown criteria has been developed and implemented in the FLAC 3D numerical code. Mechanical parameters came from lab tests performed on core samples. For the first meters, model provides consistent displacements. Beyond 4 meters, a time dependent convergence takes place and has to be integrated in the model to take into account creep and/or hydromechanical behaviour.

RÉSUMÉ

Un important programme scientifique a été conduit par l'Andra (l'agence française de la gestion des déchets radioactifs) afin d'étudier la faisabilité d'un stockage de déchets de haute activité et à vie longue dans une formation géologique profonde. Un laboratoire de recherche souterrain est actuellement en cours de construction dans le Nord-Est de la France pour évaluer les propriétés d'une roche indurée argileuse (argillite du callovo-oxfordien). Cette roche se situe entre 420 m et 550 m de profondeur. Des expérimentations géomécaniques ont été mises en oeuvre lors de la construction des puits et des galeries en particulier au niveau principal (-490 m). Une méthode classique de creusement au marteau piqueur a été utilisée pour excaver les galeries de type 'section fer à cheval' d'une surface de 17 m 2 environ avec un soutènement se composant de cintres métalliques et de boulons aciers. Des sections de mesures ont été équipées au plus tôt près du front avec des plots de convergence et des extensomètres radiaux. Dans cet article, une comparaison entre modèles mécaniques et mesures in situ est présentée. Les résultats d'une approche purement élastique ne sont pas en accord avec les déformations observées. Une loi rhéologique élastoplastique utilisant les critères Hoek & Brown a été développée et implémentée dans le code numérique FLAC 3D . Les paramètres du modèle ont été ajustés sur des essais sur échantillons. Le modèle prédit bien les déplacements mesurés lors des 4 premiers mètres d'excavation. Après, la convergence est dominée par les effets différés, nécessitant l'intégration d'une loi de fluage ou d'un comportement hydromécanique dans le modèle. . 1.

INTRODUCTION

In November 1999, having completed the preliminary work phase, Andra started construction work of an underground research laboratory (figure 1) in the district of Bure (Meuse département), located in the Northeastern of France. From 2000 to 2005, the construction of the experimental site has allowed to study radioactive waste storage possibilities in deep geological formation [START_REF] Delay | The French Underground Research Laboratory at Bure as a model precursor for deep geological repositories -IGC symposium -Florence[END_REF]Andra, 2005a).

The target horizon for the laboratory is a 130 m thick layer of argillaceous rocks that lies between about 420 and 550 meters below the surface at the URL site. From a lithological view point, the depositional period straddles the Callovian and Oxfordian subdivisions of the middle to upper Jurassic. Argillaceous rocks contain a mixture of clay minerals and clay-sized fractions of other compositions. The clays, which constitute 40 % -45 % on average of the Callovo-Oxfordian argillaceous rocks, offer groundwater isolation and radionuclides retention. Silica and carbonate-rich sedimentary components strengthen the rock to contribute to stability of the underground construction.

The stratigraphy of the URL is one of alternating limestone-rich and clay-rich units. On the upper part, the Oxfordian limestones lie from about 150 to 400 meters depth. Between the surface and the Oxfordian limestones is a 150 m thick sequence of mixed argillaceous rocks, [START_REF] Vigneron | Apport des investigations multi échelles pour la construction d'un modèle conceptuel des plateformes carbonatées de l'Oxfordien moyen et supérieur de l'est du Bassin de Paris[END_REF].

The state of in situ stress at the Meuse/Haute-Marne site has been measured by comprehensive combined methods. The vertical stress profile is presently well known on the site. The orientation of the σH stress (N155°E) is consistent with the regional stress field. The horizontal stress anisotropy is estimated between 1.1 < KH = σH/σh < 1.3 and the vertical stress and minimum horizontal stress have been directly measured close to the main level (-500 m depth) respectively equal to 12.7 MPa and 12.4 MPa [START_REF] Wileveau | Complete in situ stress determination in an argillite sedimentary formation[END_REF].

The main purpose of the geomechanical in situ investigation is to understand the rock response to the excavation of underground engineered structures and to the development of the damaged zone. The damaged zone characterization during shafts and drifts excavations will not be developed in this paper. Mechanical measurements are grouped in drift sections and within specific shaft excavation monitoring experiments socalled "mine by tests". These geomechanical experiments include a set of boreholes or convergence sections designed to monitor the behaviour of rock when openings are restarted.

Figure 2 presents the overall layout of the underground network of excavated rock in the Meuse/Haute Marne laboratory and the location of the experimental drifts constructed in the clay formation. The first geomechanical mine by experiment is located in the -445 m experimental drift, corresponding to the upper layer of the Callovo-Oxfordian layer, where the mechanical response is mainly elastic [START_REF] Wileveau | Complete in situ stress determination in an argillite sedimentary formation[END_REF]. After an important instrumentation carried out from the -445 m level, the vertical mine by test has been monitored between -465 m and -480 m during the main shaft sinking [START_REF] Souley | Hydromechanical response to a mine by test experiment in a deep claystone -SGC Sea to Sky conference -Vancouver -Oct[END_REF]. The third geomechanical experiment, dealing with this paper, is the drift excavation tests at the main level of the URL, which is around -490 m beneath the surface. SMR1.1 and SMR1.

EXPERIMENTAL RESULTS

The two instrumented sections have been put in place very close to the front face (around 1.5 meter) in order to investigate the maximal deconfinement from drift excavation. The sections are composed of radial extensometers for which the end point is fixed at 20 m far from the wall, and convergence measurements with 6 points on the section (see figure 3).

One notice that the feature of the drifts is "horseshoe section" with about 17 m 2 in area mainly supported by metallic sliding arches composed of three parts and rock bolts of 2.4 m length. The floor is also covered by a concrete slab of 0.7 m of thickness. Classical pneumatic hammer has been used to dig the galleries.

The convergences have been measured manually using a system of invar wire along 9 directions of bases. The accuracy of this method is +/-0.2 mm. The reading frequency has been adapted to the excavation advance rate in order to obtain a high density of measurements within a distance of 12 meters from the section, corresponding approximately to 3 times the excavation diameter. The convergence measurements are given on figure 4 for the two sections SMR1.1 and SMR1.3 until early October 2005. The convergence is still monitored. It is not presented in this paper.

Obviously, the behaviour of these two perpendicular sections is very different and strongly linked with the in situ stress anisotropy. The evolution of convergence in the SMR1.3 is very similar in the vertical and horizontal direction (respectively, measured on bases 6-3 and 1-5) (Fig. 4b). On the SMR1.1 section (Fig. 4a), the vertical convergence is much higher than the horizontal one what is in good agreement with the stress concentration due to the maximum horizontal stress σH acting on the walls. These values are given below (the convergence is reset to zero just before the excavation starting). Figure 5 shows the measurements for the cases of vertical downward and horizontal extensometers. Several names are used (e.g. GMR, GLE, GKE, GNI) for the drifts dug in the experimental area. PM indicates the distance between the workface and the axis of the previous gallery. Only 4 curves by extensometer are presented (0 m, 2 m, 5 m, 10 m relatively to the anchor installed at 20 m which is considered as a fixed point.

One observes the deformation rate reacts gradually with the progress of the face. Moreover, in the particular case of SMR1.3 section where the history of excavation is more complex, the effect of the others openings is clearly measured. Such interaction between drifts is mainly due to the general layout of the URL designed to have a rapid access to the facilities for the time schedule constraint. The smallest distance between parallel galleries is equal to times the diameter of the gallery between the GLE and GKE drifts. This effect has not been identified on the SMR1.1 section. The magnitude of extension at the wall relatively to the reference point taken at the 20 m anchor is comparable to the convergence measurements, even if the starting date of measurement are differed of few days. The table 1 shows the value for both instrumentations.

The values obtained by convergence and extension make up a consistent set of data, even though the deformations obtained by the convergence method give in most of the case a larger deconfinement. This difference can be explained by the delay to install the extensometer compared to the convergence section or also by the non measurable part of deformation in place up to 20 m from the wall of the openings. In the following chapter, one takes the convergence values to compare the results obtained by modelling.

INTERPRETATION OF CONVERGENCES

Results of elastic model

The first analysis has been made using a classical approach in the framework of linear elasticity assuming a plane strain approach developed by [START_REF] Panet | Contribution à l'étude du soutènement derrière le front de taille[END_REF]. More complex calculations including a 3D simulation of the drift excavation is presented in the next chapter. The results are not in agreement with the observations as it is shown in figure 6 for the case of SMR1.3 where the in situ stresses are nearly isotropic around the gallery. A better agreement is obtained while reducing Young modulus up to one order of magnitude. This assumption is inconsistent with the mechanical behaviour observed usually on samples subjected to lab tests and can not be validated. Moreover, the significant elastic deconfinement in the first meters from section predicted by plane strain model is not reproduced.

To better represent the complex behaviour of the argillite, one also needs to consider a damageable rock. The fracture network has clearly detected within the first 2 meters from the gallery wall by several direct methods (geological survey on cores samples, resin injection within the fracture network followed by overcoring, borehole camera) and indirect methods (velocity measurement, tomography). The application of elastic model, combined with in situ observations, leads us to consider (as it was previously forecasted) an elastoplastic approach for the argillite lying at this depth.

Results of elastoplastic model

Numerical calculations were carried out in 3D to simulate the drift progress (in 5 phases) of a horseshoe gallery at 490 m of depth (figures 3 and 7) according to the two orientations of the galleries with respect to the stress tensor : σh and σ Η . A first phase allows to reach the fine grid zone. Then, 3 phases of drift advancing (1 meter by 1 meter) are simulated. Finally, the drift advancing is continued to have a complete deconfinement until the end meshed zone.

The geometrical model extends on 49.4 m in X direction, 25.3 m in the direction Y and on 50.6 m height (Z direction). It consists of 627183 gridpoints and 605784 zones. The used mesh is sufficiently fine for highlighting the characteristics of the zones exceeding the damage or failure criteria. In the same way, nodal points were selected in various directions (in front of the face, at the walls side, in vault and under floor) to follow the evolution of displacements (figure 8) and stresses during the phases of drift advancing. These points also allowed easy confrontation with in situ measurements on the two sections SMR1.1 and SMR1.3.

Modelling assumptions

The commercial computation software used for this study is delivered by Itasca ( 2002 The state of natural stresses (at -490 m depth) is as follows: σv = 12.7 MPa, σh = 12.4 MPa and σH = 1.3 xσh. From the assumptions on natural stress field, the worst case has been considered (KH = 1.3).

Geomechanical properties

Numerical calculations are carried out with a damageable elastoplastic model with hardening (as developed by Hoek and Brown). Table 2 indicates the parameters of this model used as reference parameters for studied zone (named B&C zones) of argillite [START_REF]Dossier 2005 -Référentiel du site Meuse/Haute-Marne[END_REF].

Figure 8 illustrates the isovalues of total displacement obtained by the numerical calculation at the end of the 3 steps of one meter excavation. The higher extrusion on the workface is clearly represented, as well as the corner effect. In order to compare the SMR1.1 and SMR1.3 sections to the modelling results, we have put the tracked point at one meter before the initial face. The zero displacement is then considered when initial face is reached. The calculation results corresponding to 3 m of excavation with respect to the initial face show a value of vertical convergence of about 10 mm in the direction of the major stress and of 25 mm in minor stress direction, that is to say a ratio of 2.5 between these two directions. We notice that this tendency is also observed for the measurements, even if the ratio of displacements is not exactly the same.

The differences between the displacements calculated in the four configurations depicted on Figure 9 show a relative good agreement of modelling with the measurements on the 4 first meters of face progress.

In spite of the fact that the results of displacement calculation are rather well correlated with the measurements in the case of drift advancing for a gallery oriented according to σH, the model predicts displacements higher than the measured values for the other direction (σh). This is natural, since the model does not integrate neither creep, nor hydraulic coupling. Therefore, long-term convergences cannot be reproduced by the model.

CONCLUSION

The main purpose of this paper is to assess the classical approach of tools used in geomechanics to interpret the in situ displacements measured on experimental mine by test conducted by Andra in the Meuse/Haute-Marne URL.

For such hard clay lying at 490 m of depth in the Callovo-Oxfordian formation, a complex behaviour is observed, including different phenomena as elastoplastic behaviour, time dependent effect, and hydro-mechanical coupling.

On one hand, the elastic approach has not successfully provided reasonable comparison when the global elastic mechanical response of the clay has been confirmed at the 445 m of depth [START_REF] Wileveau | Complete in situ stress determination in an argillite sedimentary formation[END_REF]. On the other hand, a damageable elastoplastic constitutive model using Hoek & Brown criteria has been used. Main conclusion of this analysis is that for the first meters only, displacements observed are consistent with the elastoplastic model. After, the time dependent mechanisms take effect as a predominant part of deformations.

One notices that the elastoplastic modelling presented here has been carried out in the framework of studies on 3D complexity of URL (real geometry of drifts, anisotropy of horizontal stress, and working phases). Thus, we voluntarily simplified the argillite behaviour by neglecting the effects related on creep and hydraulic couplings. Some other numerical modelling are in progress to better understand the strong hydro-mechanical coupling observed in this clay, the real geometry and possible interaction between drifts, the role and link between plasticity and creep.

Figure 1 .

 1 Figure 1. Localization of the underground research laboratory at the Meuse/Haute-Marne site marls, and limestones of the Kimmeridgian. Underlying the Callovo-Oxfordian argillaceous rocks are the Bathonian and Bajocian-age Dogger limestones and dolomitic limestones[START_REF] Vigneron | Apport des investigations multi échelles pour la construction d'un modèle conceptuel des plateformes carbonatées de l'Oxfordien moyen et supérieur de l'est du Bassin de Paris[END_REF].

Figure 2 .

 2 Figure2presents the overall layout of the underground network of excavated rock in the Meuse/Haute Marne laboratory and the location of the experimental drifts constructed in the clay formation. The first geomechanical mine by experiment is located in the -445 m experimental drift, corresponding to the upper layer of the Callovo-Oxfordian layer, where the mechanical response is mainly elastic[START_REF] Wileveau | Complete in situ stress determination in an argillite sedimentary formation[END_REF]. After an important instrumentation carried out from the -445 m level, the vertical mine by test has been monitored between -465 m and -480 m during the main shaft sinking[START_REF] Souley | Hydromechanical response to a mine by test experiment in a deep claystone -SGC Sea to Sky conference -Vancouver -Oct[END_REF]. The third geomechanical experiment, dealing with this paper, is the drift excavation tests at the main level of the URL, which is around -490 m beneath the surface. SMR1.1 and SMR1.3 sections have been respectively installed in april 2005 and august 2005 during the works

Figure 3 .

 3 Figure 3. Example of instrumented section SMR1.1 with combined extensometer and convergence measurement at the same location. Base "n" indicates the reference number of convergence point. The extensometers, on the same section than the convergence measurements, have monitored the deformations of the rock mass automatically (frequency : 4 data/hour) during the excavation advance of the drifts.Figure5shows the measurements for the cases of vertical downward and horizontal extensometers. Several names are used (e.g. GMR, GLE, GKE, GNI) for the drifts dug in the experimental area. PM indicates the distance between the workface and the axis of the previous gallery. Only 4 curves by extensometer are presented (0 m, 2 m, 5 m, 10 m relatively to the anchor installed at 20 m which is considered as a fixed point.

  Convergence measurements on sections SMR1.1 (a) and SMR1.3 (b)

Figure 5 .

 5 Figure 5. Extension of the rock mass versus time around the drift during the drifts progress -5a) and 5b) SMR1.1 section -5c) and 5d) SMR1.3 section -the values are calculated considering the fixed point at 20 m.

Figure 6 .

 6 Figure 6. Comparison between the results of convergence measurements of SMR1.3 and plane strain modelling Considering isotropic mechanical behaviour of the argillite, the elastic parameters Young modulus (E = 4.0 GPa) and Poisson's ratio (ν = 0.3) are taken from the Andra's labtests on samples coming from deep boreholes drilled on the Meuse/Haute-Marne site[START_REF]Dossier 2005 -Référentiel du site Meuse/Haute-Marne[END_REF].

  are as follows: • null normal displacements are prescribed on the lower face of the model (Z = -515.3), on the face corresponding to the gallery symmetry plane (Y = 0) and on the "left" face of the model (X = -23.0); • stress conditions are prescribed on the upper face of the model (Z = -464.7), on the "back" face of the model (Y = 25.3) and on the right face of the model (X = 26.4).

Figure 7 .

 7 Figure 7. Mesh of the model including horseshoe shape and refined mesh around the drift (see zoom)

Figure 9

 9 Figure 8. Variation of total displacement for a 3 m advance 3.2.3Comparison between convergence measurements SMR 1.1 and SMR 1.3 and modelling Figure9illustrates the comparison between the results of numerical modelling and the convergence measurements at sections SUG1350, SUG1360 and SUG1170 (whose axis is parallel to σH) and sections SUG1150, SUG1160 and SUG1180 (whose axis is parallel to σh) according to the face advance.

Figure 9 .

 9 Figure 9. Measurements comparison between horizontal and vertical convergences and modelling results

Table 1 .

 1 Comparison between convergence and extension measurements (SMR1.1 and SMR1.3).

		Ref. number	Drift axis	Period (days) 1	type	2 (mm) Value
		SUG1350	σH	163	base 1-5 horiz. base 6-3 vertical	35.8 39.7
	SMR1.3	SUG1360	σH	163	base 1-5 horiz. base 6-3 vertical	42.7 39.8
		SUG1301	σH	163	horiz. extens.	33.8
		SUG1303	σH	163	vertical extens.	27.0
		SUG1150	σh	45	base 1-5 horiz. base 6-3 vertical	8.6 33.2
	SMR1.1	SUG1160 SUG1103	σh σh	45 45	base 1-5 horiz. base 6-3 vertical extens. horiz.	8.9 53.0 5.7
		SUG1105	σh	45	vertical extens.	37.2
		SUG1118	σh	45	vertical extens.	23.6
	1 Period of observation calculated until the 1 2005 2 The values given are the half-convergence measured st October between two bases

Table 2 .

 2 Reference values of B & C zones (used in the modelling (S and m: parameters of the Hoek & Brown criterion, α and β: parameters characterising the residual strength evolution)

	Elastic parameters
	Young's modulus (MPa) Poisson's ratio	4000 0.3
	Criteria of the Hoek & Brown model
	Failure criterion
	S (rup)	0.128
	m (rup)	2
	σc (rup) in MPa	33.5
	Damage initiation
	S (dam)	1
	m (dam)	1.5
	σc (dam) in MPa	9.6
	Residual strength
	α β in MPa	2.8 3