Paolo Ballarini

Marco Beccuti
email: beccuti@di.unito.it

Enrico Bibbona

Andras Horvath
email: horvath@di.unito.it

Roberta Sirovich

Jeremy Sproston

Analysis of Timed Properties Using the Jump-Diffusion Approximation

Keywords: diffusion approximation, stochastic differential equations with jumps, statistical model checking

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Introduction

Context. Advances in modelling lead to increasingly complex models of concurrent systems whose analysis, consequently, has become a critical issue. In particular the analysis of quantitative aspects of these systems by means of stochastic models (e.g., Markov chains) may be impaired by the combinatorial explosion of their state space. To cope with this problem several approaches have been proposed in the literature including, e.g., decomposition and aggregation, bounding techniques, compact representations of the state space. However, when the model accounts for large groups of individuals (e.g., Internet users, molecule populations) these techniques may turn out to be insufficient, meaning that discrete event simulation (DES) is the most practical option for analysing the system's performance. Indeed, DES based approaches do not require the explicit storage of the state space, but instead exploit a set of sample executions (i.e., traces) in order to devise arbitrarily accurate (statistical) estimates of relevant indicators of a model's behaviour. Fluid approximations. In the case of large interacting populations, an alternative to simulation is to use a deterministic approximation in which the behavior is represented by a set of ordinary differential equations (ODE) [START_REF] Kurtz | Solutions of ordinary differential equations as limits of pure jump Markov processes[END_REF]. However, this approach is not suitable for the study of models where stochasticity (bimodality, high variance) plays an important role even for large population counts. To analyze a model's stochastic nature, in [START_REF] Kurtz | Limit theorems and diffusion approximations for density dependent Markov chains[END_REF] a diffusion approximation was proposed, based on a set of stochastic differential equations (SDE), that can be applied up to the first visit of the boundary of the state space. Both the deterministic and the diffusion approximation are such that every state variable is approximated by a continuous variable, i.e., the state variables are made "fluid". Since in real systems the boundaries of the state space often can be visited many times, in [START_REF] Beccuti | Analysis of Petri net models through stochastic differential equations[END_REF] we proposed an extension, namely, a jump diffusion approximation, to properly approximate the original model at the boundaries as well. A further extension was made in [START_REF] Angius | Approximate analysis of biological systems by hybrid switching jump diffusion[END_REF] that uses partial fluidification of the state space, which results in a switching jump diffusion approximation, allowing us to mimic better the original process in the case of low population counts. Contribution. Starting from [START_REF] Beccuti | Analysis of Petri net models through stochastic differential equations[END_REF], in this paper, we propose a new statistical model checking method based on jump diffusion approximation. This method takes as input a DDMC and, following [START_REF] Bortolussi | Model checking single agent behaviours by fluid approximation[END_REF][START_REF] Kolesnichenko | A logic for modelchecking mean-field models[END_REF][START_REF] Bortolussi | Model checking Markov population models by Central Limit Approximation[END_REF][START_REF] Bortolussi | Fluid model checking of timed properties[END_REF], a formal description of a property, in this paper described as a (deterministic) timed automaton [START_REF] Alur | A theory of timed automata[END_REF]. The jump diffusion approximation of the DDMC is used to generate trajectories of the system and the deterministic timed automaton is used to accept or reject each trace. Based on the proportion of the accepted traces confidence intervals are derived for the probability that the system exhibits the property in question. Applicability. The applied theoretical framework requires a sequence of DDMCs indexed by a parameter N [START_REF] Kurtz | Solutions of ordinary differential equations as limits of pure jump Markov processes[END_REF]. The sequence is such that the state space, the transition intensities and also the vector describing the initial state increase as N increases. Four possibly overlapping ranges of values can be identified for N . N can be so small that the corresponding DDMC can be analyzed by analytical approaches. As N grows, analytical analysis becomes unfeasible, but the DDMC can still be evaluated efficiently by simulation. By further increasing N , even simulation of the DDMC becomes impractical but the model can still exhibit important stochastic behavior. This is the range in which the approach we propose is convenient to use: in this range the diffusion approximation provides results with reasonable precision in much shorter time than dealing with the original DDMC. For even larger values of N , the stochastic behavior disappears and the model can be analyzed with a deterministic approximation. Organization. The paper is organized as follows. Section 2 introduces DDMCs and Section 3 discusses their approximations. In Section 4 we provide the definition of the applied timed automata. We discuss the issues related to assessing properties through a diffusion approximation in Section 5. In Section 6 a case study is presented. Conclusions are drawn in Section 7.

Continuous time Markov chains (CTMC) are often used to describe the interaction of groups of identical objects. Informally, such CTMCs are called density dependent if the intensities of the interactions can be expressed as a function of the density of the objects present in the area (or volume) described by the model (as opposed to being expressed as a function of the number of objects itself).

Definition 1. Consider a sequence of CTMCs, denoted by X [N ] (t), indexed by N ∈ N \ {0} and with state space S [N ] ⊆ Z k (i.e., every state is identified by a vector of k integers), that describe the interaction of k groups of identical objects. The sequence X [N ] (t) is called density dependent if the associated transition intensities, given any two states r ∈ S [N ] and r + m ∈ S [N ] that are connected by a transition, can be written in the form

q [N ] r,r+m = N f r N , m (1)
where f : R k × Z k → R ≥0 is a bivariate function whose first argument is a vector that provides the density for each group of objects in state r and its second argument is the change in the state due to the transition from state r to state r + m ( R ≥0 is the set of non-negative real numbers).

The indexing parameter N can represent the size of the considered area or volume, or the total number of objects in the model (in this case the vector r/N is a vector of proportions). Note that in Definition 1 a single function, namely f , provides the intensity of every transition of every CTMC of the sequence of CTMCs. This implies that in every CTMC the transitions have the same effect on the state. The above definition can be relaxed by substituting [START_REF] Alur | A theory of timed automata[END_REF] with

q [N ] r,r+m = N f r N , m + N g r N , m, N (2)
where g : R k ×Z k ×N → R ≥0 is a trivariate function and g(r/N, m, N ) ∈ O (1/N ). Sequences of CTMCs in which the transition intensities are in the form given in (2) are referred to as nearly density dependent. The rationale behind the definition is the following. As N grows, thanks to g(r/N, m, N ) ∈ O (1/N ), the term N g(r/N, m, N ), which is not density dependent, remains in the order of a constant. The other term instead grows proportionally to N . Accordingly, as N grows the density dependent nature of the process prevails. Indeed, density dependent and nearly density dependent processes can be studied with the same approximations.

As for notation, the set of possible changes in the state due to a transition will be denoted by C. Formally, a vector m ∈ Z k is in C if and only if there exist two states r ∈ S [N ] and r + m ∈ S [N ] such that there is a transition from r to r + m. Note that, like the function f , also the set C is shared by every member of a given sequence of DDMC.

Example 1. As an example we consider a simple epidemic model in which two groups are involved, namely, susceptible and infected individuals. Accordingly, each state is described by a pair (i, j) providing the number of susceptible and infected people, respectively. We assume that the modelled individuals are uniformly distributed over an area split into N equally sized cells and that three kinds of events are possible. The number of susceptible individuals grows with an intensity proportional to the number of cells: q

[N ] (i,j),(i+1,j) = N λ 1 . Due to the contact of two infected and one susceptible person in one of the cells, one susceptible individual becomes infected; this happens with intensity q

[N ] (i,j),(i-1,j+1) = ij(j-1)

2 1 N 3 N λ 2 ,
where the first term is the number of ways the three individuals can be selected, the second term is the probability that the three selected individuals are together in a given cell, and the multiplication by N is due to the fact that the contact can occur in any cell. Infected individuals can become immune independently of each other and independently of the number of cells; the associated intensity is q

[N ] (i,j),(i,j-1) = jλ 3 . The intensity of the first type of event is independent of the actual state and proportional to N and thus it is a special form of [START_REF] Alur | A theory of timed automata[END_REF]. The intensity of the other two kinds of events can be rewritten as

q [N ] (i,j),(i-1,j+1) = N λ 2 2 i N j N 2 -N 1 N λ 2 2 i N j N , q [N ] (i,j),(i,j-1) = N λ 3 j N
where the first intensity is nearly density dependent while the second is density dependent. The set of possible state changes is C = {(1, 0), (-1, 1), (0, -1)}.

Approximations of nearly density dependent CTMCs

All approximations we describe in the following use a process with a continuous state space and thus are considered "fluid" approximations. In order to proceed we need to introduce the sequence of normalized CTMCs given by Z [N ] (t) = X [N ] (t)/N , called also the density process. The reason to use Z [N ] instead of the original process is that normalization brings all CTMCs of a given density dependent sequence to the same scale, making them comparable.

The first approximation we consider uses a set of ODEs in which there is one equation per group. Accordingly, the original stochastic behavior is approximated by a deterministic process. The set of ODEs used in the approximation is provided by the following result of Kurtz [START_REF] Kurtz | Solutions of ordinary differential equations as limits of pure jump Markov processes[END_REF]. Given a nearly density dependent sequence of CTMCs X [N ] (t) with initial state that tends to z 0 as N tends to infinity, i.e., lim N →∞ Z [N ] (0) = lim N →∞ X [N ] (0)/N = z 0 , if the function l∈C lf (y, l) satisfies some relatively mild conditions, then the density process Z [N ] (t) converges to a deterministic function z(t). The function z(t) is the solution of the following set of ODEs

dz(t) = l∈C lf (z(t), l) dt, z(0) = z 0 . (3)
We note that (3) is equivalent to the more familiar form dz(t) dt = l∈C lf (z(t), l); however we prefer the form in [START_REF] Angius | Approximate analysis of biological systems by hybrid switching jump diffusion[END_REF] because it has more in common with the other approximations that we introduce later.

The approximation given by z(t) has the following property:

lim

N →∞ P sup t≤T Z [N ] (t) -z(t) > δ = 0, (4)
for every δ > 0 and where T is the upper limit of the considered finite time horizon. Moreover, it was shown in [START_REF] Kurtz | Solutions of ordinary differential equations as limits of pure jump Markov processes[END_REF] that the difference between the deterministic approximation and the original stochastic behavior is characterized by

sup t≤T Z [N ] (t) -z(t) = O 1/ √ N (5)
The practical meaning of ( 5) is that the error of the deterministic approximation decreases as 1/ √ N . Another approximation of a density dependent sequence X [N ] , which is based on stochastic differential equations and thus it preserves the stochastic nature of the original process, was proposed in [START_REF] Kurtz | Limit theorems and diffusion approximations for density dependent Markov chains[END_REF][START_REF] Kurtz | Strong approximation theorems for density dependent Markov chains[END_REF]. This approximation, denoted by Y [N ] (t), is obtained by the following set of SDEs:

dY [N ] (t) = l∈C lf Y [N ] (t), l dt + l∈C l √ N f Y [N ] (t), l dW l (t) (6)
where the W l (t) with l ∈ C are independent standard one-dimensional Brownian motions. The approximation holds up to the first time Y [N ] (t) reaches a boundary of the state space. In (6) the first term is the same used by the deterministic approximation in (3), while the second term is a noise that mimics the stochasticity of the original CTMCs.

For what concerns the relation of the diffusion approximation and the original density process, in [START_REF] Kurtz | Limit theorems and diffusion approximations for density dependent Markov chains[END_REF] it has been proven that, for any finite N , we have

sup t≤T Z [N ] (t) -Y [N ] (t) = O (log N/N ) (7)
In practice, one uses

N •z(t) or N •Y [N ] (t) to approximate the original CTMC X [N ] (t). The difference between N • z(t) and X [N ] (t) according to (5) is in the order of N (1/ √ N ) = √ N . Between N • Y [N ] (t) and X [N ] (t) according to (7) it is instead N (log N /N ) = log N which is much lower than √ N .
A limitation of the previous approach based on SDEs is that it can be applied only to models where the probability of reaching a boundary of the area of the process is negligible. In order to overcome this limitation, in [START_REF] Beccuti | Analysis of Petri net models through stochastic differential equations[END_REF] we introduced a jump diffusion process in which the jumps are used to capture the behavior of the process at the boundaries. We provide here a brief description of the jump diffusion process, denoted by J [N ] (t); for a detailed treatment, see [START_REF] Beccuti | Analysis of Petri net models through stochastic differential equations[END_REF][START_REF] Angius | Approximate analysis of biological systems by hybrid switching jump diffusion[END_REF].

The main idea is to split the transitions of the model into two sets depending on the current state. In particular, we denote by C • (y) the set of transitions that change one or more components of the state which are at the boundary in state y. The jump diffusion process is defined then by

dJ [N ] (t) = l∈C-C • (J [N ] (t)) lf J [N ] (t), l dt+ (8) l∈C-C • (J [N ] (t)) l √ N f J [N ] (t), l dW l (t) + l∈C • (J [N ] (t)) l N dM [N ] l (t)
where the first two terms are analogous to those in [START_REF] Ballarini | COSMOS: A statistical model checker for the hybrid automata stochastic logic[END_REF] but are restricted to those transitions that change components away from the boundaries. If none of the components are at the boundary of the state space then J [N ] (t) behaves exactly as Y [N ] (t). The term M

[N ] l (t) corresponds to Poisson counting processes that gives rise to jumps that mimic the behavior of the original CTMC at the boundaries. In other words, when the process reaches a boundary then discrete jumps regulated by a Poisson process make it jump back eventually to the inner part of the state space. The intensity associated with dM

[N ] l (t) is µ l (t) = N f J [N ] (t), j , i.e.
, it is taken directly from the original CTMC (note that J [N ] (t) provides directly a vector of densities as required by f ). Then dM

[N ] l (t) is multiplied by l/N because that is the effect of the transition in the normalized state space.

Recent studies [START_REF] Bibbona | Strong approximation of density dependent Markov chains on bounded domains by jump diffusion processes[END_REF] have shown that the jump diffusion approximation has similar characteristics to those of the pure diffusion approximation and, in particular, that the approximation it introduces is as good as that of the "pure" diffusion process, that is:

sup t≤T Z [N ] (t) -J [N ] (t) = O (log N /N ) (9)
Numerical evaluation of the goodness of the jump diffusion approximation has been illustrated instead in [START_REF] Beccuti | Analysis of Petri net models through stochastic differential equations[END_REF][START_REF] Angius | Approximate analysis of biological systems by hybrid switching jump diffusion[END_REF].

Timed Automata

In this section, we introduce a timed automata-based formalism for the specification of timed properties of CTMCs. As is standard when using timed automata for the specification of properties of stochastic systems (e.g., [START_REF] Donatelli | Model checking timed and stochastic properties with CSL TA[END_REF][START_REF] Chen | Model checking of continuoustime Markov chains against timed automata specifications[END_REF][START_REF] Mikeev | On-the-fly verification and optimization of DTA-properties for large Markov chains[END_REF][START_REF] Bortolussi | Model checking Markov population models by Central Limit Approximation[END_REF][START_REF] Bortolussi | Fluid model checking of timed properties[END_REF]), we use deterministic timed automata (DTA): that is, each input sequence of the timed automaton (which in our context is a trajectory, i.e., a function from time to the state space of the CTMC, representing a particular behavior of the CTMC or of its diffusion approximation) corresponds to a single run of the timed automaton. In order to provide a uniform framework for timed properties interpreted on CTMC and on jump diffusion approximations, our DTA are labeled with constraints both on clocks and on variables characterizing the state space, but are not labeled with actions corresponding to individual CTMC transitions (which have no meaning in the jump-diffusion diffusion approximation setting).

Edges of our variant of DTA are urgent: they are taken as soon as they are enabled. Urgency of edges allows for a natural interpretation of our DTA not only on behaviors of CTMC, but also on trajectories of their diffusion approximations. We denote by S ⊆ Z k the state space and by V = {ϑ 1 , ..., ϑ k } a set of k variables, where we interpret ϑ i as a variable corresponding to the i-th element of the vector representing a state. Let C be a finite set of variables called clocks. Definition 2. A constraint is defined by the following grammar:

Φ ::= ϕ ≤ ϕ | c ≤ λ | c ≥ λ | Φ ∧ Φ , ϕ ::= ϕ + ϕ | ϕ -ϕ | ϕ * ϕ | ϕ/ϕ | ϑ i | λ ,
where c ∈ C is a clock, ϑ i ∈ V and λ ∈ Q is a rational constant. A guard constraint is a constraint Φ such that, for each a ∈ V ∪ C, there is at most one subformula of Φ featuring a. An invariant constraint is a guard constraint Φ in which there is no subformula of the form c ≥ λ. We write Guards(V, C) and Invariants(V, C) to denote the set of guard constraints and invariant constraints, respectively, over V and C.

Examples of invariant constraints include

ϑ 1 ≤ 10∧ϑ 2 ≥ ϑ 3 and ϑ 1 ≥ 3∧c 1 ≤ 15, whereas c 1 ≥ 3 ∧ c 2 ≤ 10 ∧ ϑ 1 ≥ 3 is an example of a guard constraint that is not an invariant constraint (due to the conjunct c 1 ≥ 3). A function v : C → R ≥0 is
→ Invariants(V, C); (4) a set E ⊆ L × Guards(V, C) × 2 C × L of edges,
where each edge ( , Φ, C, ) ∈ E comprises a source location , an enabling condition Φ, a set C of clocks to be reset to 0, and a target location . A timed automaton is deterministic if, for any location ∈ L and for any pair

( , Φ 1 , C 1 , 1 ), ( , Φ 2 , C 2 , 2 ) ∈ E, we have that Φ 1 ∧ Φ 2 is unsatisfiable.
We use DTA to determine whether a trajectory X : R ≥0 → S satisfies a timed property. More precisely, the DTA reads the trajectory X and traverses edges between locations on the basis of (1) the states visited by the trajectory as time passes and (2) the current values of the clocks. The values of the clocks increase at the same rate as real-time. The DTA must leave its current location without letting time pass if there exists an edge ( , Φ, C, ) ∈ E such that the enabling condition Φ is currently satisfied (hence, the DTA can be regarded as having an "urgent" semantics in which an enabled edge must be taken as soon as possible): this satisfaction of the enabling condition of the guard may occur, for example, because the value of a state variable falls below some threshold, or the value of a clock reaches a particular value. Furthermore, an additional constraint on the trajectory is imposed by the invariant conditions: during a period in which the DTA is in a particular location , the invariant condition Inv( ) must be satisfied by the states visited by the trajectory and by the current value of the clocks during that period, otherwise the trajectory will be regarded as not satisfying the timed property. A set of clocks can be reset to 0 when an edge is taken. If the DTA, starting from the initial location, reaches a final location when reading the trajectory X, then we say that the trajectory is accepted by the DTA (which, intuitively, corresponds to the trajectory X satisfying the timed property represented by the DTA), otherwise it is rejected.

In the following, we describe formally the acceptance of trajectories by a DTA. Let ( , Φ, C, ) ∈ E be an edge of a DTA A. Then we write source( , Φ, C, ) = , guard( , Φ, C, ) = Φ, reset( , Φ, C, ) = C, and target( , Φ, C, ) = . Let ∈ L be a location of A, and let y ∈ S be state and v ∈ Val(C). We write (y, v) |= Guards( ) if and only if (y, v) |= guard(e) for all e ∈ E such that source(e) = . A pair ( , v) ∈ L × Val(C) is called a configuration. We write ( , v)

---→ ( 1 , v 1 ) γ1,e1 ---→ • • • γm-1,em-1 -------→ ( m , v m ). Let Λ π = {λ π 0 , λ π 1 , .
. . , λ π m } be the set of constants such that λ π 0 = 0 and λ π i = i-1 k=0 γ k for all i such that 1 ≤ i ≤ m.

Definition 4. Let A be a DTA. We say that X : R ≥0 → S is accepted by the DTA if there exists a path π = ( 0 , v 0 )

γ0,e0 ---→ ( 1 , v 1 ) γ1,e1 ---→ • • • γm-1,em-1 -------→ ( m , v m ) of A such that 0 = init , v 0 = 0, m ∈ F
and, for all 0 ≤ i < m, the following conditions are satisfied:

for all 0 ≤ γ < γ i , we have (X(

λ π i + γ ), v i + γ ) |= Inv( i ) and (X(λ π i + γ ), v i + γ ) |= Guards( i ); -(X(λ π i + γ i ), v i + γ i ) |= guard(e i ).

Assessing timed automata based properties by diffusion approximations

In this paper we limit our attention to illustrating the practical applicability of the approach. According to (9), there is a correspondence between the trajectories of the CTMC and those of the approximating jump diffusion process.

Moreover, the larger N is, the tighter the relation gets. It is natural hence to expect that over a certain threshold for N , which depends on the considered model, one can safely use trajectories of the diffusion process instead of trajectories of the CTMC to assess DTA-based temporal properties.

There is, however, a fundamental difference between a CTMC and a diffusion process. A diffusion process exhibits extreme oscillatory nature along its drift in any infinitesimal interval. This means that if a diffusion exceeds a given limit L for the first time then it goes below L with probability 1 afterwords in any infinitesimal interval. Consider now a diffusion process X(t) and a DTA with three locations. The initial location is with invariant X(t) ≤ L and has a transition enabled if X(t) ≥ L. The second location is with invariant X(t) ≥ L, it does not have an enabled transition associated with the situation X(t) ≤ L and it has a transition enabled when X(t) ≥ 2L that leads to the third location which is a final one. Due to the oscillatory nature of the diffusion process, with probability 0 a trajectory is accepted by the DTA. Note however that such a situation is coherent with what happens in a CTMC as N grows large. For large values of N , the trajectories of a CTMC are more and more similar to those of a diffusion. Consequently, the probability of the set of those trajectories of the CTMC that are accepted by the above described DTA tends to 0 as N tends to infinity. The characteristics of the diffusion process and that of the CTMCs with large N must be taken into account during the definition of the DTA in order to avoid results that are consequences of these characteristics and not the properties of the studied phenomenon. In practice, the problem is alleviated by using piecewise constant abstractions of the trajectories of the diffusion process. We consider now the three kinds of approximations errors that occur during the analysis of CTMCs based on diffusion processes. First, the diffusion process is an approximation of the original CTMC. The goodness of this approximation was discussed in Section 3. Second, the analysis is carried out based on traces generated by approximate simulation. Indeed, exact simulation can be carried out only in special cases of diffusion (for example, in case of a Wiener process without drift) but in general the process is multidimensional and it includes a state dependent drift and a state dependent noise that cannot be simulated exactly in general. Third, a diffusion process fluctuates in any infinitesimal interval which means that it is not possible to obtain a complete representation of a trace. Indeed, the temporal properties are assessed based on a constant piecewise approximation of an infinitely fluctuating trace.

Let us illustrate this third source of approximation error in some detail. In Figure 1 we plotted two versions of the same trajectory with two different time steps (δ). Consider now a DTA which accepts only those traces along which

name description init. values A Axin2 protein 0 Am Axin2 mRNA 0 G GSK3 protein 50 • N L LRP5/6 coreceptor 20 • N B free β-catenin 0 AL Axin2-LRP5/6 complex 50 • N GA GSK3-Axin2 complex 0 C GSK3-Axin-β-catenin complex 0
Table 1. Species of the Wnt pathway.

R1 :C k 1 --→ B + GA R7 : Am k 7 --→ A + Am R2 :C k 2 --→ GA R8 : A + L k 8 --→ AL R3 :B + GA k 3 --→ C R9 : AL k 9 --→ A + L R4 :A + G k 4 --→ GA R10 : 2B k 10 ---→ 2B + Am R5 :GA k 5 --→ G + A R11 : Am k 11 ---→ ∅ R6 :∅ k 6 --→ B R12 : AL k 12 ---→ L
Table 2. Reactions of the Wnt pathway.

the process never exceeds level 12 in the time interval [0, 10]. Clearly, using the piecewise constant abstraction of the trace, the trace with δ = 1 is accepted while the other is rejected.

The previous example indicates that the choice of the time step during the generation of the traces is of fundamental importance to achieve good approximation of the original behavior. The same problem, i.e., not knowing the fluctuation between two consecutive time points, is present to a somewhat lesser extent also when jump diffusion processes are used to obtain approximations of more classical measures, like transient probabilities. In that case, it is of crucial importance to find with sufficient precision the time instants when the process reaches the boundary, i.e., the time instants when the change from pure diffusion process to jump diffusion process has to be made. When assessing temporal properties described by DTA, the problem appears also inside the state space around the thresholds present in the automaton. The choice of the time step was discussed to some extent in [START_REF] Angius | Approximate analysis of biological systems by hybrid switching jump diffusion[END_REF][START_REF] Beccuti | Analysis of Petri net models through stochastic differential equations[END_REF]. In theory, it is possible to add intermediate points given a trace but this can be done only in very special cases, like the one used before, i.e., the pure Wiener process.

All the three kinds of approximation error decrease as the indexing parameter N increases. Numerical experiments suggest that in the situation when it is reasonable to use the diffusion approximation, i.e., when the CTMC is too large for the analysis but there are still important stochastic behaviors in the system, the approximation errors are in an acceptable range.

Experimental results

The experimental results described in this section were carried out using a prototype implementation integrated in the GreatSPN suite [START_REF] Amparore | Stochastic) model checking in GreatSPN[END_REF], for the SDE part, and with the COSMOS statistical model checker [START_REF] Ballarini | COSMOS: A statistical model checker for the hybrid automata stochastic logic[END_REF] (which uses a generalisation of the DTA formalism [START_REF] Ballarini | HASL: A new approach for performance evaluation and model checking from concepts to experimentation[END_REF]), for the CTMC part.

Case study: A model of the Wnt pathway. We consider a model of the Wnt/βcatenin pathway, an intracellular signalling pathway involved in neuroinflammation, a key mechanism in numerous brain diseases [START_REF] Gressens | Microglial Wnt signaling inhibition promotes microglia activation and oligodendrocyte maturation blockade[END_REF]. Such model [START_REF] Jensen | A Wnt oscillator model for somitogenesis[END_REF] accounts kinetic rates k1 7 k4 0.2/N k7 0.7 k10 0.7/N k2 200 k5 1.2 k8 10/N k11 0.025

k3 0.1/N k6 0.4•N k9 0.08 k12 0.1 initial population G N • 50 L N • 20 AL N • 50
Table 3. Kinetic rate constants and initial populations for the DDMC model of the Wnt pathway (both dependent on index N ). for 8 biochemical species (Table 1) regulated through 12 reactions (Table 2). It consists of three main actors: the β-catenin (denoted B) and Axin2 proteins (A), forming a negative feedback loop, and the Wnt protein (here subsumed by the LRP5-6 membrane receptor, i.e., L), representing the extracellular signal. The behavior can be summarised as follows. With scarcity of extracellular Wnt molecules (low L), a degradation complex (C, a trimer resulting by β-catenin binding to previously formed GSK3-Axin2 dimer, i.e., GA) causes the phosphorylation and subsequent destruction of β-catenin located in the cell's cytosol. On the other hand with an abundant Wnt signal (high L), the degradation complex is deactivated (as Axin is degraded through reversibly binding with receptor L, i.e., forming the AL complex) resulting in an accumulation of β-catenin which in turn activates (through transcription of the Axin2 messenger RNA, i.e., A m ) the expression Axin2, and therefore determining its own destruction (i.e., negative feedback loop).

In [START_REF] Jensen | A Wnt oscillator model for somitogenesis[END_REF] the model is given in ODE form and it is shown to exhibit sustained oscillations (Figure 2) for specific parameter settings. Here we consider a sequence of DDMCs indexed by N (here proportional to the volume) and of the parameters of the ODE model in [START_REF] Jensen | A Wnt oscillator model for somitogenesis[END_REF] 5 . Table 3 depicts the kinetic rate constants and the initial populations of the Wnt-pathway DDMC 6 whereas Figure 2 compares species A's projection of a sample path of the CTMC for various values of N with the deterministic trajectory of the corresponding ODEs (notice that for readability the CTMC paths have been normalised, i.e., the molecule count of A is divided by N ). Furthermore observe that ODEs exhibit sustained oscillations, which, after the second period, have almost constant amplitude, and that, for increasing values of N , CTMC trajectories approximated quite accurately the ODE's. The choice of N when analyzing a real scenario depends on the consid- ered volume; in wet-lab experiments typically molecular value is usually greater than 500.

A  L mid0 L A  H
Automaton. Inspired by Mikeev at al. [START_REF] Mikeev | On-the-fly verification and optimization of DTA-properties for large Markov chains[END_REF] we propose a DTA (Figure 3) to measure the duration of the period exhibited by the population of the Axin2 protein. The rationale for noisy period detection [START_REF] Mikeev | On-the-fly verification and optimization of DTA-properties for large Markov chains[END_REF] is to split the domain of the observed species, i.e., A, in three subintervals: low (i.e., A ≤ L), mid (i.e., L < A ≤ H) and high (i.e., A ≥ H) 7 . A noisy period realisation [START_REF] Ballarini | Analysing oscillatory trends of discrete-state stochastic processes through HASL statistical model checking[END_REF] corresponds to the time interval occurring between two successive entries to the low region of the state-space interleaved by a visit to the high region. The single clock DTA in Figure 3 is indeed designed to detect the first noisy period realisation of species A. It consists of two parts: the first one processes the initial low 0 -high 0 -low 1 traversal (representing a spurious period), at the end of which (low 1 -mid 2 transition) the clock x is reset to start timing the realisation of the first non-spurious period whose termination corresponds with the low 3 -mid end edge. Note that ignoring the first spurious period (through the first part of the DTA) is necessary since to detect a complete period we need to identify the actual starting point (i.e., the first low-mid crossing that follows a visit to high) which we cannot do from the initial state because the system starts at A = 0. Furthermore note that trajectories are accepted on condition that the observed duration of the first period is within T min ≤ x ≤ T max which, by choosing different values for T min and T max , allows us to assess the probability density of the period duration (Figure 4). Observe that for any element N > 1 of the Wnt-DDMC sequence the probability of non-sustainably oscillating paths (i.e., paths non-perpetually traversing the low-mid-high regions) is negligible, therefore, given that L and H are properly chosen (so to be above, resp. below, the average height of minimal, resp. maximal peaks of oscillations), the DTA accepts all trajectories of the model.

Experiments. We compare the proposed approach by assessing of DTA-based oscillation-period properties on both a few CTMCs (of the Wnt-DDMC se- 4) and employs the DTA of Figure 3. The second one is devoted to comparing both runtime and accuracy of the two approaches w.r.t. estimating the duration of the oscillation period (Table 4) and employs a slightly modified DTA 8 . Table 4 compares, as a function of N , the execution times and the confidence intervals for the mean duration of the first non-spurious oscillation period with confidence level set to 0.99 in case of generating 10000 traces. Columns two and three depict the value of the L, resp. H, parameter of the DTA; the fourth and fifth (resp. seventh and eighth), columns show the runtime and estimated confidence-interval computed through our SDE prototype (resp. COSMOS); the sixth column shows the proportion of the number of jumps occurred because of hitting the border during the simulation of the SDE; finally the ninth column gives the speed up obtained by our SDE approach. In Figure 4 the probability density functions (pdf) of the length of the first non-spurious period are plotted for N equal to 100, 500, 1000 and 2000. The pdf for N equal to 5000 (not shown for the lack of space) confirms the trend toward a closer correspondence between the SDE and COSMOS results.

Discussion. As expected, the SDE approach becomes more convenient, in terms of runtime and precision, as N increases. 9 In particular, for N = 100 the precision is strongly affected by the SDE approximation error, moreover the SDE execution time is greater than COSMOS since for each SDE trace the process hits the 8 I.e., we use the DTA in Figure 3 but without the Tmin ≤ x ≤ Tmax conjunct on the edge from low3-mid end , which allows us to obtain the value of the clock x at the moment of reaching the final location of this modified DTA: this value gives the length of the first non-spurious oscillation period. 9 Observe that the dimension of integration step is dynamically computed through a heuristic function which provides a good trade-off between speed-up and precision of the solution boundaries, on average, along 90% of a trace length (col. 6, Table 4). With N = 1000 the SDE based analysis is about twice faster than that based on the CTMC (col. 9, Table 4) while the precision is acceptable: indeed the SDE approach is able to reproduce the multimodal behavior of the pdf generated by COSMOS (Figure 4 bottom-left plot, i.e., N = 1000). Such trend is confirmed by experiments with N = 2000, (speedup ∼6x, even closer approximation) and with N = 5000 (exhibiting a 63x speedup obtained with the SDE approach).

Conclusions

In this paper we presented a framework that allows for assessing temporal properties, described in terms of DTA, of DDMCs through their jump diffusion approximation. The applicability of the approach was illustrated through a case study regarding a biological oscillator. As future work we aim to study the theoretical limits of assessing DTA-based temporal properties of diffusion processes. Furthermore, the approach can be extended to hybrid jump diffusion processes, which are obtained by partial fluidification of DDMCs, that are useful to study systems in which not all population counts are high and thus fluidification of all state variables would lead to large approximation errors.

  referred to as a clock valuation, and the set of all clock valuations is denoted by Val(C). For any v ∈ Val(C), γ ∈ R ≥0 and C ⊆ C, we use v+γ to denote the clock valuation that increments all clock values in v by γ (that is, (v+γ)(c) = v(c)+γ for all c ∈ C), and v[C:=0] to denote the clock valuation in which clocks in C are reset to 0 (that is, v[C:=0](c) = 0 for c ∈ C, and v[C:=0](c) = v(c) for c ∈ C \ C). The clock valuation that assigns 0 to all clocks in C is denoted by 0. Let Φ be a constraint, let y ∈ S be a state and let v ∈ Val(C) be a clock valuation. Then we write (y, v) |= Φ if and only if substituting ϑ i by y i (where y i is the i-th element of the vector y) and c by v(c) in Φ results in Φ resolving to true. For example, for y such that y 1 = 4 and v such that v(c 1 ) = 12.1, we write (y, v) |= ϑ 1 ≥ 3 ∧ c 1 ≤ 15. Definition 3. A timed automaton is a tuple (L, init , F, C, Inv, E) comprising: (1) a finite set L of locations, with an initial location init ∈ L and a set F ⊆ L of final locations; (2) a finite set C of clocks; (3) an invariant condition Inv : L

  ( , v ) to denote the DTA-transition from configuration ( , v) to configuration ( , v ) after γ > 0 time units have elapsed and by taking the edge e. The transition ( , v) γ,e --→ ( , v ) exists if (1) source(e) = , (2) v = (v + γ)[reset(e):=0], and (3) target(e) = . A path of A is a finite sequence of DTA-transitions π = ( 0 , v 0 ) γ0,e0

Fig. 1 .

 1 Fig. 1. Two versions of the same trajectory of a pure Wiener process with different time steps.

Fig. 2 .

 2 Fig. 2. Sample paths of A (Axin2 protein) for various values of N and the behavior obtained by the ODE description.

  ) ^(Tmin  x  Tmax),; detecting the first "unspurious" periodlow2 A  L mid2 L A  H

Fig. 3 .

 3 Fig. 3. DTA to study the oscillation period of Axin2 proteins.

Fig. 4 .

 4 Fig. 4. Probability mass functions of the length of the first non-spurious oscillation period with bin length equal to 0.1 and with N equal to 100 (top left), 500 (top right), 1000 (bottom left), 2000 (bottom right).

Table 4 .

 4 Comparing SDE and COSMOS results considering 10000 traces.quence) and on their SDE approximation. The experiments were executed on a server with 48 core AMD Opteron(tm) Processor 6176 by considering five CTMCs of the Wnt-DDMC sequence corresponding the following values of N , i.e., N ∈ {100, 500, 1000, 2000, 5000}, and while the SDE results were computed with the GSPN prototype the CTMCs results were computed with COSMOS[START_REF] Ballarini | COSMOS: A statistical model checker for the hybrid automata stochastic logic[END_REF] which, to the best of our knowledge, is one of the most efficient statistical model checkers. We have run two families of experiments. The first one is devoted to assessing the density function of the oscillation period (Figure

				SDE		COSMOS
	N	L	H	Time Average period	Jump Tot.	Time Average period Speedup
	100 50 1×10 4 180h. [103.174, 103.344]	0.90	30h. [112.628, 113.133]	0.17
	500 50 5×10 4 88h. [111.021, 111.115]	0.76 167h. [112.654, 112.784]	1.9
	1,000 500 1×10 5 54h. [111.977, 112.089]	0.60 344h. [112.381, 112.499]	6,37
	2,000 500 1.5×10 5 39h. [112.408, 112.474]	0.45 705h. [112.504, 112.578]	18.08
	5,000 500 5×10 5 28h. [112.693, 112.725]	0.26 1763h. [112.708, 112.749]	62.96

I.e., for N = 1 we assumed the discrete initial populations and reaction intensities being equal to the continuous ones as given in[START_REF] Jensen | A Wnt oscillator model for somitogenesis[END_REF], note that this is in agreement with a cell volume V = 10 9 /nA where nA is the Avogadro number given that species concentrations and kinetic rate constants of the ODE model are expressed in nM.

Notice that zero-order and second-order reactions' rates are dependent on N because for these conversion from continuous to discrete rates depends on cell's volume.

L and H, where L < H, are two thresholds chosen so that the minimal, resp. maximal, peaks of oscillation are most likely to fall below L, resp. above H.


A Boukelia

H Eslami

F Masrouri

Effect of temperature and initial state on variation of thermal parameters of fine compacted soils

Introduction

Soil thermal properties are required in some engineering applications such as the design of high-level radioactive waste disposals [START_REF] Rutqvist | A modeling approach for analysis of coupled multiphase fluid flow, heat transfer, and deformation in fractured porous rock[END_REF], buried power transmission [START_REF] De Lieto Vollaro | Thermal analysis of underground electrical power cables buried in non-homogeneous soils[END_REF], energy geostructures [START_REF] Pahud | Geothermal energy and heat storage[END_REF][START_REF] Brandl | Energy foundations and other thermo-active ground structures[END_REF] and thermal energy storage [START_REF] Navarro | Thermal energy storage in building integrated thermal systems: A review. Part 2. Integration as passive system[END_REF][START_REF] Giordano | Borehole thermal energy storage (BTES). First results from the injection phase of a living lab in Torino (NW Italy)[END_REF][START_REF] Giordano | Borehole thermal energy storage (BTES). First results from the injection phase of a living lab in Torino (NW Italy)[END_REF]. The study of heat flow in soil is based on the thermal properties and temperature gradient. The thermal parameters governing the transfer of heat are the thermal conductivity (), which is the ability of the material to conduct heat, the volumetric heat capacity (C), which describes the ability of the material to store thermal energy while undergoing a given temperature change, and the thermal diffusivity ( =/C), which describes the ability of a material to conduct thermal energy relative to its ability to store thermal energy.

A variety of measurement techniques are available for natural materials with a broad temperature range. Recently, Zhang, Cai,, Liu & Puppala (2016) presented most commonly used measurement techniques such as the steady-state method, the transient hot-wire method, the laser flash diffusivity method and the transient plane source method. The hot-wire method also known as the needle-probe method (ASTM, 2000) is a transient technique that measures temperature rise at a known distance from a linear heat source embedded in the test sample.

This method is widely used in natural soil and compacted soil characterization for its accuracy, speed and simple application.

The thermal parameters (, C and ) depend on soil parameters such as the mineralogy, water content, bulk density, particle size distribution and structural arrangement [START_REF] Abu-Hamdeh | Measurement of the Thermal Conductivity of Sandy Loam and Clay Loam Soils using Single and Dual Probes[END_REF], [START_REF] Abu-Hamdeh | Thermal Properties of Soils as affected by Density and Water Content[END_REF], [START_REF] Tang | Effet de la température sur le comportement des barrières de confinement[END_REF], [START_REF] Brandl | Energy foundations and other thermo-active ground structures[END_REF], [START_REF] Ehdezi | Enhancing Pavements for Thermal Applications[END_REF]). [START_REF] Abu-Hamdeh | Soil Thermal Conductivity: Effects of Density, Moisture, Salt Concentration, and Organic Matter[END_REF] measured the thermal conductivity of four soils as a function of their water content and density. The authors noted that for each soil,  increased with increasing density.

Similarly, when the water content increased , increased as well. The results of [START_REF] Tang | Effet de la température sur le comportement des barrières de confinement[END_REF] showed the same trends for a clayey material. [START_REF] Barry-Macaulay | Thermal conductivity of soils and rocks from the Melbourne (Australia) region[END_REF] presented the same type of results as part of their important database of Australian natural materials. For instance, as the thermal conductivity of solid particles exceeds those of the air and the water, an increase in the dry density results in an increase in . Furthermore, increasing the contact surface between solid particles increases the heat flux and results in an increase of . In the same way, as the thermal conductivity of water is higher than that of the air, an increase in the saturation rate results in an increase in .

Few studies have been conducted on the thermal conductivity of compacted soils. [START_REF] Ekwue | Thermal Conductivity of Some Compacted Trinidadian Soils as affected by Peat Content[END_REF] studied the combined effects of soil density and water content on soil thermal conductivity of three soils: a sandy loam, a clayey loam and a clay. They measured the thermal conductivity for different values of water content and dry density on the compaction curve of these soils. For each soil,  reaches a maximal value near the Proctor optimum water content. The study of [START_REF] Ekwue | Thermal Conductivity of Some Compacted Trinidadian Soils as affected by Peat Content[END_REF] was also focused on the impact of soil mineralogy on  and the material with a high sand content showed the most important impact. However, the coupled effect of soil density and water content on the volumetric heat capacity and thermal diffusivity was not well studied. The monotonic and cyclic heat effect on thermal parameters was also not studied.

Several prediction models of soil thermal parameters, taking into account the soil properties, are available to evaluate the thermal conductivity [START_REF] De Vries | Thermal properties of soils[END_REF], [START_REF] Johansen | Thermal conductivity of soils[END_REF] and [START_REF] Kersten | Thermal properties of soils[END_REF]). For example, [START_REF] Farouki | Thermal properties of soils[END_REF] and, more recently, Dong, John, McCartney, and Lu (2015) presented an extensive review of the numerous available prediction models.

However, these models are perfectible [START_REF] Dong | Critical Review of Thermal Conductivity Models for Unsaturated Soils[END_REF]) and need to be improved to take into account new environmental or industrial issues such as soil temperature.

In civil engineering, the temperature variation range evolves according to the process; for example, cyclic variations from 4 and 30°C were recorded for heat exchange piles (Peron, Knellwolf, & Laloui, 2011) a maximum temperature of 70°C was estimated for thermal energy storage [START_REF] Giordano | Borehole thermal energy storage (BTES). First results from the injection phase of a living lab in Torino (NW Italy)[END_REF], 100°C was used for the design of high-level radioactive waste disposal [START_REF] Tang | Effet de la température sur le comportement des barrières de confinement[END_REF], and 90°C was used in the design of buried power transmission cables [START_REF] Hanna | Thermal analysis of power cables in multi-layered soil. Part 1: Theoretical model[END_REF]. Temperature variations affect the physical properties of soil (solid and water particles volumetric variation and change in water status), inducing changes in the thermal properties. Considering the impact of temperature on the thermal proprieties of soils between 0° and 70°C, only a few studies are available. [START_REF] Hiraiwa | Temperature dependence of thermal conductivity of soil over a wide range of temperature (5-75°C)[END_REF] measured the thermal conductivity of two soils, a clay loam and a light clay, at different volumetric water contents and different temperatures between 5 and 75°C.

Their results showed that  increased with increasing temperature and volumetric water content. [START_REF] Smits | Temperature dependence of thermal properties of sands across a wide range of temperatures (30-70°C)[END_REF] measured the thermal conductivity and diffusivity of two sands prepared at various saturation rates at different temperatures between 30 and 70°C. Their results showed that the thermal properties increased dramatically for temperatures above 50°C, but small changes in thermal properties were observed at temperatures between 30 and 50°C. This literature review shows that numerous studies are available on the impact of soil parameters on the soil thermal conductivity; some of these studies address compacted soils, but fewer focus on the compacted soils used in civil engineering that are submitted to cyclic temperature variations from 1 to 70°C. Furthermore, few studies and little data are available on the variation of the volumetric heat capacity and the thermal diffusivity as a function of soil properties, despite their great importance in the study of heat flow and heat storage in soils.

The aim of this study was to better understand the coupled effect of water content (w), dry density (d) and temperature variations (T) on the thermal parameters (, C and ) of compacted soils. The following issues were addressed:

 whether the mineralogy and the size distribution of soil particles affect their thermal properties,  the coupled effect of w and d on the thermal parameters, and  the effect of monotonic and cyclic thermal variation on soil thermal characteristics.

In the following sections, the preparation of the materials and the experimental device are described first. Then, the results are presented and analysed to explain the main evolutions of the thermal properties according to the initial state parameters of compacted soils and the impact of thermal variations.

Materials and methods

In this section, the properties of the five studied materials, the compaction of the materials at various water contents and dry densities, and the apparatus used to measure the thermal properties are presented.

The materials properties

Five different soils were studied. The mineralogical compositions of these soils are presented in Table 1. The illitic soil (I) named Arginotech® came from eastern Germany. The Plaisir loam (PL) was extracted from the Paris region and was dried, pulverized and sieved through a 2 mm sieve before being quartered and used for various experiments [START_REF] Boukelia | Modélisation physique et numérique des géo-structures énergétiques[END_REF]. Two other loams from the Parisian basin, namely, the Jossigny loam (JL) and the Xeuilley loam (XL), were also studied. The characteristics of each material including grain size distribution, Atterberg limits (AFNOR, 1993), specific surface (AFNOR, 1999a), carbonate content [START_REF] Afnor | Essai Proctor Normal-Essai Proctor Modifié [Soils: Investigation and testing. Determination of the compaction characteristics of a soil[END_REF] and Proctor compaction parameters (AFNOR, 1999b) are listed in Table 2.

The particle size distributions were determined using a Laser diffraction particle size analyser (Malvern Mastersizer 2000®) [START_REF] Afnor | Essai Proctor Normal-Essai Proctor Modifié [Soils: Investigation and testing. Determination of the compaction characteristics of a soil[END_REF] for the illitic material (Eslami, Rosin-Paumier, Abdallah, & Masrouri, 2015) and a wetting sieve method for the Plaisir loam. For the Jossigny and Xeuilley loams, the particle size distribution curves were found in the literature [START_REF] Fleureau | Comportement du limon de Jossigny remanié soumis à une pression interstitielle négative[END_REF]Inderto, 1993 and[START_REF] Blanck | Effet d'un traitement non traditionnel acide sur le comportement mécanique de trois limons. 20 ème Congrès Français de Mécanique[END_REF][START_REF] Blanck | Effet d'un traitement non traditionnel acide sur le comportement mécanique de trois limons. 20 ème Congrès Français de Mécanique[END_REF]. The Proctor optimum water contents (wOPN) and maximum dry densities (dmax) were obtained from the standard Proctor curve performed for each mixture (AFNOR, 1999b) (Figure 1 and Table 3).

Sample preparation

Six test series were performed (Table 5). In the first series, the effect of the dry density was studied on an illitic soil (I) and a sand-illitic mixture (S+I). In the second series, the effect of water content was studied on the same soils (I and S+I). In the third series, the effect of particle size and mineralogy were evaluated for each soil. Then, the coupled effect of dry density and water content was analysed by measuring the thermal properties of five materials compacted at various water contents and dry densities using constant compaction energy (4 th series). Then, the temperature effect was studied on PL, I and S+I by measuring the thermal properties of different points on a compaction curve at varying temperatures within the range of 1 to 70°C (5 th series). Finally, the effect of cyclic temperature variations was studied on PL (6 th series).

To prepare samples at the desired water content and dry density, powdered material was first mixed with water to reach the target water content and then packed into hermitic bags to homogenize over at least 24 h. Two types of samples were prepared. For the 1 st and 2 nd series, samples 70 mm in height and 35 mm in diameter were statically compacted. For the 3 rd through 6 th series, samples 116 mm in height and 152 mm in diameter were dynamically compacted in three layers in a CBR (Californian Bearing Ratio) mould. The standard Proctor compaction energy was applied. The samples were then heated or cooled to different temperatures in a climatic chamber, and the thermal properties of each sample were measured.

Thermal parameter measurements

The thermal properties were measured using a KD2 Pro thermal properties analyser®. Two specific sensors were used: a dual-needle SH-1 and a single needle TR-1. The dual-probe SH-1 consists of two parallel probes (30 mm long and 1.3 mm diameter with 6 mm spacing). One of these probes comprises a thermistor, and the other comprises the heater element. This sensor measures thermal conductivity (), thermal resistivity (R), thermal diffusivity () and volumetric heat capacity (C) by employing the dual needle heat pulse method. The single needle TR-1 (2.4 mm diameter and 100 mm long) measures only thermal conductivity (); this is used when  is higher than 2 W.m -1 .K -1 . The measurement range and the precision of both sensors are summarized in Table 5.

To measure the thermal parameters of sample, the probe was covered by a thin layer of grease (Arctiv Silver® 5 -High-density polysynthetic silver thermal compound) and then was placed in the sample after soil drilling at the same diameter. In this case, the use of the grease is recommended to improve the contact between the sensor needle and the soil. A waiting time of 15 minutes was imposed before each test to reach an equilibrium temperature between the probe and the soil. The presented value is a mean value of 4 tests in different locations of the sample.

Experimental results and discussion

In the following sections, the experimental results and discussions for each series are presented successively.

Dry density effect on thermal parameters (1 st series)

The effects of dry density on thermal parameters are studied on both illitic material (I) and a sand-illitic mixture (S+I). Illitic samples were prepared at an initial water content in the range of 28.8 to 33.4% and dry densities varying from 1.20 to 1.52 Mg/m 3 . Sand-illitic (S+I) samples were prepared at an initial water content (w) in the range of 16.9 to 20.4% and dry densities (d) varying from 1.54 to 1.79 Mg/m 3 . The variations of water content and dry density were chosen in the same range of the standard Proctor curve for both materials. The samples were thus prepared at water contents and dry densities compatible with their use in geotechnical engineering. The real density (h) of the samples can be determined from Eq. 1.

d h w      ) 1 ( (Eq. 1)
The results (Figure 2a,b,c) showed that the thermal conductivity, volumetric heat capacity and thermal diffusivity increased with increasing dry density for both materials regardless of the water content. Linear increases in  C and  as functions of increasing dry density were shown, in agreement with literature.

The thermal diffusivities (=/C) measured on the illitic samples were significantly lower than those of the sand-illitic samples.

Water content effect on thermal properties (2 nd series)

The effect of water content on the thermal proprieties was showed by [START_REF] Tang | Effet de la température sur le comportement des barrières de confinement[END_REF][START_REF] Brandl | Energy foundations and other thermo-active ground structures[END_REF] and [START_REF] Barry-Macaulay | Thermal conductivity of soils and rocks from the Melbourne (Australia) region[END_REF]. In Figure 2, the range variations of water content and dry density of I and S+I samples were the same as those in part 3.1. The enhancement of thermal conductivity as a function of water content increase was clearly observed (Figure 2a). Figure 2b showed that the volumetric heat capacity of both materials varied in the same range. The range variation of w has a negligible impact on the thermal diffusivity of both materials. The results obtained in the 1 st and 2 nd series are in agreement with literature. They therefore validated the experimental procedure used in this study.

Mineralogy and particle size effect on thermal properties (3 rd series)

The mineralogy of the sample has an effect on its thermal conductivity [START_REF] Ekwue | Thermal Conductivity of Some Compacted Trinidadian Soils as affected by Peat Content[END_REF][START_REF] Zhang | Investigation on thermal characteristics and prediction models of soils[END_REF]. For example, [START_REF] Brigaud | Mineralogy, porosity and fluid control on thermal conductivity of sedimentary rocks[END_REF] denoted that the of quartz (quartz = 7.7 W.m -1 .K -1 ) is higher than that of illite (illite = 1.85 W.m -1 .K -1 ). As a consequence, the addition of sand to illitic material provided a higher for the S+I samples in comparison with the illitic samples (Figure 2a).

Coupled effect of water content and dry densities on thermal properties (4 th series)

The coupled effect of water content and dry density was studied by measuring the thermal parameters of samples compacted on several points of the compaction curves of five materials (illite (I), illite+sand (S+I), Plaisir loam (PL), sand+Jossigny loam (S+JL), sand+Xeuilley loam (S+XL)) of varying size distribution and mineralogy. Measurements of thermal parameters (C and αwere carried out on samples prepared with the same compaction energy.

The results (Figure 3) showed that the thermal conductivity of each material increased on the dry side of the compaction curve until reaching a maximum near the Proctor optimum. On the dry side of the compaction curve, the dry density and the water content both increased, resulting in an increase of . In contrast, on the wet side of the compaction curve, the evolutions are different according to the sample mineralogy. For the silicious materials (S+I, PL, S+JL, S+XL),  decreased, whereas for the silicate material (I),  remained at its highest values. These evolutions are consistent with the physical properties of the studied samples. On the wet side of the compaction curve, the water content continues to increase, whereas the dry density decreases. As the saturation rate remains approximately the same, the water molecules took the place of solid grains. As quartz (7.7 W.m -1 .K -1 ) is higher than water (0.61 W.m -1 .K -1 ) [START_REF] Brigaud | Mineralogy, porosity and fluid control on thermal conductivity of sedimentary rocks[END_REF], the thermal conductivity of samples containing quartz decreased quite quickly, whereas the thermal conductivity of samples containing silicates (illite =1.9 W.m -1 .K -1 ) remained at an approximately constant value.

The volumetric heat capacity of soils increased on the dry side of the compaction curve until reaching a maximum near the Proctor optimum. Then, the values remained constant or decreased slightly. The variation range of the volumetric heat capacity was identical for the five materials studied.

The thermal diffusivity followed the same variation as the thermal conductivity in accordance with its definition (D=/C). Consequently, the thermal diffusivities of sand-loam mixtures (S+JL and S+XL) were three times higher than that of the illitic soil.

Combined effect of water content, dry density and temperature variations on thermal properties (5 th series)

According to the potential use of compacted soils near a heat source or heat sink (energy storage, buried cables, or waste storage), the effect of temperature on the thermal properties was studied within a maximum temperature range of 1 to 70°C. Samples were prepared at various w and d under the same compaction energy (standard Proctor). Three materials were investigated: the illitic material (I, Figure 4), the sand-illitic soil mixture (S+I, Figure 5) and the Plaisir loam (PL, Figure 7 and Figure 8). Thermal diffusivity measurements for the I samples were quite scattered due to their very low values compared with the measurement range (Table 4). In the studied temperature range, the main evolutions obtained in the previous part were observed for each material:

-thermal parameters reached their maximum values near the Proctor optimum; -thermal parameters increased on the dry side of the compaction curve; -thermal parameters remained at their maximal values on the wet side of the compaction curve for illitic samples; and -thermal parameters decreased on the wet side of the compaction curve for the S+I and PL samples.

Following this general trend, variations were noticed according to the sample temperature. An extensive experimental study was carried out on the illitic material, which was expected to be the material most sensitive to temperature variation. The uncertainty of this method (10%) is presented with a double arrow for one point of each series (Figure 4a, b andc). The variation of  in the temperature range of 1 to 40°C was within this uncertainty interval, but for 70°C results have clearly shown an increase of  with increasing temperature. Measurements performed on S+I and PL over a maximal temperature range of 1-50°C confirmed this trend.

The increase of thermal conductivity linked to the sample temperature was more important on the dry side than on the wet side of the compaction curve. This phenomenon is in addition to the increase of  linked to the dilatation of the components (water and minerals) with increasing temperature.

In a saturated soil, the heat flux moves through solids and liquids by diffusion. Convection movements may aid the transfer, but the liquid convection may be limited, especially in low permeability materials. In an unsaturated soil, at low saturation rates, as is the case on the dry side of the compaction curve, the material contains solids, water and air. The thermal conductivity of air is very low unless it contains water vapour. The water vapour moves through the pores, increasing the thermal conductivity of the material. At higher temperatures, the air is able to reach higher moisture contents, which increases its participation in thermal conduction.

The impact of temperature variations on the volumetric heat capacity was not as clear as that on  especially for the I and S+I samples. Figure 4b and 5b do not show a clear trend for the evolution of this parameter. The impact of temperature variations seemed to be below the measurement sensitivity. For PL samples, C measurements at 50°C were higher than values at 20°C in the case of L1 samples (Figure 8). The thermal diffusivity increased with increasing temperature in accordance with the evolution (Figure 4c and5c).

Cyclic temperature variation effect on thermal properties (6 th series)

Several samples of Plaisir loam were submitted to cyclic variations of temperature. Different cycles of 20/50°C (Table 6) were applied to the samples that were hermetically closed to keep w constant. The measurements were performed at various steps of the temperature programmes, as defined in Figure 6.

The experimental results showed that heating the samples increased the thermal conductivity and the volumetric heat capacity. These results confirmed the trends obtained in the previous part.

The measurements after 60 cycles showed that the thermal conductivity PL at 20°C after the thermal cycles (C3) was lower than PL at 50°C (C2). Nevertheless, the value of PL at 20°C measured after the cycles (C3) was slightly higher than PL at 20°C measured at the beginning of the test (C1) (Figure 7). The thermal conductivity reflects the capacity of a material to conduct a heat flux, this difference can be due to a change in the soil structure after several cycles.

Measurements of the volumetric heat capacity at 20°C (C3) remained very close to those obtained in test (C1) (Figure 8). The volumetric heat capacity reflects the ability of a material to store energy, it may be considered as the sum of the participation of soil different component. The ratio between solid, liquid and air did not vary in the sample and the volumetric heat capacity remained unchanged.

Conclusions

The effects of dry density, water content, mineralogy, size distribution and temperature on the thermal conductivity, volumetric heat capacity and thermal diffusivity of five materials were studied in the laboratory. In accordance with previous studies, the thermal parameters of the studied materials increased with increasing dry density and water content. The samples with wider granularity had a higher density and better solid contacts that improved their thermal conductivity. The effect of water content on thermal conductivity was clearly observed in loose materials, whereas in denser materials, the effect was negligible. The thermal conductivity of materials increased with increasing quartz percentage and the spread of the granulometric curve, whereas the volumetric heat capacity of materials seemed less sensitive to variations in the mineralogy and particle size. In the compacted soils, the thermal conductivity, the volumetric heat capacity and the diffusivity increased on the dry side of the compaction curve until reaching a maximum near the Proctor optimum.

The effect of temperature variation on the thermal properties was studied within a maximum temperature range of 1 to 70°C. The thermal conductivity and thermal diffusivity increased significantly on the dry side of the compaction curve and at high temperatures (illitic soil at 70°C), while the effect of temperature on the specific heat capacity was not significant. The increase in thermal conductivity induced by temperature variation was more important on the dry side than on the wet side of the compaction curve due to the water vapour movement.

Heating the samples increased the thermal properties, but this modification is partially reversible after several cycles for thermal conductivity and totally reversible for volumetric heat capacity. Thermal diffusivity (mm 2 /s) 0.1 -1 ± 10 % SH-1 Volumetric heat capacity (MJ.m -3 K -1 ) 0.5 -4 ± 10 %

Figure 1 .Figure 2 .Figure 3 .Figure 4 .

 1234 Figure 1. Compaction curves of the studied materials.

Figure 5 .Figure 6 .

 56 Figure 5. Evolution of the thermal properties of the sand-illitic soil mixture (S+I) according

Figure 7 .

 7 Figure 7. Warming effect on thermal conductivity of PL (P1, P2, 6 th series).

Table 1 .

 1 Mineralogical composition of the raw materials

	Material	Ref. Quartz calcium	Feldspar	clay minerals	Others
				carbonate
	Illitic soil	I	Traces 12% Calcite	Traces	77% Illite
						10% Kaolinite
	Plaisir Loam	PL	81%	5% Calcite	3%	5%
				7% Dolomite
	Jossigny Loam JL	98%	Traces	1%	1%
	Xeuilley Loam XL 83%	2%	3%	11%	1% Goethite
	Hostun Sand	S	97.4% Traces			2.6%

Table 2 .

 2 The characteristics of the studied materials

	Properties	Illitic soil I	Plaisir loam PL	Jossigny loam JL Fleureau & Inderto (1993)	Xeuilley loam XL Blanck et al. (2011)
	Grain size distribution
	Passer -by 80 m	100	41	80	95
	Passer -by 2 m	85	20	28	25
	Atterberg limits
	Plastic limit (%)	34	20.6	16 -19	28
	Liquid limit (%)	65	27.3	37	37
	Plasticity index	31	6.7	18 -21	9
	Specific surface
	MBV(g/100g)	5.41	1.85	-	3.1
	Carbonate content
	CaCO3	-	0.8	-	1.3
	Proctor compaction
	wOPN (%)	31.3	16	15.5	18.5
	dmaxw	1.43	1.81	1.75	1.71
	maxw	1.88	2.10	2.02	2.03
	Soil Class
	GTR Classification	A3	A1	A2	A2
	USCS Classification	MH	CL	CL	ML

Table 3 .

 3 Composition and parameters of the standard Proctor curve for the mixtures

	Mixture Composition	wOPN (%)	dmax	hOPN
				(Mg/m 3 )	(Mg/m 3 )
	S+I	50% Hostun sand and 50% Illitic soil	18.6	1.71	2.03
	S+JL	50% Hostun sand and 50% Jossigny Loam	13.6	1.89	2.15
	S+XL	50% Hostun sand and 50% Xeuilley Loam	13.7	1.88	2.14

Table 5 .

 5 Range and precision of sensors SH-1 and TR-1

	Sensor Propriety	Measure range	Precision
	TR-1	Thermal conductivity (W.m -1 .K -1 )	0.1 -0.2 0.2 -4	± 0.02 ± 10 %
	SH-1	Thermal conductivity (W.m -1 .K -1 )	0.02 -0.2 0.2 -2	± 0.01 ± 10 %
	SH-1

Acknowledgements :

The authors acknowledge C. Fontaine from IC2MP laboratory (Poitiers, France) for the mineralogical analyses.


The Aquitaine Basin is the retro-foreland basin associated with the Pyrenean orogen. It developed from Campanian to Mid-Miocene by flexure of the upper (European) plate. To the south the orogen comprises the North Pyrenean Zone, the Axial Zone, the South Pyrenean Zone and the Ebro basin on the lower (Iberian) plate. The North Pyrenean Zone is a narrow thrust belt comprising both nonmetamorphic and metamorphic Triassic to mid-Cretaceous strata that record both Pyrenean shortening and Aptian-Cenomanian extension locally associated with HT/LP metamorphism. This latter zone is separated from the underlying non-metamorphic series by a basal tectonic contact along which lherzolite massifs and breccias represent remnants of a zone of exhumed mantle.

This study aims to understand the relationship between the deformation history of the central North Pyrenean Zone and the evolution of the Aquitaine foreland basin, from the initial rift architecture to the present-day structure. Based on geophysical, structural, seismic, borehole and paleotemperature data (RSCM) a new crustal cross-section of the retro-foreland system constrains the structure of the North Pyrenean Zone and the central Aquitaine Basin. The detailed foreland basin evolution is constrained by chrono-and litho-statigraphy and subsidence analyses from the literature. The section is sequentially restored to an Early Cretaceous rift template allowing us to correlate the timing of deformation and distribution of loads with foreland basin evolution.

We propose a kinematic model for the progressive inversion of the rift system. During early convergence (Campanian-Maastrichtian) the closing of the exhumed mantle domain is associated with the thrusting of the metamorphic unit over the distal European margin to the north. This episode generated the first tectonic loading of the European lithosphere, from Campanian to Late Maastrichtian, contemporaneous with post-rift thermal subsidence. The major continental collision, from Thanetian to Mid-Miocene, associated with the emplacement of a mantle body at depth, generated a second phase of tectonic subsidence. These two events are recorded as two distinct tectonic phases in the tectonic subsidence of the Aquitaine foreland basin, separated by a quiet phase. We propose that the rift architecture has a strong influence on the deformation style of the retro-wedge and that major normal faults that delimit the rift domain are inverted during the compression.

  PROGRESSIVE INVERSION OF A HYPER-EXTENDED RIFTED MARGIN: THE CENTRAL NORTH PYRENEAN ZONE, FRANCE P. Angrand 1 , M. Ford 1 , A.G. Bader 2 1 CRPG, UMR 7358, 15, rue Notre-Dame-des-Pauvres, 54501 Vandoeuvre-les-Nancy, France. 2 , avenue aude-u e n, P 00 450 0 r ans ede 2, France


H Eslami

Impact of temperature cyclic variations on bearing capacities of a sensitive soil

Keywords: Geothermal energy, Laboratory tests, Pressuremeter, Temperature, Compacted soils, Illite

The incorporation of heat exchangers in geostructures produces a cyclic variation of the temperature in the adjacent soil. Therefore, there are important questions about the effect of temperature variations on hydro-mechanical soil parameters. In this study, the mini-pressuremeter tests were conducted in laboratory on a homogeneous material submitted to different thermal loading (1 to 40 °C). The tested material, an illitic soil, is compacted in a thermo-regulated container at optimal water content (31.3 %) and 90% of maximal dry density (1.29 Mg/m 3 ). Only six tests are performed in each container to prevent edge effects and the influence between the tests. The impact of temperature variation on creep pressure (pf), limit pressure (pl) and pressuremeter modulus (Ep) were determined. A decrease in creep pressure and limit pressure with increasing temperature was observed, while the variation of pressuremeter modulus was less pronounced. The first cycle induced more important parameter variations than the subsequent cycles.

Introduction

The incorporation of heat exchangers in geostructures leads to a temperature variation in the adjacent soil over a range from 4 to 30 °C [START_REF] Péron | A method for the geotechnical design of heat exchanger piles[END_REF].Consequently, there are important questions about the effect of temperature variations on hydro-mechanical soil properties and load transfer in the pile.

In the case of energy piles, on the one hand, the thermal expansion or contraction of the pile causes additional thermal stress in the pile which can influence the load transfer, and consequently the skin friction and the end bearing [START_REF] Bourne-Webb | A framework for understanding energy pile behaviour[END_REF], [START_REF] Arairo | Experimental and numerical investigations of temperature effects on the pressuremeter parameters of a compacted illitic soil[END_REF]. On the other hand, the cyclic thermal variation imposed to the soil may affect its hydro-mechanical properties. Thus, the behaviour of an energy pile is directly related to the coupling of thermal, hydraulic and mechanical phenomenon.

Different in situ and laboratory studies were conducted to study the behaviour of energy piles and their thermo-mechanical response [START_REF] Laloui | Comportement d'un pieu bi-fonction, fondation et échangeur de chaleur[END_REF]- [START_REF] Akrouch | Thermo-mechanical behavior of energy piles in high plasticity clays[END_REF]. In all these studies, the effect of temperature on the soil parameters has been neglected. However, a thermal variation may have an impact on the shear strength, yielding and critical-state soil properties [START_REF] Hueckel | Thermoplasticity of saturated clays: experimental constitutive study[END_REF]- [START_REF] Uchaipichat | Experimental investigation of thermo-hydro-mechanical behaviour of an unsaturated silt[END_REF]. This phenomenon, named the thermal softening, may have an impact on the bearing capacity of the pile. The bearing capacity calculations of deep foundations are actually based on pressuremeter or penetrometer testing results. In this study, the mini-pressuremeter tests were conducted in laboratory on a solid submitted to different thermal loading (1 to 40 °C). Thus, in this study, a mini-pressuremeter test adapted from the pressuremeter test, is used to quantify the temperature effect on the mechanical behavior of the soil.

In the following sections, the material and the developed experimental device are first described. Then, the results are detailed and the impact of thermal variations on the pressuremeter parameters is discussed.

Material and method

In this part, the studied material is first presented, then experimental device is described and finally the mini-pressuremeter tests are detailed.

Basic characterization of the material

The tested material is an illitic soil named Arginotech® that contains 77% illite, 10% kaolinite, 12% calcite and traces of quartz and feldspar according to the furnisher. Mini-penetration tests, realised on this soil by Eslami et al. [START_REF] Eslami | Impact of temperature variation on penetration test parameters in compacted soils[END_REF] showed the sensitivity of this material to thermal variations.

The particle-size distribution of the soil was determined using a laser Malvern Mastersizer 2000® device (Figure 1a). Almost 85% of the particles of the illitic material are smaller than 0.002 mm (clay particles). Its liquid limit (LL), its plastic limit (PL) and its plasticity index (PI) are 65%, 34%, and 31% respectively [START_REF] Afnor | -Sols : reconnaissance et essais ; Détermination des limites d'Atterberg -Limite de liquidité à la coupelle-Limite de plasticité au rouleau[END_REF]. A optimum water content of 31.3% and A maximum dry density of 1.43 Mg/m 3 are obtained standard Proctor curve for the material [START_REF] Afnor | -Sols : Reconnaissance et essais Détermination des références de compactage d 'un matériau[END_REF] (Figure 1b).

Experimental device and methods

To perform mini-pressuremeter tests at controlled temperatures, an homogeneous solid with a large volume is required as well as an experimental device to impose thermal variations to the solid. In this part, the developed methodology to obtain a meter-scale solid with homogeneous properties (temperature, water content and dry densities) is described.

First of all, water is added to the material in a large capacity fixed-speed mixer (38 rpm) in order to reach the target water content (wOPN = 31.3%). The wet material is

(a) (b)
stored in 8 plastic drums for a minimum period of 5 days to ensure a good homogenization. A pneumatic compactor is used to compact the material in a cylindrical stainless steel container of 800 mm in height and 600 mm in diameter (Figure 2a). The compactor applied dynamic forces on a metallic plate of 4 mm in thick and 600 mm in diameter to allow a homogeneous compaction. The results of the preliminary series lead to the choice of a compaction in 11 layers of 70 mm in thickness.

A stainless steel tube is welded to the outside of the container. Insulating sleeves are placed on the tube in order to reduce heat exchange with the surrounding atmosphere. The top of the massif is insulated with a plastic film to preserve the initial water content. Finally, the assembly is placed in a box made of extruded polystyrene plates of 40 mm thick to reinforce the insulation. The stainless steel tube is connected to a thermoregulator to allow the circulation of an ethylene-glycol water solution at the target temperature in the tube (1 to 40 °C). Figure 2b shows the total parts of the developed experimental device in order to perform the mini-pressuremeter tests. Seven thermal sensors PT100 are positioned within the solid at various depths and various distances from the wall of the container. The sensors are plugged to a data logger in order to monitor the temperature evolution inside the compacted soil during the temperature variations.

Mini-pressuremeter tests

Only six mini-pressuremeter tests are performed in each solid to prevent edge effects and cross-influence between the tests. The test points are positioned on a concentric circle twice smaller than the container (Figure 3). The pressuremeter tests are conducted with an APAGEO® mini-pressuremeter probe of 380 mm in height and 28 mm in diameter. Before each pressuremeter test, a core with a diameter equal to the mini-pressuremeter probe is made using a core sampler. The 630 mm core is divided into smaller pieces to measure the water content and the density of the material as a function of the depth. The center of the probe is positioned in the middle of the height of the compacted soil (385 mm). The probe is connected to a pressure-volume controller brand GDS®. The pressure controlled test consists in applying increasing pressure with equal increments of 25 kPa for at least 1 minute. The equilibrium volume is measured for each increment and the volume versus pressure curve is plotted. The test is stopped when the imposed volume reached 140,000 mm 3 . Immediately after the pressuremeter test at a given temperature, the borehole is filled with the same material, at the same water content, to avoid influencing the following tests. The effect of temperature variations on the resistance of the minipressuremeter membrane is measured by placing the probe in a climatic chamber at a given temperature during the calibration test of the membrane resistance.

Results

In the framework of our project, a preliminary series and four test series (six pressuremeter tests for each series) are conducted. In this paper only the two last test series, which aim to quantify the effect of a cyclic thermal variation on the pressuremeter parameters, are presented. The performed tests in each solid are numbered as follows Xx; with X, the massif temperature when performing test (i.e. 1, 20 or 40 °C) and x, the test index that increments for each new test at the same temperature.

The results of the preliminary test series show a good repeatability of the tests and allow estimating the uncertainty of each measured parameter according to the Student's t-distribution for a 90% accuracy: Ep: ±10%, pf: ±5%, pl: ±5%.

The selected test series, named series I and series II, consists in tests at 20, 1 and 40 °C. The initial temperature of the massif is 20 °C and the massif was successively brought to 1 and 40 °C, each step lasts at least 70 hours (Figure 4 and Figure 5). The thermal equilibrium is reached for each step before performing the mini-pressuremeter test.

The Table 1 and the Table 2 present all the obtained parameters and properties. The average water contents and the average dry densities variations according to the thermal solicitation are lower than 1%: the solids are homogeneous.

The mini-pressuremeter tests are conducted at the end of the selected steps as shown in the Figure 4 and Figure 5. The pressuremeter curves (Figure 6) are consistent with the three pressuremeter domains. A softening takes place after heating the solid at 40 °C and a hardening is observed after cooling it at 1 °C, whatever the order of testing. The pressuremeter parameters are calculated for each step of each series and presented in Figure 7. In series I, pf increases slightly with cooling the massif from 20 to 1 °C, then it remains almost constant with cycles of cooling/heating (1-40-1 °C). pl increases with heating and decreases with cooling. In series II, the first value at 20 °C was not acquired and the test 1f was inoperable; in Figure 7, the values are estimated according to the similarity between series I and series II. The hypothesis states that the relative difference between 40c parameters and 1c parameter is the same than between 40e parameters and 1f parameters. The same method was used to place 20j parameters.

The parameters evolution these two series are compatible. After the first thermal cycle, the evolution of pf over the imposed thermal variation is under the parameter uncertainty; this parameter reaches a stable value (Figure 7a,b). The evolution of pl is slightly different, the thermal impact remains measurable even if its impact is totally reversible. Its evolution rate is estimated at 1.5 kPa/°C in the studied temperature range (1-40 °C) (Figure 7c,d). These results are consistent with the contraction of the elastic domain (yield locus) with the increasing temperature is shown in the literature with oedometric or triaxial tests for various clay materials [START_REF] Hueckel | Thermoplasticity of saturated clays: experimental constitutive study[END_REF]- [START_REF] Uchaipichat | Experimental investigation of thermo-hydro-mechanical behaviour of an unsaturated silt[END_REF].

The variation of Ep with temperature is lower than the other pressuremeter parameters (Figure 7e,f). The Ep evolution for series I solid seems not relevant because its evolution is erratic. On the contrary, a slight Ep decrease, consistent with the previous tests, is measured when series II is warmed.

Conclusions

The objective of this paper is to quantify the effect of cyclic temperature variations on parameters linked to the pile design. A protocol allowing the realization of minipressuremeter test in laboratory conditions was developed. The evolution of the pressuremeter modulus (Ep), the creep pressure (pf) and the limit pressure (pl) were measured according to the temperature variations on a soil selected for its high sensitivity to temperature variations. The results of the mini-pressuremeter tests on the compacted illitic soil showed a decrease in the creep pressure and the limit pressure as temperature increases (thermal softening), while the variation of the pressuremeter modulus is less significant, probably due to the sensitivity of this parameter to the test conditions. For the following cycles, the effect of the temperature variation is lower. After more than one cycle, pf seemed to reach an equilibrium value independent of the temperature. The pl keeps its temperature dependence beyond the 1 st cycle but it evolution is reversible and the evolution rate is 1.5 kPa/°C in the studied temperature range (1-40 °C). The limit pressure is the key parameter of calculating ultimate load of piles, so the evolution of this parameter with temperature variation would be taking into account to calculate the bearing capacity of geothermal piles built in sensitive soils.

Figure 1 .

 1 Figure 1. Characteristic of the studied illitic soil: a) particle size distribution and b) compaction curve, Sr: The degree of saturation.

Figure 2 .

 2 Figure 2. (a) Thermo-regulated metric scale container; (b) Total parts of developed experimental device.

Figure 3 .

 3 Figure 3. Positions of the test points and thermal sensor in the solid.

Figure 4 .

 4 Figure 4. Temperature variation at 150 mm from the edge of the solid and the chronology of series I.

Figure 5 .

 5 Figure 5. Temperature variation at 150 mm from the edge of the solid and the chronology of series II.

Table 1 .

 1 Pressuremeter parameters for the series I: temperature cycling (20-1-40-1-40-1-40-1 °C).

Figure 6 .

 6 Figure 6. Pressuremeter curves: (a) series I; (b) series II.

Figure 7 .

 7 Figure 7. Variation of creep pressure (pf) (a-b), limit pressure (pl) (c-d) and pressuremeter modulus (Ep) (e-f) with temperature for series I and II. Dotted lines correspond to estimated parameters.

Test Temperature (°C) w (%) ρd (Mg/m3) pf (kPa) pl (kPa) Ep (MPa)

	20i	20	31.4	1.25	103	240	5.51
	1c	1	31.4	1.26	124	303	5.73
	40c	40	32.1	1.25	125	236	7.68
	1d	1	31.7	1.25	123	337	4.16
	40d	40	31.5	1.28	143	270	6.85
	1e	1	31.6	1.24	133	310	5.66

Table 2 .

 2 Pressuremeter parameters for the series II: temperature cycling (20-1-40-1-40-1-40 °C).

	Test	Temperature (°C)	w (%)	ρd (Mg/m 3 )	pf (kPa)	pl (kPa)	Ep (MPa)
	40e	40	31.2	1.30	174	305	6.24
	1g	1	31.5	1.24	170	357	6.97
	40f	40	31.2	1.28	170	307	6.18
	1h	1	31.0	1.24	160	344	6.16
	40g	40	31.1	1.25	153	276	5.76

ACKNOWLEDGEMENTS

This study was part of the GECKO research program funded by ANR.