EA Claire Margolinas Acté

Floriane Wozniak
email: floriane.wozniak@iufm.unistra.fr

Early construction of number as position with young children: a teaching experiment

Keywords: Number line, cardinal, ordinal, position, didactic design, kindergarten

Our study is motivated by the importance of the number line in the visualisation of elementary properties of operations and numbers. For the number line to be an aid for visualisation, first it must be conceived as a coordinate system (origin, orientation and unit). This paper focuses on the construction of the concept of position on a line, and on the number as a means of remembering position, for pre-elementary school children aged 5-6 years. Using didactic design as a methodology, we conducted an experiment to observe how knowledge was constructed while carrying out a certain kind of task in a situation where position had to be designated. Our study shows that pupils, despite recognising number as a quantity, fail to apprehend number as something which represents position. They are, however, capable of understanding the characteristic elements of the number line in suitable situations. Our research suggests that it is possible to teach the use of ordinal numbers to young pupils.

Introduction

This investigation, as part of our work on numbers in kindergarten [START_REF] Margolinas | Le nombre à l'école maternelle. Une approche didactique[END_REF], focuses on how children aged 5-6 years can learn to perceive position on a line and the use of a number to remember that position.

A teaching experiment

Task design lies at the very core of research in mathematics education [START_REF] Sierpinska | Research in mathematics education: through a keyhole[END_REF][START_REF] Wittman | Mathematics education as a 'design science[END_REF], not only as a goal in order to better teach mathematics but also as a way to research pupils' knowledge in meaningful situations: 'in which mathematical ideas are useful, using the linked constructs of purpose and utility' (Ainley 2008, p. 1), similar to what [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF] calls a 'fundamental situation' and Wittman a 'substantial learning environment ' (1995, pp. 365-366). This study is consistent with the didactic design approach, which is conceived for the purposes of research [START_REF] Artigue | Didactique engineering[END_REF][START_REF] Artigue | Didactic engineering, research and development tool: some theoretical problems linked to this duality[END_REF][START_REF] Chevallard | Teaching mathematics in tomorrow's society: a case for an oncoming counterparadigm[END_REF][START_REF] Margolinas | En amont et en aval des ingénieries didactiques[END_REF]. We can also regard our research as clinical teaching, described by Wittman (1995, pp. 367-368) very clearly:

[…] 'clinical teaching experiments' in which teaching units can be used not only as research tools, but also as objects of study.

The data collected in these experiments have multiple uses: they tell us something about the teaching/learning processes, individual and social outcomes of learning, children's productive thinking, and children's difficulties. They also help us to evaluate the unit and to revise it in order to make teaching and learning more efficient. In this paper, we focus on 'children's productive thinking and children's difficulties' with ordinal numbers: we consider, like [START_REF] Tsamir | Multiple solution methods and multiple outcomes -is it a task for kindergarten children?[END_REF], that pupils aged 5-6 years are capable of solving problems using several methods. Therefore we seek to contribute to question 4 of this issue of ZDM: What does progressive visualisation mean, including how it comes about, in classroom activity? In particular, we want to answer the two following research questions: How does the visualisation of the coordinate system evolve during the learning unit? How do pupils use numbers in order to memorise and represent a position?

The number line

The number line plays an important role from the beginning of primary school in the visualisation of the properties of numbers [START_REF] Csikos | The effects of using drawings in developing young children's mathematical word problem solving: a design experiment with third-grade Hungarian students[END_REF], even if the way in which this visualisation is used in class can be questioned [START_REF] Ernest | The number line as a teaching aid[END_REF][START_REF] Van Den Heuvel-Panhuizen | Learning from 'Didactikids': an impetus for revisiting the empty number line[END_REF]. [START_REF] Ernest | The number line as a teaching aid[END_REF] regards number lines not as a teaching aid but 'as direct contributors to components of the mathematics curriculum ' (p. 423). We concur with him: 'In view of the widespread use of number line diagrams and models in school mathematics, the area needs further investigation and research ' (p. 423). The construction of number has often been researched (e.g. [START_REF] Comiti | Les premières acquisitions de la notion de nombre par l'enfant[END_REF][START_REF] Fuson | Children's counting and concepts of number[END_REF][START_REF] Gelman | The child's understanding of number[END_REF][START_REF] Piaget | La genèse du nombre chez l'enfant[END_REF]. Most of these studies focus on oral number strings or on the cardinal aspect of numbers [START_REF] Urbanska | On the numerical competence of six-years-old children[END_REF][START_REF] Wright | A study of the numerical development of 5-year-olds and 6-year-olds[END_REF]. However, this is not the case in the paper by [START_REF] Bruce | One, two, three and counting[END_REF], which focuses on the ordinal aspect of numbers, and more precisely the oral expression of numbers: 'as a trigger to ordinal understanding' (p. 7). Their results show a great difference between the acquisition of number in a cardinal context and in an ordinal one. Thus, even if the pupils knew how to count, 'the use of that skill in an ordinal context was just not considered' (p. 21). Only one child in their sample of 93 attempted to determine position using counting as a strategy. They conclude 'that the relatively tardy development of ordinal number may be mostly the result of the attention given by adults, and the nature of the common experiences of young children in a pre-school setting' (p. 24). We have therefore designed a sequence of tasks involving the position on a line, which is one way of defining an ordinal number, in order to focus young pupils' attention on this aspect of numbers. The aim of this exploratory research is to understand how pupils are able to visualise and represent position on a line and how this visualisation evolves during the designed sequence. In order to build a sequence of tasks on this theme, we first describe cardinal and ordinal numbers in a material setting.

Cardinal numbers and ordinal numbers in our setting

The material setting we have chosen is based on the following idea. Take two identical bags in each of which there are nine white beads and one red bead. There are two identical sets from a quantitative point of view. Now take two bits of thread, each knotted at one end, and thread the beads from the two bags separately. There is only one chance out of ten for these two 'necklaces' to have the red beads in the same place with respect to the knot (one-to-one comparison of the position). Threading the beads has created an order: the red bead is in a particular position, which may not be the same in the two necklaces. The number which allows the necklaces to be differentiated no longer refers to the quantity of beads but to the position of the red bead among the white ones: for instance in second or third position. Thus a shift in meaning has been created: from the number as a remembered quantity -the cardinal number -to the number as a remembered position -the ordinal number. Our design is based on this use of the position of one coloured bead among neutral beads in necklaces of ten beads. What defines a 'one-dimensional coordinate' remains implicit: a coordinate system in a onedimensional space (a line) comprises an origin and an orientation. Conceiving the number in order to remember a location on a line leads to conceiving an implicit linear coordinate system. 1

A 'formulation situation' to lead pupils towards the representation

In contrast to the work of [START_REF] Bruce | One, two, three and counting[END_REF], this study did not consider oral language as the only criterion for measuring the learning of the ordinal number. We wanted to explore the possibilities of the pupils producing written formulations in order to remember a position. Therefore, according to Brousseau's Theory of Situations (1997), we have to build a situation in which it is necessary to use a written representation in order to succeed. That is the purpose of 'formulation situations'. In an 'action situation' [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF], the pupil seeks to carry on an action which gives tangible results; while in a formulation situation, the immediate action is delayed, making the recording of information necessary. To make a formulation situation clearer, we can look at 1

In the rest of this paper we write 'coordinate system' but here we are dealing only with one dimension. We are aware that in English it is sometimes called a 'number line' but this term is not suitable for our paper, since we have to consider the question of the coordinate system ("repère" in French) and the use of numbers separately. moments in real life. We thus illustrate the three commonest types of formulation situation, which were used in our didactic design. The first impediment to immediate action corresponds to a distance between two places. Let us assume that you are doing repair work upstairs in your house but you need tools that are in your garage. You have to stop your action and ask yourself what you have to get. By contrast with an immediate action, where all the tools are present, it is necessary to become aware of the action to be fulfilled in order to find the right tool. The distance between the two places leads to postponing the action and transforms an activity of 'doing' into an activity of 'thinking about what to do'. The second category of formulation situation corresponds to a distance between two moments. Let us take the case of shopping: you notice that you are lacking something but you will do the shopping later. It is in this type of situation that writing (making a shopping list) becomes necessary. Formulation situations with different distant moments thus play an important role in the development of mental images, which are themselves parts of visualisation [START_REF] Arcavi | The role of visual representations in the learning of mathematics[END_REF]. The last category of formulation situation corresponds to a communication with others. Let us assume that you are in a bakery. You have to ask for a baguette because you are not allowed to take it yourself. This communication can be oral or written (on-line shopping for example). Writing requires the transmitter to represent what is wanted (using words, diagrams or signs). When the roles of transmitter and receiver alternate, the receiver in becoming aware of the difficulty in understanding the message received can then become inspired to enrich his or her own representations. In the experiment we carried out, we were able to observe the catalyst role of this didactic situation for the pupils' representations. The study of the pupils' written productions showed the way pupils graphically visualised the position and how this visualisation evolved over the situations.

Written representation and conceptualisation of numbers as coordinates on the number line

As Duval (1999, pp. 3-4) says, 'representation and visualization are at the core of understanding in mathematics' and 'there is no other way of gaining access to the mathematical objects but to produce some semiotic representation'. This point of view is shared by [START_REF] Bosch | La sensibilité de l'activité mathématique aux ostensifs[END_REF]: in Anthropological Theory of the Didactic, every object which has a materiality and which is perceptible, such as sound, gesture, diagram, geometrical instruments, is by definition an ostensive. These authors distinguish the instrumental valence of the ostensive -what is made possible by the ostensive -and the semiotic valence of the ostensive -what it allows to be evoked [START_REF] Wozniak | Instrumental value and semiotic value of ostensives and task design[END_REF]. Thus, 'All these repeated observations show that semiotic representations constitute an irreducible aspect of mathematical knowledge and that wanting to subordinate them to concepts leads to false issues in learning' (Duval 1999, p. 8).

In the present paper, we look at the written productions (drawings, written numbers, etc.) by very young pupils, in situations with some materials (necklaces which had to be reproduced in various situations). The object (the necklace) was therefore present and its material characteristics, essentially visual, are important (size and colour of the beads, length of the necklace, shape of the knot, etc.). Pupils of this age know the written representation of numbers (in this case, up to at least 10), yet many authors insist not only on the importance of visual representations and visualisation but also on the relationship between several representations in learning mathematics [START_REF] Douady | Tool, Object, Setting, Window Elements for analysing and constructing didactical situations in mathematics[END_REF][START_REF] Rivera | Toward a visually-oriented school mathematics curriculum: Research, theory, practice, and issues[END_REF]. We therefore consider that written representations (drawings, written numbers, the association of both) produced by pupils in a functional context may be analysed from two points of view. On the one hand from the point of view of their efficiency in a situation (do they allow the task to be successfully accomplished?), but also on the other hand from the point of view of what they reveal of the conceptualisation of numbers as coordinates on the number line. Involving very young pupils, we try to understand, according to our research questions, 'how signs are produced already in elementary, initial mathematical learning processes' (Steinbring 2006, p. 157).

Task design 2.1 Presentation of the design

The design was based on six phases which are presented briefly here:

-Phase 1: Installation of the milieu -Phase 2: Action: replication of a model necklace -Phase 3: Formulation with model and replication in two different places -Phase 4: Formulation with model and replication in two different moments -Phase 5: Formulation with communication with others -Evaluation. The sequence began with the discovery of the milieu. A necklace was made by each pupil with 10 beads, including a coloured one, and a thread with a knot at one end. Afterwards the pupils had to compare all the necklaces made the same way. In this first phase, the comparison of the necklaces allowed the teachers to introduce the method of one-to-one comparison (Fig. 1): knot-knot; white bead-white bead and so on) as a means of comparing the position of the coloured bead in the two necklaces. During this phase, teachers and pupils agreed on the meaning of the words 'identical necklace', 'place', 'position', and 'location'.

Fig. 1 Two identical necklaces

During the second phase, the teacher asked the pupils to make a necklace identical to a model. The one-to-one comparison became the validation procedure of the task. These first two phases were action situations. They were followed by three formulation situations. In the third phase, the model was no longer visible during the reproduction. The place where the model was put and the place where the pupils had access to the beads and strings were distant. During the validation, model and reproduction were compared. The teachers only said, 'We will see if your necklace is the same as the model.' Teachers used one-to-one comparison and noted without comment the successes and failures. Teachers did not ask pupils which technique had been effective, nor did they try to identify the effective techniques themselves. In the fourth phase, the pupils were asked to reproduce a model necklace with thread and beads given at a later time (not before the afternoon on the same day). This was the first time pupils were asked to produce a written message. The validation was carried out in the same way as before: an attitude of neutrality on the part of the teachers who noted the successes and failures using one-toone comparison. The fifth phase was the production of a message for another pupil, enabling that pupil to make a necklace identical to the model. Each pupil was placed first in the role of transmitter then in the role of receiver of a message. During validation, the messages were analysed collectively to extract the elements of the technique allowing the task to be carried out. This was the moment when the teachers institutionalised the characteristic elements for the message to be effective. The message was expected to contain three pieces of information: the origin and the orientation of the necklace and the position of the bead compared to the origin. During the evaluation, the pupils were invited to solve three problems: given a model, find an identical necklace in a basket of necklaces placed at a distance; devise a message (for themselves or for others) to reproduce a model necklace; and select 'the necklace with the red bead in 7 th position from the knot' from among others.

Specific features of the 'école maternelle' in France

Education systems which concern children younger than 6 years old vary from country to country. In France, the 'école maternelle' is a school and not a pre-school. Even though attendance is not compulsory before the age of 6 years, in practice nearly all children of 3 years of age are enrolled at school. Teachers who teach at the 'école maternelle' can also teach at elementary school. The teaching objectives are defined by a national curriculum, as for other school levels,2 and the section about mathematics is included in the 'discover the world' part. The teaching programme insists on the 'discovery of the functions of the number, especially as a representation of quantity and means of identifying positions of ordered objects in a list' and emphasises work on 'various forms of representations (drawings, diagrams)'. However, the documentation available for teachers does not focus much on ordinal numbers. Teachers use number lines (at least from 1 to 10, and often more) or calendars (for example in Fig. 2 a part of the number line and a calendar are visible). These objects are used at least once a day when the date is written. For many teachers, ordinal numbers are limited to activities about the calendar. When the pupils are required to use a written number it is generally to express a cardinal, or the day's date.

Experimental conditions and ethical considerations

This study was carried out in two classes (5-6 years old) in May and June 2012. These teachers did not know each other and only communicated with one of us. One teacher teaches a class of 27 pupils in a socially mixed area (class A). The other teaches a class of 24 pupils in a deprived urban area (class B). The social differences of the pupils were not a concern of our research. On the basis of the research design, the teachers were free to choose the timing of the different phases of the situations as they wanted. This flexibility allowed both teachers to specify constraints to be taken into account -the usual period of the activity, working conditions, the classroom itself, etc. From an ethical point of view, although the design had a research objective, it took place during the normal activities of the class, since it concerned knowledge present in the national curriculum. The experiment took place at the end of the school year so that it did not interfere with the teachers' programmes. Each session was videotaped. All the pupils' written work was kept for research purposes. The anonymity of the participants was strictly respected.

Oral and written numbers: differences between the two classes

The pupils could speak with their classmates during phases 1 to 4 and usually they commented on what they were doing or what they were going to do. They were encouraged to do so by the teachers (who did not pass any judgment). Then, the numbers were presented orally during the two first phases in which no writing was necessary. Indeed, to succeed in the task of reproducing a model necklace, the pupils had to count the beads, and some of them counted aloud; they could also refer to the position of the coloured bead using an ordinal number (e.g. 'the coloured bead is the 5 th '). Although in our study in both classes numbers were orally present, this was not the case in the spontaneous writing. In class B, at the beginning of phase 4, one pupil (Harut) announced that numbers were to be used. So, the pupils around him tried to use numbers. This was not surprising: the pupils were trying to understand what the teacher expected them to do. Since written numbers are generally highly valued by the teacher during non-experimental teaching, the pupils interpreted Harut's comment as a good idea. Therefore, some pupils in class B attempted to use numbers in their messages (see below for the analysis of these written productions). In class A, no pupil wrote a number spontaneously during phases 4 and 5. This is why in this class we added a sixth phase similar to phase 4. The pupils were asked to write a message for themselves in order to duplicate the necklace another day, writing only digits (they already knew what 'digit' meant).

Roles of teachers and researchers in the design

In the clinical teaching of a substantial teaching unit 'the teacher has to follow the student's ideas in trying to solve the problem' (Wittman 1995, p. 367). The two teachers involved who agreed to take part in the experiment are very experienced (they are part-time teacher trainers), and their expertise allowed them to become involved in the experiment and understand its implications, which is essential in order to define the 'division of responsibilities' between researcher and teacher [START_REF] Brousseau | Notes on the observation of classroom practices[END_REF]. We asked the teachers to stay in the background because we were interested in the observation of the pupils' procedures in precise situations. The teachers were asked to have a neutral attitude. This means that their intervention and the sharing of information about the pupils' work was intended to be limited to picking up on the expressions of the pupils and the techniques that pupils put in place themselves. That is why, during the validation, the teachers used one-to-one comparison and noted without comment successes and failures. The neutral position of the teacher was an important methodological choice if we were to observe how the representations of the pupils evolved by themselves in the designed situations. Our objective in this experiment was not first and foremost to teach pupils what an ordinal number is, but to know the pupils' various representations better. That is why we asked the teachers not to intervene in the pupils' work but this did not prevent them from facilitating the sharing of representations. The researcher provided the teachers with the task sequence and precise instructions. During the sequence they videotaped the interactions without intervening. They administrated the individual evaluations at the end of the sequence, asking the pupils to explain their written production when there were only numbers.

Data analysis

Since our paper focuses on the pupils' written production, we only mention the way this kind of data was processed. We refer to oral interactions only so that the written productions are better understood. We distinguished two main types of written productions: drawn representations and numerical representations. Then, within each of these types, we established categories and subcriteria of qualitative analysis based on the following questions: What are the different ways of representing position? What are the indications of the construction of ordinal numbers? What are the written markers of the position? We studied each piece of written material made by the pupils using these criteria. In this way, we obtained an overview of the different representations as well as a finer understanding of their progressive construction.

How is it possible to communicate the position through a drawing?

In order to explain a position, it is necessary to define the implicit coordinate system: the origin and orientation of the line. In the case of the necklaces, the threads on which the beads are strung have two ends. It is therefore possible to choose either one of the ends as the origin of the system. The choice of the origin determines the orientation which corresponds to the sense of movement from one end to the other. This is why we use the expression 'information about orientation' to refer to the elements that allow us to define a direction on the curve of the thread of the necklace. Thus, in the situation we studied, representing the knot or representing the end of the necklace was enough to communicate both the origin and the orientation. Also, when the necklaces were held vertically, the beads could fall if the knot had not been at the lower end of the thread. However, in order to recognise that the necklace was drawn vertically, information which communicated the orientation in relation to the sheet of paper, such as, for example, a letter or a number on one side of the drawing, had to be given. Yet, representing the orientation was not sufficient to tell the position. Theoretically, a point on a curve is located thanks to its curvilinear abscissa. However, the characteristic of the material comes into play again: the coloured bead is not a point on the curve. It is a single element among 9 similar elements; it is one of the 10 on the list. For [START_REF] Goody | The domestication of the savage mind[END_REF], the list is the typical form of the first anthropological function of writing. It is an organised set where the origin and orientation are given. With the necklaces, none of the elements was differentiated except for the coloured bead. Therefore, a way to indicate the coloured bead and the neutral beads had to be found. The representation of the necklace was therefore a way of naming the position of the coloured bead, in the same way as writing out a whole list of names is a way of putting the position of one particular name in the list into words. We therefore classify the drawing according to three main questions:

(1) Does the drawing give any information about orientation (e.g. a knot)?

(2) Does the drawing give any information about the actual position of the coloured bead?

(3) If the information about orientation and position are given, is the position of the coloured bead correct or not in the coordinate system? 2.6.2 How is it possible to communicate the position with numbers?

In our situations, where the number of beads remained the same (9 uncoloured, 1 coloured), only the position of the coloured bead needed to be given to be able to reproduce the necklace. An efficient way is to provide a single number to represent the position if the origin and the orientation are known. We can refer to necklace 6 to indicate that the coloured bead was in 6 th position; here 6 is an ordinal number, even when we write 6 and not 6 th . Another possibility is to name the quantity of beads that make up the necklace in the order in which they should be threaded. In this case necklace 6 could be described as 5 1 4, which should be interpreted as 5 uncoloured beads, 1 coloured bead, then 4 uncoloured beads. It is even possible to give only the number 5 to represent the 5 uncoloured beads to be threaded at the beginning of the necklace, and leave the other data implicit. In these cases, the relevant numbers are always cardinal numbers. It is the sense that is given to the number that determines whether an ordinal number or a cardinal number is being used. Indeed, the same necklace may be designated by the number 6 as the position of the coloured bead among the beads of the necklace (ordinal sense) or by the number 5 as a quantity of the first uncoloured beads to be placed before the coloured bead (cardinal sense). Such a sense may be implicit in the self-formulation (phases 3, 4 and 6), but must be explicit for communication with others for it to be efficient (phase 5). We therefore classify the written numbers according to two main questions:

(1) Do the numbers give some information about the quantity involved in the beadworks?

(2) Can the numbers be considered in an ordinal perspective?

Results and analysis

We examine the written material made by the pupils without distinguishing the classes or the phases when the representations were made. We consider the drawn representations followed by the numerical representations.

Messages with drawing

How did pupils represent orientation?

The effect of the material was noticeable on the types of representation. In class A the thread is rigid and the beads do not slide easily along it. Thus the representations of the necklaces were more like segments as in Fig. 3. By contrast, in class B, the thread was not rigid and some pupils placed the necklaces in a vertical position for comparison. Thus, there were representations which used this physical phenomenon to represent the orientation of the necklace. For instance, Melania (Fig. 4), used the verticality, thanks to writing her first name and drawing a smiley which indicated the orientation of the sheet of paper. These led to a certain consistency in the representations in class A: segments with a sign to represent the knot. In class B, however, there was great variety in the representations of the orientation: if one considers the presence (or not) of the knot, the presence (or not) of the end of the necklace and the use (or not) of the verticality, we observe that out of the eight possible cases,4 seven are represented. Nevertheless, some drawings, very well drawn besides, showed that the pupils do not always try to represent the real object identically. This is the case for example with Kanaë (Fig. 5) whose necklace shows the orientation (the knot) and also the position of the coloured bead, but whose colours were completely different from the real ones (the model necklace had nine white beads and one blue). This is also the case of Tesnime (Fig. 6) whose drawing shows no mark of orientation but where the beads were alternately coloured (one red, one blue) although the model necklace had nine white beads and one green. These productions (Figs. 5 to 8) reveal that the pupils did not simply draw what they saw. They tried to represent, sometimes with difficulties, what they perceived as the main characteristics of the necklaces. By adding up all the representations drawn during the experiment, we obtained 113 drawings. Among these, 82% include some indicators about the orientation of the drawing. It is possible to question whether such an indication was intentional: it could be the result of an attempt to draw the necklace identically. This orientation was given by the presence of the knot in 66% of cases, the end of the necklace (31%), and the use of vertical height (12%) or by another process (such as the association with writing) in 8% of representations. It should be noted that many pupils combined several methods to indicate orientation. Thus, the majority of the drawings show a representation of the knot. The thread that was given to all the pupils already had a tied knot and this probably led them to favour this method to indicate orientation.

How did pupils represent position?

We have already seen some drawings where it is impossible to determine the position of the coloured bead, for example Sanaa's (Fig. 7) or Tesnime's (Fig. 6). Tesnime's drawing shows that it is not necessarily a question of graphic impossibility but a lack of conceptualisation of the pertinent elements. Nevertheless, of the 113 drawings, 83% contained indications of the position (one distinct bead in a set of ten beads). The position of the coloured bead is well understood by the pupils as an important element in passing on information to enable the reproduction of the necklace. Many pupils did not hesitate to use colours other than those of the model necklaces (or beads for the reproduction). This shows that they were quite capable of abstracting information about the position of other material characteristics, such as colour.

To locate the position of the bead, one has to create an orientation on a line and coordinate this orientation with the information of the position of the coloured bead. While 82% of the representations gave pertinent information about orientation,73% gave pertinent information about position and only 59% succeeded in combining these two aspects to produce complete information that we will call 'complete messages' from now on. The mistakes, for pupils who succeeded in drawing orientation and position, were about the use of the coordinate system. The pupils did not take into account the coordinate system implied by the elements of the drawing (knot, end, verticality), for example Melvin (Fig. 9). The messages produced by the pupils were not intended to represent the necklace but to retain relevant information in order to reproduce a necklace identical to the model. Therefore, we crosschecked this analysis with the necklaces produced.

Incomplete messages are rarely efficient Forty-three drawings were produced in phase 4 of self-formulation. Among these, 38% were complete messages and 95% of them (19 drawings out of 20 complete messages) led to correct reproduction of the model necklace. The exception was Kanaë's message (Fig. 5). Among the 62% of the drawings which were incomplete, 50% led nonetheless to success. The pupils had probably memorised the model necklace or else they remembered how to interpret elements that were not visible for us. It is not surprising that a pupil decodes the implicit in his own representation. On the other hand, in phase 5 of communication with others, there was only 1 incomplete message out of 17 which led to success (11%). That was Melis's message, in which there are both drawing and written numbers (Fig. 10). We assume that the numbers contribute to this success, which is explored below.

Complete messages are generally understood

In phase 5 of communication with others, 45 drawings were produced and 62% of them were complete messages. However, of the 28 complete messages, only 23 led to the correct reproduction, that is, 82% efficiency. Thus only five pupils who received a complete message did not manage to interpret the complete representation. The messages of Lucas, Raphaël and Zoubir were not sufficiently well-drawn: there were some ambiguous-coloured beads, crossed-out beads, or an ambiguous knot. Eliot's message was read by Noah, who had not yet produced any message including an orientation or a position. Noah read the position, since he produced a back-to-front necklace (3 instead of 7); see below for a study of Noah's case. Mahmut, who was able to produce a complete drawing from phase 4, did not understand Nawel's message (Fig. 11). However, Nawel's message included both a complete message (vertical, bead coloured green in 8 th position) and numbers. It is possible that Mahmut did not interpret the orientation with the verticality since he represented the knot and the end of the necklace in his drawings. It is also possible that he tried to interpret the mirrored written numbers unsuccessfully. To sum up, the great majority of complete messages were well interpreted by the receivers. When this was not so, it could be due to the receiver not knowing the code used by the sender -Eliot's and Nawel's messages; or to the ambiguity of the drawing -Lucas's, Zoubir's and Raphaël's messages.

In conclusion, what do the complete messages tell us?

In most cases, the complete representation allowed the message to be read, even when this message was far from realistic. This cross-study of the nature of the representations and the success in reading the message, especially in the case of a message to someone else, shows that it is an interpretation which requires implicit knowledge of the visual coordinate system present in the message. It seems to us, therefore, that these situations allowed the pupils to produce and use visual representations of the coordinate system.

Messages with numbers

How did pupils use written numbers?

By adding up all the representations with written numbers in class A (phase 6, evaluation) and in class B (phases 4 and 5, evaluation) we obtained 62 productions. In class B, 58% of the messages with numbers were associated with drawings, while in class A there was no association of numbers with drawings. Some productions used only cardinal numbers or only ordinal numbers and others used both.

Pupils' use of cardinal numbers

Pupils used a combination of numbers, and sometimes drawings and numbers, to express the order of the quantities of uncoloured beads in respect of the necklace: 76% of the messages with written numbers used quantities. Among them, less than 14% were non-relevant representations (e.g. writing of the total number of beads, as in Fig. 6). The relevant representation combined quantities with the order of writing. This is particularly clear in Nawel's drawing (Fig. 12): the necklace is oriented vertically according to the order in which the numbers were read (even if these numbers were mirror written, something that is frequent at this age). The numbers were written beside the quantity of corresponding beads, in the same colour and giving the same information as in the drawing. The sequences written by Melis (Fig. 13) and Aymen (Fig. 14) are a list. The numbers are to be considered in the order in which they were written. It should be noted that these pupils managed to interpret their message, when it was given to them in the afternoon, including Melis whose handwriting, if it were read from left to right, would lead to necklace 8 rather than necklace 3.

Pupils' use of ordinal numbers

The messages using a number to express position were much rarer than the messages using quantity (26% of 62 productions) and, as we will see, they were rarely pertinent: 44% of the writings using some kind of ordinal number were non-functional writings. When the teacher (class A) asked the pupils to write only numbers, with only one pencil, some of them wrote the list of numbers from 1 to 10, for example, without any mark to show which is the coloured bead, such as Loanne (Fig. 15). If we consider the potentially pertinent written messages, some did not show the pupils' intention well. For instance, Yanis (Fig. 16) could not read his own message nor reproduce his model necklace. On the contrary, right from phase 4, Valmir produced writing (Fig. 17) which combined cardinal and ordinal numbers in an original way. Fig. 17 B1, Valmir, phase 4, necklace 6. The digit 6 is red By using colour to distinguish the number of the coloured bead ( 6) and the other beads, Valmir made good use of the ordinal number. He associated it with two written number strings (1, 2, 3, 4, 5 and 1, 2, 3, 4) thereby giving the cardinal number of the uncoloured beads situated before and after the coloured bead. Despite the efficiency of this message, Valmir no longer used numbers after this. Eva (Fig. 18) used a similar representation, in phase 6 (without colour since the teacher only gave one pencil to each pupil). Thus, in the number line, she wrote the number 6 in bold to represent the position of the coloured bead. To sum up, we have only 4 messages which truly give the position of the coloured bead using an ordinal number.

Numerical representations are often correct in the coordinate system

In our corpus, only 12% of the representations with written numbers contain an error with regard to the coordinate system of the necklace. And 6 errors out of 7 were inversions, for example 712 for necklace 3 in Melis's representation (Fig. 13). Success was therefore higher in the marking by numerical representations (89%) than by drawn ones (58%). Perhaps the numerical representations led the pupils to concentrate on what is essential for the reproduction of the necklace, that is the numbers involved and the order of fabrication. The production of a drawn representation, which resembles a drawing, can focus on other criteria (aesthetics and realism with regard to the material).

Numerical representations are often efficient formulations

This analysis is based on data gathered from 43 messages using numbers excepted during evaluation. In class B, during phase 4 (self-formulation), 5 pupils managed to reproduce the necklace out of the 10 pupils who produced messages including numbers. However, this success rate (50%) seemed low to the pupils and led them to think that 'numbers don't work'. This may have prevented the spread of the use of the written number. Then, in phase 5 (communication with others), only 8 pupils used numbers with a success rate of 62.5%.

In class A, where numbers were imposed in phase 6 (self-formulation), all of the 14 pupils who produced a complete message managed to reproduce the model. The complete numerical message was therefore a representation of the necklace which allowed it to be replicated (89% successfully).

Discussion

At the end of this study, we think that pupils did not simply do drawings of the necklace but sought to represent a coordinate system, using numbers in two ways: the usual (cardinal) way and an inventive (using the order of writing) way. These thoughts lead us to return to the initial question of progressive visualisation. Thus we will analyse the evolution of the representations by studying the paths taken by three pupils during the progression of the different phases.

4.1 How does the visualisation of the coordinate system evolve during the learning unit?

The evolution of the representations produced by some pupils shows the relation between the change of representations and the evolution of knowledge about the number line. We will follow two pupils, Walid and Noah, whose clear evolution seems to be revealing about the phases of the process.

A slow positive evolution: the case of Noah and Walid

For many pupils it is not easy to produce a representation (drawn or written) which is accurate with regard to the necklace's coordinate system. For instance, in phase 4, Walid and Noah produced very incomplete drawings (Noah rubbed out some of his). Nevertheless, in both cases, some distinctive elements were represented; Walid in particular showed the end of the necklace, a little less clearly the knot and a coloured bead. Both Noah and Walid failed to reproduce the model necklace. When they were the transmitters of a message in phase 5, there was a greater realism in Noah's drawing (Fig. 21): ten distinguishable beads on a thread, but still no position and no orientation; whereas Walid produced a scribble in which there were no elements of position or orientation (Fig. 22). The receivers of these messages, whether that of Noah (Eliot) or of Walid (Betul), refuse to make a necklace: they explain that they cannot tell where to put the coloured bead. This strong feedback, not only of the material milieu (phase 4) but also from another pupil (phase 5), undoubtedly affected the evolution of the representations of Noah and Walid. Besides this, as receivers, they both received a complete message. Noah produced a necklace 4, back-to-front compared to the model necklace: he understood the position of the coloured bead but not the orientation given by the knot. Walid was successful in producing the expected necklace, showing that he knew how to interpret the knot and thread as an orientation and the coloured bead in position 4 (Figs. 23 and24). The evolution of these two pupils shows the great impact of the different phases on the evolution of the representations. The understanding of the main characteristics of the necklace arises when the pupils have to reproduce a necklace without direct access to the model. The phase 5, which contained a strong feedback for the pupils, has been a major factor leading to the evolution of the representations and to the perception of the elements useful to memorise a position.

Towards the ordinal number: the case of Manon

Only a few pupils used the ordinal number clearly to give the position of the coloured bead, in the last phases of the experiment. We give now some elements of the evolution of one of them, Manon. From the start, Manon understood how to use a drawn representation successfully (Fig. 27 and Fig. 28). In phase 6 (Fig. 29), she explained to the other pupils that she was using the number 4 in order to give the position of the red bead which was in the fourth place. The following numbers expressed the quantities representing the uncoloured beads on each side of the coloured bead. In the evaluation (Fig. 30) she adopted a slightly different code. Manon explained, 'First there are four white beads. It is fifth. And behind there are five white beads.' The ordinal number was thus between two cardinal numbers. Orally, she distinctly used the ordinal (fifth) for the position and the cardinal (five) for the quantity. Nevertheless, she did not go so far as to use the ordinal number alone as a message.

The evolution of the representations used by Manon seems to show that the evolution towards the ordinal number needs a good understanding of the different elements of a coordinate system. But it is not enough. The relation between cardinal number and ordinal must be clarified. The experiment did not permit her to realise that the cardinal numbers are not necessary to memorise the position and that the ordinal number alone is sufficient.

4.2 How do pupils use numbers in order to memorise and represent a position?

In the proposed situations, pupils had to give information allowing the point to be situated on a line in order to succeed in replicating the necklace. The pupils' initial difficulties and their evolution towards more and more efficient representations of the coordinate system confirm our hypothesis. They did not only perceive the necklace visually but they also understood that some characteristics were important where others were not: they perceived with their mind's eye [START_REF] Rivera | Toward a visually-oriented school mathematics curriculum: Research, theory, practice, and issues[END_REF]. Their drawings then played the role of a sign linked at the same time to the object (the necklace) and to the concept (coordinate system), in the epistemological triangle [START_REF] Steinbring | What makes a sign a Mathematical Sign? -An epistemological perspective on mathematical interaction[END_REF]. We recognise here the duality between instrumental valence (the drawings are instruments to memorise information about position) and semiotic valence (they reveal the construction of the concept of a coordinate system and the role that the number can play in it). The relevant characteristics of the number line (origin, orientation and position) emerge from their interactions in the different situations. Some pupils favour orientation first, others favour position, but the difficulty for most of them was to coordinate information concerning orientation and position. Using necklaces seemed particularly pertinent here because they provided the material characteristics of the information and therefore there was a possible milieu for validation (by one-to-one comparison). Other authors (Van den Heuvel-Panhuizen 2008; Whitney 1985) also use necklaces as a possible start to the study of the number line but their necklaces are not oriented (there is no knot), and they were designed for older pupils, to model the empty number line.

The pupils knew how to write numbers, at least up to 10. Some of them used number writing spontaneously, associated with drawings or not, to represent a position on a line. When they were obliged to use numbers, nearly all of them understood that it was possible to use numbers to determine the position. They all showed inventiveness since they mostly produced writing that they had never seen, using three figures in the order of linguistic writing. We can connect this inventiveness to the work of [START_REF] Brizuela | The role of punctuation marks while learning about written numbers[END_REF], who noted that pupils of the same age were capable of giving meaning to punctuation marks in numerical writing. Outside this context, the written productions could be considered ambiguous since 316 could be read three hundred and sixteen, but in the situation these writings are perfectly readable as threeone-six. We can ask ourselves about the meaning pupils give spontaneously to numerical writing and the order of the digits in written numbers. The pupils' use of this writing echoes certain wellknown difficulties met at other levels in school, for example writing 1005 for one hundred and five. That is, using the linguistic order to write numbers, like some numbers of the Roman writing system (CV). The inventiveness the pupils show in using the linguistic order associated with numerical writing is efficient in our situations but could be an obstacle thereafter.

For almost all of the pupils, numerical writing represented quantities. The order in which these quantities should be considered to make the necklace was given by the order of the writing. The pupils had a use which conforms to the usage that they are asked to have in most situations met at school: they used the number as a reminder of the quantity. It should be noted that for those who understood the importance of coordinating orientation and position (as shown in their drawings), ordered numerical writing seemed to have the same function as the complete drawing. The pupils seemed to perceive the mathematical concept of the coordinate system in the context of the given situations. There was already a beginning of abstraction in the sense used by Steinbring (2006, p. 138): 'In processes of developing and learning mathematical knowledge, concrete objects are more and more replaced by mental objects and structures on the side of the reference object.'

The ordinal number was little present in speaking, apart from the use of the term 'first' in association with 'last' and this is consistent with the results obtained by [START_REF] Bruce | One, two, three and counting[END_REF].

The ordinal number was also little present in written form, although it represents one of the best solutions to the problem (a single number unambiguously gives position). It is interesting to note that the few pupils who used the ordinal number associated it with other numbers: either the sequence of numbers from 1 to 10, or the quantities of neutral beads before and after the coloured bead, the ordinal then being the number associated with the coloured bead. These pupils saw quite well that there was a dissymmetry in the role of the different beads. Thus, the coloured bead can be given by its number, but the others cannot. In writing, the absence of a mark between the number which gives quantity and that which gives position could have been a further difficulty: the use of the superscript in 4 th to mean fourth was not taught. Therefore the use of a single number did not allow the pupils to be sure of giving an unambiguous piece of information: ostensives are necessary to accomplish a mathematical task and evoke the underlying concepts.

Conclusion

Our study was motivated by the importance of the number line for the visualisation of elementary properties of operations and numbers. For the number line to be an aid for visualisation, it must be conceived as a coordinate system (origin, orientation and unit). Our study shows that pupils who are going to enter elementary school, despite knowing the number very well as something which represents quantity, fail to understand the number as something to represent position. However, they are capable, in suitable situations, of understanding the characteristic elements of the number line. We think therefore that it should be possible (as the French curriculum demands) to construct young pupils' knowledge to build the number line, through work which would bear specifically on the position-number. However, without effective teaching, it seems to us that it is not surprising that the number line is not of any help in thinking through operations, as [START_REF] Ernest | The number line as a teaching aid[END_REF] and Van den Heuzel-Panhuizen (2008) showed. Indeed, for such a progressive visualisation of the number line to be possible, a construction of the concept of the coordinate system would have undoubtedly to take place simultaneously.

Fig. 2

 2 Fig. 2 A poster in one of the classrooms

Fig. 3

 3 Fig. 3 A1, Eliot, phase 5, necklace 7 3 Fig. 4 B1, Melania, phase 4, necklace 4

Fig. 5

 5 Fig. 5 A2, Kanaë, phase 4, necklace 5 Fig. 6 B2, Tesnime, phase 4, necklace 8. The beads alternate: one red, one blue, one red, one blue, and so on Some drawings show the difficulty of the pupils in producing a representation of the orientation of the necklace, as in Fig. 7. Nevertheless, in most of the drawings without orientation, the position is respected for one of the possible orientations of the necklace, as in Fig. 8.

Fig. 7

 7 Fig. 7 A2, Sanaa, phase 4, necklace 8. Some beads are pink (one big and some small ones) and others violet

Fig. 8

 8 Fig. 8 B4, Melvin, phase 4, necklace 6

Fig. 9

 9 Fig. 9 B4, Melvin, phase 5, necklace 4 3.1.3 Were the messages efficient?

Fig. 10

 10 Fig. 10 B2, Melis, phase 5, necklace 8. The seven beads at the bottom are pink, then one bead is violet, the last two are green

Fig. 11

 11 Fig. 11 B1, Nawel, phase 5, necklace 8. One bead is green; the others are brown or red

Fig. 12

 12 Fig. 12 B1, Nawel, phase 4, necklace 3. One green bead, the digit 1 is green Fig. 13 B2, Melis, phase 4, necklace 3. One green bead, the digit 1 is green Fig. 14 B2, Aymen, phase 4, necklace 6. The digit 1 is green

Fig. 15

 15 Fig. 15 A2, Loanne, phase 6, necklace 7

Fig. 16

 16 Fig. 16 B1, Yanis, phase 4, necklace 5Fig.17B1, Valmir, phase 4, necklace 6. The digit 6 is red By using colour to distinguish the number of the coloured bead (6) and the other beads, Valmir made good use of the ordinal number. He associated it with two written number strings (1, 2, 3, 4, 5 and 1, 2, 3, 4) thereby giving the cardinal number of the uncoloured beads situated before and after the coloured bead. Despite the efficiency of this message, Valmir no longer used numbers after this. Eva (Fig.18) used a similar representation, in phase 6 (without colour since the teacher only gave one pencil to each pupil). Thus, in the number line, she wrote the number 6 in bold to represent the position of the coloured bead.

Fig. 18

 18 Fig. 18 A2, Eva, phase 6, necklace 6

Fig. 19

 19 Fig. 19 A3, Noah, phase 4, necklace 4. Two beads are red, three are white and the rest is erased

Fig. 20

 20 Fig. 20 B3, Walid, phase 4, necklace 7. One bead has red borders

Fig. 21

 21 Fig. 21 A3, Noah, phase 5, necklace 8 Fig. 22 B3, Walid, phase 5, necklace 5

Fig. 23

 23 Fig. 23 A1, Eliot, phase 5, necklace 7 Fig. 24 B3, Betul, phase 5, necklace 4In phase 6, Noah produced '4213' in writing for necklace 7, which could be read successfully from left to right as 4 and 2 neutral beads at the beginning, 1 coloured bead then 3 neutral beads. Several days later, Noah was successful in reproducing the necklace. At the evaluation, Noah and Walid succeeded in making a complete representation of the necklace (Fig.25and Fig.26).

Fig. 25

 25 Fig. 25 A3, Noah, phase 5, necklace 8. The drawing is done upon two folds of papers Fig. 26 B3, Walid, evaluation, necklace 8

Fig. 27

 27 Fig. 27 A4, Manon, phase 4, necklace 8. The coloured bead (pale pink) has been emphasised Fig. 28 A4, Manon, phase 5, necklace 8

http://www.education.gouv.fr/bo/2008/hs3/programme_maternelle.htm. Accessed 10 October 2013.

For this figure, as for those that follow, we use the following key: A1 refers to class A, group 1 (customary mixed groups in each class, not corresponding to an academic level), the pupil's first name, phase number; necklace 7 means that the coloured bead which the pupil had to reproduce was in 7 th position from the knot. Since there is no colour in the printed version, the arrow means that one bead is different from the others.

The only case which is not represented is the presence of the knot with the absence of the representation of the end of the necklace and the verticality.


Amélie Neuville
email: amelie.neuville@eost.u-strasbg.fr

Renaud Toussaint

Jean Schmittbuhl

Effects of the roughness of a fracture on thermal exchange

Keywords: fracture, roughness, lubrication, heat exchange, Soultz-sous-Forêts fracture, rugosité, lubrification, échange de chaleur, Soultz-sous-Forêts

Heat exchange during laminar flow in an open fracture is studied numerically on the basis of the Stokes equation in the limit of hydro-thermal lubrication. We examine the influence of fracture roughness on hydraulic permeability and heat flux through the fracture sides when a cold fluid is injected into a homogeneous hot host rock. Spatial temperature fluctuations inside the fluid are studied assuming the temperature of the rock to be constant and the fracture aperture to be self-affine. An application to the case study at the deep geothermal reservoir of Soultz-sous-Forêts, France, is presented. Finally, a thermal model based on sparse spatial information of the geometrical aperture is successfully proposed to reproduce the response of the fracture.

INTRODUCTION

Modeling of the fluid transport in low permeable crustal rocks is of central importance for many applications [START_REF] Neuman | Trends, prospects and challenges in quantifying flow and transport through fractured rocks[END_REF]. Among them is the monitoring of the geothermal circulation in the project of Soultz-sous-Forêts, France, [START_REF] Bachler | Impact of graben parallel faults on hydrothermal convection -rhine graben case study[END_REF] where the heat exchange especially occurs through open fractures in granite [START_REF] Gérard | The deep EGS (Enhanced geothermal System) project at Soultz-sous-Forêts (Alsace, France)[END_REF]. Numerous hydro-thermal models have already been proposed. For simple geometries some analytical solutions are known: e.g. the cases of parallel plates (e.g. [START_REF] Turcotte | Geodynamics[END_REF]) or flat cylinders [START_REF] Heuer | Mathematical model of a Hot Dry Rock system[END_REF]. More complex models exist as well like the models of three dimensional (3D) networks of fractures reproducing geological observations and possibly completed with stochastical distributions of fractures (e.g. in Soultzsous-Forêts, France, [9; 24] or in Rosemanowes, UK [START_REF] Kolditz | Numerical simulation of flow and heat transfer in fractured cristalline rocks: application to the hot dry rock site in Rosemanowes (U.K.)[END_REF]). Nevertheless, the geometry of each fracture is generally simple. [START_REF] Kolditz | Numerical simulation of flow and heat transfer in fractured cristalline rocks: application to the hot dry rock site in Rosemanowes (U.K.)[END_REF] [START_REF] Kolditz | Numerical simulation of flow and heat transfer in fractured cristalline rocks: application to the hot dry rock site in Rosemanowes (U.K.)[END_REF] have however suspected that differences between heat models and field observations could be due to channeling induced by the fracture roughness or the fracture network. Channeling of the fluid flow owing to fracture roughness has indeed already been experimentally observed (e.g [16; 21;31]).

Here, we limit our study to the fracture scale. The specificity of our hydro-thermal model is to take into account the different scale fluctuations of the fracture morphology. We aim at bringing out the main parameters which control the hydraulic and thermal behavior of a complex rough fracture. The perspective is to propose a small set of effective parameters that could be introduced within simplified elements for an upscaled network model. We first describe our geometrical model of the fracture aperture thanks to self-affine apertures. Then using lubrication approximations, we obtain the bidimensional (2D) pressure and thermal equations when a cold fluid is injected through the fracture in a stationary regime. The temperature within the surrounding rock is supposed to be hot and constant in time and space. The fluid density is also supposed to be constant. We locally assume that the equation for heat flux is identical to the one dimensional (1D) basic one for parallel plates, but with an adjusted characteristic thermal length: we define a thermal aperture which could significantly differ from the local geometrical aperture. We apply our numerical model to the case study at Soultzsous-Forêts and we show for this case an example of computed hydraulic and thermal behavior. Finally we aim at bringing out what is the minimal geometrical information needed to get the dominant behavior of the hydraulic and thermal fields. This last approach is based on spatial Fourier filtering of the geometrical aperture field.

MODELING

Roughness of the fracture aperture

We consider that the mean fracture plane is described by the ( ) z , x ˆ coordinates and the perpendicular direction is y ˆ

(Figure 1) -where the hat notation refers to unit vectors along the (x,y,z) axis. It has been shown that a possible geometrical model of natural rough fractures consists in selfaffine surfaces. A surface described by a function y=f(x,z) is self-affine if it is statistically invariant under the scaling transformation x→ λx, z→ λz and y→λ ζ y, where ζ is called the roughness exponent or Hurst exponent. Such surfaces are therefore statistically invariant upon an isotropic scaling within their mean plane while along the perpendicular direction, the scaling is anisotropic (e.g. [7; 22; 8; 29; 30]. Most fracture surfaces in heterogeneous material exhibit a Hurst exponent equal to ζ =0.8 [5; 29; 30; 27]. Sandstone fractures however show ζ =0.5 [4; 15]. It is important to note that a self-affine surface having a roughness exponent smaller than one is asymptotically flat at large scales [START_REF] Roux | Some physical properties of self-affine rough surfaces[END_REF]. Accordingly, the self-affine topography can be seen as a perturbation of a flat interface. When the lubrication approximation [START_REF] Pinkus | Theory of hydrodynamic Lubrication[END_REF] holds, in particular with smooth enough self-affine perturbations or highly viscous fluid, only the local aperture controls the flow and not the slope of the fracture. Under this assumption, the only required geometrical input is the aperture field (also called the geometrical aperture); there is especially no need to know the geometry of each facing fracture surfaces. The aperture between two uncorrelated self-affine fracture surfaces having the same roughness exponent is as well self-affine [START_REF] Méheust | Scale effects related to flow in rough fractures[END_REF]. Thus we generate the numerical apertures by using self-affine functions.

Several independent self-affine aperture morphologies can be generated with the same roughness exponent chosen equal to ζ = 0.8. They exhibit various morphology patterns according to the chosen seed of the random generator [START_REF] Méheust | Ecoulements dans les fractures ouvertes[END_REF]. The mean geometrical aperture A and the root-mean square deviation σ (RMS) of an aperture a(x,z) are defined as

( ) ( )

Physics of hydraulic flow

The hydraulic flow is obtained under the same hypotheses and solved in the same way as in Méheust & Schmittbuhl, (2001) [START_REF] Méheust | Geometrical heterogeneities and permeability anisotropy of rough fractures[END_REF]. We use finite differences, and the system of linear equations is inverted using an iterative biconjugate gradient method [START_REF] Press | Numerical Recipes[END_REF].

We impose a pressure drop across the system and study the steady state flow of a Newtonian fluid at low Reynolds number, so that the viscous term dominates the inertial one in the Navier-Stokes equation [33; 3]:

D u P 3 ! Δ = ∇ η
, where η is the dynamic viscosity, D u 3 the velocity of the fluid and P is the pressure deviation from the hydrostatic profile (or the hydraulic head equal to the pressure corrected by the gravity effect). To be in the framework of the lubrication approximation [START_REF] Pinkus | Theory of hydrodynamic Lubrication[END_REF], we consider fractures with constant enough apertures together with a small Reynolds number. In doing so the velocity vector of the fluid flow has negligible components normal to the mean fracture plane. We consider that the macroscopic pressure gradient is imposed along x ˆ; z ˆ is therefore perpendicular to the mean flow direction. Accordingly the fluid velocity follows a parabolic law (e.g. [START_REF] Iwai | Fundamental Studies of Fluid Flow through a single fracture[END_REF]) (Figure 2):

) y y )( y y ( P ) z , y , x ( u D 2 1 2 3 2 - - ∇ = η ! [Eq. 1],
where y1 and y2 are the local fracture sides coordinates and 2 ∇ is the gradient operator in the fracture plane. The hydraulic flow through the fracture aperture follows a cubic law:

P a dy ) z , y , x ( u ) z , x ( q a D 2 3 3 12 ∇ - = = ∫ η ! [Eq. 2 ],
and the bidimensional (2D) velocity u ! is defined from the average of the velocity

Physics of thermal exchange

On the basis of a classical description (e.g. [START_REF] Turcotte | Geodynamics[END_REF]), we aim at modeling the fluid temperature when cold water is permanently injected at the inlet of a hot fracture at temperature T0. As the conduction inside the rock is not taken into account (hypothesis of infinite thermal conduction inside the rock), the fracture sides are supposed to be permanently hot at the fixed temperature Tr. This hypothesis should hold for moderate time scales (e.g., months), after the fluid injection stabilized, and before the rock temperature significantly changed. The fluid temperature is controlled by the balance between thermal convection and conduction inside the fluid, which reads [START_REF] Landau | Physique théorique mécanique des fluides[END_REF]:

T T u D Δ = ∇ ⋅ χ 3 !
where χ is the thermal diffusivity of the fluid and T the fluid temperature. We extend the local lubrication approximation by considering that the slopes of the fracture morphology are small enough to limit the conduction only along the y-axis.

. y x .

y . y

We suppose that the leading terms are the conduction along the y-axis and the in-plane convection (since there is no fluid velocity component along y ˆ). Indeed, the off-plane free convection has been shown to be negligible (its magnitude is of the order of km/year [START_REF] Bataillé | Modeling the coupling between free and forced convection in a vertical permeable slot: Implications for the heat production of an Enhanced Geothermal System[END_REF]). So the previous equation reduces to:

z T u x T u y T D z D x ∂ ∂ + ∂ ∂ = ∂ ∂ χ χ 3 3 2 2 ! ! [Eq. 4]
where

D z D x u , u 3 3

! !

are the in-plane components of the fluid velocity. The fluid is supposed to be at rock temperature along the fracture sides, and sufficiently far from the inlet. When we integrate Eq. 4 along the fracture aperture we assume that

z T q x T q z x ∂ ∂ + ∂ ∂ = β is independent of y,
where qx and qz are the in-plane component of q defined in Eq. 2. Accordingly, we find that the temperature solution has a quartic profile (Figure 2) along the fracture aperture ‡ :

r T ) y y )( y y )( y y )( y y ( a T + - - - - - = 2 1 2 1 3 5 5 2 χ β [Eq.5],
where y1 and y2 are the local fracture sides coordinates.

Similarly to what is done for the hydraulic flow, we solve the thermal equation by integrating it along the fracture aperture (following the lubrication approximation extended to the thermal field). In particularly, when doing the balance of the energy fluxes, we express the advected free energy flux as

[ ] ∫ - a D dy T ) z , y , x ( T ) z , y , x ( u c 0 3 ρ .
Accordingly we introduce:

∫ ∫ = a D a D dy ) z , y , x ( u dy ) z , y , x ( T ) z , y , x ( u ) z , x ( T 3 3 [Eq. 6]
which is an average of the temperature profile weighted by the local norm of velocity. We also use the Nusselt number

( ) 0 2 2 = - + ∇ ⋅ r T T Nu a T q χ [Eq. 7],
Boundary conditions are:

0 ) , 0 ( T z T = at the inlet and r x T ) z , l ( T =
at the outlet (with lx large enough). Any boundary condition for the temperature along z=0 or z=lz can be used as the hydraulic flow q ! is null there. ‡ We compared our method to another algorithm based on a Lattice Boltzmann (LB) method, which does not reduce Navier-Stokes to a Stokes equation and does not hypothesize any lubrication approximation, in order to solve the velocity and temperature fields. From those results it appears that the analytical parabolic and quartic approximations (with the proper coefficients) of the respective fields are indeed consistent within a 5% error bar with the LB results.

We discretize this equation by using a first order finite difference scheme and finally get T by inverting the system using a biconjugated gradient method [START_REF] Press | Numerical Recipes[END_REF].

It is finally possible to get the three-dimensional temperature field T anywhere within the fluid by using the previous β expression and the quartic profile (Eq. 5). Figure 3 illustrates an example of temperature field for a given z=z0, T(x,y,z=z0), obtained in that way from the bidimensional field T (shown in Figure 7) computed inside a variable aperture field (see Figure 4). Along any given cut at x=x0, the temperature (represented by the color scale) follows a quartic law. The boundaries between the colors are isotherms and their shape does not follow any particular law.

2 0 / ) z z , y , x ( a = ±
, computed from T shown in Figure 7, in z 0 =700m. Exemple de température T(x,y,z=z 0 ) à travers l'ouverture variable entre

2 0 / ) z z , y , x ( a = ±
, calculé d'après T illustré Figure 7, en z 0 =700m. L'échelle de couleur est linéaire.

Definition of characteristic quantities describing the computed hydraulic and thermal fields

Comparison to modeling without roughness

If we consider a fracture modeled by two parallel plates separated by a constant aperture A, then the gradient of pressure is constant all along the fracture as well as the hydraulic flow which is equal to:

x  l P q x // η 12 3 Δ - =
, where the subscript // is for parallel plate conditions and ΔP = Plx -P0. Under these conditions the analytical solution of Eq. 7 is:

r // r // T R x exp ) T T ( T + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - = 0 [Eq. 8],
where R// is a thermal length describing the distance at which the fluid typically reaches the temperature of the surrounding rock. We have:

Nu APe Nu A l P Nu q A R x // // = Δ - = = χ η χ 24 2 4 [Eq. 9],
where Pe is the Péclet number defined by χ 2 // q Pe = .

Pe expresses the magnitude of the convection with respect to the conduction. For rough fractures, we want to study whether the temperature profiles along x at a coarse grained scale can still be described by Eq. 8 and if so, what is the impact of the fracture roughness on the thermal length R.

Hydraulic aperture

The hydraulic flow can be macroscopically described using the hydraulic aperture H [6; 36], defined as the equivalent parallel plate aperture to get the macroscopic flow x q under the pressure gradient

x l P Δ : 3 1 12 P l q H x x Δ - = η [Eq. 10],
where the quantity under bracket is the spatial average over x and y. Note that the hydraulic aperture H is an effective measure that can be estimated from hydraulic tests whereas the geometrical aperture A is deduced from a direct measurement of the fracture geometry. If H/A is higher than 1, then the fracture is more permeable than parallel plates separated by A. Hydraulic apertures can also be defined locally as:

3 1 12 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ - = P l ) z , x ( q ) z , x ( h x x η [Eq. 11].
Local geometrical and hydraulic apertures are denoted here with small letters while the corresponding macroscopic variables (mean geometrical and hydraulic aperture) are in capital letters.

Thermal aperture

For the thermal aspect, once T is known, we aim at defining a thermal length R like in Eq. 8. To do that, we define T , a z- average temperature which varies only along the forced gradient direction x, and weighted by the 2D fluid velocity x u to fulfill energy conservation:

∫ ∫ = z z l x l x dz ) z , x ( u dz ) z , x ( T ) z , x ( u ) x ( T [Eq. 12].
Then, based on the flat plate temperature solution (Eq. 8), we do a linear fit of

( ) ( ) [ ] r r T T T T ln - - 0
plotted as a function of x, and we use the slope of this fit to get the characteristic thermal length R. This fit is computed over the zone where the numerical precision of the fitted quantities is sufficient (larger than ln(2.10 -6 )). Similarly to the parallel plate case (Eq. 9), the thermal length R can be used to define a thermal aperture Γ:

χ ηNu l P R x 24 4 Γ Δ - =
, which means that a fracture modeled by parallel plates separated by a distance Γ provides the same averaged thermal behavior as the rough fracture of mean geometrical aperture A.

CASE STUDY AT SOULTZ-SOUS-FORETS (FRANCE)

Computation of apertures, hydraulic and thermal fields

Let us consider the GPK3 and GPK2 wells of the deep geothermal drilling near Soultz-sous-Forêts (France), which are separated by a distance of about 600m at roughly 5000 m of depth. From hydraulic tests [START_REF] Sanjuan | Tracer testing of the geothermal heat exchanger at Soultz-sous-Forets (France) between 2000 and[END_REF], it has been shown that the hydraulic connection between both wells is relatively direct and straight. [START_REF] Sausse | Characterization of fracture connectivity and fluid flow pathways derived from geological interpretation and 3D modelling of the deep seated EGS reservoir of Soultz (France)[END_REF] [START_REF] Sausse | Characterization of fracture connectivity and fluid flow pathways derived from geological interpretation and 3D modelling of the deep seated EGS reservoir of Soultz (France)[END_REF] showed that actually a fault is linking GPK3 (at 4775m) to GPK2. This fault zone consists of a large number of clusters of small fractures which probably lead to complex hydraulic streamlines and heat exchanges. We study here a simplified model of this connecting fault zone between the wells using one single rectangular rough fracture. The size of the studied fracture is lx x ly=680x1370 m 2 . Individual fracture apertures are typically of the order of 0.2 mm (Genter & Jung, private communication) while the fracture zone is rather thicker (10 cm) [START_REF] Sausse | Characterization of fracture connectivity and fluid flow pathways derived from geological interpretation and 3D modelling of the deep seated EGS reservoir of Soultz (France)[END_REF]. To account for this variability of the fault zone aperture, we use a probabilistic model of the geometrical aperture but with macroscopic properties that are consistent with in-situ observations: a mean aperture A equal to 3.60 mm and its standard deviation to σ =1.23 mm. Figure 4 shows an example of a self-affine aperture randomly generated with the required parameters.

34). La barre de couleur représente les valeurs d'ouverture en m et les valeurs sur les bords sont les coordonnées spatiales (x,z), aussi en m.

With little knowledge about the pressure conditions along the boundaries of this model, we assume that the two facing sides along x of this rectangular fracture correspond to the inlet and outlet of the model where the pressure is homogeneous, respectively P0 and Plx. In other words, we assume the streamlines to be as straight as possible between both wells. The pressure gradient is chosen as ΔP/lx=-10 -2 bar/m, which corresponds to about 6 bars between the bottom of both wells. The dynamic viscosity is chosen to be 4x10 -4 Pa.s (reference value for pure water at 10 Pa and 100°C from the table in Spurk & Aksel, 2008 [32]). The Reynolds number rescaled with the fracture characteristic dimensions of the fracture [START_REF] Méheust | Geometrical heterogeneities and permeability anisotropy of rough fractures[END_REF] is equal to Re'=(ρux a 2 )/(η.lx)=0.026 and the Péclet number is Pe=3.8x10 4 . Then we solve the hydraulic flow in the fracture domain and obtain the 2D velocity field, u ! defined in Eq. 3. Figure 5 shows the spatial fluctuations of u !

. For information, a parallel plate model separated by the chosen aperture A would predict a homogeneous fluid velocity of 3.6 m/s and a thermal length R//=33.3m. As we see in Figure 5, the 2D velocity field exhibits interesting features: the fluid is rather immobile along the upper and lower borders of the fracture (close to z=0 and z=lx) while most of the fluid flows very quickly through a channel in the middle of the fracture. The macroscopic hydraulic aperture is deduced from the local hydraulic flow estimate (Eq. 10): H= 3.73 mm, which is slightly higher than the mean mechanical aperture A=3.60mm. Therefore, this fracture is more permeable than parallel plates separated by A. In other words the fracture is geometrically thinner than what one would expect from the knowledge of H possibly inverted from an hydraulic test. However, the local hydraulic apertures h (Eq. 11) range from nearly 0 to 5.43 mm (see in Figure 6). From the average estimate of the fluid velocity, we can go back to our approximation of a linear inlet, even if the fracture is not vertical and does not intersect the well on a very long distance. We might estimate this distance from the following argument. The flowrate observed at Soultz is about Q=20L/s. Thus, using a velocity of about u=3.6m/s and a fracture aperture equal to 3.6mm implies that the well crosses such fractures over a cumulated length of about (neglecting the well radius):

m . . . uA Q L 5 1 10 6 3 6 3 10 20 3 3 ≈ × × × = = - - ,
which is effectively much smaller than the boundary size. However we expect the presence of connecting fractures between the well and the fault zone to be sufficiently permeable to define an effective linear inlet of constant effective pressure. All the results presented here are valid under any dimensioning which keeps the ratio lx / R// constant (here equal to 20.5): for instance the results apply for lx=690m, A=10mm and ΔP/lx=-1.7x10 -4 bar/m (using the same fluid parameters). As we see, T is very inhomogeneous and also exhibits channeling. The chosen inlet temperature is T0=60°C, the rock temperature is Tr=200°C and the fluid diffusivity is χ=0.17mm 2 /s (corresponding to water at T=100°, from the table in Taine & al., 2003 [34]). Note that the rock temperature will evolve over time in contrast to the one of our approximations. Indeed the rock thermal diffusivity is about 1mm 2 /s which is larger than the fluid diffusivity (χ=0.17mm 2 /s) but not sufficiently to be fully neglected.

However, T is rather different from a parallel plate solution. Indeed the solution is not invariant in z ˆ. Different temperature profiles function of x are shown in Figure 8. Two end-member types of behavior are plotted: temperature profiles at z=960m (curve iv) and z=700m (curve v). The temperature difference can be larger than 100°C in the inlet region. Even rather far from the inlet, for example at x=200m, the temperature difference can still be of the order of 17°C (200.0°C for z=960m, and 183.4°C for z=700m). ). The temperature field T(x,y,z=700m) is as well shown in Figure 3, where we see how the temperature evolves along the x-axis and across the aperture. Temperature profiles can be compared to the one obtained for a parallel plate model where plates are separated by the aperture A (curve iii) which reads from Eqs. 8 and 12:

// // T T = . for respectively z=960m and z=700m (see Figure 7). Température 1D en degrés en fonction de x. La courbe continue noire (i) est le profil calculé T . La courbe bleue tiretée (ii) est le fit de la courbe (i) avec une fonction exponentielle. La courbe magenta (iii) est la température 1D obtenue en négligeant la perturbation auto-affine (modèle de plaques parallèles). Les courbes (iv) et (v) sont les profiles de température T to the averaged observed temperature T (defined in Eq. 12) (Figure 8, curve i), we see that // T is not representative of the end-member types of behavior. Therefore, we model T by using an adapted parallel law mod T (curve ii) which is an exponential law with a suitable thermal length R:

r r mod T R x x exp ) T T ( T + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - - = 0 0 [Eq. 14],
where R=97m (i.e. 2.9xR//) and x0=-10m. Due to the choice of the minimization to obtain parameters R and x0 (least square applied on the semi-log plot), the beginning of the fit is not accurate. We see that the distance between wells (600m in our case study) is about 6 times larger than R. However owing to channeling the fluid temperature will not necessarily be in full equilibrium with the rock temperature at the out well. The thermal aperture is finally equal to Γ=4.7 mm, which is rather different from the geometrical aperture A=3.6 mm. A larger thermal aperture (compared to the geometrical one) means an inhibited thermalization on average.

Temperature estimation with few parameters

The knowledge of the spatial correlations rather than all the details of the geometrical aperture seems to be a key parameter to evaluate the hydraulic flow and the temperature of the fluid in a rough fracture. Indeed the macroscopic geometrical aperture A brings too little information to characterize the heat exchange at the fracture scale. By contrast, it is impossible in particular for field measurements to know in detail the spatial variability of the local geometrical aperture a. Therefore we propose to characterize the macroscopic geometrical properties with more than a single value, using several parameters describing the largest spatial variations. Numerically, it is possible to obtain them by filtering the fracture aperture field in the Fourier domain. Let us assume that we only have these data available to evaluate the hydraulic flow and heat exchange. Using the same method and the same parameters as previously, we compute the pressure field corresponding to the filtered aperture field. In Figure 10 we show the hydraulic aperture field we obtain. As we see, the high hydraulic aperture channel in the middle of the figure remains, while high frequency variations are removed. These large scale fluctuations of the hydraulic flow, computed from the knowledge of a very limited set of Fourier modes of the geometrical aperture, might be obtained from field measurements. Then the corresponding temperature field shown in Figure 11 is computed. The main features of the thermal field (Figure 7) are still visible: the main channel is at the same position and the values are of the same order of magnitude. Despite small local differences, this substitution model gives a relevant description of what thermally happens. Figure 11: Map of the temperature field obtained using the previous coarse-grained aperture and its corresponding hydraulic results (Figure 9 and Figure 10). The color scale is in °C and it changes exponentially. The spatial coordinates are in m. Carte de température obtenue en utilisant les ouvertures filtrées et les résultats hydrauliques correspondant (Figure 9 and Figure 10). Les coordonnées spatiales sont en m.

CONCLUSIONS

We propose a numerical model to estimate the impact of the fracture roughness on the heat exchange at the fracture scale between a cold fluid and the hot surrounding rock. We assume the flow regime to be permanent and laminar. The numerical model is based on a lubrication approximation for the fluid flow (Reynolds equation). We also introduce a "thermal lubrication" approximation which leads to a quartic profile of the temperature across the aperture. It is obtained by assuming that the in-plane convection is dominant with respect to the in-plane conduction (i.e. high in-plane Péclet number). The lubrication approximation implies also that the off-plane convection is neglected; subsequently the heat conduction initiated by the temperature difference between the rock and the fluid is supposed to be the major off-plane phenomenon.

Our model shows that the roughness of the fracture can be responsible for fluid channeling inside the fracture. In this zone of high convection, the heat exchange is inhibited, i.e. the fluid needs a longer transport distance to reach the rock temperature. Spatial variability of the temperature is characterized on average by a thermal length and a thermal aperture. In this article, we illustrate our modeling with a case study at the geothermal reservoir of Soultz-sous-Forêts, France, with a rough aperture which leads to inhibited thermal exchanges owing to a strong channeling effect. With other compatible aperture fields (not illustrated here) we observe that the roughness may also highly reduce the convection because of hydraulic barriers perpendicular to the main flow. With these later configurations, the heat exchange is locally enhanced with a high conduction compared to convection. In any case, we notice that the temperature distribution is strongly affected by the space variability of the hydraulic flow. From the numerical solutions, we see that the mean geometrical aperture provides too little information to characterize the variability of the fluid flow and fluid temperature. In contrast, the knowledge of the dominant spatial variation of the geometrical aperture field (here obtained by keeping only the largest scale fluctuations using Fourier filtering) provides interesting information about the spatial pattern of the hydraulic and thermal fields. The macroscopic spatial correlation of the aperture is shown to be an important parameter ruling the hydro-thermal behavior. Note that we considered a self-affine model for the aperture roughness, but other types of geometrical descriptions of this roughness (given either by constraints from field measurements, or other kind of geometrical models), could be evenly considered using the type of simulations described here.

  the length and lz the height of the fracture. To keep the boundary geometry of the domain as simple as possible, we do not allow any contact area (i.e. no local aperture equal to zero). This is obtained by considering a large enough aperture average to get strictly positive aperture fields. It has to be noted that our hydro-thermal model can be applied to other geometrical models (i.e. different from a self-affine model) which might be more relevant depending on the geological context.

Figure 1 :

 1 Figure 1: Schematic fracture with variable aperture a(x,z); ρ, c, χ, η are respectively the following fluid properties: density, heat capacity, thermal diffusivity and dynamic viscosity. Schéma de fracture d'ouverture variable a(x,z); ρ, c, χ, η sont les propriétés respectives suivantes du fluide: densité, capacité thermique, diffusivité thermique et viscosité dynamique.

  the fluid to be incompressible, the Reynolds equation is obtained: conditions of this equation, we impose the pressure at the inlet and outlet of the fracture (if x = 0, P = P0 and if x = lx, P = Plx, with P0 > Plx) and consider impermeable sides at z = 0 and z = lz.

Figure 2 :

 2 Figure 2: Local velocity quadratic profile (dotted line) and temperature quartic profile (dashed line) inside a fracture across the aperture at the mesh scale; arbitrary abscissa units. Along the fracture sides, 0 3 = D u and T=T r , and the roots of the polynoms given by equations 1 and 5 are respected. Profil local parabolique de vitesse (ligne pointillée) et profil local quartique de température (ligne tiretée) dans la fracture, à travers l'ouverture. Le long des bords, 0 3 = D u et T=T r , et les racines des polynômes donnés par les équations 1 et 5 sont respectées.

  compares the efficiency of the heat flow along the fracture boundaries:

  heat flow at the fracture aperture scale without convection: polynomial expression of T (in Eq. 5) and the definition of T , Equation 4 leads then to:

Figure 3 :

 3 Figure 3: Example of temperature T(x,y,z=z 0 ) inside a variable aperture between

Figure 4

 4 Figure 4 Aperture field with mean aperture A=3.60 mm and variability of the aperture σ=1.23 mm (σ/A=0.34). The color bar represents the aperture in m, the side units are plane spatial coordinates (x,z), also in m. Champ d'ouverture de moyenne A=3.60mm et de RMS σ=1.23 mm (σ/A=0.34). La barre de couleur représente les valeurs d'ouverture en m et les valeurs sur les bords sont les coordonnées spatiales (x,z), aussi en m.

Figure 5 :

 5 Figure 5: Color map of u, the 2D velocity field norm in m/s. Dark areas correspond to very high velocity while light areas show static fluid. A linear pressure gradient is imposed between the left and right of the fracture. The spatial coordinates are in m. Carte de u, la norme du champ de vitesse 2D en m/s. Les zones sombres correspondent à une forte vitesse tandis que les zones claires indiquent un fluide immobile. Un gradient de pression linéaire est imposé entre les bords gauches et droits de la fracture. Les coordonnées spatiales sont en m.

Figure 6 :

 6 Figure 6: Color map of the local hydraulic aperture in m computed from the variable aperture and velocity field shown in Figure 4 and Figure 5. The spatial coordinates are in m. Carte de l'ouverture hydraulique locale en m, calculée d'après l'ouverture variable (Figure 4) et le champ de vitesses (Figure 5). Les coordonnées spatiales sont en m.

Figure 7 :

 7 Figure 7: Map of the averaged temperature field T in Celsius degrees (°C). The color bar changes exponentially; thus small variations slightly below the temperature rock (200°C) are highly visible. The dotted lines indicate the location of the profiles of temperature ) z , x ( T shown in Figure 8, for z=960m and z=700m. The spatial coordinates are in m. Carte du champ de température moyenné T en degrés Celcius (°C). L'échelle de couleur change exponentiellement; les petites variations en dessous de la température de la roche (200°C) sont donc très visibles. Les lignes en pointillées indiquent la position des profiles

Figure 8 Fluid

 8 Figure 8 Fluid 1D temperature in °C as function of x. The continuous black curve (i) shows the computed temperature T . The blue dashed curve (ii) is the fit of curve (i) with an exponential function. The dot dashed magenta curve (iii) is the fluid 1D temperature by neglecting the self-affinity perturbation (inside flat parallel plates). The curves (iv) and (v) are the profiles of temperature

  et z=700m (voir Figure 7) Following this model (curve iii) the fluid should be at 199.7°C at x=200m. If we compare //

Figure 9 :

 9 Figure 9: Map of the coarse-grained fracture aperture in m, obtained by filtering the aperture (Figure 4) keeping only the zero and 1 st Fourier modes along x and z. The spatial coordinates are in m. Carte de l'ouverture géométrique à faible résolution spatiale, en m, obtenue en filtrant l'ouverture (Figure 4), en ne gardant que la moyenne et les 1 ers modes de Fourier sur x et z. Les coordonnées spatiales sont en m.

Figure 9

 9 Figure9shows the aperture field displayed in Figure4once it has been filtered with the following criterion: only the Fourier coefficients fulfilling:1 2 2 2 2 ≤ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

Figure 10 :

 10 Figure 10: Map of the local hydraulic aperture in m, obtained from the filtered geometrical aperture shown in Figure 9. This figure has to be compared to Figure 6 (same color bar). The spatial coordinates are in m. Carte de l'ouverture hydraulique en m obtenue d'après les ouvertures géométriques filtrées de la Figure 9. Ce champ est comparable à celui de la Figure 6 (même échelle de couleur). Les coordonnées spatiales sont en m.

We thank Albert Genter, Reinhard Jung Marion, Patrick Nami, Marion Schindler, Eirik G. Flekkøy, Stéphane Roux, Jose S. Andrade Jr. and Yves Méheust for fruitful discussions. This work has been supported by the EHDRA project, the REALISE program and the French Norwegian PICS project.