Nilo Nascimento
email: niloon@ehr.ufmg.br

Brigitte Vinçon-Leite

Bernard De Gouvello
email: bernard.degouvello@leesu.enpc.fr

Lorena Gutierrez

Massimiliano Granceri
email: grancerm@leesu.enpc.fr

Talita Silva
email: talita.silva@ehr.ufmg.br

Heloisa Costa
email: heloisasmcosta@gmail.com

Keywords: Green blue infrastructure, land use classification, metropolitan areas, retrofitting green blue infrastructure in built areas, urban water management

Cet article décrit les résultats préliminaires d'un projet de recherche en cours qui a comme objectif d'examiner si les infrastructures vertes et bleues (IVB) peuvent être considérées comme un moyen efficace pour la protection des milieux aquatiques et pour le retour vers un cycle plus naturel de l'eau dans une région métropolitaine. Il vise à évaluer les bénéfices qui peuvent être obtenus par une approche IVB en termes hydrologiques, écologiques et urbanistiques dans le but de protéger les lacs et les réservoirs important à l'échelle métropolitaine. Le projet de recherche porte sur la région métropolitaine de Belo Horizonte (RMBH), au Brésil, une grande agglomération pour laquelle un processus de planification stratégique est en cours d'élaboration depuis 2010. Les résultats obtenus par une analyse préliminaire des caractéristiques et de l'occupation des sols d'un bassin versant ainsi que d'une proposition de mise en oeuvre d'IVB sont présentés et discutés brièvement.

Green blue infrastructure at metropolitan scale: a water sustainability approach in the Metropolitan Region of Belo Horizonte, Brazil Les infrastructures vertes et bleues à l'échelle métropolitaine : une approche de la gestion durable de l'eau dans la région métropolitaine de Belo Horizonte (Brésil)

INTRODUCTION

Green infrastructure is frequently reported as an approach able to decisively contribute to the sustainability of urban water management [START_REF] Maes | More green infrastructure is required to maintain ecosystem services under current trends in land-use change in Europe[END_REF]. The same can be stated for the emerging concept of green blue infrastructure (GBI). Although the understanding of green-blue approaches may vary regionally, it usually refers to the association of urban water management concepts such as water sensitive urban design (WSUD) and low impact development (LID with more regional approaches encompassing blue green corridors providing connectivity between protected areas and promoting biodiversity, among other functions. GBI is, therefore, a new concept that aims to protect the environment, both the "green" (e.g. trees, grass, bushes, etc.…) and the "blue" (e.g. ponds, creeks, lakes, reservoirs, rivers, etc.). In practice, it is an interconnected network of natural and artificial green spaces and water-bodies, within and between urban areas. GBI seems to be perfectly adequate for use in metropolitan regions where a considerable variety of land uses occurs, including highly dense urban as well as low-density peri-urban areas, industrial and agricultural land uses, natural parks and reserves, big reservoirs, etc. In this kind of environment, GBI can play an important role for the protection of water resources and the mitigation of at least part of the environmental impacts due to urbanisation, industrial and agricultural activities, among others.

In a metropolitan region, GBI may play other important roles such as:

• Protecting areas of particular environmental interest at metropolitan scales as it is the case of catchments which are strategic for the production of drinking water; • Reducing natural risks as flooding and land sliding;

• Mitigating environmental impacts of agricultural, industrial and mining activities;

• Mitigating impacts of urbanization on the local climate, as it is the case of heat islands, through a considerable increase on the presence of threes and green spaces in urban areas;

• Promoting connectivity, integrating urban and protect areas within metropolitan regions;

• Restoring and protecting riparian areas, water sources and recharge areas and steep hilltops;

• Promoting biodiversity by improving the ecological state of urban and metropolitan areas and recovering ecological continuity.

Nevertheless, one of the main challenges of adopting GBI approaches relates to the fact that the core of metropolitan areas is frequently composed by already built and densely occupied urban areas (e.g.: [START_REF] Romnée | Methodology for a stormwater sensitive urban watershed design[END_REF][START_REF] Baek | Optimizing low impact development (LID) fir stormwater runoff treatment in urban area, Korea: Experimental and modelling approach[END_REF]. Actually, when it comes to adopting green blue strategies in a metropolitan scale, large cities may be seen as territorial discontinuities in a network of blue green corridors connecting protected areas and other land uses of high environmental interest.

Particularly in the developing world, existing water infrastructure has frequently been built according to conventional approaches. This leads to low adaptability in a changing world where pressures due to climate change and population growth as well as positive socioeconomic evolutions such as a more balanced income distribution followed by demands for quality of life improvements require more flexible and adaptable approaches on urban development and natural resources management.

The present extended abstract describes preliminary results of an on-going research project aiming at investigating whether GBI implementation may be considered as an effective way to protecting water bodies and to recover the natural water cycle in a metropolitan region. It focuses on assessing the benefits that may be provided by a GBI approach in hydrological, ecological and urbanistic terms with the aim of protecting lakes and reservoirs that are relevant at a metropolitan scale. The research project focuses on the Metropolitan Region of Belo Horizonte (RMBH for the acronym in Portuguese), in Brazil, a large conurbation for which a strategic planning process is being developed since 2010.

MATERIAL AND METHODS

In this topic, we briefly describe the methods adopted in this research project and case studies.

Case studies

Belo Horizonte (BH) is an important hub in the mining and steel industry fields for the entire Brazil. Located in the South East region of the country, along with Sao Paulo and Rio de Janeiro, it belongs to the economic and industrial Brazilian triangle. Designed to be the capital of the state of Minas Gerais, the construction of BH began just at the end of the nineteenth century. The metropolitan region of Belo Horizonte (RMBH), actually formed by 34 municipalities, has an area of 9,179 km 2 and gathers 5,800,000 inhabitants whose 2,500,000 are dwellers of its largest city, Belo Horizonte (IBGE 2014). This research is conducted through two case studies in the RMBH involving two catchments and three municipalities: Pampulha catchment (100 km 2 ) and Vargem das Flores catchment (120 km 2 ), both located over two municipalities (Figure 1). The methodological approaches is based on a set of integrated tools including land use assessment through GIS, catchment hydrological modelling and ecological modelling of lakes and reservoir (Figure 2) .A fourth dimension also considered, the social one, focuses on participation processes aiming at building up together a scenario of GBI validation and implementation.

Figure 2. The Methodology scheme.

The GIS analysis, based on a detailed cartographic dataset, is a key aspect in terms of landscape ecology, land use and hydrological analysis and as a tool to be employed in collaboration with stakeholders. Land use (LU) is assessed for the whole catchment territory, classifying it according to categories, knowing that some combinations of these categories may possibly exist. Parameters such as lot size, impervious surface, building typology (houses, apartment buildings) and vegetation cover, road categories and others allow identifying and gathering similar land use patches. Other land characteristics (topography, depth of the water table, etc) make it possible to identify types of GBI that can be implemented for each LU patch.

The GIS assessment phase combined with literature review, field visits and participatory processes lead to the conceptualization of three scenarios that will be the base of the hydrological and ecological modelling processes, as follows:

a) Scenario 0: refers to the current environmental status of the case study catchments. It is the base for the calibration and validation of the hydrological and ecological models.

b) "Business as usual" scenario represents a conventional approach on land use and water management in the catchments as stated in the current municipal land use regulations.

c) GBI scenario corresponds to the adoption in the catchment case studies of the GBI approach.

For the hydrological modelling we selected the XP Solution version of the Stormwater Management Model [START_REF] Rossman | Storm Water Management Model User's Manual[END_REF], the XP-SWMM. The physical and ecological modelling of Lake Pampulha and the Vargem das Flores reservoir are performed according to [START_REF] Granceri | Sustainable management of water resources in metropolitan areas. Which role for Green and Blue Infrastructures in the Brazilian metropolitan region of Belo Horizonte?[END_REF].

GBI objectives, as stated in the introduction are quite large and probably difficult to be achieved all together at local scale. Additionally, it is relevant to highlight that in a large metropolitan area, the diversity of catchment physical characteristics, ecological state and land uses imply that GBI priorities will vary according to local conditions. The present project mainly focuses on the protection of strategic sources for drinking water supply at metropolitan scale, contributing to runoff and diffuse pollution abatement and to recovering, as much as possible, the former natural water cycle. Flood control is also a strategic objective for many of the RMBH catchments, considering that reducing flood risk is an important target in itself and may decisively contribute for GBI acceptance by the population, in general, For each catchment in the RMBH, GBI objectives will be more precisely stated through the diagnoses provided by the scenario 0, previously described, combined with the assessment of catchment characteristics, land use and urban patterns which will allow to devise the more suitable GBI devices to be implemented. The GBI alternatives will then be submitted to the participatory process in order to select those that may be of higher acceptance.

The participatory process with stakeholders will be structured in different phases which will include (i) the selection of participants among developers, civil society representatives, urban planners and other staff members of the city halls, city council representatives…, (ii) the assessment on their former knowledge about GBI and on their perception of the environmental qualities and problems of the case studies and (iii) detailed and iterative discussions on GBI alternatives to face those problems.

RESULTS AND DISCUSSIONS

In this topic, we briefly describe and discuss partial results obtained for the Ressaca catchment, a subcatchment of the Pampulha stream catchment and direct tributary to Lake Pampulha. These results focus on physical and land use assessments and on preliminary proposals for GBI adoption in this area. Figure 3 illustrates the analysis performed in this catchment encompassing topography and hydrography (Fig. 3 In Table 1, one can find a suggested of GBI strategies that would be implemented in this area. Land use is classified in 8 different categories. According to the sub-catchment characteristics, some of these categories may also overlaps (i.e. Vulnerable Social Areas, Hydrographic Vulnerable Area, Pristine Riverbed). This overlapping layers aim to highlight those areas where people and the natural environment cycle should be protected and managed with more sensitivity. This predominant characteristic put the challenge of GBI implementation in the vision of retrofitting.

Table 1. Ressaca creek catchment: proposed GBI (Legend: +++ highly recommended; ++ recommended; + can be employed; -not recommended; / not pertinent)

Next steps will include modelling of current, business as usual and GBI scenarios as well as meeting with stakeholders to assess and debate about viability and acceptability of the suggested GBI interventions.

Figure 1 .

 1 Figure 1. The Pampulha and Vargem das Flores catchments with the respective reservoirs (Source: Granceri et al, 2015).

  a) and urban fabric and open spaces (Fig.3 b). Figure3c is a result of combining the topographic and the urban fabric maps is order to elaborate a preliminary proposal for LU classification and GBI implementation in this area. With the exception of green areas, this catchment is densely and fully occupied by residential and commercial areas. Most of the streams in the catchment are lined and culvert. Therefore, fully applying GBI concepts in this area leads to proposals for stream recovery as well as the adoption of a variety of LID devices, illustrating the challenges of retrofitting this kind of infrastructure in a built environment.

Figure 3 .

 3 Figure 3. Ressaca creek catchment: (a) Topography and hydrography; (b) Urban fabric; (c) LU classification

Acknowledgments The authors would like to thank the research funding agencies FAPEMIG, CNPQ and CAPES for their financial support to the development of this research. The first and the seventh authors are CNPq fellows under the programme CNPq-PQ. M. Granceri, PhD student in a double degree program between Université Paris-Est and Universidade Federal de Minas Gerais was supported by a doctoral grant from Ecole des Ponts ParisTech, Ecole doctorale SIE, in France, and by Capes, in Brazil. This research is supported by XP Solutions Student Leasing Program.


Georgios Bouloukakis

Nikolaos Georgantas

Siddhartha Dutta

Valérie Issarny

Integration of Heterogeneous Services and Things into Choreographies

Keywords: Middleware, Service Bus, Protocol Interoperability, Binding Component

protocols are constantly increasing in the research and industrial landscape. However, the current standardization efforts limit the incorporation of Things as first-class entities into choreographies. To tackle this interoperability barrier, we propose and demonstrate the eVolution Service Bus (VSB), a middleware solution targeted to enable the interaction between Things-based and businessoriented services. Particularly, we demonstrate the incorporation of a service/Thing into the following choreographies: i) temperature sensors interacting with a business-oriented service, and ii) business-oriented services interacting with a route planner service.

Introduction

Service Oriented Architecture (SOA) allows software entities to interact in standard ways. Choreographies [START_REF] Barker | Choreographing web services[END_REF] enable large scale integration of such entities via SOA in a peer-to-peer fashion. Interactions are realized using well known protocols such as SOAP and REST; and each service exposes its functionalities (operations, messages, etc.) by relying on XML-based standards (WSDL/WADL). The existence of standards, facilitates the development of frameworks and the wrapping of systems for interoperability. However, with the advent of paradigms such as the Internet of Things [START_REF] Guinard | Interacting with the SOA-based internet of things: Discovery, query, selection, and on-demand provisioning of web services[END_REF], major tech industry actors have introduced their own APIs to support the deployment of devices (Things). The resulting lack of standards limits the incorporation of Things as first-class entities into choreographies. To remedy this, heterogeneous protocols employed by Things, such as CoAP [START_REF] Shelby | IETF RFC 7252: The Constrained Application Protocol (CoAP)[END_REF] supporting client-service interactions, MQTT [START_REF] Banks | Mqtt version 3.1. 1[END_REF] based on the publish-subscribe interaction paradigm, SemiSpace 1 . offering a lightweight shared tuple space, or Websockets [START_REF] Lubbers | Html5 web sockets: A quantum leap in scalability for the web[END_REF] based on streaming interactions, should be made interoperable.

In this paper, we introduce and demonstrate the eVolution Service Bus (VSB, proposed by the CHOReVOLUTION project2 . Its objective is to seamlessly interconnect services and Things that employ heterogeneous interaction protocols at the middleware level, e.g., SOAP and REST for Web services; CoAP, MQTT and WebSockets for Things. This is based on runtime conversions between such protocols, with respect to their primitives and data type systems, while properly mapping between their semantics. This also includes mapping between the public interfaces of services/Things, regarding their operations and data, from the viewpoint of the middleware. More specifically, middleware-level operations and data are converted, while their business semantics remains transparent to the conversion.

VSB follows the well-known Enterprise Service Bus (ESB) paradigm [START_REF] Chappell | Enterprise Service Bus[END_REF]. In this paradigm, a common intermediate bus protocol is used to facilitate interconnection between multiple heterogeneous middleware protocols: instead of implementing all possible conversions between the protocols, we only need to implement the conversion of each protocol to the common bus protocol, thus considerably reducing the development effort. This conversion is done by a component associated to the service/Thing in question and its middleware, called a Binding Component (BC), as it binds the service/Thing to the service bus.

We introduce a generic architecture for VSB, which relies on the Generic Middleware (GM) abstraction. GM identifies and supports the basic interaction styles found in most middleware protocols: one-way, two-way asynchronous, twoway synchronous, and stream interactions. It also distinguishes between the two roles participating in an interaction, such as: sender/receiver, client/server and consumer/producer. In VSB, we define the GM API primitives, which rely on two main actions: i) a post action for sending a piece of data; and ii) a get action for receiving a piece of data. We then implement the primitives of every concrete middleware protocol by extending the GM API in the Protocol Pool.

In the following section, we present our solution for synthesizing lightweight BCs. Then, we provide details about the VSB implementation followed by the way it is to be used by developers. Finally, we demonstrate the incorporation of services/Things to choreographies.

System Overview

To enable the incorporation of any service/Thing into a concrete choreography, we elaborate a generic architecture for BCs. Particularly, a Generic BC specifies all the aforementioned interaction styles using our GM API. We determine the supported interactions and the middleware concrete protocol of each service/Thing through a generic interface description, which we call GIDL (similar to WADL/WSDL, but general enough to support a variety of protocols). To synthesize a new BC and bridge two concrete protocols (the choreography protocol and the service's Thing's protocol), we use the implementations defined in the Protocol Pool. Implementation. VSB has been implemented using Java 8 and the Maven software project management tool. A GIDL interface is in JSON format, inspired by the Swagger3 specification. BCs are synthesized using the JCodeModel API (code generator). Regarding the Protocol Pool, REST has been implemented using the Restlet API, SOAP using the JAX-WS API, MQTT using the Paho project and CoAP using the Californium framework. The implementation is available as open source 4 . Use by the developer. We present the architecture of the VSB development environment in Fig. 1. VSB can be used with the following ways: i) an application developer incorporates a new service/Thing to the choreography ii) a middleware developer introduces a new protocol. New service/Thing : A service/Thing is incorporated to the choreography through the following steps: i) the VSB Manager requires the service's/Thing's GIDL interface and the choreography protocol (e.g., SOAP); ii) if the service's/Thing's protocol (e.g., SemiSpace) is not included in the Protocol Pool, the VSB Manager cannot create the corresponding BC. In such case, the middleware developer must enrich the Protocol Pool with the service's/Thing's concrete protocol. iii) in case the protocol is supported (e.g., CoAP), the VSB Manager requires the corresponding BC from the BC Generator ; iv) an artifact with the BC is synthesized as described above; v) through this artifact, the BC can be deployed, configured, started or stopped its execution using the BC manager. New protocol : To add a new protocol (e.g., SemiSpace) in our framework, a middleware developer should follow the steps below: i) identify the protocol's primitives with respect to the GM API; ii) implement the above primitives; iii) implement the methods for the deployment (startup, shutdown, etc) of each component role (client, server, etc); and iv) bind the protocol to the Protocol Pool. To facilitate middleware developers introducing new protocols, we intend to develop an eclipse plugin.

Demonstration. We consider two choreographies of services interacting using two different protocols: MQTT and SOAP. Regarding the 1st, developers desire to use a REST service to gather all the information published by temperature sensors (MQTT publishers). Sensor data should be published using the method POST (one-way interaction) to the resource named sensors through the URI: /sensors/{id}/temperature. We map the topic of MQTT publishers to the REST service resource and we demonstrate this communication using a synthesized BC. Concerning the 2nd choreography (SOAP protocol), developers wish to get information from a legacy route planner service, provided by Google. Thus, a SOAP request, requestRoute (origin, dest), should be sent to the REST service which accepts requests to the URI: /directions/{origin}/{dest} using the GET method (two-way synchronous interaction). For this, we synthesize a BC that exposes a WSDL interface and is able to accept the requests sent. SOAP operations are mapped to the REST resources and XML data is converted to JSON for both the requests and responses. The above use cases are demonstrated in the following video: https://youtu.be/UgfM3810RS8

Conclusion

To allow the integration of both services and Things in choreographies, we introduce a lightweight and fully distributed ESB, which allows interconnection among smart devices and services in a peer to peer way. Different protocols can be introduced as VSB's common bus protocol with the same easiness as for integrating support for a new middleware protocol of a service/Thing; additionally, there is no need for relying on a full-fledged ESB platform. Using our approach, legacy services can be integrated into choreographies without the need for editing. VSB BCs are built and deployed as necessary; hence, no BC is needed when a service/Thing employs the same middleware protocol as the one used as the choreography protocol. We intend to extend the functionality of BCs, e.g., under certain aspects of IoT and mobile applications, to support dynamic changes in choreography scenarios.

Fig. 1 .

 1 Fig. 1. VSB architecture

The work is supported by the collaborative research associate team ACHOR (inria.fr/en/associate-team/achor) and the EU-funded H2020 project CHOReVO-LUTION (chorevolution.eu). 1 http://www.semispace.org

http://www.chorevolution.eu

http://swagger.io/

https://tuleap.ow2.org/plugins/git/chorevolution/evolution-service-bus


domain: scco info

text:
Olivia Guest

Nicolas P Rougier

What is computational reproducibility?

Computational modelling is the process by which phenomena found in complex systems are expressed algorithmically. The creation of such simulations is useful because it allows us to test whether our understanding is sophisticated enough to create credible working models of the phenomena we are studying. In neuroscience and cognitive science especially, computational modelling comprises more than just capturing a single phenomenon, it also implements a theory. It gives scientists a method of allowing their ideas to be executed, i.e., for emergent properties to appear when they are implemented and run [START_REF] Mcclelland | The place of modeling in cognitive science[END_REF]. In this context, a model is said to be replicable if experiments within it can be carried out successfully using the original codebase, with the implicit assumption that such a codebase is available.

However, for models to be evaluated it is mandatory to ensure they are reproducible [START_REF] Topalidou | A long journey into reproducible computational neuroscience[END_REF]. That is, that they can be recreated based on their specification -the details deemed important enough to be included in the accompanying article [START_REF] Hinsen | Writing software specifications[END_REF]. Ideally, this should be possible without contacting the authors for advice, and critically, without referring to the original code [START_REF] Cooper | Implementations are not specifications: Specification, replication and experimentation in computational cognitive modeling[END_REF]. If the specification is sufficient to successfully recreate the codebase from scratch, then the model is said to be reproducible. This adds further credence to both the model and its overarching theoretical framework. If not, and the model cannot be recreated, then even if the experiments can be carried out successfully within the original codebase, the model is not reproducible [START_REF] Crook | 20 years of computational neuroscience[END_REF].

How to share computational research?

Access to the original codebase is not always straightforward. There have been few substantial changes within scholarly communication and research dissemination since 1665, when the first academic journals (Le Journal des Sçavans and Philosophical Transactions of the Royal Society) were published. Dissemination of scientific discoveries via publishers continues to consist primarily of static text and figures. However, most research is underpinned by, if not wholly comprised of, code, which is inherently dynamic.

Given code forms the backbone of modern scientific research, it is perhaps unusual that its position within this framework is not clear. For example, it is not straightforward where codebases should be placed: in a footnote (with code assured to be available upon request), in supplementary materials, or in an online repository? Even though more journals are requesting code, as well as raw data, few publisherbacked repositories exist. It is striking that an overwhelming number of journals make no provisions for and offer little guidance on hosting these files or indeed facilitating access to them.

Is it time for progress?

The open source and open science communities proposed solutions to some of the aforementioned problems without publishers' aid nor mediation. Firstly, a set of new innovative software tools (e.g., the binder project) make modelling work more accessible. Secondly, some researchers have taken matters into their own hands and created resources for best practice (e.g., version control: [START_REF] Blischak | A quick introduction to version control with git and github[END_REF][START_REF] Eglen | Towards standard practices for sharing computer code and programs in neuroscience[END_REF][START_REF] Wilson | Software carpentry: lessons learned [version 2; referees: 3 approved[END_REF]. While others lead by example: [START_REF] Ogrean | Frontier fields clusters: Deep chandra observations of the complex merger macs j1149.6+2223[END_REF] published an article with an interactive figure; and the LIGO Open Science Center released extensive amounts of data and code (LIGO Open Science Center: Tutorials, 2016). In the same vein, the ReScience journal encourages the reproduction of modelling work.

Is the scientific community ready to embrace and facilitate changes with respect to: associating articles with original codebases in a transparent way and, more broadly, making sure computational theories are well-specified and coherently implemented?


Alexandre Muller
email: alexandre.muller@esstin.uhp-nancy.fr

Philippe Weber Abdeljabbar
email: philippe.weber@esstin.uhp-nancy.frbesalem@inrets.fr

Ben Salem

Jean Lamour

Process model-based Dynamic Bayesian Networks for Prognostic

The degradation checking of critical components is one of the efficient means to minimize the non expected stops of production system and to reduce its costs. In this purpose, a methodology is proposed to design a prognosis process taking into account the behaviour of environmental events (exogenous inputs). The main contribution is the development of the probabilistic model which is based on the translation of a preliminary process model into Dynamic Bayesian Network (DBN). Specifications are given to assure the coherence both of the translation and the resulted model. These requirements are formalized according to the NIAM/ORM method towards its semantic strength, clarity and stability. An application is proposed on a water heater system.

I. INTRODUCTION

In today's market, companies have to well-maintain their production system to face fierce competition and satisfy customers which requires low-cost products or services of high quality to be delivered quickly [START_REF] Lee | Tele-service engineering in manufacturing: challenges and opportunities[END_REF]. Indeed maintenance plays a critical role in a manufacturing organisation's ability to maintain its competitiveness by contributing to the reduction of work-in-process inventory, the improvement of equipment reliability, and improved productivity, quality and product service [START_REF] Weinstein | Integrating maintenance and production decisions in a hierarchical production planning environment[END_REF]. In that way, maintenance is an essential factor to dynamically optimise the performances by reducing the "inefficiencies" … but only if maintenance evolves from traditional preventive strategies to just-in-time or proactive ones. The prognosis process which is the support of the anticipative role is often considered as the Achilles heel within proactive strategy. Most of the existing prognostic methods are component-oriented and without a real formalization in the modeling. It is now not sufficient to face the expected performances optimization with regard to the complexity of the plant where the degradation and deviation modeling is really difficult due to the economic or stochastic degradation dependencies among components.

In that way, this paper deals with the deployment of prognosis process based on a methodology described in section 2 which combines both a probabilistic approach for degradation mechanism modelling and an event one for dynamical monitoring. The implementation of the methodology is constrained by the DBN formalism which forbids the creation of cycle in the network as explained in section 3A. Thus, additional specifications are required to overcome this limitation. Theses specifications are formalized in section 3B according to the NIAM/ORM method. Then, a case study for a water heater system illustrates this work in section 4 before the brief conclusion drawn in section 5.

II. PROGNOSIS PROCESS DEVELOPMENT

This development is supported by the global methodology, more detailed in [START_REF] Muller | Probabilistic vs. dynamical prognosis process-based emaintenance system[END_REF], and which is structured in three steps:

A. Process & Flow-based approach

The industrial system to be considered is modeled through functional analysis by using process approach [START_REF]2000 Quality management systems-Requirements[END_REF]. One process is broken-up into several sub-processes, and the same mechanism is applied from the subprocesses until the expected level of abstraction is reached (e.g. component). The proposed definition of a process relies on four concepts interpreted as follow: Support which implements the process, the Goal which represents the purpose of the process, the Function which is a desired action and the Behavior which explains how a system does what it is intended to do. The behavior can be identified as: nominal (function realized), degraded (partial loss of the function) and failing (total loss of the function). The process behavior is described by the causal relationships between its inputs flows, the support, and its outputs flows. The concept of Flow-based performance evaluation means that the performance (goal's achievement) of a process can be directly measured on the produced flows and more precisely, on the value of its attributes.

B. Elaboration of the probabilistic network

The second step consists in transforming the previous process model into a probabilistic one which represents the causal relationships and temporal degradations in a unified way. The probabilistic model is developed by means of Bayesian Networks (BN) which offer interesting perspectives on knowledge representation and decision support system [START_REF] Weber | System approach-based Bayesian Network to aid maintenance of manufacturing process[END_REF]. The model goal is to support the mechanisms of inference which can propagate, in the future, the process degradation and determine at every time the impact of this degradation on the whole system. The temporal degradation mechanisms are integrated to the probabilistic model by MULLER A., WEBER P., BEN means of dynamic nodes [START_REF] Weber | Dynamic Bayesian Networks modelling the dependability of systems with degradations and exogenous constraints[END_REF]. Hence, the resulting network is a Dynamic Bayesian Network (DBN) which represents the joint probability distribution over all variables.

C. Design of the monitoring architecture.

The development of the monitoring architecture materialising the event approach is directly deducted from probabilistic model: each observable variable is associated to an indicator. The values of these indicators are refreshed with the current status of the shop floor data and controlled in real time by means of algorithms [START_REF] Iung | Proactive maintenance strategy for harbor crane operation improvement[END_REF]. These events are displayed and stored in an operating knowledge database adapted to the refinement of the probabilistic model's parameters.

III. PROBABILISTIC MODEL DEVELOPMENT

The system modeled by Process & Flow-based approach is converted into a DBN according to several translation rules. The process model provides the information necessary to define the structure of the model. But it isn't enough to create the DBN. Indeed, additional sources of knowledge are required to implement the translation Process/DBN:

A FMEA (Failure Mode and Effects Analysis) provides the potential degradation of each support. In addition, this analysis provides the consequences of degradations on the system. A HAZOP (HAZard and OPerability) study (which consists in analysis the flows and the possible deviations of their attributes) allows, from a deviation on a flow property, to define its link with one degradation or failure of the process and/or system. Knowledge of the physical relationships which links input flows, support and output flows for each process.

An operational database for the parameters learning if there is enough sufficient usage data. Otherwise, expert's knowledge is used to elicit the parameters in terms of subjective estimates.

A. Modeling stages

In fact, the translation process unfolds in three stages: the definition of nodes & states, the creation of the structure and the definition of the network's parameters.

1) Nodes and states:

The dynamic Bayesian network nodes represent the supports and the flows of processes. A support is associated to several nodes which correspond to its potential degradation modes. A flow contains as many nodes as its related attributes. The basic idea is to identify each flow attribute with a "static" variable and each degradation mode with a "dynamic" variable. A support state is defined as a reachable combination of the states of its degradation modes M i . A state of M i is defined either by its physical meaning (e.g. oxidation or corrosion level) or by the consequence of the degraded state on the process performance (e.g. % loss of conductivity). The states of attributes correspond to the process performance allocation fixed by the company.

2) Structure: An edge is synonym of a conditional dependency between the variable that it links (e.g. the impact of the support's degradation on a produced flow). As the processes are linked together by their common flow (an output flow for the upstream process can be an input flow for the downstream process), the model structure consists in specifying a set of edges which represents the causal relationships between the inputs flows, the support, and the outputs flows. By default, each output flow attribute can have for parents every attribute of the input flow and every degradation mode of the related process.

3) Network's parameters: Each node has a Conditional Probability Table (CPT) that quantifies either causal or temporal relationships. Root nodes have only prior probability distributions modeling the statistics of extern input flows. For causal relationship, a conditional probability is affected to each instance of the output flow attributes considering each configuration of input flows and support. For a temporal relationship, the elicitation of conditional probability defines the deterioration (stochastic processes). In our case, these processes are represented by DBN model.

B. Process model specifications

The Dynamic Bayesian Network used is based on an inference algorithm which forbids the cycle creation in the network. As it is presented in 2.B., the main structure of the DBN comes from the structure of the preliminary process model. Therefore, there are two solutions to prevent the creation of cycle.

The first one consists in defining particular restrictions on the process modeling in order to avoid that the designer can define a model with cycle.

Output Flow

Input Flow

Support (k+1) Support (k) DBN

Function

Support

Input Flow Output Flow

Operational data FMEA Physical relations

HAZOP

Translation Rules

Process model

The second solution is based on the abolition of causality loop in the Bayesian network thanks to a set of rules which authorize arc deletion or change direction of some arrows [START_REF] Castillo | Expert Systems and Probabilistic Network Models[END_REF]. However this manipulation leads necessary to the questioning of several dependence relationships stated by the process model. Consequently, this last strategy has been rejected.

The process approach we proposed is based on the concepts introduced by popular methods for functional modeling such as IDEF0 (called as well SADT). These methods describe and decompose the organization of systems in a structured graphical form. Only, the application of a method in a specific domain requires some adaptation, modification of the method to take into account the specificities of the domain. In our case, it implies that the process model must be established in regards of the prognostic finality and by considering the constraints induced by the DBN formalism.

1) The cycle problem: Whatever the situation, the creation of a cycle in the DBN results from a loop which already existed in the process model. So, it is important to understand when and why a cycle creation is possible in the process model. This situation occurs each time that a control loop is required to correct discrepancies between a measured process variable and the desired set point. As it is shown in Fig. 2, the loop "To control -Order -To transform -finished product -To measure -Product report -To control" provide irremediably the cycle "Order -finished product -Product report -Order" during the translation into DBN. 2) The control loop interpretation: The process approach describes the manufacturing system functioning unless time consideration. In the previous example, an input product is transformed in a finished product according to an order which has been established thanks to the result of a measure done on the previous finished product. In fact there are two different finished product flows:

The first one is the "current finished product". This flow represents the prognostic i.e. the expected state of the finished product taking into account to the behavior of the rest of the system The second one is the "previous finished product" which belongs to the previous production cycle. This flow represents the state of the finished product which has been measured by the sensor during the previous cycle product. Thus, the process model contains flows which don't belong to the same temporal slice (the time unit considered here is the manufacturing lead time). These flows are different and so, they can't be represented and interpreted in the same way.

3) Necessary specifications:

To distinguish the flows according to their temporal dimension, the more simple solution is to define a new flow characteristic: -If the flow is related to the current manufacturing lead time, it is associated to the following designation: "flow (k+1)" or "flow" by simplification.

-Else, a flow subjected to a measure process and used to provide information about the previous state of the flow is defined as "flow (k)". In fact, this manipulation consists in adding the temporal dimension not included in process approach.

C. Formalization of the complementary specifications

The specifications of the process approach are formalized according to the NIAM/ORM method. Normally used on data modeling [START_REF] Halpin | Database Modeling with Microsoft Visio for Enterprise Architects[END_REF], NIAM/ORM provides intuitive diagrams which express the information in terms of elementary relationships between the objects of the domain. Its main advantages are the semantic strength, clarity, relevance and stability [START_REF] Halpin | Object-Role Modelling (ORM/NIAM)[END_REF]. Actually, the strong point of NIAM/ORM is the possible validation or checking of the model by verbalization. The method has been implemented for

Product report

Cycle instance to formalize normative knowledge in safety standards [START_REF] Blaise | Formalisation of normative knowledge for safe design[END_REF]. The tool VisioModeler 1 is used to develop the specifications. The formalization of the process approach results in the elaboration of several ORM diagrams based on the general concepts described in section 2. The flow specifications diagrams represented Fig. 3 is the only one included in this paper. Its interpretation reveals that a flow (typed as Mass, Energy or Information) is composed of objects characterized by attributes. In addition, each flow is characterized by its product-cycle temporal specification: either (k+1) or (k). The proposed method is applied to a water heater process depicted in Fig. 5. The goal of the process is to assure a constant water flow rate Qo with a given controlled temperature To. The function of the process "Water heater" is fulfilled by controlling the valve position and by heating the water in a tank. The subprocess is itself broken up into seven sub-processes implemented respectively by a component of the system (Fig. 7). The decomposition of the flows "Water to distribute" and "Water distributed and heated" according to their temporal dimension avoid the loop creation. In this way, the translation process is implemented without problem. It results in the DBN model described in Fig. 8. This model is a unified representation of all the knowledge formalised from the process model towards the dysfunctional analysis FMEA, HAZOP and the operational data. In a strict sense, the DBN model contains one dynamic variable associated to each support i.e. it is assumed that each component is subjected to a single deterioration process. Take, as an example, the Heating resistors model. The dysfunctional analyses lead to identify 4 different states for this component. These states are defined according to their consequences on the process performance i.e. the maximum water temperature reachable by the system. The failing state (state 4) is achieved when To=Ti. The deterioration of the heating resistors is modelled by a Markovian process represented similarly by the DBN and the MC (Markov Chain) depicted in Fig. 6. The transition rates λ are fixed a priori according to the operating knowledge available for similar resistors (it is assumed that these parameters are time-invariant).

N.B.

What is very interesting in the DBN modelling is that the representation of complex system with dependant components is possible [START_REF] Weber | Reliability modelling with dynamic Bayesian networks[END_REF]. In fact, DBN offers a factorised form of the traditional MC which normally suffers from a combinatory explosion of the states number. In this case study, as each component have 2, 3 or 4 states, the MC model of the entire system would contain 1728 states while the DBN model is only composed of 25 nodes.

Prognostic issued from prognosis process

The prognosis process is started in order to evaluate the expected temperature To and water flow rate Qo in the future given that all the components are new at time k=0. A maintenance intervention is simulated at k=1000 time unit. This action is assumed to be perfect i.e. the component is re-initialized in the nominal state. The temporal evolution of the probabilities related to the "Heating resistors" states is depicted in Fig. 9. This figure points out the impact of the maintenance action on the component behaviour. The propagation through the entire DBN model allows emphasizing the effect of this component degradation on the system performances. In this application, the performances of the system are evaluated directly on states of the two flow attributes To and Qo. Consequently, the temporal evolution of these parameters (shown in Fig. 10) can be considered as the forecasting of the system performances. The impact of the maintenance action done on the heating resistors is visible on the temporal evolution on To but not on Qo which is not sensitive to this action because the level of water is controlled independently from the water temperature.

Finally, the concrete result of the prognosis process is a probability that the system reaches its finality at time k= 2000. The prognostic said that there is 21% chance that the Qo will be correct and 33% that To will be correct. This information can be propagated to another process (as an input data) or directly used by a decision process. In regard of the work done in the maintenance and safety academic communities, the approach introduced is an added value on the systemic and dynamic visions of the degradation modeling. The work presented in this paper explains why a methodology supporting prognosis process modelling should be formalized to take into account the characteristics and the constraints induced by its modelling tool. Moreover, the integration of the NIAM/ORM metamodel under MEGA Process2 is in progress in our laboratory in order to develop a tool to assist the designer in developing its process & flow-based model. As a result any user would design a process model of his production system translatable automatically into DBN.

Fig. 1 .

 1 Fig. 1. Translation Process model / dynamic Bayesian network

Fig. 2 .

 2 Fig. 2. Illustration of the cycle problem

Fig. 3 .

 3 Fig. 3. Flow specifications

Fig. 4 .Fig. 5 .

 45 Fig. 4. Illustration of the cycle problemThe implementation of the specification on the previous example is described in Fig.4. Contrary to Fig.2, the DBN which results from the translation of the new process model doesn't contain cycle. The finished product flow is decomposed into two flows: finished product (k) which expresses the state of this flow during the previous cycle (k) and finished product (k+1) which is related to the present cycle (k+1).IV. CASE STUDY

Fig. 6 .

 6 Fig. 6. Heating resistors DBN/MC model The flows are represented by their attributes (Qo, To, H…). The prior probabilities attached to the input flow are fixed by default. For the other flows, the definition of the CPT relies on the elicitation of a conditional probability p to each instance of the output flow attributes, e.g. To, in each configuration of its parents variables (E.E, Heating resistors, Heating order and H level for To). The Software BayesiaLab is used for implementing the inferences mechanisms (http://www.bayesia.com).

Fig. 7 .

 7 Fig. 7. "Water heater" process model

Fig. 9 .

 9 Fig. 9. Temporal degradation of the heating resistors

Fig. 10 .

 10 Fig. 10. Temporal evolution of the expected performances of the system

  SALEM A. Process model-based Dynamic Bayesian Networks for Prognostic. Fourth International Conference on Intelligent Systems Design and Applications (ISDA 2004), Budapest, Hungary, August 26-28, 2004.

http://www.mega.com/us/product/megaprocess/


Jean-Raynald De Dreuzy

Philippe Davy

Olivier Bour

come

Percolation parameter and percolation-threshold estimates for 3D random ellipses with widely-scattered distributions of eccentricity and size

Corresponding author: O. Bour. E-mail: bour@univ-rennes1.fr

ABSTRACT

In fractured materials of very low matrix permeability, fracture connectivity is the firstorder determinant of the occurrence of flow. For systems having a narrow distribution of object sizes (short-range percolation), a first-order percolation criterion is given by the total excluded volume that is almost constant at threshold. In the case of fractured media, recent observations have demonstrated that the fracture-length distribution is extremely large.

Because of this widely-scattered fracture-length distribution, the classical expression of the total excluded volume is no longer scale invariant at the percolation threshold and has no finite limit for infinitely large systems. Thus, the classical estimation method of the percolation threshold established in short-range percolation becomes useless for the connectivity determination of fractured media. In this study, we derive a new expression of the total excluded volume that remains scale invariant at the percolation threshold and that can thus be used as the proper control parameter, called parameter of percolation in percolation theory. We show that the scale-invariant expression of the total excluded volume is the geometrical union normalized by the system volume rather than the summation of the mutual excluded volumes normalized by the system volume. The summation of the mutual excluded volume (classical expression) remains linked to the number of intersections between fractures, whereas the normalized geometrical union of the mutual excluded volume (new expression) can be basically identified to the percolation parameter. Moreover, fluctuations of this percolation parameter at threshold with length and eccentricity distributions remain limited within a range of less than one order magnitude, giving in turn a first rough percolation criterion. We finally show that the scale-dependence of the percolation parameter causes the connectivity of fractured media to increase with scale, meaning especially that the hydraulic properties of fractured media can dramatically change with scale.

hal-00135821, version 1 -9 Mar 2007

In fractured rocks of very low matrix permeability, such as crystalline rocks, fluid flow is often restricted to a few fractures, as shown by hydraulic field experiments [START_REF]National Research Council, Rock Fractures and Fluid Flow[END_REF]. The occurrence of flow and transport in such fractured media is thus conditioned by the fracturenetwork connectivity at the system scale. The connectivity of fracture networks, which may be defined through the probability of connection of a system, has been determined for a variety of two-and three-dimensional elements. The early models using sticks in two dimensions [START_REF] Robinson | [END_REF]3] were followed by plates and polygons in three dimensions [4,5], in order to account for more complex shapes of elements. According to more recent observations of fractured rocks, fractures are not only characterized by a variety of shapes but also by a broad range of lengths. Their length distribution is currently modeled by a power-law such as:

n(l)=α.l -a (1)
where n(l).dl is the number of fractures having a length in the range [l,l+dl], α is the coefficient of proportionality, and a is an exponent varying generally between 1 and 3 [6,7].

The consequences of this wide and non-limited range of fracture lengths on connectivity have been studied in two dimensions on off-lattice stick networks [8] as well as in three dimensions on on-lattice stick networks and on off-lattice networks of orthogonal planes [9]. In this paper, we propose an enhanced analysis of the connectivity properties of multi-scale fracture networks. Fractures are modeled by more realistic elliptic shapes [START_REF]National Research Council, Rock Fractures and Fluid Flow[END_REF] and the connectivity is analyzed for off-lattice networks of ellipses with widely-scattered distributions of eccentricity and length (Fig. 1).

In the scope of percolation theory, there is a single control parameter -the percolation parameter -that statistically measures the state of connection [START_REF] Stauffer | Introduction to percolation theory[END_REF]. In short-range percolation, i.e. when all elements are much smaller than the system size, the control parameter is the density of objects. When systems are made up of elements of different shapes, the total excluded volume gives a better percolation criterion than the density of object and can thus be used as a better parameter of percolation [START_REF] Balberg | [END_REF].

However, because the natural fracture length distribution appears to be a power-law (1), fractures of the size of the system appear with a non-negligible probability and introduce long-range correlations. As a consequence, some systems may be connected at all scales by a single crossing fracture. The probability of including in the system such a fracture increases with scale provided that the length distribution is broad enough, implying that the density of objects at threshold, averaged over a large number of simulations, decreases with the system hal-00135821, version 1 -9 Mar 2007 size. The density is thus no longer suitable for defining alone the connection state of the system. In this study, we show that there still exists a scale-invariant parameter for such longrange percolation problems, which is no longer the average of the mutual excluded volumes but their geometrical union.

I. CLASSICAL EXPRESSION OF THE EXCLUDED VOLUME FOR A WIDELY-SCATTERED LENGTH DISTRIBUTION OF OBJECTS

Because of its key role in this study, we give the detailed definition and expression of the total excluded volume as it has been classically derived firstly in short-range percolation and secondly in long-range percolation. In short-range percolation when all elements are identical, Balberg [START_REF] Balberg | [END_REF] defines the excluded volume of an object (V e ) "as the volume around an object into which the center of another similar object is not allowed to enter if overlapping of the two objects is to be avoided". The total excluded volume is this volume multiplied by the number of elements at threshold N c . For systems made up of non-identical elements, the total excluded volume <V ex > is modified by replacing the mutual excluded volume V e by its average over all possible pairs of elements <V e >. The variations of <V ex > at the percolation threshold with respect to the element shape remains limited so that <V ex >=0.7-2.8 in three dimensions [12,13]. In long-range percolation, the total excluded volume has been classically derived according to the same two-stages method: (i) the calculation of the mutual excluded volume V e for two ellipses of different size and (ii) the average over all possible pairs of ellipses to get the total excluded volume <V ex >.

(i) The mutual excluded volume V e for any two convex overlapping objects can be derived theoretically as it depends only on the mean radii of curvature R 1 and R 2 , on the surface areas A 1 and A 2 and on the volumes of the objects V 1 and V 2 [14]:

π 4 .R A .R A V V V 1 2 2 1 2 1 e + + + = (2)
The expression of the mutual excluded volume of two discs of radii R has been analytically calculated and is: V e =π 2 .R 3 [4]. In the more complex case of ellipses of eccentricities e 1 and e 2 and of major-axis length l 1 and l 2 , we computed the excluded volume numerically following the procedure of [13] and found: (ii) When averaged over all possible pair of ellipses having both a distribution of eccentricities e and major axes l, expression (3) leads to

2 V
> < > < > < >= < l l e V e . . . 2 2

π

. For a network made up of N elements, the total excluded volume is simply the mutual excluded volume multiplied by N [START_REF] Balberg | [END_REF]:

> < > < > < >= < l l e N V ex . . . . 2 2 π (4)
In order to test our algorithms, we have computed <V ex > for systems of volume V made up of discs having all the same radius much smaller than the system size, i.e. when the exponent a in equation ( 1) tends toward infinity. We find a normalized total excluded volume <V ex >/V of 2.2 which is in close agreement with [5] but different from [13,15]. The discrepancy of 20% with Charlaix's result is likely to result from the different way of generating the Poisson distribution of the disc centers.

When applied to systems having widely-scattered distributions of element size such that their power-law length exponent a in expression ( 1) is lower than 4, Equation (4) normalized by the system volume was found to vary over several orders of magnitude at the percolation threshold (Fig. 2). This conclusion confirms the results previously obtained for perpendicular plans networks [9]. Because of its scale-dependence, the normalized total excluded volume cannot be used as a single order parameter for defining the state of connection of the system as it is in short-range percolation theory. However it is still exactly proportional to the density of intersections I (Fig. 3). The number of intersections per ellipse is given by the probability of intersection between two ellipses <V e >/V times the number of ellipses N, which leads to I=<V ex >/V. As a consequence, at threshold, the number of intersections per object is no longer scale invariant as in short-range percolation [START_REF] Baudet | NATO Conference on "Scaling in Disordered Systems[END_REF]; more precisely it decreases by orders of magnitude. It especially implies that the density of interconnected objects at percolation threshold decreases with scale.

II. EXPRESSION OF THE PARAMETER OF PERCOLATION FOR A WIDELY-SCATTERED LENGTH DISTRIBUTION OF OBJECTS

Previous numerical results obtained on orthogonal plans seem to indicate that a scaleinvariant expression is given by the third moment of the length distribution <l 3 > rather than by the multiplication of the first moment by the second moment <l>.<l 2 > [4,9]. We denote this new expression by <V*> and derive it by replacing <l>.<l 2 > by <l 3 > in (4):

hal-00135821, version 1 -9 Mar 2007

> < > < >= < 3 2 . . . * l e N V π (5)
We have checked that, whatever the length distribution, <V*>/V remain scale-invariant for systems made up of discs (Fig. 4a) or of ellipses having either constant eccentricity or a uniform distribution of eccentricity (Fig. 4b). Because of its scale-invariance, V*/V can be used as the parameter of percolation p, proving meanwhile that the state of connection of the system can still be characterized by a single order parameter. We thus propose the following expression for the parameter of percolation p:

3 3 2 . . . L l e N p > < > < = π (6)
where L is the system size.

We note that the two expressions ( 4) and ( 5) normalized by the system volume V, i.e.

<V ex >/V and <V*>/V, lead approximately to the same estimate, as long as the length distribution remains bounded and narrow (i.e. for power-law length distributions such that a>4). On the other hand, when the radius distribution is widely-scattered -i.e. when a<4 in the case of the power-law distribution-, the two formulae lead to very different estimates. It especially means that the right theoretical expression can only be found by studying systems with widely-scattered length distribution.

III. RELATION BETWEEN THE CONCEPT OF EXCLUDED VOLUME AND THE EXPRESSION OF THE PARAMETER OF PERCOLATION

We have seen in the previous section that the classical total excluded volume is no longer connected to the parameter of percolation. In this section, we show that the parameter of percolation p as given by ( 6) has still a sense in terms of mutual excluded volume. We argue that the volume <V*> used in the derivation of the parameter of percolation for a set of N ellipses is the union of the mutual excluded volume of the N ellipses, whereas the total excluded volume <V ex > is the average of the mutual excluded volume over all possible pairs taken in the set formed by the N ellipses.

We calculate the contribution of the N ellipses to the excluded-volume union by ordering them in decreasing size. The contribution of the largest ellipse to the excludedvolume union, along with all other discs, is given precisely by the excluded volume of the two largest elements of sizes l and l-dl, because all other mutual excluded volumes are imbedded in the excluded volume of these two largest elements. From the expression of the mutual excluded volume (3) and when dl<<l, the contribution of the two largest elements is given by hal-00135821, version 1 -9 Mar 2007 π 2 .<e>.l 3 . We then remove the largest element from the system in order to avoid redundancy and calculate the contribution of the remaining ellipses to the excluded-volume union. This process iterated over all elements of decreasing size corresponds to the integration over the length distribution of ellipses, so that the excluded-volume union is equal to

l l n l d ). ( . 3 2 ∫ π ,
which is precisely <V*>. We thus demonstrate that the parameter of percolation is the geometrical union of the mutual excluded volume.

The geometrical derivation of the excluded-volume union is closely related to the theoretical concept of percolation according to which the percolation threshold is reached when the available volume in terms of excluded volume vanishes. Moreover, the excludedvolume union <V*> can be dominated by the largest elements, reflecting the fact that the network can be connected by the largest elements. By averaging the mutual excluded volume over all possible pairs of elements, the total excluded volume <V ex > remove this possibility and thus underestimates the connection probability of the system.

IV. ESTIMATES OF THE PERCOLATION THRESHOLD P c AND GEOLOGICAL IMPLICATIONS

In this section, we use the expression of the percolation parameter p (6) to determine the percolation threshold p c as a function of the eccentricity e and of the power-law length exponent a (Figs. 5a and5b). In the case of short-range percolation-i.e. for power-law length exponents a larger than 4-, the values of p c remain within the range 0.7-2.8 found by Balberg [12] (Fig. 5a). More generally, whatever the length and eccentricity distributions, the values of p c remain restricted to a range of width smaller than one order of magnitude, giving in turn a first-order criterion of percolation for systems made up of widely-scattered length and aspect-ratio distributions.

The variations of p c according to e and a display two common characteristics: (i) an increase when decreasing a at fixed eccentricity e (Fig. 5b) and (ii) a maximum when varying

the eccentricity e at fixed power-law length exponent a (Fig. 5a). As this topic is still under investigations, we report here the most likely origins of these variations. (i) The increase of the percolation threshold p c when the power-law length exponent a decreases (Fig. 5b) may be due to truncation effects. Indeed ellipses truncated by the sides of the system (see Fig. 1) have an internal characteristic length smaller than the original one, and an aspect ratio larger than the generic eccentricity e. Since the probability of occurrence of large truncated fractures increases when a decreases, we expect deviation due to truncation effect to increase when a decreases. (ii) For networks made up of elements having all the same length much smaller hal-00135821, version 1 -9 Mar 2007 than the system size (a=∞), the presence of the maximum may be an effect of the varying aspect ratio. The value of p c (e,a=∞) is thought to increase when the local anisotropy decreases from that of sticks to that of spheres [START_REF] Balberg | [END_REF]. Applying this reasoning to ellipses in 3D, the maximum value of the percolation parameter at threshold is attained when the local anisotropy is at a minimum. The local anisotropy is a measure of the dispersion of the typical lengths of the ellipse in the three dimensions:1,e,0. Since the minimum variance of this triplet is reached when e=0.5, the total excluded volume is expected to reach a maximum value for ellipses of eccentricity around 0.5.

The parameter of percolation p as defined by expression ( 6) can be derived analytically in the case of a power-law length distribution such as [START_REF]National Research Council, Rock Fractures and Fluid Flow[END_REF]. As long as a<4, the parameter of percolation depends on the system size as L 4-a for a given network (i.e. for fixed values of a and fracture density). This case is relevant to natural fractures whose power-length exponents a are in the range 2.5-4 [START_REF] Nicol | [END_REF]. The percolation parameter is thus expected to increase with system size, implying that systems are on average unconnected at small scales, and connected at large scales. The crossover scale above which fracture networks are always well connected is defined by p(L)=p c . This has a strong impact on the hydraulic properties of geological media. If, at low scales, the crystalline rocks permeability is very low because it is controlled by the matrix (non fractured rock), at scales larger than the previously identified crossover scale, there is an interconnected network of fractures, whose permeability can be larger by several orders of magnitude as observed by Clauser [18]. Similar conclusions have been drawn in clayey media [19]. The widely-scattered fracture-length distribution entails an increase of connectivity that in turn may change completely the hydraulic properties of the fractured media.

In conclusion, we have shown that the state of connection of a system made up of elements having a widely-scattered and non-limited length distribution is still determined by a single scale-invariant control parameter that we call, like in short-range percolation, the parameter of percolation. This parameter of percolation has a meaning in terms of excluded volume as it is precisely the geometrical union of the mutual excluded volumes. On the other hand, the total excluded volume -average of the mutual excluded volume over all pair of elements-, previously proposed as a parameter of percolation, decreases by orders of magnitude at threshold but still relates to the number of intersections. Fluctuations of the percolation threshold with the eccentricity and length distributions remain limited to a range smaller than one order of magnitude, giving thus a first rough percolation criterion.

  Percolation parameter and percolation-threshold estimates for 3D random ellipses with widely-scattered distributions of eccentricity and size J.-R. de Dreuzy, P. Davy and O. Bour Géosciences Rennes (UPR 4661) Campus de Beaulieu, 35042 Rennes Cédex, France

  hal-00135821, version 1 -9 Mar 2007This last expression (3) is of the form of (2) and generalizes the expression of the mutual excluded volume of two discs.

Figure captions FIG. 1 :FIG. 2 :FIG. 3 :FIG. 4 :

 captions1234 Figure captions FIG. 1: Infinite clusters at threshold for a=3.25 and (a) e=1 (563 discs) for the network on the left and (b) e=0.1 (4900 ellipses) for the network on the right. The grey scale shading is proportional to the size of the element.

FIG. 5 :

 5 FIG. 5: (a) Percolation threshold p c against the eccentricity e for a=2.5 (open discs), a=3 (upward open triangles), a=3.5 (downward open triangles), a=5 (open squares) and a=∞ (solid squares). (b) Percolation threshold p c against the power-law length exponent a for different values of the eccentricity e. Dashed lines indicate the limits found by Balberg [12] for three-dimensional networks.

Acknowledgements

-This work was supported by the European Community through the project SCALFRAC (Scale Dependence of Groundwater Flow and Contaminant Transport in Fractured Rock, Contract ENV4-CT97-0456) and the GdR ForPro. hal-00135821, version 1 -9 Mar 2007