domain: info math

text:
A P Tchangani
email: ayeley.tchangani@iut-tarbes.fr

Bayesian Networks in Risk Informed Decision-Making

Keywords: Risk management, Decision-Making, Uncertainties, Bayesian Technology

We live in an era where every human entity, from a simple citizen to the head of an entity as large as a country, is forced to make risky decisions. Indeed, risk is omnipresent in human activities or subject to humans today. We can take the example of the COVID 19 pandemic which surprised everyone; of course in this case we are completely helpless given the fact that there is no feedback. Fortunately, there are cases of risk where feedback can be used to formalize and model future events in order to control or manage the risk. This requires adequate modeling, analysis and decision support tools that can be used for decision support purposes for industrial systems managers, policy makers or others. The objective of this chapter is therefore to present the possibilities of risk assessment and management assistance built around Bayesian tools. The Bayesian networks have this faculty to take into account the uncertainty linked as well to the characterization of the elements of the decision problem as to the relations linking these elements. In this chapter, we will therefore show how Bayesian technology can be effectively exploited to construct models for risk evaluation and management in almost all socio-econonomic and even environmental fields. The principle we adopt is one of presentation -illustration in the sense that each Bayesian technology will first be presented followed by an illustration of its use in solving concrete problems. We will try to diversify the problems of illustrations through the socio-economical fields as much as possible in order to demonstrate the possibilities around the Bayesian network technology, which go from prediction, diagnosis, prognosis, to the optimization of assisted decision 1 making.

Introduction

In reality, any human decision is risky in the sense that the outcomes of these decisions are subject to uncertainty of all kinds (random, epistemic, fuzzy, imprecise, etc.). This assertion, confronted with the requirement for transparency, for the scientific quasi-truth of any decision, whether particular or public, on the part of citizens, consumers, or users of the tools made available to them by researchers, calls for a treatment or construction of a decision-making framework that inspires a certain degree of confidence regarding the outcome of the decisions taken. The construction of any framework that should help human beings to make the most appropriate decisions is based on the use of appropriate scientific tools. The choice of these tools is based on the attributes of the decisions to be made and thus of the framework to be built such as: the multiplicity of actors involved in the decision-making process; the multiplicity of objectives targeted by the decisions; the multiplicity of attributes or criteria characterizing the alternatives of decisions in relation to the targeted objectives; and especially the uncertainty that characterizes both the elements of the decision problem (objectives approximately expressed) and the relationships between these elements (attribute-objective relationships that are uncertain or not clearly defined); etc.

In the perspective of integrating uncertainty into decisions of all kinds in order to minimize the risk that the decision-maker runs, researchers are constantly looking for the most efficient tools for modeling and treating uncertainty. To this end, Bayesian networks, since their introduction in 1988 by Judea Pearl [START_REF] Pearl | Probabilistic Reasoning in Intelligent Systems[END_REF], have proven themselves as a tool for modeling and treating uncertainty and thus the possibility of using them to build a decision-making framework where risk is an important dimension. Thus in this chapter we will try to show how the tools built over time and still under construction around Bayesian technology can be effectively exploited in risky decision support processes.

Bayesian technoloogy

A Bayesian Network is a directed acyclic graph (DAG) that represents relationships (in general causal relationships) between elements in a certain knowledge domain; each element represents a random variable associated with a probability table characterizing its parameters, the random variable is also called a node in BN representing. The causal relationship is represented by an adge which is a directed link from one variable called parent to the other variable known as a child. Propagation of local evidence or information through the network relies on Bayes theorem [START_REF] Bayes | An Essay Towards Solving a Problem in the Doctrine of Chances[END_REF], see [START_REF] Pearl | Probabilistic Reasoning in Intelligent Systems[END_REF] [START_REF] Jensen | Lecture Notes on Bayesian Networks and Influence Diagrams[END_REF]. The strength of relationships called parameters of the BN are collected in conditional probability tables for nodes that do have parents and/or a priori probability tables for nodes without parents. Thanks to their capacity to apprehend uncertainty and complex relationships and to take into account other dimensions involved in risk management such as time, the large size of the systems to be managed, and multidimensionality, extensions of BNs have been introduced such as: dynamic Bayesian networks (DBN) to take into account the effect of time [START_REF] Murphy | Dynamic Bayesian Networks: Representation, Inference and Learning[END_REF]; object oriented Bayesian networks (OOBN) to apprehend the large size of the systems ; multi-dimensional Bayesian networks or extended object-oriented Bayesian networks (EOOBN) [START_REF] Liu | Object Oriented Bayesian Network for complex system risk assessment[END_REF] These are the tools that we propose to present in a risk management perspective.

Bayesian Network Meta-Model for Risk management

This part is based on [START_REF] Tchangani | A Model to support Risk Management Decision-Making[END_REF] and its purpose is to establish a meta-model for risk management; the Meta-Model for Risk management will consists in an influence diagram (ID) which is an extension of Bayesian networks to allow evaluating alternative decisions and not only relationships as in BN. They are simple visual representation of a decision problem under uncertainty. Influence diagrams offer an intuitive way to identify and display the essential elements, including decisions, uncertainties, and preferences, and how they influence each other. It shows the dependencies among the variables more clearly than decision tree. The subsequent paragraphs will present all the different variables that will be used by the ultimate influence diagram model in the established framework.

Variables identification To identify and define all the variables to be used in the risk management model (the ultimate influence diagram), we propose to follow the risk management flow chart depicted on Figure 1 that is explained in the following.

First of all, the analyst or decision maker must identify all the risk factors, in fact all the events that may have a negative impact on the performance of the entity by using risk assessment approaches evoked previously. We consider that this process will lead to a finite discrete set E of events.

The second stage consist in assessing the variables defining the state of the system that is identifying all the things (economic, social, technological, institutional, cognitive, cultural conditions, etc.) that influence the vulnerability or resiliency (capacity of the entity to resist or not to an adverse event) of the entity given an undesirable event; we consider that a finite discrete set S has been identified.

In the third stage, one will evaluate the consequences (characterization of negative impact on the entity; complete failure of the system, approximate running of the system, dangerous situation for users, etc.) on the entity if some of the previous events do occur; these consequences depend also on the state of the entity. We consider that a finite discrete set C of consequences is identified.

The fourth stage is dedicated to defining desires by decision maker; desires are things one want to affect through management decisions and actions; they define the criteria on which managements decisions will be based and consist most of the time in putting conditions or constraints over consequences (or aggregate indicators) such as damage cost during earthquake must be low, avoid power supply failure during an earthquake, etc. This process will generally lead to identify the previously mentioned desires set D.

Finally control variables or management actions are defined; these are things that can be realized in order to achieve desires. Examples: respect anti-seismic norms when constructing; educate population with regard appropriate reaction to adopt during an earthquake; build modern facilities, etc. Once again, we consider that a finite discrete set A of actions is available to decision maker.

Relationships identification

To identify all the relationships that may exist between previously defined variables, we propose to use a meta-matrix analysis. The entry (I, J) of such a meta-matrix is a directed graph (see Figure 4) describing the influence of variables of set I on the variables of set J. The meta -matrix of our model is a 5Ö5 matrix of causality, influence, correlation, etc. graphs between previously identified sets E (events), S (entity state), C (consequences), D (desires) and A (actions) as shown by the following Figure 5 where blank entries mean no direct influence of the corresponding sets.

These graphs are presented and explained in the following points.

Events graph (E -Egraph): this graph defines causal relationships that may exist between events; to identify these relationships one must answer questions such as "which event may lead to which one?" For instance an earthquake may cause a tsunami or stones fall in mountains regions.

State graph (S -Sgraph) represents potential influence that may exist between the variables defining the state of the entity. Consequences graph (C -Cgraph) defines relationships between consequences. For instance, human consequences during an earthquake may be decomposed into economic consequences, infrastructures consequences, cultural consequences, etc. To define this graph one can use a bottom up analysis, going from a particular consequence and identifying all the consequences that lead to it.

Desires graph (D -Dgraph) is similar to consequences graph.

Actions graph (A -Agraph): this graph defines how one action may influence another one or how the success of an action may depend on another one.

Events-Consequences graph (E -Cgraph) defines how uncontrollable variables representing events will impact the consequences.

Sate-Consequences graph (S -Cgraph): this graph signifies that the importance of consequences depend on the state of the entity in terms of vulnerability or resiliency.

Consequences-Desires graph (C -Dgraph) is straightforward as desires are defined as conditions or constraints over consequences.

Actions-Events graph (A -Egraph): this graph represents the risk prevention actions as for some events there may exist actions that reduce their likelihood.

Actions-State graph (A-Sgraph) describes how some actions influence the state of the entity; indeed this graph represents the risk mitigation actions effects.

From the meta-matrix, defining a meta-model in terms of meta-Bayesian network is straightforward and is given by Figure 3 to control sequence where we add a constraints graph that has an influence on actions in order to take into account unavoidable resources limitation and other physical and feasibility requirements for actions. Of course when facing a real problem, the meta-model of Figure 3 must be instantiated with knowledgeable variables to obtain a real Bayesian network to support making decision.

Usage of the model

The overall model can be used in two senses: deductive or inductive. In deductive sense, by specifying some local evidence or occurence of an event X in the model, one can propagate it using inference algorithms of Baysian networks to estimate the resultant risk R d (X) on each desire d and then aggregate them (using appropriate aggregator according to pursued objective) to obtain the overall resultant risk R(X). In inductive sense, by giving some requirements concerning acceptable risk for each desire, one can back propagate this information to determine the most appropriate actions to set up. One must notice that this model can be used by portion in the sense that the user can be interested in only a subset of variables and do the propagations processes. Illustrative case study in natural risk management Let us consider (interested reader can find details of this application in [START_REF] Tchangani | A Model to support Risk Management Decision-Making[END_REF]) a problem of developing a model that can be used by authorities of a country or a region (that may face an earthquake events) to support making sound decisions before, during and after an earthquake. Using steps and tools presented previously a Bayesian Meta-Model for Risk Management as sketched by Figure 3 has been obtained arround following variables.

Event: eartnque that can take two status (yes/no) Entity state characterization: it is characterized through four variables:

-Population awareness of the phenomenon: this is a qualitative appreciation of how well the concerned population know that an earthquake can occur in the region; -Infrastructure conditions: a region where infrastructures (buildings, roads, dams, etc.) are built when respecting anti-seismic norms will probably resist better during an earthquake than a region that does not respect these norms; -Emergency systems: this variable describes the quantity and qualities of resources developed by the region authorities to monitor adverse events (network of sensors to detect tectonic movements, geographic information systems, communication systems, etc.) and to efficiently react during an event (emergency equipment, quality and quantity of emergency trained agents, etc.); -Education level : a well educated population will be more receptive to prescribed behaviour during an earthquake than a non educated one

Consequences characterization: Consequences taken into account are economic consequences, social consequences, infrastructures consequences, and environnemental consequences.

-Economic consequences: four variables characterize economic consequences * Loss of jobs and know how : this can be considered to be a mean/long term consequence; * Macro-economic consequences: destruction of industrial infrastructures and others will lead to negative macro-economic consequences; * Relocation (of population) cost: this is an immediate consequence that will be influenced by some variables related to emergency systems and infrastructure conditions * Evacuation (of population) cost: this is also an immediate consequence as the previous one and will be characterized almost in the same way, etc.;

-Social consequences: five variables were identified as social consequences * Lives loss: It will depend on some state of the entity variables and other consequences (consequences on buildings for instance); * Impact on the revenue: economic consequences such as jobs loss will lead to a reduction in revenue of the population that will increase social dependency for instance; * Impact on the social dependency: reduction in the revenue may increase social dependency among the population * Impact on education level : lives loss, revenue reduction and social dependency may lead to a negative impact on the education level; * Post event social consequences: these consequences could consist in changes in cultural habits (migration of people from rural area to towns that will raise some problem such as criminality) or an impact on the structure of the population (reduction of active members of the population), etc.

-Infrastrucure consequences: three main variables describ these consequences * Loss of energy infrastructures: this variable, will be a descendant of variables such as dams, power plants, power lines, etc.; * Consequences on buildings: damages caused to building will depend on the intensity of the earthquake as well as the nature of the buildings (are the buildings constructed when respecting anti-seismic norms or not?) and they will impact on lives loss that will depend on the usage of the building (office, home, industrial building, etc.); * Loss of communication resources: the loss of communication infrastructures such as roads, bridges, airports, ports or electronic communication infrastructures will result in negative socioeconomic consequences;

-Environmental consequences: they are viewed through three variables * Impact on water resources: contamination of rivers and underground water by dangerous products from an exploded chemical or nuclear plants; * Impact on agriculture resources: flooded area may become impracticable for agriculture or crops may be destroyed by fire, floods or a tsunami will ultimately affect economic and social consequences; * Climate consequences: destruction of forests by fire resulted from an earthquake or by a tsunami can have a long term consequences on the climate;

Actions identification: four main actions have been identified that can be used to manage earthkaque risk.

-Prepare population: this action could take different forms: educate the population; inform and train population to have a good behaviour in the case of an earthquake;

-Set up and organize emergency systems: create a network of sensors to pre-detect an earthquake event in order to alert population by different media for instance;

-Prepare after event: subscribe insurance to face after events problems;

-Take legislative decisions: vote laws and norms to be respected when constructing some infrastructures (buildings, dames, power plants, roads, bridges, etc.);

Constraints specification: contraints may be financial, technical, geographical or time of the day.

-Financial constraints: the considered country or region may face serious financial resources limitation in order to undertake actions defined previously;

-Technical constraints: the region or country may lack technical skills to train emergency agents; to construct and organize an efficiency emergency system;

-Geographic constraints: the accessibility of a region that face a natural disaster by emergency resources may be very difficult (mountains region for instance);

-Period of the day: according to the period of the day an earthquake takes place it will be more or less easy to organize emergency systems and rescue people;

Desires formulation: desires may be defined by thresholds on consequences level or constraints satisfaction by some consequences (have low level lives loss, prevent infrastructures collapse, prevent occurrence of hunger, etc.).

Once variables are identified and their relationships sketched, one can consider building the entire model. This can be done by implementing existing Bayesian network learning and inference algorithms [START_REF] Pearl | Probabilistic Reasoning in Intelligent Systems[END_REF] [START_REF] Jensen | Lecture Notes on Bayesian Networks and Influence Diagrams[END_REF] to construct one's own decision support system or one can use existing decision support software based on Bayesian network technology such as that of [START_REF] Expert | [END_REF] and [START_REF] Netica | [END_REF], the principal ones in our knowledge. Figure 4 shows an extract of a model that could be set up to support decision making and planning regarding risk related to an earthquake event in a building; the focused consequence in this extracted model is human damage. Notice that variables in this model can be considered as macro variables that can be decomposed into more elementary variables depending on the level of abstraction decision makers accept. We did not find it necessary to consider specifying modalities of variables nor conditional probability tables as this is just an illustration of what can be obtained as structure of a risk management decision model in a particular case using the developed approach. When necessary, by using a team of experts, specification of these parameters can be done without major difficulties and value nodes can be added so that one can optimize or prioritize actions by simulation.

BN in industrial application

To illustrate the possibility of using Bayesian networks for risk assessment and management in the industrial field, let us consider (an academic problem that author uses in his risk management class) the launch monitoring system for an Ariane rocket. The success of an Ariane launch is based on the transmission of flight parameters from the satellite antennas in Libreville to the graphics station or TMAE console at the Kourou space center (see Figure 5 below).

Suppose we are interested in the possibility that the system may fail (i.e. data are not available in Kourou) after 5 years of operation and that the technical data of the elements of the transmission chain are given in the table in Figure 6. It is assumed that each element can take only two modes OK (the element is functionning) or OFF (the element failed). The quantitative data MTTF (Mean Time To Failure, it is assumed that the time to failure follows exponential laws) of the elements are shown in the following table. Note that these data are just indicative. The Bayesian network model (built using Netica software) of the combinations of the basic events (failure of the elements of the transmission chain) to the dreaded event, which is the absence of flight parameter data at Kourou after 5 years of operation, is given in Figure 7. In terms of risk management, the previous Baysian model of Figure 7 can be used for prediction (determining the probable state of the system given the state of certain basic elements), diagnosis (determining the probable state of the elements given the observed state of the system) and certain indicators that enable decisions to be made regarding intervention on systems (maintenance, replacement, improvement, predictive maintenance, risk analysis, etc.). To this end let us define following parameters and variable:

x i (t) is the inoperability status of component i at time instant t that is binary in this case and given by equation ( 1)

x i (t) = 1 if component C i is OF F at time t 0 if component C i is OK at time t ; (1)
so that the inoperability status of the overall system can be resumed by its inoperability vector x(t) as shown by equation ( 2)

x(t) = x 1 (t) x 2 (t) ... x n (t) . (2)

BAYESIAN TECHNOLOOGY

The ultimate analysis goal is to determine ϕ(x(t)) that represents the inoperability status of the system at time t defined by equation ( 3)

ϕ(x(t)) = 1 if system S is OF F at time t 0 if system S is OK at time t . (3)
The main risk indicator in terms of risk management is the probability of faillure of the system (launch faills) given by equation ( 4)

I S (t) = Pr {ϕ(x(t)) = 1} . (4)
Besides I S (t) as indicator of risk, some parameters can be usefull for risk management such as following ones:

Criticality of a component: a vector x(t) is critical at time t for component C i if it verifies equation (5) ϕ((1 i , x(t))) = 1 and ϕ((0 i , x(t))) = 0 (5)
where (× i , x(t)) means that the component C i is in its status × (0 or 1) at time t. It means that if component C i is inoperable at time instant t then the system will be inoperable and if it is functioning the system will be functioning.

Risk augmentation factor RAF i (t) of component i: relative increase of the probability of inoperability of the system knowing that the basic component C i is inoperable given by equation ( 6)

RAF i (t) = I S/xi(t)=1 (t) -I S (t) I S (t) (6)
where I S/xi(t)=1 is the probability of inoperability of the system given that the component C i is totally inoperable.

Risk diminution factor RDF i (t) of component i: relative decrease of the probability of inoperability of the system knowing that the component C i is operating given by following equation ( 7)

RDF i (t) = I S (t) -I S/xi(t)=0 (t) I S (t) (7)
where I S/xi(t)=0 is the probability of inoperability of the system given that the component C i is totally operable.

Vessely-Fussel or diagnosis importance factor V F i (t) of component i: probability that the basic component C i is inoperable knowing that the system is inoperable defined by equation ( 8)

V F i (t) = Pr {x i (t) = 1/ϕ(x(t)) = 1} (8)
Birnbaum's factor BF i (t) of component i: probability that vector x(t) is critical for component C i at time instant t defined by ( 9)

BF i (t) = I S/xi(t)=1 (t) -I S/xi(t)=0 (t) (9)
Lambert or critical component factor LF i (t) (diagnosis) of component i: probability that the vector x(t) is critical for component C i and the system is inoperable or the probability that the inoperability of component C i is the cause of the inoperability of the system that is given by equation ( 10)

LF i (t) = Ψ i (t) Pr {x i (t) = 1} (10)
where Ψ i (t) is given by ( 11)

Ψ i (t) = I S/xi(t)=1 (t) -I S/xi(t)=0 (t) I S (t) (11)

Dynamic Bayesian Networks (DBN)

Dynamic Bayesian networks [START_REF] Murphy | Dynamic Bayesian Networks: Representation, Inference and Learning[END_REF] derive from an extension of Bayesian networks (see [START_REF] Nielsen | Bayesian networks and decision graphs[END_REF] [18] and references therein) that describe probabilistic relationships between variables of a knowledge domain in order to take into account time behavior. Dynamic Bayesian networks (DBNs) are directed acyclic graphical models of stochastic processes, see [START_REF] Langseth | Fusion of domain knowledge with data for structural learning in object oriented domains[END_REF], and they generalize Hidden Markov Models (HMMs) and Linear Dynamical Systems (LDSs) by representing the hidden and observed state in terms of state variables, which can have complex interdependencies. The graphical structure provides an easy way to specify these conditional interdependencies, and hence to provide a compact parameterization of the model. A dynamic Bayesian networks is completely defined by two components: its structure that is a directed acyclic graph (DAG) where nodes represent variables and directed arcs represents influential relationships between these variables and its parameters that represent conditional probability density (CPD) functions in the case of a continuous variable (the allowed values of the variable belong to a continuous set) or conditional probability table (CPT) in the case of a discrete variable (the allowed values of the variable belong to a discrete set that will be in general a finite set). A dynamic Bayesian network structure consists of an intra slices directed acyclic graph and an inter slices directed graph; slices represent time instants to describe dynamic behavior of the system. Intra slice graph models the instantaneous relationships of nodes (a Bayesian network) and the inter slice graph represents the dynamics of the nodes. Intra slice parameters are conditional probability density functions and/or conditional probability tables of the corresponding Bayesian network and inter slice parameters represent the dynamics of variables on one hand and their relationships with the variables that influence their behavior on the other hand. The advantage of the Bayesian network model over the Markov chain representation for instance, besides the fact that the model is more compact and/or the possibility to consider the influence of the history up to some complexity, is that the transition matrix P can be learnt (estimated) from the expert knowledge and/or experimental data or parameters depending on external dynamic signals.

BN and DBN have been widely used to assist decision making processes in domains such as: dependability, product heath management and maintenance, see for instance [START_REF] Liu | Object Oriented Bayesian Network for complex system risk assessment[END_REF], [START_REF] Tchangani | Modeling Dynamic Reliability using Dynamic Bayesian Networks[END_REF], and references therein. Dynamic Bayesian networks are of particular interest for modeling interactions whose effects are delayed as is usually the case for physical systems. Indeed, in this case the interactions are modeled by the inter-slice relations in the Dynamic Bayesian network while the functional relations form the static Bayesian network at the level of a slice. Of course, the indicators or measures of importance defined in the safety of operation literature such as those in equations ( 4) to [START_REF] Langseth | Parameter learning in objectoriented Bayesian networks[END_REF] are easily calculable in a DBN model by running the model.

Illustrative case study Consider a power supply system for a server consisting of a power supplier (P), a circuit breaker (C) and two parallel circuits (active redundancy) each consisting of a cable (C1/2) and a transformer (T1/2) as shown in Figure 8.

The main objective is to model this system in order to prognosis the possibility of inoperability of the server due to a lack of electrical energy. FTA analysis of this system leads to 6 minimal cut sets: two of order 1 (the number of elements in the cut set) in terms of P (main power supplier) and C (circuit bricker) and four of order 2 consisting in T 1T 2 (transformers), T 1C2 (transformer 1 and cable 2), T 2C1 (transformer 2 and cable 1) and finally C1C2 (cables) and the DBN model of the system is given by Figure 9.

Let us consider the following dynamic scenario: it is admitted that nominal failure rate of all components are considered to be λ 0 i = 10 -3 /T U where TU stands for time unit; but the real failure rate of the two transformers depend on a time varying disturbance signal w(t), with a nominal value w 0 , according to the law given by equation ( 12)

λ Ti (t) = λ Ti (t -1) + β Ti (w(t) -w 0 ), λ Ti (0) = λ 0 i (12)
and the main purpose is to study the influence of this signal on the I S (t) (see equation ( 4)) of the system. Dynamic Bayesian Network model of this system is shown on Figure 9 where S(t) represents the status of whether the server is supplied of power or not. In terms of prognostics, I S (t) is a good indicator to determine for instance the remaining useful life RU L α (t 0 ) at each time instant t 0 at caution or boldness rate α (probability that the system being operational); indeed by setting up the rate α, RU L α (t 0 ) is given by equation ( 13)

RU L α (t 0 ) = I -1 S (1 -α) -t 0 ( 13
)
where I -1 S (1 -α) is the inverse of I S (t) defined by ( 14)

I -1 S (1 -α) = {T : I S (T ) = 1 -α} (14)
the main challenge therefore is to calculate this indicator I S (t). By running the DBN model given by Figure 9, we obtain result shown on Figure 11 for I S (t) where the red curve corresponds to perturbed case whereas the blue one corresponds to nominal behavior of external disturbance signal w(t).

Supplier

From this Figure 11 we can see that at caution or boldness index of α = 10%, the nominal and the disturbed RULs at time instant t 0 = 20 are given by RU L 0.1 w0 (20) ≈ 750 -20 = 730 T U and RU L 0.1 w(t) (20) ≈ 250 -20 = 230 T U. In terms of decision making, given the b ehavior of external signal w(t), decision maker can either maintain the caution or boldness level of 10% and then shorten the mission time from 730 to 230 or maintain the mission time of 730 by diminishing the caution from 10% to almost 0%.

Object Oriented Bayesian Network

Using BN techniques for modelling risk assessment processes becomes increasingly complex when the size of the system increases. For a large-scale system with many interacting elements, constructing a BN to represent its behaviour may be very tricky. Meanwhile, when the size of network grows, the model visibility reduces and the update of parameters becomes burdensome. For this reason, Object Oriented (OO) techniques might be a relevant alternative to reduce the complexity by highlighting a generic pattern representative of the various dimensions of the problem. An Object Oriented Bayesian Network (OOBN) is a direct application of the object paradigm [START_REF] Bangso | Object oriented Bayesian networks a framework for topdown specification of large Bayesian networks and repetitive structures[END_REF] [START_REF] Koller | Object-oriented Bayesian networks[END_REF]. The OOBN takes advantage of classic BN, but introduces the concept of instance nodes. An instance node is an abstraction of a part of a network that can be used as an elementary component to represent the whole structure. The notion of encapsulation allows the transmission of all properties of the network fragment. An OO network can be viewed as a hierarchical description/model of a problem. This makes the modelling easier since the OOBN fragments at different levels of abstraction are more readable. An OOBN model can be built by asking experts' opinions or using learning techniques. Authors of [START_REF] Langseth | Fusion of domain knowledge with data for structural learning in object oriented domains[END_REF] and [START_REF] Ph | Structured probabilistic inference[END_REF] give some insights into OOBN structure learning. The construction of such a model can be facilitated by an ontology representation [START_REF] Liu | Modelling a large scale system for risk assessment[END_REF]. Once the structure of the system is defined, the CPTs (also called parameters) have to be parameterized. The paper [START_REF] Langseth | Parameter learning in objectoriented Bayesian networks[END_REF] extends the parameters learning algorithm to the objects that have the same structure based on OO assumptions. The parameters being identical reduce the number of parameters to be specified or learnt. Modelling a complex system by an OOBN allows not only reducing the design work but also updating calculations. However, most of the existing works dealing with this topic consider that parameters do not change from an object to another, which most often is not a realistic assumption in a real-world problem modelling context. As our main goal is to use OOBN for the representation of a large, repeatable and inherited system, this shortcoming must be remedied. In the following section, we will describe a proposed extension of OOBN paradigm that we refer to as EOOBN.

Extended Object Oriented Bayesian Network

Most of existing works dealing with OOBN consider that parameters do not change from an object to another, which most often is not a realistic assumption in a real world problem modeling context. To overcome this shortcoming the notion of Extended Object Oriented Bayesian Network or multi-dimension BN (MBN) is introduced; the basics of this model are briefly recalled in the following paragraphs, see [START_REF] Liu | Object Oriented Bayesian Network for complex system risk assessment[END_REF] and [START_REF] Liu | Modélisation bayesienne des interactions multidimensionnelles dans un système com[END_REF] for details.

Construction of an EOOBN

In this section, we present an EOOBN approach which introduces much more flexibility, such as the possibility of instantiating parameters according to the considered object and taking into account dynamic behaviour of the system.

Basic definitions

To overcome the limitations of classical OOBN associated with the structure building mentioned in [START_REF] Ph | Structured probabilistic inference[END_REF] and [START_REF] Koller | Object-oriented Bayesian networks[END_REF] and the difficulty to take into account dynamic interactions pointed out in [START_REF] Koller | Object-oriented Bayesian networks[END_REF], we propose here an extended OOBN to ease parameter variation and dynamic consideration process. The original contribution in this article is to develop an EOOBN that simulates the large-scale system with different parameters for objects having the same structure. Moreover, our approach is easy to adapt by collaboration mechanism and to propagate observations in a large system. The main components of an EOOBN are listed as follows.

Class, object, communication channel and virtual node Below, we will consider two levels in EOOBN: definition of a class is made at structure level (i.e. the nodes and their connexions in the object) and the object itself will be instantiated both through the input values and with respect to its parameters at the object level, which are likely to evolve with context or time.

Class. A class (C) is the structure part (S) in a BN independent of the CPT parameters values. It has three kinds of nodes, namely, input nodes, output nodes and internal nodes. Only the input and output nodes are visible from outside the class.

Object. An object (OS,P) in the EOOBN is an instantiation of the corresponding class. There are two parts in an object: the structure (S) which inherits from the class and the parameters (P) which will be defined by experts or learning processes. Instantiating a class consists in setting its parameters through a learning process or an expert opinion.

Communication channel. The input and output build the communication channel for an EOOBN. It supports the process of exchanging information within the object and requires the following conditions to be satisfied:

-Input nodes cannot have parents inside the class;

-Input node is a reference node which is the projection of an output;

-Internal nodes cannot have neither parents nor children outside the class;

-Output nodes cannot have child inside the class.

Figure 12 presents the class which follows the four conditions above for an EOOBN with input nodes V in1 and V in2 , output nodes V out1 and V out2 , and internal nodes V 1 and V 2 .

The class characterizes only the structure of the network. A class can be used only after its instantiation. The EOOBN not only inherits all the advantages of the classical OOBN such as hierarchy or encapsulation, but also offers more flexibility in terms of quantification. The possibility of having different parameter values from an object to another offers the possibility for different objects of the same category to have different parameters. The necessity to follow the behaviour of real-world systems appeals for introducing dynamicity in the model. This can be done by extended classical DBN in OO frame as shown in the following paragraph. Although in [START_REF] Bangso | Object oriented Bayesian networks a framework for topdown specification of large Bayesian networks and repetitive structures[END_REF] a DBN simulation approach is given based on a self-reference node in an object, the dynamic simulation is done by adding a reference link (from dynamic output node to dynamic input node) inside the class; a confusion might appear when trying to add the dynamic part within a large OOBN. To overcome this issue, we introduce the virtual nodes in the EOOBN to simulate the dynamic part.

Virtual node. The virtual node is a communication channel for the class/object. It usually stands for the temporal node.

-The virtual node is either an input node or an output node in the class/object;

-It is associated with the dynamic node in the class/ object as a communication channel with the next time-slice;

-The transition model represents the parameters between the virtual input and dynamic node; Conditional probabilities between a dynamic node and its virtual output are equalled to 1.

After the instantiation, the class in of Figure one obtain a dynamic object that can be used to study the dynamic behaviour of the considered system. The instantiation of a class requires to be able to quantify its parameters. This can be obtained from expert opinions or learning processes. Due to the spatial characteristics of EOOBN such as the repeatable structure of certain objects, the parameter learning maybe different from the classical learning process. The learning problem for EOOBN will be addressed in a future work. The EOOBN flexibility is mainly explained on one hand by the concepts of class and objects allowing, as previously mentioned, parameter variations but also, in another hand, by its aptitude to accept the introduction of system dynamic behaviour. The definition of class not only enables the independence of its constitutive variables but also local independent computation for each object. The EOOBN models proposed in this article will be always structured according to global and object levels. In order for existing inference algorithms to be used for EOOBN assessment, some adjustments are required that will be presented in the next section.

A specific propagation method has been developped for EOOBN. In terms of usage of this model for risk informed decision making, following type of decision can be considered:

Pre-active deci sions: preparing things to prepare the system to face potential threats; the model can be used to identify weak points with regard to identified threats in a large-scale system to search for appropriate defence mechanism.

Reactive decisions: real-time decision-making when the adverse event takes place. Indeed by simulating, the system decision-makers can deduce what will happen at a particular place knowing that something abnormal does happen somewhere else.

Pro-active decisions: the model can be used to look for actions that maybe set up to avoid some catastrophic situations.

Case study

Main developpments of this part are in [START_REF] Liu | Object Oriented Bayesian Network for complex system risk assessment[END_REF]; in this section, the construction method has been applied to build an EOOBN within the framework of a cement manufacture process (Figure 13).

The cement production process is made of seven steps:

1. Obtaining the raw material from a quarry through a blasting machine;

2. Proportioning, blending and grinding;

3. Preheating the material in a tower;

4. Heating in a kiln;

5. Cooling the clinker; 6. Grinding finish;

7. Bagging thanks to a packaging machine.

Every step of the process is supported by operation machines and operators. It is assumed that every step relies on a single machine and that only the qualified intermediate product can be transferred to the next step. Every machine has its own operation conditions. The fabrication process in each step is a complex physical chemistry dynamic reaction, and the use of an EOOBN is relevant for its simulation. It will enable among other things to take into account of the uncertain environment, considering individually each material or human resource without simplifying the system with too many hypothesis while embracing the whole system whatever its size.

Step 1: Formalize the structure of a system (by splitting the system into different classes). According to the previous description, the manufacturing process is split into seven blocks corresponding each to the different steps of Figure 9. A BN-based general machine model is proposed for each asset; the general model will be treated further as a class in the EOOBN.

Step 2: Design the structure of each class with respect to system and without considering the dynamic part. A machine involves seven basic variables which are listed as follows:

-Machine state (MS)

-Upstream state -Operator skill (OS) -Controller -Product quantity
-Amount of quality product (also called downstream product (DP)) The machine yield relies on the capacity of the asset to produce a quantity of product with the required quality. Initial quantity of products (P) likely to be generated will depend on the OS, the MS and the upstream product (UP) input. The final quantity or DP will be a proportion of this initial amount of products (P) respecting the quality criteria that will depend not only on OS and MS, but as well on the aptitude of the system to detect possible defects a control (C).

Step 3: Identify the dynamic node in the class and add the virtual input and output nodes associated with it around the dynamic variable. this case study, the MS variable is considered as dynamic since it loses efficiency over time. The corresponding virtual input and output nodes are then introduced to obtain the dynamic class.

Step 4: Instantiate the class by introducing the parameters corresponding to the object. To characterize the specificity of each machine, different parameters are identified and used to instantiate the objects. At this level, the knowledge comes generally from learning processes or expertise; but in the present case, the machine parameters are arbitrarily decided by the authors. To characterize the specificity of each machine, different parameters are identified and used to instantiate the objects. At this level, the knowledge comes generally from learning processes or expertise; but in the present case, the machine parameters are arbitrarily decided by the authors.

Step 5: Connect the objects through their communication channels. Two communication channels appear in the case study corresponding respectively to the: intermediate product flow and Dynamic of machine state (MS) model.

From the model established (see [START_REF] Liu | Object Oriented Bayesian Network for complex system risk assessment[END_REF] for details), various key performance indicators can be estimated such as the probability associated with the modalities of the variables of machine 7. In particular, the form of degradation over time of the MS combined with a time-dependent productivity slowdown is observed. The selected case study considered here only few modalities for each variable, but obviously the results could be significantly refined by taking into account more discrete variation intervals for each variable. It is also possible to introduce various evidences whose propagation in the network will help reducing uncertainties and ease the decision-making. The EOOBN simulates the system as a whole without any restrictive assumption. Splitting a big system into different homogeneous sub-models helps reducing the calculation and consequently the computation time while considering the unicity of each asset. As a benefit, one can also consider the advantage of dissociating the different dimensions characterizing the behaviour of a system. Here, the model is based on two dimensions representing the spatio-temporal evolution of a process, but the methodology we propose is not limited and can accept an infinite number of dimensions to be handled simultaneously.

Bayesian networks in risky decision-making 0.3.1 Decision-Making by mathematical programming

Any organization, from a single person to a multinational company through small companies, factories, etc., make always decisions (investment, resources acquisition, production planning, etc.). Decisions are made by choosing numerical values of some free variables called decision variables or control variables (production level for a given period, stock level, investment budget level, etc.) in order to optimize (minimize or maximize) some objectives (maximize profit, minimize cost, minimize risk, etc.) when respecting some constraints due to necessarily limitation of available resources (budget limitation, lake of qualified manpower, etc.) and some technical specifications. For instance, a production manager of a given company must establish a production plan by choosing the level of each item to be produced for a given period (a day, a week or a month) using informations given by commercial and/or marketing department ; a project manager must establish the planning of its crews respecting social and labor laws constraints ; a logistic engineer must do the planning of reception dates of the resources to be used in a production process according to some requirements ; and so on. These problems are well formulated when all the parameters are specified by the so-called mathematical programming (also known as optimization) problem in the literature. Basically, a mathematical programming problem consists in the following :

     min / max x f (x) G(x) ≤ 0 x ∈ D ⊆ R n (15)
where x ∈ R n is decision variables vector, G(x) : R n → R m is constraints vector, f (x) : R n → R is the objective function to be optimized and D is the admissible set of decision variables. Here R and R d denote the set of real numbers and the real space vector of dimension d respectively. System (15) can represent a wide range of practical problem in engineering, social sciences, economics, management sciences, etc. According to the nature of functions f , G, the set D, and decision variables x, the problem is known as linear programming, nonlinear programming, integer programming, mix integer programming, convex optimization, quadratic optimization (see [START_REF] Dantzig | Linear programming and extensions[END_REF] [31] and references therein for more details on theory, classification, algorithms, and applications).

The efficient of the solution produced by produced by these algorithms will depend on the quality of the parameters of function G(x) and f (x) that generally uncertain and are provided most of the time by experts from different domains (production managers, specialists of marketing, logistic engineers, etc.). These specialists are not able, most of time, to give precise values of these parameters without relating them to some external random event E. For a production system where x consists in quantities of different items to be produced, the objective vector, f (x), may be provided by commercial or marketing specialists according to their knowledge of the market in terms of: the prize of product P will be high, low, or medium, etc. if the event E is observed. This formulation is useless for the decision maker in terms of solving the problem [START_REF] Murphy | Dynamic Bayesian Networks: Representation, Inference and Learning[END_REF] and so it is necessary to have a model that translate these uncertainties in numerical values. There are two main ways to take into account the problems of uncertainties: fuzzy logic theory approach that deals with imprecise (vagueness) variables or events and probability theory that is concerned with precise events but subjected to randomness. An intelligent framework based based on Byesian network technology can be suitable to estimate these parametrs in order to use them in mathematical programming model, see [START_REF] Tchangani | Decision-Making with Uncertain Data: Bayesian Linear Programming Approach[END_REF] where a similar such framework were constructed arround four units.

Bayesian Networks Unit (BNU) In BNU, one will construct causal relationships between external random events from the set E and parameters of unction f (x) and G(x). The main purpose of this unit is to propagate the information of occurrence of particular events E i ∈ E in the bayesian network in order to modify the belief on parameters. This knowledge in terms of probabilities will be used to select in some manner appropriate parameters to be used in the optimization process. The following paragraph will consider some procedures to choose these parameters.

Parameters Selection Unit (PSU) Once the previous conditional probabilities are computed, the parameters selection unit will use these informations to select particular value of these parameters. This unit may need an expert knowledge in order to derive a meaningful procedure of parameters selection. Let us present some possible procedures for parameters selection.

Optmization Unit (OPU) This unit will use the previous selected parameters to perform the optimization process. As mention in the introduction using existing or own built optimization software

Decision making in uncertain and conflicting environment

State of the world model (SWM) for decision

In many decision-making problems, the reward or outcome of a given decision depends on the state of the world. Classically, such decision-making problems are formulated in terms of state-of-the-world decision-making model [START_REF] Winston | Operations Research: Applications and Algorithms[END_REF].

The principle (roughly) is as follow: there are n possible actions or decisions {a 1 , a 2 , ..., a n } and m possible states of the world {x 1 , x 2 , ..., x m } . If an action a i is decided and the observed state of the world is x j then decision maker receives a reward r ij . Different algorithms are used in the literature to solve this decision-making problems, that is deciding the best action (or optimal action) knowing that the state of the world will be observed a posteriori ; most popular algorithms are: Maximin Reward, Maximax Reward, Minimax Regret and Maximum Expected Value approach [START_REF] Winston | Operations Research: Applications and Algorithms[END_REF] that are presented below.

Maximin Reward: the optimal action a * is selected as

a * = arg max ai min xj r ij . (16)
Maximax Reward: in this case, we have

a * = arg max ai max xj r ij . (17)
Minimax Regret: the regret matrix is defined as follow. Let call i * (j) be the optimal action index if we know that the state of the world is x j , then the regret of taking action a i when the state of the world is x j is given by r i * (j)j -r ij and so the minimax regret algorithm selects as optimal action a * defined by

a * = arg min ai max xj (r i * (j)j -r ij ) . (18)
Maximum Expected Value: here it is supposed that each state x has probability p(x) of occurrence and the optimal action is selected as

a * = arg max ai   m j=1 p(x j )r ij   . (19)
Previous algorithms are easy to implement and computationally efficient ; their main drawback is that they give an off-line (open loop) type solution. They do not consider a possible evidence on environmental events [START_REF] Tchangani | Modeling for Reactive Control and Decision Making in Uncertain Environment In: Control and Learning in Robotic Systems[END_REF] in order to have an adaptive (reactive) behavior during decision process. If the state of the world were known in advance, the decision would be obvious. In most practical cases decision makers try to estimate the state of the world by using some sensors (to be understood in large context including humans) which information are used to estimate the most probable state of the world before making decision leading to an influence diagram see [START_REF] Tchangani | Modeling for Reactive Control and Decision Making in Uncertain Environment In: Control and Learning in Robotic Systems[END_REF].

BN approach to deal with SWM

The concern regarding the classical approaches or algorithms mentioned above to solve decision problems in front of the state of world model is the absence of an intelligence system that could allow by an observation or a prediction to improve the quality of the decision. In this perspective, Bayesian networks can be used to build an intelligence system in the sense of the possibility of predicting the most probable state of nature which makes it possible to take a decision on the basis of other criteria than the average of an indicator which is a very strong measure of uncertainty, see [START_REF] Savage | The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty[END_REF]. Thus we propose the model BN Model to predicte actual state of the world is a BN (Intelligent framework) which allows to estimate the most probable state of nature that allow to choose the final decision reflecting the confidence that the decision-maker that is summarized on Figure 14.

Conflicting decision making: game theory

Game theory is a branch of mathematical analysis developed to study decision making in conflict situations. Such a situation exists when two or more decision makers who have different objectives act on the same system or share the same resources. Game theory provides a mathematical process for selecting an optimum strategy (that is, an optimum decision or a sequence of decisions) in the face of an opponent who has a strategy of his own. In traditional game theory one usually makes following assumptions.

Each decision maker (agent, player) has available to him well-specified choices (decisions) or sequences of choices.

Every possible combination of choices available to the players leads to a well-defined state.

A specified outcome(reward, payoff or consequences) for each player is associated with each state. In the case of zero-sum game, payoffs are numerical values and sum of payoffs to all players is zero in each state.

Each decision maker has perfect knowledge of the game and of his opposition ; that is, he knows in full detail the rules of the game as well as payoffs and strategies of all other players. All decision makers are (substantive) rational ; that is, each player, given two alternatives, will select the one that yields him the greater payoff.

The last two assumptions, in particular, restrict the application of game theory in real-world conflicting situations. Nonetheless, game theory has provided a means for analyzing many problems of interest in economics, management science, and other fields. Substantive rationality suppose that the decision makers seek maximum utility and have all cognitive resources to this end and this is not the case in majority of cases. Noncooperative games are those games where players can not cooperate. One solution concept in the case of non constant sum game is the so-called Nash equilibrium, named after John Nash, that is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. For games in which players randomize (mixed strategies), the expected or average payoff must be at least as large as that obtainable by any other strategy. In a constant sum game (noncooperative game), the sum of all players' payoffs is the same for any outcome. Hence, a gain for one participant

Learning process

Evaluation process

Inference process

Post observation

Figure 14: BN Intelligent Model of the state of the world decision-making model is always at the expense of another, such as in most sporting events. Given the conflicting interests, the equilibrium of such games is often in mixed strategies. Since payoffs can always be normalized, constant sum games may be represented as (and are equivalent to) zero sum game in which the sum of all players' payoffs is always zero. Maximin optimization in this case is a rational solution concept. For simplicity let us consider a two persons, players, decision makers or agents, A and B, noncooperative game that can model practical situation such as two armies in military operations, a government fighting against a terrorist group. Let suppose that A and B dispose of finite set strategies A and B respectively. To any couple (a, b) where a ∈ A and b ∈ B is associated a gain or reward R(a, b) for player A that is loss for player B. A main drawback of this model is the absence of interaction between players because of perfect information assumption that is difficult to be realized in practice. For instance on can not reasonably talk about perfect information when two armies are fighting each other or when a state is fighting a terrorist group or in concurrence on a market. In fact, in the point of view of given player, let say A, the behavior of other player(s) B can be considered as a state and so by using some "sensors" including spying for instance, player A can guest possibly behavior of the opponent B.

Model for Reactive Decision Making

Presentation

The purpose of this section is to construct an integrated framework for making decision with state of world decision model and noncooperative game model. Though this decision support system is to be used in dynamic environment, the constraint of real time and emergency conduce to construct a system that reacts in one step ; the sequentiallity is just induced by information arrival process. This approximation is acceptable in our point of view because we believe that in emergency situation, decisions are made to be just "good enough", avoiding catastrophic situation for instance, instead of looking for a long term performance as is stated in introductory section.

We consider building a decision support system in the spirit of a decision maker A, with actions or decisions set A, that is acting in an uncertain environment determined by a random variable X which takes its values called states in a set X . Moreover there are other decision makers whose interests are in conflict with A's interests ; in the spirit of A their behavior is summarized in a random variable B taking its values in a set B. This model of decision making is known in the literature as cybernetic system [START_REF] Moïsséev | Problèmes mathématiques d'analyse des systèmes[END_REF]. Let suppose that the goal of decision maker A is to act in order to optimize (to be understood in a broad sense) a function of form O(a, b, x), where a ∈ A, b ∈ B and x ∈ X , that takes its values in a set O (to each outcome o ∈ O and each decision a ∈ A one may associate a numerical value R(a, o) that permit to numerically order elements of O) ; this function is the outcome for decision maker A when he/she makes decision a, B makes decision b and X is in state x. In practical situation actual state of the world or the strategy (behavior) of opponent can not be known. But as it was done in former section, here we consider that, decision maker A dispose of sensors (in a broad sense including humans) that will give informations to estimate the actual state of the world X and actual strategy of opponent B. Each sensor will have a partial representation of X or B and send this information asynchronously to A decision center. We suppose that there are n X sensors to sense state of world represented by

X = X 1 , X 2 , ..., X n X (20)
and n B sensors for sensing opponent strategies determined by

B = B 1 , B 2 , ..., B n B . (21)
This problem can be compactly described by an influence diagram ; in terms of influence diagram we have:

n X + 1 chances nodes to represent the state of world X and its images X 1 , X 2 , ..., X n X given by the n X sensors ; Figure 15 summarizes this model in terms of an influence diagram. Notice that one may avoid using nodes X and B by letting sensors directly influence the decision node A by considering estimation as part of decision process.

The purpose of the reactive decision support system constructed so far is to efficiently react to any new evidence sensed by sensors by making the appropriate (in some sense) decision. Let denote by 2 X and 2 B the collection of all subsets of X and B respectively and let S X ∈ 2 X and S B ∈ 2 B be actual sensors information. We define a function δ A called decision rule or policy for decision maker A that associate to any collection of sensors information the appropriate action or decision in A.

Definition 1 A decision rule (or policy) of A is the function π

A defined by π A : 2 X × 2 B → A, (S X , S B ) → π A (S X , S B ) . (22)
According to the nature and application domain of the decision problem, different criteria may be used to define policy function π A . Here is some criteria that can be used. Failure avoidance/Goal attainment For some applications in domain such as maintenance, medical treatment, air traffic control, and mainly for any system related to security of humans, the principal objective of a decision support system is to permit to act in the sense that avoid catastrophes. So it is supposed that the outcome set O contain a particular subset F ⊂ O (possibly single element) to be avoided known as failure subset ; then the "appropriate" action a * given actual sensors information is determined by

Opponents

π A (S X , S B ) = a * = min a∈A {Pr {O ∈ F/S X , S B }} . (23)
In other applications the main purpose will be to act so that the outcome O attain a particular subset G ⊂ O (possibly single element) known as the goal. In fact it is complementary to failure avoidance criterion because avoiding particular subset is equivalent to trying to attain its complementary. In this case we have

π A (S X , S B ) = a * = max a∈A {Pr {O ∈ G/S X , S B }} . ( 24
)
Maximum expected reward/Maximin reward When outcome O is convertible to numerical value through R(a, o) ; for instance in application such as production management, marketing and any application where the outcome can be expressed in money like scale, criteria presented in section 3 can be used to select appropriate action. For instance when using maximum expected reward criterion, π A (S X , S B ) is determined by

BN approach for country risk reduction

The illustrative application considers the problem of classifying countries in terms of risk some actors such as investors may face by investing in these countries. In international trade context, risk classification of countries relies mainly on the Worldwide Governance Indicators (WGI) that determine the level of risks related to Governance in sourcing countries. There are 6 WGI, attributes, dimensions, or indicators of governance identified by the World Bank for this purpose that are briefly described in the following points [START_REF] Tchangani | Bipolar fuzzy nominal classification (BFNC) framework[END_REF].

1. Voice and Accountability (VA): perceptions of the extent to which a country's citizens are able to participate in selecting their government, as well as freedom of expression, freedom of association, and a free media.

2. Political Stability and Absence of Violence/Terrorism (PV): perceptions of the likelihood that the government will be destabilized or overthrown by unconstitutional or violent means, including politically-motivated violence and terrorism.

3. Government Effectiveness (GE): capturing perceptions of the quality of public services, the quality of the civil service and the degree of its independence from political pressures, the quality of policy formulation and implementation, and the credibility of the government's commitment to such policies.

4. Regulatory Quality (RQ): perceptions of the ability of the government to formulate and implement sound policies and regulations that permit and promote private sector development.

5. Rule of Law (RL): capturing perceptions of the extent to which agents have confidence in and abide by the rules of society, and in particular the quality of contract enforcement, property rights, the police, and the courts, as well as the likelihood of crime and violence.

6. Control of Corruption (CC): perceptions of the extent to which public power is exercised for private gain, including both petty and grand forms of corruption, as well as "capture" of the state by elites and private interests.

Data concerning 209 countries were evaluated and registered in 2014 by Foreign Trade Association(FTA) (a Brussels based organization working for European enterprises) and their statistics were analyzed in [START_REF] Tchangani | Bipolar fuzzy nominal classification (BFNC) framework[END_REF] and DEMATEL analysis [START_REF] Fontela | The DEMATEL observer, DEMATEL 1976 Report[END_REF] leads to an influential network relation map (INRM), see [START_REF] Wang | DEMATEL-based model to improve the performance in a matrix organization[END_REF] [21] for details. There is a direct isomporphism between this INRM and the corresponding BN that could be derived; Figure 16 is obtained from that INRM that can be used for decision making toward risk management.

In terms of BN establishement from INRM (see [START_REF] Tchangani | Bipolar fuzzy nominal classification (BFNC) framework[END_REF]) by taking into account inherent uncertainty that may affect influence relationship, structural as well as parameters learning processes have to be set up. Structural learning may build on this INRM as an initial model and progressively refine the structure by suppressing some irrelevant edges or adding some new edges using existing data and/or experts' knowledge. Once the structure has been learnt, parameters that represent conditional probabilities describing the strength of edges can be learnt in the same way by using existing data or experts' knowledge. Techniques for optimizing these learning processes by using specific properties of EOOBN can be found in [?] and [START_REF] Liu | Modélisation bayesienne des interactions multidimensionnelles dans un système com[END_REF]. As there are two main stakeholders (investors on one hand and country authorities on other hand), risk management process using BN model of Figure 16 can be viewed in two directions according to the profile of decision maker. For instance authorities of a given country may like to invest or to act to improve some of the 6 WGI in order to enhance their position regarding the perception of risk by potential investors; investors on other hand, may like to know most influencing WGI toward risk for sequential preemptive decision making process for instance.

RQ VA GE CC

RL PV

Conclusion

This chapter has provided an overview of how to use different Bayesian technology tools to model and analyze risky decision problems. In terms of Bayesian technology, three main variants have been presented, namely Basic Bayesian Networks (BN), Dynamic Bayesian Networks (DBN), Object Oriented Bayesian Networks (OOBN) and their extension (EOOBN) recently developed in the author's research team. On the applications side, case studies in various domains such as industry, decision problems based on mathematical programming, decision problems in the face of nature (state of world decision model) and in a conflict environment, or risk analysis in international trade have been visited. The work presented in this chapter shows the possibilities of Bayesian technology in the processes of analysis, evaluation and risk management. The models established in this chapter, have in common, how to obtain the information or knowledge necessary to solve risky decision problems. Risk is a topic of utmost importance for decision makers, whether it is the risk related to unknown diseases, the risk related to industrial activities, the risk related to malicious activities, etc., and therefore this topic must be considered by scientific methods. This chapter has shown, in some respects, that tools built around Bayesian technology can effectively contribute to understanding these risk-related decision problems. In a decision-making process, there is no worse enemy than uncertainty because it leads to feverishness, fear and apprehension on the part of decision-makers. The adequate treatment of this uncertainty is therefore a commendable contribution to the decision support processes and this chapter has tried to answer this imperative through Bayesian technology.

Figure 1 :Figure 2 :

 12 Figure 1: Risk management process flow chart

Figure 3 :

 3 Figure 3: Bayesian Meta-Model for Risk Management

Figure 4 :

 4 Figure 4: Example of an instance of Bayesian Meta-Model for Risk Management

Figure 5 :Figure 6 :

 56 Figure 5: Structure of considered illustrative system

Figure 7 :

 7 Figure 7: Bayesian network model risk assessment of Kourou launch problem

Figure 9 :

 9 Figure 9: Dynamic Bayesian network model of considered illustrative application

  (a): w(t) (b): failure rate of transformers

Figure 10 :Figure 11 :

 1011 Figure 10: Simulated behavior of external signal w(t) and transformers failure rate

Figure 12 :

 12 Figure 12: Basic structure of a class in EOOBN

Figure 13 :

 13 Figure 13: Cement manufacture process

n B + 1

 1 chances nodes to represent opponent B and its images B 1 , B 2 , ..., B n B given by the n B sensors ; two chances nodes X and B resulting of information fusion from each category of sensors ; a decision node to represent decision maker A ; a chance node O to represent the outcome O(a, b, x) ; and finally a possible value node to represent R(a, o).

Figure 15 :

 15 Figure 15: Schematic representation of intelligent reactive decision making environment

π

  A (S X , S B ) = a * = max a∈A {E o {R(a, o)/S X , S B }} ; (25) and in the case of maximin reward criterion it is determined by π A (S X , S B ) = a * = max a∈A min o∈O {R(a, o)/S X , S B } .(26)

Figure 16 :

 16 Figure 16: Bayesian network equivalent of INRM

Technical Skills Financial Resources Anti_Sismic Laws Emergency System Prepare Population Anti_Sismic Rspect Building Age Emergency Resources Bulding Usage Education Level Anti_Sismic Training Population Awereness Building Damage Infrastructure Damage Emergency Efficiency Human Damage Event (Erthquake) Constraints Actions State of the system Consequences


F De Los Mozos

K Ballerat-Busserolles

B Liborio

Y Coulier

J-Y Coxam

Calorimetric and densimetric data to help the simulation of the impact of annex gases co-injected with CO 2 during its geological storage

    Calorimetric and densimetric data to help the simulation of the impact of annex gases co-injected with CO 2 during its geological storage F de los Mozos, K Ballerat-Busserolles, B Liborio, Y Coulier, J-Y Coxam

Introduction

Most of the industrialized countries have now started to significantly reduce their greenhouse gases emissions. Capture and geological storage processes entered an operational phase with the emergence of several pilot-sites, notably in France, where several technologies are currently being tested.

During the storage of CO 2 in geological formations, four main mechanisms have to be considered: (i) Structural trapping, which is the presence of an impermeable cap-rock which prevents CO 2 from escaping from the outset, (ii) Residual CO 2 trapping, where CO 2 is trapped by capillary forces in the interstices of the rock formation, (iii) Solubility trapping, where the CO 2 dissolves in the water found in the geological formation and sinks and (iv) Mineral trapping happening when dissolved CO 2 chemically reacts with the rock formation to produce minerals.

Considering the whole treatment network of CO 2 storage in geological repositories, the major problem in the separation step is the elimination of some recalcitrant impurities during the treatment of industrial effluents. The gas mixture composition can vary considerably both qualitatively and quantitatively (varying from 80 to 99% of CO 2 ) [START_REF] Wang | Effects of Imputities on Geological Storage of Carbon Dioxide[END_REF], based on the origin of CO 2 and the chosen process of capture (oxycombustion, post-combustion, pre-combustion). The presence of annex gases and the purity level of CO 2 during its separation are essential aspects of capture and are well documented [START_REF] Broutin Paul | CO2 Capture : Technologies to Reduce Greenhouse Gas Emissions[END_REF]. In addition to CO 2 , several components are of concern to the energy industry and can be present at various concentration levels, including O 2 , N 2 , SO x , H 2 S, N y O x , H 2 , CO, and Ar. Some of these impurities (SO x , H 2 S, N y O x and CO) are toxic for human health and the environment, even at low concentrations. [START_REF] Toxicology | Emergency and Continuous Exposure Limits for Selected Airborne Contaminants[END_REF] The presence of impurities in the CO 2 stream may have an effect on all types of geological storage scenarios, especially in terms of changes in storage capacity and injectivity due to changes in phase behavior with respect with pure CO 2 . In addition, impurities could have a significant effect on injectivity through geochemical reactions in the vicinity of injection wells. Geochemical effects such as dissolution of CO 2 and reactions with minerals may determine the long term fate of injected CO 2 , especially in deep saline formations; the impurities may affect the risk profile of storage sites, as geochemical reactions are widely seen as a key mechanism for the stabilization of pressure and brine displacement. Geochemichal reactions can also affect the integrity of caprock sequences above storage complexes.

Until recent years, the role of gas impurities on the geochemical behavior of storage sites is not well documented. Knauss et al. [START_REF] Knauss | Evaluation of the impact of CO2, cocontaminant gas, aqueous fluid and reservoir rock interactions on the geologic sequestration of CO2[END_REF] Evaluated effects of H 2 S and SO 2 on the evolution of a mineralogical assemblage that is characteristic of a geological reservoir at the Frio pilot site (USA). They observed that the effects of H 2 S are negligible, while SO 2 , in oxidizing conditions, maintains a low pH that is not compatible with carbon sequestration by carbonates precipitation. The authors then concluded that laboratory data are lacking and it is consequently hard to validate the numerical approaches even if simulations tend to improve in those contexts. [START_REF] Xiao | The effects of gas-fluid-rock interactions on CO2 injection and storage: Insights from reactive transport modeling[END_REF][START_REF] Xu | Numerical modeling of injection and mineral trapping of CO2 with H2S and SO2 in a sandstone formation[END_REF] Determining equilibrium conditions between various phases is crucial when investigating a fluid with several components. Dissolution, transport, and precipitation of chemical species will occur differently depending on the state of the fluid phase (monophasic liquid, biphasic liquid or liquid + vapor). The well-known systems of gas mixtures or water/salt/gas mixtures are highly simplified because of the limited number of species that are considered (binary or ternary mixtures; [START_REF] Carroll | Phase diagrams reveal acid-gas injection subtleties[END_REF][START_REF] Carroll | Acid-gas injection encounters diverse H2S, water phase changes[END_REF][START_REF] Carroll | Density, phase behavior keys to acid gas injection[END_REF] Moreover, Data on solubility of gas mixtures used to construct phase equilibrium diagrams are limited to conditions of temperature and pressure much lower than those typical of the geological storage sites. In addition, the diagrams generally do not take into account high ionic strength aqueous fluids present within geological formations considered for CO 2 storage.

Concerning risk assessment, the impacts of impurities on vulnerable assets (human health, aquifers, ecosystems etc.) in case of leakage are poorly known. One of the main reasons for this lack of knowledge is the absence of data concerning the expected composition of leakage, first studies consequently focused on pure CO 2 leakages. To overcome this issue, usual approaches for impact assessment [START_REF] Loschetter | Modular toolbox for assessing the impact of CO2 leaking from a geological storage reservoir into a building[END_REF] assume that the leaking gas has the same composition as the injected gas, which is a very strong hypothesis. Furthermore, the toxicity of impurities is strongly dependent on the chemical form and thus on the aqueous speciation of pollutants in the reservoir waters.

To answer this question, the use of models for simulation of the system has to be improved. Different levels are considered, starting with geological models to simulate the rock-aquifer interactions [START_REF] Karay | Modelling of Groundwater Flow in Fractured Rocks[END_REF] and the influence of injected gases on those interactions. It is then obvious that the understanding of the rock-aquifer behavior is driven by the physicochemical, and moreover by the thermodynamic properties of the brine loaded with gas. More specifically, the acidity of the solutions, the solubility of the gases and the energetic properties during mixing are key data in the simulations.

In order to start with thermodynamic modeling, the study of dissolution of pure gas in aqueous solutions of electrolytes present in the brine has to be carried out. These simplified ternary systems {gas-salt-water} allow determining interaction parameters, hardly obtained in more complex mixtures.

Gas dissolution can be represented by a thermodynamic model that takes into account physical and chemical equilibria. For physical dissolution, the equation for gas-liquid equilibrium is usually based on a dissymmetrical approach . [START_REF] Corvisier | Simulations of the impact of co-injected gases on CO2 storage, the SIGARRR project: first results on water-gas interactions modeling[END_REF][START_REF] Hajiw | IMPACT OF IMPURITIES ON CO 2 STORAGE IN SALINE AQUIFERS: MODELLING OF GASES SOLUBILITY IN WATER[END_REF] The general equation of the model is reported in eq.1, below.

Eq. (1)

Where is the molality of the dissolved gaseous component i, is the activity coefficient, is Henry's law constant at water saturation pressure, is the molar volume at infinite dilution, is mole fraction in the gas phase and is fugacity coefficient. Non ideality in liquid phase is represented by an activity coefficient, taking into account interactions between the gas molecules, water and ions. The activity coefficient is estimated using thermodynamic electrolyte models, such as e-NRTL or Pitzer modified models. These models include interaction parameters which need to be adjusted using experimental solubility data and enthalpy data; the enthalpy of solution can be derived from equation 2 using van't Hoff equation. Henry's constant and volume at infinite dilution are determined from solubility and experimental volume data obtained for low gas dilute solution. Non ideality in gas phase is represented by the fugacity coefficient, derived from an equation of state such as Viriel or Peng-Robinson. The interaction parameters in gas phase are adjusted on vapor-liquid equilibrium data. [START_REF] Hajiw | IMPACT OF IMPURITIES ON CO 2 STORAGE IN SALINE AQUIFERS: MODELLING OF GASES SOLUBILITY IN WATER[END_REF] In order to obtain the experimental data required for optimizing the thermodynamic models, different experimental methods have been developed in our laboratory. This paper focuses on the determination of volumes at infinite dilution of gas, namely CO 2 , using a new experimental methodology for measurements of densities of gas loaded solutions in aqueous solutions. Enthalpies of solution and gas solubility of CO 2 and SO 2 will be also presented, and a comparison of the effect of the two gases on energetic properties and solubility will be described.

Material and Methods

Densimetric measurements

The original methodology of the vibrating tube densimeter was modified to allow density measurements for aqueous solutions loaded with controlled amount of gas. Measurements can be performed between 293.15 K and 423 K at pressure up to 15 MPa. Different vibrating tube densimeters have been used, depending on the domain of temperature investigated. At low temperature, the measurements can be carried out using an Anton Paar MPDS 5 densimeter, while a home-made densimeter is used for the highest temperatures. [START_REF] Perfetti | Densities and heat capacities of aqueous arsenious and arsenic acid solutions to 350°C and 300 bar, and revised thermodynamic properties of As(OH)3°(aq), AsO(OH)3°(aq) and iron sulfarsenide minerals[END_REF] The density of the solution is derived from the period of vibration of the vibrating tube following eq. ( 2). eq. ( 2)

 and are the density and the period of vibration related to the solution;  0 and  0 are the density and the period of vibration related to a reference fluid (water) and K the calibration constant of the densimeter, experimentally determined as function of temperature and pressure. An original set-up has been designed in order to obtain the densities of gas loaded solutions with well-known composition. The schematic view is shown in figure 1. The densimeter worked in flowing mode. This method makes it possible to prepare the mixture flowing through the vibrating tube, with a controlled gas loading charge. This preparation of solution in flowing mode prevents bubble formations and avoids corrosion problems.

Firstly, a well known quantity of gas is carefully introduced in a first high pressure syringe pump. The quantity of gas injected in the pump is calculated from the volume indicated by the pump, previously calibrated and the pressure inside the pump, read with a pressure transducer (precision 0.025% full scale). The temperature of the pump is maintained constant using a julabo thermostatic bath (precision 0.1 K). Then, a precise quantity of liquid is added using a second high pressure pump. The injection pressure of liquid is slightly higher than the initial pressure of the first pump filled with the gas. The quantity of liquid is obtained from the volume variation, the temperature and pressure of the second pump. In this experiment the main issue is to prepare homogeneous solutions loaded with gas but avoiding any gas phase. It is then necessary to prepare a solution with a composition below the limit of gas solubility, estimated at temperature and pressure of the experiments, to insure no degassing.

The experimental procedure consists in measuring the period of vibration of a reference fluid (in our case water) and of the gas loaded solution. Different decreasing gas loading charges are obtained by dilution of the initial gas loaded solution, prepared as describe before, with a solution which does not contained gas. The dilution occurs at a mixing point located on the flow lines, as shown in figure 1. The new gas loading charge is calculated from pump flow rates and solution densities. The apparent molar volume V ,CO2 is calculated from the density, using the equation (3).

eq. (3)

Where M s is the molar mass of the solute (the gas in our case) and m CO2 the molality of the solution. The subscript 0 refers to water.

In order to validate the procedure of preparation of the gas loaded solution, the experiments are repeated in same conditions of temperature and pressure, but starting with two different initial solutions (different CO 2 compositions). Experimental results for densities and apparent molar volume for {CO 2 -water} system are given in figures 2. The results are in good agreement with the literature values. [START_REF] Crovetto | Solubility of Co2 in water and density of aqueous Co2 near the solvent critical temperature[END_REF][START_REF] Hnědkovský | Volumes of aqueous solutions of CH4, CO2, H2S and NH3at temperatures from 298.15 K to 705 K and pressures to 35 MPa[END_REF][START_REF] Mcbride-Wright | Viscosity and Density of Aqueous Solutions of Carbon Dioxide at Temperatures from (274 to 449) K and at Pressures up to 100 MPa[END_REF] Figure 3 shows that the pressure has a very small impact on the volume. The molar volume at infinite dilution of CO2 in water is increasing with temperature, as expected for non-electrolyte molecule in water [START_REF] Slavík | Heat Capacities of Aqueous Solutions of Acetone; 2,5-Hexanedione; Diethyl Ether; 1,2-Dimethoxyethane; Benzyl Alcohol; and Cyclohexanol at Temperatures to 523 K[END_REF] and will diverge when reaching the critical temperature.

This study will now be extended to other gases in water and aqueous solutions of electrolytes, to give robust data for adjusting models representative of dissolution of annex gases in brines.

Calorimetric measurements

Enthalpy of solutions and solubility of gas in aqueous solutions have been determined using a method largely used in the group. The enthalpies are obtained as a function of gas loading charge,  (moles CO 2 /mole amine), from the experimental heat flux measured by a Setaram C-80 calorimeter during absorption of CO 2 . The method is detailed elsewhere. [START_REF] Arcis | Enthalpy of solution of CO2 in aqueous solutions of methyldiethanolamine at T= 322.5 K and pressure up to 5 MPa[END_REF][START_REF] Coulier | Thermodynamic Modeling and Experimental Study of CO 2 Dissolution in New Absorbents for Post-Combustion CO 2 Capture Processes[END_REF][START_REF] Koschel | Enthalpy and solubility data of CO2 in water and NaCl(aq) at conditions of interest for geological sequestration[END_REF] A schematic overview of the experimental set up is given in figure 4. The experiments are performed at constant temperature (±0.01 K) and pressure (±0.025 MPa) in a flow mode. The aqueous solution and the carbon dioxide are injected into a mixing cell located inside the calorimeter. The two fluids flow at controlled volume flow rate using two high-pressure syringe pumps (Isco Model 260 DM). These two pumps are temperature regulated using a thermostatic bath in order to maintain a constant molar flow rate. The gas loading charge is determined from volume flow rates and fluids densities. The enthalpy of solution is calculated from the heat flux signal measured during gas absorption following equation (4). eq. ( 4)

Where  SOL H is the enthalpy of solution, F the heat flux signal during gas absorption, F BL the heat flux signal of the baseline recorded when only one fluid is flowing in the mixing cell, K the calibration constant and the molar flow rate of compound i, namely gas or water. The enthalpy of solution (eq. ( 4)) can be expressed in joule per mole of gas or per mole of solvent. The figure X shows an example of enthalpy of solution per mole of CO 2 or per mole of water as function of gas loading charge. When the enthalpy of solution is expressed per mol of gas (fig. 5, blue points), a constant  SOL H value is observed at low loading charges. This behavior makes it possible to determine an enthalpy of solution at low loading charges that could be compared to enthalpy value derived from Henry's constant. At highest loading charge, where the liquid is gas saturated, the calculated  SOL H is decreasing because CO 2 considered in equation ( 4) is not fully absorbed CO 2 . The absolute uncertainty on the enthalpy of solution is estimated to be about 5%.

When expressed per mol of water (fig 5, red points), the enthalpy of solution increases linearly with CO 2 loading charge. The slope corresponds to the enthalpy value on the plateau previously obtained. For the highest loading charges, when he liquid is gas saturated, the calculated  SOL H is more or less constant. The first point of this plateau corresponds to the gas solubility. The gas solubility can then be graphically determined from calorimetric measurement. The main difficulty with this method is to determine accurately low gas solubilities. The experiments have to be carried out at low gas loading charges that mean to work with small gas flow rate. The global uncertainty on the determination of gas solubility will then increase with the relative uncertainty on volume flow rate. In this work, the relative uncertainty on the limit of gas solubility is estimated to be ± 10%.

Absorption of CO2 in aqueous solutions

The results obtained for dissolution of CO 2 in water and in aqueous solutions of electrolytes at 323 K and 5 MPa are reported in figure 6. The enthalpy of solution in pure water, expressed per mole of CO 2 (fig. 6,left) is exothermic and the value of the plateau is about -13 kJ.mol -1 . This value is in good agreement with reported literature data. [START_REF] Koschel | Enthalpy and solubility data of CO2 in water and NaCl(aq) at conditions of interest for geological sequestration[END_REF] When adding a salt, the enthalpy slightly decreases.  SOL H in CaCl 2 aqueous solution is close to -12.5 kJ.mol -1 , and reaches -11 kJ.mol -1 in Na 2 SO 4 aqueous solutions. These enthalpy values are consistent with a mechanism of physical dissolution, and with no chemical absorption. The models used to describe these mixtures will then consider only the physical absorption term.

The solubility of CO 2 is determined graphically from the plot of the enthalpy per mole of water (fig. 6, right). The value obtained in pure water, expressed in CO 2 molar fraction, is close to 0.014. This result is in good agreement with literature data. [START_REF] Koschel | Enthalpy and solubility data of CO2 in water and NaCl(aq) at conditions of interest for geological sequestration[END_REF][START_REF] Takenouchi | The binary system H 2 O-CO 2 at high temperatures and pressures[END_REF][START_REF] Takenouchi | The solubility of carbon dioxide in nacl solutions at high temperatures and pressures[END_REF] This solubility decreases when adding salts, as it can be seen on figure 6 (right figure). This behavior corresponds to the salting-out effect, which can be quantified using the equation [START_REF] Xiao | The effects of gas-fluid-rock interactions on CO2 injection and storage: Insights from reactive transport modeling[END_REF]. eq. ( 5) Where x 0 and x are the solubilities in pure water and salted solution, respectively. The salting out effect calculated at 323 K is about 35% for NaCl; this value does not depend on pressure. The salting out effect is due to the structuration of water in presence of ions. In salted solutions, water is less available for the gas solvation because of electrostatic interactions with the ions. The salting out effect slightly increases in Na 2 SO 4 solutions, with however a difference compared to water within the experimental uncertainty. This value decreases more significantly for CaCl 2 (25 %). This may indicate that the charge of the cation has a larger influence on the salting out effect than the anion.

Absorption of SO2 in aqueous solutions

The same experimental work has been performed with sulfur dioxide instead of carbon dioxide. The experiment were carried out at lower pressure than for CO 2 (namely 0.3 MPa) for safety and experimental constraints. The results are reported in figure 7, for dissolution in water and in aqueous solutions of NaCl and Na 2 SO 4 . The enthalpy of solution of SO 2 in water per mole of gas is close to -28 kJ.mol -1 . This value is consistent with literature value reported at 298.15 K. [START_REF] Goldberg | Thermodynamics of solution of SO2(g) in water and of aqueous sulfur dioxide solutions[END_REF] The mechanism of gas dissolution includes here chemical reactions such as ionizations of SO 2 to form HSO 3 -and SO 3 2-

. The addition of salt decreases significantly the enthalpy of solution (fig. 7,left).

When plotting the enthalpy per mole of water (figure 7, right) for the calculation of gas solubility, it can be observed that this graphical determination of a first point on the plateau is more difficult than for CO 2 . The solubility of SO 2 in water reported in literature [START_REF] Hales | Solubility of sulfur dioxide in water at low concentrations[END_REF] is about 0.022 in molar fraction. Our calorimetric data shows no clear plateau around this value. However, close to this value, it can be observed a change in the slope of the  SOL H, probably linked to this solubility. The residual slope has to be attributed to intermolecular reorganization in the solution in presence of ions. When adding salt in the solution, it can be observed that the change in the slope related to the gas solubility increases exhibiting an opposite behavior compared to CO 2 ; the presence of salt leads to a salting-in effect of the gas. A deeper study must be carried out to explain how interactions with ionic species can affect the chemical reactions during gas dissolution.

Comparison between the two gases

The experimental enthalpies of solution of gas in water and aqueous solutions of NaCl and Na 2 SO 4 has been reported in figure 8. Comparison between the different salted solutions is performed at a same ionic strength. This work shows that for CO 2 dissolution, the enthalpy of solution does not vary when adding electrolytes; the physical dissolution of gas is not so much sensible to the nature and to the electrical charge of ions. For SO 2 dissolution,  SOL H decreases significantly when adding NaCl. This change is larger when adding Na 2 SO 4 , probably because of the presence of divalent anion. The chemical reactions in water leading to the formation of hydrogen sulfite increases both the enthalpy of solution and gas solubility compared to physical dissolution of CO 2 . The presence of salt decreases the enthalpy. A thermodynamic representation of SO 2 dissolution, including chemical reactions, will permit to explain the influence of the presence of ions, in term of ionpair formation with the sulfite or structuration of the solution due to electrostatic forces. The thermodynamic modeling could also help to explain the salting-in effect observed with SO 2 , while the presence of salts generates a salting-out effect in presence of CO 2 .

Aknowledgement

This work is part of the French ANR program SIGARRR (ANR-13-SEED-0006): Simulation of the Impact of Annex Gases (SO x , NyO x , O 2 ) co-injected with CO 2 during its geological storage on the Reservoir-Rocks Reactivity.

Figure 1 :

 1 Figure 1: schematic view of the densimetric device for measurements on gas loaded solutions.

Figure 2 :

 2 Figure 2: density (left) and apparent molar volumes (right) of aqueous solutions of CO 2 at 314 K and 2.6 MPa. The molar compositions of the initial gas loaded solutions are: ,x CO 2 = 0.0065 ;, x CO 2 = 0.0086.

Figure 2

 2 Figure 2 show a good consistency between the measurements obtained from with 2 different initial solutions of CO 2 mole fractions 0.065 and 0.086. This technique makes it possible to obtain molar volume at infinite dilution required in equation (1), by extrapolation of at m CO2 = 0.

Figure 3 :

 3 Figure 3 : molar volume at infinite dilution of CO 2 in water. Left : results at 298.15 K, versus pressure; Right: results versus temperature. : our data, : literature[15-17]

Figure 4 :

 4 Figure 4: schematic representation of the calorimetric measurements.

Figure 5 :

 5 Figure 5: Enthalpy of mixing of CO 2 in water, at 323 K and 5.1 MPa.

Figure 6 :

 6 Figure 6: Enthalpy of solution of CO 2 in water and aqueous solutions of electrolytes, at 323 K and 5 MPa. Left : enthalpy per mole of CO 2 ; right : enthalpy per mole of water. : pure water; : water + Na 2 SO 4 ; : water + NaCl; : water + CaCl 2 ;

Figure 7 :

 7 Figure 7: Enthalpy of solution of SO 2 in pure water and aqueous solutions of electrolytes, at 323 K and 0.3 MPa. : pure water; : water + Na 2 SO 4 ; : water + NaCl;

Figure 8 :

 8 Figure 8: Enthalpy of solution of gases as function of the composition of the solution; : SO 2 ; : CO 2 .