Jean-Claude Benet
email: jean-claude.benet@umontpellier.fr

Frédéric Jamin

Moulay Saïd

El Youssoufi

J C Bénet

M .S. El F J Amin

Youssoufi

Chemical potential of a two-component liquid in porous media: The case of unsaturated soil

Keywords: Chemical potential, Soil water retention curve, Porous media, Unsaturated soil

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Thermodynamic modelling of unsaturated soils in areas such as environmental geomechanics, agronomy, etc., implies knowledge of the state of the various components of the liquid phase. In soil, the liquid phase may be multi-

The introduction of the chemical potentials of the various components makes it possible to describe the thermodynamic equilibrium and transfers in porous media. [START_REF] Anoua | The use of chemical potential to describe water transfer in complex media with strong solid-liquid bonding[END_REF] This thermodynamic function is defined as the derivative of the internal energy with respect to the mass change of a component with entropy, volume and mass of other component constants. 8,[START_REF] Job | Chemical potential-a quantity in search of recognition[END_REF] In a reversible evolution, it represents the mechanical work to be done (or received) when extracting (or introducing) a unit mass of the component in the system. This quantity is therefore representative of the action of the rest of the system on the component.

In a previous publication, [START_REF] Bénet | Measurement of the chemical potential of a liquid in porous media[END_REF] it was shown that the experimental techniques available, i.e. tensiometry and the use of saturated salt solutions, make it possible to establish the relationship between chemical potential and water content when the liquid phase of the porous medium is composed of pure water. This relationship has been established in the isothermal case for media having various structures: soils, gels and food products. We intend to extend this result to the case where the pore space contains a binary solution of two immiscible liquids and a gas phase. In this case measuring the pressure of the solution is not sufficient to characterize the thermodynamic state of the two components.

The aim of this paper is to propose a method to determine the chemical potential of the components of a solution of two miscible liquids in porous media, particularly in the case of unsaturated soil. It is limited to the capillary state; the equilibrium of the liquid phase in the soil is determined by gas/liquid interfaces whose geometry obeys Laplace's law. The water-alcohol solution will be considered as a case study.

Chemical potential of a single liquid in porous media

Measurement method

Measurement of chemical potential is based on its fundamental property: if in a system α, a component k is in thermodynamic equilibrium with the same component in a system β, through a membrane that is permeable to component k only, the second principle of thermodynamics implies [START_REF] Callen | Thermodynamics and An Introduction to Thermostatistics[END_REF] :

µ αk = µ βk (1)
where µ αk and µ βk (J/kg) are the chemical potentials of component k in the two systems.

In the case of a single liquid in a porous medium, three different circumstances may be distinguished related to different configurations of the liquid with respect to the solid in the porous medium. The first concerns the circumstances where gas-liquid meniscus equilibrium is governed by Laplace's law, 8 the liquid phase being continuous. This configuration is sometimes called funicular. The surface tension at the liquid-gas interface creates a pressure difference between the gas and the liquid:

p g -p ℓ = σ  1 R 1 + 1 R 2  (2)
where p g and p ℓ are the pressures of the gas and liquid phases, σ is the surface tension of the liquid, and R 1 and R 2 are the principal radii of curvature of the liquid-gas interfaces. Gagneux and Millet [START_REF] Gagneux | Analytic calculation of capillary bridge properties deduced as an inverse problem from experimental data[END_REF] give extensive details on different configurations of this curvature. In Eq. ( 2), the radii are considered as positive when the curvature is turned into the interior of the gas phase.

Relationship (2) determines the geometry of the interface. In order to measure the chemical potential in the liquid in this case, the semi permeable membrane used is a high air entry disk (HAED) which is impermeable to air but not to liquid. The chemical potential of liquid in this configuration is evaluated by measuring the pressure of the single liquid p ℓ (whose specific density is ρ * ℓ ) on the other side of the HAED:

µ ℓ = µ 0 ℓ (T , p atm ) + p ℓ ρ * ℓ ( 3
)
where µ 0 ℓ (T , p atm ) is a constant that depends on tempera- ture and a reference pressure. The latter is taken equal to the atmospheric pressure p atm ; the result is that p ℓ is the effective pressure of the liquid phase and p g is the effective pressure of the gas phase equal to 0 kPa. In the case of water, the measurement range of tensiometry is restricted, first, by the difficulty of ensuring perfect continuity of the phase of water through the porous plate and, second, by the possibility of the water cavitation occurring at low pressure. Therefore, to widen the measurement range towards higher values of chemical potential, an over-pressure can be applied in the gasphase of the medium. [START_REF] Salager | Effect of temperature on water retention phenomena in deformable soils: theoretical and experimental aspects[END_REF] This technique regularly leads to experimental results within a wide pressure range from 1×10 2 to 5×10 5 Pa (Leong and Rahardjo 14 , Salager et al. [START_REF] Salager | Effect of temperature on water retention phenomena in deformable soils: theoretical and experimental aspects[END_REF] ).

However, liquid relative pressure values below -10 5 Pa should be carefully interpreted. [START_REF] Baker | Unsaturated soil mechanics critical review of physical foundations[END_REF] The second circumstance concerns the state of the liquid when it is adsorbed on the solid surface. This configuration is often called hygroscopic. This state is characterized by a substantial deviation of the liquid vapour pressure from its saturation vapour pressure. A boundary layer, between the porous medium and the atmosphere, allows to pass the water molecules and so it plays the role of a semi-permeable membrane for water, allowing to apply the relation (1). Hence the chemical potential is estimated from the relative humidity HR of the vapour in equilibrium, with the relationship:

µ ℓ = µ 0 ℓ (T , p atm ) + RT M ℓ ln (HR) ( 4
)
where R is the gas constant and M ℓ is the molar mass of the liquid. The reference state corresponds to pure water at a flat surface, at atmospheric pressure and in equilibrium with its vapour at temperature T ; the reference chemical potentials in (3) and (4) are identical. Thereafter the chemical potential reference is taken as zero.

Between funicular and hygroscopic state, there is a set of intermediate configurations that includes the presence of thin film surface layers and gas-liquid inter-granular meniscus sometimes called pendular. HAED method is not efficient enough to characterize the relationship between the pressure in water in the porous medium and water in the tensiometer tank. That results from the fact that the meniscus radii in porous stone are too large to significantly lower the vapour activity below 1. For such a configuration, the two previous measurement methods are inadequate. However, the transition from hygroscopic field to funicular field is not abrupt; in certain range of the liquid content, there are bridges with a capillary configuration, for which it is possible to define a difference of pressure across the interface. Eq. ( 1) must be satisfied for the component in all its forms: capillary bridges, liquid films on the surface of the solid and liquid vapour. The thermodynamic state of the fluid can be characterized by a single chemical potential. The interpolation steps between the funicular and hygroscopic areas are generally easier for a variety of structures: powders, gels, food products. [START_REF] Bénet | Measurement of the chemical potential of a liquid in porous media[END_REF] This interpolation leads to establishing the relationship between the chemical potential and water content for the entire range of water content of the medium.

Relationship between chemical potential and pure water content in soil

The porous media used in the experimental study that follows is clayey silty sand, comprising 40% quartz, 50% calcite and 10% clay, [START_REF] Jamin | Contribution à l'étude du transport de matière et de la rhéologie dans les sols non saturés à différentes températures[END_REF][START_REF] Salager | Effect of temperature on water retention phenomena in deformable soils: theoretical and experimental aspects[END_REF] with particle size less than 2 mm. During testing, it was compacted to achieve a bulk density of the solid phase of 1500 kg/m 3 , corresponding to a porosity of 0.43.

In the funicular configuration, the chemical potential of pure water is measured using the suction plate. [START_REF] Jamin | Contribution à l'étude du transport de matière et de la rhéologie dans les sols non saturés à différentes températures[END_REF][START_REF] Salager | Effect of temperature on water retention phenomena in deformable soils: theoretical and experimental aspects[END_REF] The initial state is the saturated state. The temperature is set at 20 °C. The suction plate gives the relationship between the pressure of the water in the soil and the water content. It is possible to apply gas pressure up to 500 kPa using this method to establish the relationship in the range [1 kPa; 500 kPa]. The chemical potential of the water in the soil is evaluated using (3).

In the hygroscopic area, the chemical potential of pure water is measured with the saturated salt solution method. [START_REF] Jamin | Contribution à l'étude du transport de matière et de la rhéologie dans les sols non saturés à différentes températures[END_REF] The initial state is the saturated state. The temperature is set at 20 °C. The equilibrium between water in the soil and water in salt solutions makes it possible to calculate the chemical potential of water in the soil by using the relation (4).

Fig. 1 shows the variation in logarithmic scale of the chemical potential of water in the soil as a function of water content. It highlights the interpolated part of the curve. Measurements using liquid vapour equilibrium could be made from a water content of 5%, which corresponds (Fig. 1) to a chemical potential of 5000 J/kg. Eq. ( 4) gives an equilibrium relative humidity of 96.3%, which shows that adsorption is weak. So we are still in a state managed by capillary effects, although imperfect porous stone does not allow measurement up to this value. Subsequently, the capillary fields also dominate the funicular area and the pendulum field; in both cases, the capillary determines the chemical potential of the fluid. The study will be focused at the intermediate area of water content between 5% and 15%.

Chemical potential of a solution of two miscible liquids in porous media

System definition

We examine the case of a liquid phase consisting of pure water and a water-miscible liquid (e.g. water-alcohol, water-polyethylene glycol, etc.), referred to hereinafter as component i. The considered medium is composed of an isotropic porous non-deformable skeleton whose voids are filled by the binary liquid solution and a gas phase. The temperature is assumed to be uniform and constant. The gas phase consists of air and the vapour of the two liquids in equilibrium with the liquid solution. The total pressure of the gas phase is assumed to be uniform and constant and equal to the atmospheric pressure; the partial air pressure varies to meet this assumption. The liquid phase is assumed to be incompressible at the pore scale. The system considered thereafter is the matter in a volume containing the unit mass of solid phase (m s = 1 kg). This volume V is given by:

V = 1 ρ s (5)
where ρ s is the apparent density (kg/m 3 ). So, the masses of water and component i (respectively denoted as m w and m i ) in the system are equal to the water content and component i content defined respectively by w w = m w /m s and w i = m i /m s . The chemical potential of water and component i is defined using the free energy potential f (m w , m i ) [START_REF] Callen | Thermodynamics and An Introduction to Thermostatistics[END_REF] :

µ w =  ∂ f ∂m w  m i =  ∂ f ∂w w  w i (6) µ i =  ∂ f ∂m i  m w =  ∂ f ∂w i  w w . ( 7
)
The derivatives of the chemical potentials need to satisfy the condition (8):

 ∂µ w ∂w i  w w =  ∂µ i ∂w w  w i . ( 8
)
The number of moles per unit mass for the two components is given by:

n w = w w M w (9)
n i = w i M i , (10)
where M w and M i are respectively the molar mass of the pure water and the component i.

The molar fractions are given by:

x w = n w n w + n i ( 11
)
x i = n i n w + n i . (12)
With:

x w = 1 -x i . (13)
According to ( 10)-( 12), variable x i depends on the two state variables:

w i = M i M w x i (1 -x i ) w w . (14)
With relationships used later:

 ∂x i ∂w i  w w = -  ∂x w ∂w i  w w = M w M i (1 -x i ) 2 w w (15)
 ∂x i ∂w w

 w i = -  ∂x w ∂w w  w i = - M i M w x 2 i w i = x i (1 -x i ) w w . ( 16
)
V * ℓ denotes the volume occupied by the liquid phase in the medium at the pore scale, ρ * w and ρ * i the specific densities of water and component i. Assuming the additivity of volumes:

V * ℓ = w w ρ * w + w i ρ * i . (17)
Using (14), this relation can be written:

V * ℓ = w w ρ * w  1 + M i M w x i (1 -x i ) ρ * w ρ * i  (18) V * ℓ = w i ρ * w  ρ * w ρ * i + M w M i x w (1 -x w )
 .

(

) 19
In case the pure water saturates the medium x i = 0, the water content is denoted as w sat w , (18) gives the void volume V v of the porous medium:

V v = w sat w ρ * w . ( 20
)
For a given value of w w , the fraction x i cannot grow indefinitely (V * ℓ ≤ V v ); it must meet the following condition:

x i (1 -x i ) ≤ M w M i ρ * i ρ * w  w sat w -w w  w w = A. ( 21
)
The limit x lim i value of the fraction x i is given by the equation:

x lim i = A 1 + A . (22)

Expression of the chemical potential of water and component i

To assess the mass chemical potential of water in the solution of a porous medium, we imagined the experiment shown diagrammatically in Fig. 2.

The porous medium containing a binary solution of water and a component i in (A) is in communication, through a porous stone, with compartment (B) containing the only solution. It ensures equality between the pressure p B ℓ in B and the pressure of the liquid phase p A ℓ in the pores of A:

p A ℓ = p B ℓ . (23)
Firstly, consider an ideal semi-permeable membrane to water separated compartments B and C; C contains pure water and we denote p C ℓ = p w as the pressure of pure water in C. The pressure difference between these compartments is equal to the osmotic pressure Π w :

Π w = p B ℓ -p w . (24)
Thereafter the gas phase is at atmospheric pressure; the pressure of the liquid phase used in relations (23) and ( 24) is the effective pressure. At equilibrium, the chemical potential of the water component is the same in A, B, C. It is measured in C; using (3): Measuring pure water pressure in compartment C gives the chemical potential of the water component in the solution of the porous medium (A). If we know the partial pressure of the vapour in equilibrium with the water p vw (x i ) in compartment B and p 0 vw = p vw (x i = 0) in compartment C, neglecting fluid compressibility osmotic pressure is given by 8 :

µ w (w w , x i ) = p w ρ * w = p B ℓ -Π w ρ * w . (25)
Π w = ρ * w RT M w ln  p 0 vw p vw (x i )  (26)
where M w is the molar mass of water. From a thermodynamics point of view, there is no difference between a brine solution (salt dissolved in the water) and a solution of two miscible liquids. Thus, the relation (26) is valid in both cases. According to Guggenheim, 8 the equilibrium pressure p vw (x i ) depends little on the outside pressure. The chemical potential of water in the medium (25) is put in the form:

µ w (w w , x i ) = p B ℓ (w w , x i ) ρ * w + RT M w ln  p vw (x i ) p 0 vw  . (27)
Using Laplace's law (2):

p B ℓ (w w , x i ) = -σ (x i )  1 R 1 + 1 R 2  ( 28
)
where σ (x i ) is the surface tension of the solution.

We consider two states: the first (state 1) corresponding to pure water x i = 0 occupying a volume V * ℓ in the volume V , and the second (state 2) x i ̸ = 0 corresponding to a solution, occupying the same volume V * ℓ . Assuming that the liquid phase is incompressible, and that the solid phase is perfectly wetting for water and for the binary solution and the surface tension is uniform, the geometry of the liquid phase in the medium, as determined by capillary effects, is the same in both cases. The principal radii of curvature R 1 and R 2 of the menisci are independent of x i , which leads, using (28), to:

p B ℓ (w w , x i ) p B ℓ  w 0 w , x i = 0  = σ (x i ) σ (x i = 0) = f (x i ) (29)
where w 0 w is the water content for state 1. According to (18), equal volumes in the states 1 and 2 are reflected by a relationship between w 0 w and w w :

w 0 w = w w  1 + M i M w x i (1 -x i ) ρ * w ρ * i  . ( 30
)
For pure water, the chemical potential is given, using (3), by:

µ w  w 0 w , x i = 0  = p B ℓ  w 0 w , x i = 0  ρ * w . (31)
With ( 29) and (31), the chemical potential of water in the solution of porous media (27) is:

µ w (w w , x i ) = µ w  w 0 w , x i = 0  f (x i ) + RT M w ln  p vw (x i ) p 0 vw  . (32)
Secondly, consider an ideal semi-permeable membrane to component i separated compartments B and C; C contains the pure component i and we denote p C ℓ = p i as the pressure of pure component i in C. As pure water case, all Eqs. ( 24)-( 26) are identically applied to the component i by replacing w by i. Following the reasoning that leads to (27), the chemical potential of i in the three compartments is given by:

µ i (w i , x w ) = p B ℓ (w i , x w ) ρ * i + RT M i ln  p vi (x w ) p 0 vi  ( 33
)
where p 0 vi = p vi (x w = 0) is the equilibrium vapour pressure of pure component i in compartment C. The relation analogous to (29) is written:

p B ℓ (w i , x w ) p B ℓ  w 0 i , x w = 0  = σ (x w ) σ (x w = 0) = g (x w ) (34)
where w 0 i corresponds to the pure liquid i, this value can be calculated by ( 19).

Applying Eq. ( 29) in the special case of the two pure liquids:

p B ℓ  w 0 i , x w = 0  p B ℓ  w 0 w , x i = 0  = σ (x w = 0) σ (x i = 0) = C . ( 35
)
Molar fractions x i and x w are being linked by (13):

g (x w ) = f (x i ) C . ( 36
)
For pure i, ρ * i being the specific density of component i, the chemical potential of component i is:

µ i  w 0 i , x w = 0  = p B ℓ  w 0 i , x w = 0  ρ * i . ( 37
)
Introducing ( 31) and ( 37) in (35) gives:

ρ * i µ i  w 0 i , x w = 0  ρ * w µ w  w 0 w , x i = 0  = C . (38)
This relationship gives the variation of the chemical potential of pure i in the porous medium as a function of the content of i, knowing the variation of the chemical potential of pure water with the water content. With (37), relation (33) becomes:

µ i (w i , x w ) = µ i  w 0 i , x w = 0  g (x w ) + RT M i ln  p vi (x w ) p 0 vi  . (39)
According to (36) and ( 38), (39) can also be written in terms of the variable x i to locate µ i and p vi :

µ i (w i , x i ) = ρ * w ρ * i µ w  w 0 w , x i = 0  f (x i ) + RT M i ln  p vi (x i ) p 0 vi  . (40)
Relations (32) and (40) show the actions of the interfaces (first term of the second member) and the mutual action of the fluid (second term of the second member) on the chemical potentials of the two fluids. The second derivative of the free energy is obtained from (33) using ( 15):

∂ 2 f ∂w i ∂w w = ∂µ w ∂w i = ∂µ w ∂x i ∂x i ∂w i = (1 -x i ) 2 w w  M w M i µ w  w 0 w , x i = 0  ∂f (x i ) ∂x i + RT M i ∂ ln (p vw (x i )) ∂x i  . (41)
It can also be deduced from (40) using ( 16)

∂ 2 f ∂w w ∂w i = ∂µ i ∂w w = ∂µ i ∂x w ∂x w ∂w w = x i (1 -x i ) w w ×  µ i  w 0 i , x w = 0  ∂g (x i ) ∂x i - RT M i ∂ ln (p vi (x i )) ∂x i  . (42)
According to (36), ( 38), ( 11) and ( 12), the ratio of the first two terms in (41) and ( 42) is equal to one:

(1 -x i ) x i M w M i µ w  w 0 w , x i = 0  ∂f (x i ) ∂x i µ i  w 0 i , x w = 0  ∂g(x i ) ∂x i = x w x w M w M w ρ * i ρ * w = n w n i M w M i ρ * i ρ * w = 1. (43)
Using Duhem-Margule relationship 8 :

(1

-x i ) ∂ ln (p vw (x i )) ∂x i + x i ∂ ln (p vi (x i )) ∂x i = 0. (44)
The expressions (41) and (42) are equivalent.

Case of a water-alcohol solution in a clayey silty sand

Water-alcohol solution

The molar mass of pure alcohol is 0.046 kg/mole and its density is 789 kg/m 3 . The study is restricted to water contents between 5% and 15% and alcohol mole fractions between 0 and 0.15. In this area, satisfying the relation ( 22), the effects of the surface layers and mutual effects of the two fluids are in the same order. Variations in partial pressures p vw (x i ) and p vi (x i ) above a water-ethyl alcohol solution at 25 °C in the free state, are given in Fig. 3. 8 The chemical potentials of water and alcohol in the free state (outside the porous medium) are derived from the relations (4) with, as in Section 3, zero constant for the pure fluids at atmospheric pressure. Fig. 4 shows the variation of the two chemical potentials in relation with x i . Fig. 5 shows the functions defined by ( 29) and (34) from the surface tension of a water-alcohol solution. 8 For this solution the coefficient C defined by ( 35) is equal to 0.328.

Chemical potential of water and alcohol in the clayey silty sand

The water content adopted to set the curves is the water content w 0 w . In the water content range studied, w 0 w given by (30), varies almost linearly, the ratio w 0 w /w w is one for x i = 0 and 1.57, for x i = 0.15. Fig. 6 shows the variation of the chemical potential of water as a function of x i for w 0 w = 6%. This figure brings up the first term of (32) related to the effects of interfaces, and the second term in (32) related to the presence of alcohol. Increasing x i causes lowering of surface tension (Fig. 5).

The difference between the pressure of the gas phase and the pressure of the liquid phase decreases. The pressure of the gas phase being constant (equal to the atmospheric pressure), this translates into an increase in the pressure of the liquid phase and therefore the increase of the first term of (32). For the second term of Eq. ( 32) there is a decrease due to the decrease of the partial pressure of water in equilibrium with the alcohol content (Fig. 4). For

x i = 0, the interface effect occurs in accordance with the relationship shown in Fig. 1. When x i increases, the effect of the presence of alcohol gradually prevails over the interface effect. w of the field of study. The decrease of the chemical potential with x i is recognized for all water contents as indicated above. This figure shows the variation of the chemical potential of water for a free solution. This variation is consistent with Fig. 4 in the field of study. It can be seen that the distance between the curves increases as the water content decreases; this shows the increasing effect of the interfaces. When the water content exceeds 10%, the effect weakens; the soil solution behaves like a water-alcohol solution in the free state.

Fig. 8 shows the variation of the chemical potential of water as a function of the water content for various values of x i . The curve for x i = 0 conforms to the curve of Fig. 1, with the dotted portion corresponding to the hygroscopic field. The contribution of component i tends to lower the chemical potential. This figure also shows that, for all x i the curves tend asymptotically to the curve representing the hygroscopic behaviour when w 0 w tends to zero. This suggests that in the hygroscopic field, the effect of component i in the chemical potential of water becomes progressively weaker as does the interface effect. This reflects the fact that liquid-solid bond strength increases significantly when water content tends to zero. Fig. 9 shows the variation of the chemical potential of alcohol as a function of x i for various water contents. In the field explored, curves deviate slightly from the curve corresponding to a free solution. The interface effect (Fig. 1) is an order well below the alcohol concentration effect (Fig. 4); it begins to manifest water content of about 5%.

Conclusions

The assumption behind the assessment of chemical potentials of the components of a binary solution in a porous medium from the chemical potential of pure water is that, for the same volume occupied by the fluid phase in the porous medium, the geometry of the fluid phase is independent of the composition of the solution. This hypothesis is based on the fact of the low compressibility of the fluid on one hand, and the capillary pattern on the other: the shape of the liquid-gas interface is independent of the value of the surface tension so that it is uniform throughout the interface. The presented developments are limited to the water content range where the geometry of the gas-liquid interface is mainly controlled by surface tension.

In this context, the expressions of chemical potentials of the two components were established. They show the superposition of two effects: the effect of interfacial forces and the effect of the presence of the other component.

An interaction study of the water-alcohol solution in clayey silty sand soil was then conducted for the alcohol mole fractions between 0 and 0.15 and the water content ranging from 5% to 15%. In this range, it was found that the chemical potential of water was affected by the capillary effects below a water content of 10%. Above that, the chemical potential of water was relatively unaffected by the capillary effects and its variation as a function of the mole fraction of alcohol has shown little difference from that of the potential of water in a free solution. The chemical potential of the alcohol was little affected by the capillary effects within the range of water content studied. It varied substantially with the molar fraction of alcohol, in accordance with the variation in a free solution. The presented study emphasizes the water component, and a parallel study could be conducted taking other liquids as reference. This study helps to judge the need to introduce a correction to the values of chemical potential in the case of modelling chemical phenomena, adsorption or phase change in porous media.

The experimental information necessary to consider other solutions includes the equilibrium vapour pressures of the components of the solution in the free state, and the change in surface tension with the molar fraction of

Fig. 1 .

 1 Fig. 1. Variation of the chemical potential of water as a function of water content for clayey silty sand for T = 20 °C.

Fig. 2 .

 2 Fig. 2. Experimental device to evaluate the chemical potential in the solution of a capillary porous media.

Fig. 3 .

 3 Fig. 3. Vapour pressure variation of the water and alcohol in a solution in relation to the mole fraction of alcohol solution.

Fig. 4 .

 4 Fig. 4. Chemical potential of water and alcohol in a free state solution.

Fig. 7

 7 Fig. 7 shows the change in the chemical potential of water as a function of x i for various water contents w 0

Fig. 5 .

 5 Fig. 5. Functions f (x i ) and g (x i ) for the water-alcohol solution.

Fig. 6 .

 6 Fig.6. Variation of the chemical potential of water in relation to x i for w 0 w = 6%; interface effect and alcohol effect.

Fig. 7 .

 7 Fig. 7. Change in the chemical potential of water as a function of x i for various water contents.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Variation of the chemical potential of water as a function of the water content for various values of x i .

one of the components. For the consideration of any other porous medium, one would obviously have to establish relationship between the water content and chemical potential for the components of the solution in pure form.


Keywords: Complexity, risk interaction, risk analysis, importance measures, project management

Risk analysis and prioritization is a key process in project risk management. Its outcomes serve as input of the risk response planning process where decisions are made. Complexity of projects is characterized by the emergence of phenomena that are difficult to detect and to manage using classical methods. It may disturb risk assessment, on which priorities are further established. This paper aims at using importance measure techniques in the complex project risk management field. This involves modeling the complex project risk network and providing complementary analysis results based on risk importance measures accounting for risk interactions. These new project risk indicators allowthe managerfora morecomprehensive understanding ofthe risks. An application to a complex engineering project is provided to illustrate this approach to assess both risks and risk interactions, in order to establish priorities for further decision-making.

I. INTRODUCTION

Risk management is indispensable for the successful realization of projects. The Project Management Institute defines a project risk as -an uncertain event or condition whose occurrence affects at least one of the project objectives, e.g., scope, schedule, cost and quality‖ [START_REF] Pmi | A Guide to the Project Management Body of Knowledge (PMBOK)[END_REF]. The classical steps of Project Risk Management (PRM) involve: risk identification, risk analysis, risk response planning, risk monitoring & control and lessons learned. A common and critical issue in the PRM process is to assess and determine the relative priority of various risks. As projects are subject to more uncertainties and tight budget constraints, it is necessary for the manager to identify the most important risks to take care of, because no project can afford to manage and control all the potential risks. In this regard, risk analysis is required to prioritize risks and then the manager can allocate resources to mitigate the most F. Marle is with the Laboratoire Genie Industriel, Ecole Centrale Paris, 92290Chatenay-Malabry, France (e-mail: franck.marle@ecp.fr).

M. Xie is with the Department of Systems Engineering and EngineeringManagement, City University of Hong Kong, Hong Kong, China (e-mail:minxie@cityu.edu.hk). critical ones.

Risks are generally assessed with respect to their probability (of occurrence) and impact (of their materialization). Risk rating or criticality is used as an aggregate measure of risk importance, and it is often defined as the product of risk probability and impact scores [START_REF] Pmi | A Guide to the Project Management Body of Knowledge (PMBOK)[END_REF][START_REF] Raz | Use and benefits of tools for project risk management[END_REF][START_REF] Williams | A classified bibliography of recent research relating to project risk management[END_REF]. A widely praised and adopted as simple, effective tool for risk prioritization in PRM practice and other contexts is the ‗risk matrix' or ‗probability-impact grid' (PIG) [START_REF] Baccarini | The risk ranking of projects: a methodology[END_REF][START_REF] Ahmed | A review of techniques for risk management in projects[END_REF].However, some researchers highlight the weakness related to the fact that interdependencies among risks are not properly considered in PRM just using the risk matrix or the PIG approach [START_REF] Cox | What's wrong with risk matrices?[END_REF][START_REF] Hubbard | The failure of risk management: why it's broken and how to fix it[END_REF][START_REF] Chapman | How to manage project opportunity and risk: why uncertainty management can be a much better approach than risk management[END_REF]. For example, Salado and Nilchiani have discussed the importance of modeling risk dependencies and hence the uncertainty propagation [START_REF] Salado | Taxonomy and Categorization of Uncertainties in Space Systems with an Application to the Measurement of the Value of Adaptability[END_REF][START_REF] Salado | Assessing the Impacts of Uncertainty Propagation to System Requirements by Evaluating Requirement Connectivity[END_REF].

Indeed, projects are facing a growing complexity due to the involvement of numerous, diverse and strongly interrelated elements [START_REF] Baccarini | The concept of project complexitya review[END_REF][START_REF] Chu | Theories of complexity -Common denominators of complex systems[END_REF][START_REF] Lazzerini | Analyzing risk impact factors using extended fuzzy cognitive maps[END_REF][START_REF] Wu | A risk analysis model in concurrent engineering product development[END_REF]. This leads to an increasing number of interdependent risks, which must be taken into account in order to be able to put properly priorities to these risks. Some research have discussed how to model and understand the complexity in network structures, e.g., in [START_REF] Buldyrev | Catastrophic cascade of failures in interdependent networks[END_REF][START_REF] Sanz | Dynamics of interacting diseases[END_REF][START_REF] Xia | Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks[END_REF][START_REF] Liu | Controllability of complex networks[END_REF][START_REF] Boccaletti | Complex networks: Structure and dynamics[END_REF]. A modeling and analysis process of the risk network will be used in this paper based on previous works [START_REF] Fang | A simulation-based risk network model for decision support in project risk management[END_REF].

Complexity appears to be one of the main reasons of the unpredictability of projects, particularly in terms of problems and failures. There is a high number and great diversity of interdependent elements to manage, with also a lot of and diverse parameters that characterize them. The amount and diversity of these interactions are huge that projects rapidly become unmanageable due to the sole use of classical project management tools and methods like the mentioned risk matrix or the PIG approach. Both the complexity of the evolving environment and the internal complexity of the project justify the need for a new approach. For instance, some events may occur and eventually propagate throughout the project, which is likely to reduce the project performance if this emergent behavior is not properly anticipated and controlled [START_REF] Eckert | Change and customisation in complex engineering domains[END_REF][START_REF] Kloss-Grote | How to measure the effectiveness of risk management in engineering design projects? Presentation of RMPASS: a new method for assessing risk management performance and the impact of knowledge management-including a few results[END_REF].

The approach using importance measures (IMs) was first introduced by Birnhaum [START_REF] Birnbaum | On the importance of different elements in a multi-element system[END_REF], and a series of IMs have been proposed afterwards [START_REF] Kuo | Importance Measures in Reliability, Risk, and Optimization: Principles and Applications[END_REF][START_REF] Zhu | Importance measures in reliability and mathematical programming[END_REF][START_REF] Xie | On some importance measures of system components[END_REF][START_REF] Xie | On a general measure of component importance[END_REF]. They all aim at quantifying the contribution of individual components to the system performance and have been developed in different fields, like reliability, risk, availability, throughout [START_REF] Cheok | Use of importance measures in risk-informed regulatory applications[END_REF][START_REF] Van Der Borst | An overview of PSA importance measures[END_REF][START_REF] Vaurio | Importance measures in risk-informed decision making: Ranking, optimisation and configuration control[END_REF][START_REF] Borgonovo | Differential importance and comparative statics: An application to inventory management[END_REF][START_REF] Zio | Importance measures-based prioritization for improving the performance of multi-state systems: application to the railway industry[END_REF]. Besides, some other risk measures like mean-riskmeasure have been widely applied in systems engineering. For example, Choi and co-authors conductedmean variance analysis on inventory management and supply chain management [START_REF] Choi | Mean-variance analysis for the newsvendor problem[END_REF][START_REF] Li | Mean variance analysis of fast fashion supply chains with returns policy[END_REF][START_REF] Zhao | Mean-risk analysis of wholesale price contracts with stochastic price-dependent demand[END_REF].In this research, we propose to measure the importance of risk with regard to its contribution to the global project risk exposure.The originality of this paper is tailoring and application of some importance measures to refined project risk assessments, taking into account risk interactions. Namely, the measure and ranking of risks in terms of their importance in the risk network may differ from the classical project risk analysis results. The importance of risk interactions can also be measured and prioritized with respect to their role in the risk network. The approach can thus provide more comprehensive understanding of project risks and perhaps novel insights for decision-making. This work proposes a refined priority of risks and a complementary priority analysis of risk interactions. It is regarded as an initial step to develop dependency-driven risk assessment methods. Risk assessment or estimation involves using qualitative or quantitative approaches. In the context of projects, no previous experience can be reused as is, and the number of similar projects is not high enough to get significant statistics based on the past. The aim of this work is to give rankings, order of magnitudes where gaps between different risks are significant compared to the error made in estimates. The important thing is how data are transformed in order to analyze relatively risk importance, independently of the reliability of initial risk estimate.

The remainder of the paper is organized as follows. Section 2 introduces a project risk network to model the complexity of risk interdependencies. Section 3 proposes some importance measures to project risks considering their interactions with respect to their contribution to the overall project risk exposure. Section 4 illustrates how the proposed approach is tested on a real large engineering project in the construction industry. Finally, some conclusions are drawn and perspectives are discussed in Section 5.

II. A RISK NETWORK-BASED MODEL

The aim of this work is to combine importance measures applied on refined risk assessments, considering risk interactions. Based on classical project risk analysis methods, a risk network model is built to capture the project risks and the propagation behavior through their interactions.

A. Classical Project Risk Assessment Techniques

Risk criticality is used as an aggregate measure of risk importance, and is often defined as the product of risk probability and impact, or severity [START_REF] Pmi | A Guide to the Project Management Body of Knowledge (PMBOK)[END_REF][START_REF] Raz | Use and benefits of tools for project risk management[END_REF][START_REF] Williams | A classified bibliography of recent research relating to project risk management[END_REF].

Qualitative scales are often used to express project risk probability with 5 to 10 levels (e.g., very rare, rare, unlikely, likely, etc.), which typically correspond to non-linear probability measures (e.g., 10 -4 , 10 -3 , 10 -2 , 10 -1 , etc.) [START_REF] Cox | What's wrong with risk matrices?[END_REF][START_REF] Vose | Risk Analysis: a Quantitative Guide[END_REF]. To avoid some fundamental flaws in calculation according to [START_REF] Kujawski | Analysis and critique of the system readiness level[END_REF], the qualitative scales need to be converted to quantitative measures of risk probability [START_REF] Fang | A simulation-based risk network model for decision support in project risk management[END_REF]. Logarithmic scales have been used by statisticians for many decades [START_REF] Fleiss | Statistical methods for rates and proportions[END_REF]. They allow us todistribute probabilities unevenly. In practice, they devote more space to small values, imposing a compressed, logarithmic mapping. For our calculations, based on this principle, we use the following mappingEquation:

𝑃 = α * 10 -𝛽 /𝑠 (1)
where Pis the quantitative probability measure, s indicates the qualitative scale value, with parameters α > 0 , β > 0 .Risk impact is assessed by classical PRM methods, based upon a mix of previous experience and expert judgment [START_REF] Pmi | A Guide to the Project Management Body of Knowledge (PMBOK)[END_REF][START_REF] Chapman | Project Risk Management -Processes, Techniques and Insights[END_REF]. In practice, if risks are analyzed individually (i.e., without considering the interrelationship among risks), the classical risk criticality indicator can be used to measure the importance of a risk:

𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙𝑖𝑡𝑦 𝑖 = 𝑃 𝑖 * 𝐼 𝑖 (1)
where P i and I i indicate the estimated probability and impact of Risk i.

When considering a project with N risks, a potential objective function O of risk management, i.e., the project risk exposure, can be defined as [START_REF] Hillson | Managing project risks using a cross risk breakdown matrix[END_REF][START_REF] Rafele | Understanding project risk exposure using the two-dimensional risk breakdown matrix[END_REF]:

𝑂 = 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙𝑖𝑡𝑦 𝑖 𝑛 𝑖=1 = 𝑃 𝑖 * 𝐼 𝑖 𝑛 𝑖=1 (3)
The next section shows how the consideration of risk interactions in our approach may change values of parameters in Eq. (3).

B. Modeling and Analyzing the Project Risk Network

A risk network model is used to capture the project risks and their interactions, and to reassess their values in terms of refined probability and refined criticality. This work is an initial attempt to incorporate risk dependency in risk analysis for non-probabilistic risk assessment methods since only materialization dependency has been incorporated. Such network representation models the individual risks and their interactions which convey the risk propagation behavior, i.e., the occurrence of one risk triggering another risk in the network. Simulation technique is used to run the model and observe the occurrence of risks. In what follows, the risk network model is briefly described. Readers may refer to [START_REF] Fang | A simulation-based risk network model for decision support in project risk management[END_REF] for more details.

First, the network is built. Risk interaction is considered as the existence of a possible precedence relationship between two risks [START_REF] Marle | Potential Applications of DSM Principles in Project Risk Management[END_REF]. Multiple experts are engaged for this modeling task, after being made aware of the possible confusion between direct and indirect interactions among risks, and being asked to concentrate on direct dependencies. In the context of project risk management, there is generally not enough experience on identification and assessment of risk interactions. This is why expertise is the chosen way in this study to model them.The interrelations between project objects, such as tasks, actors and product components, can facilitate the identification of interrelations between the risks related to these objects.

Generally, an indirect cause-effect dependency involves modeling at least one additional intermediary node and additional edges between the cause and the effect. This means that the initial risk list may be enriched thanks to risk interaction identification.

We define the Risk-Risk Matrix (RR), which is a square matrix called MDM (Multi-Domain Matrix) in the Dependency and Structure Modeling (DSM) approach [START_REF] Steward | The Design Structure Matrix: a method for managing the design of complex systems[END_REF][START_REF] Browning | Applying the design structure matrix to system decomposition and integration problems: a review and new directions[END_REF] as follows:

𝑅𝑅 𝑖𝑗 = 𝑇𝑃 𝑖𝑗 if there is a cause -effect relationship from risk 𝑗 to 𝑖 𝑅𝑅 𝑖𝑗 = 0 otherwise (4)
where the transition probability TP ij is defined as the probability that the occurrence of the cause riskj triggers the occurrence of the effect risk i.

There is then an important distinction between the spontaneous probability (that a risk is caused by external events or risks which are outside the scope of the model) and the transition probability (that a risk is triggered by another risk within the model). For the example in Fig. 1, Risk 1 occurs only in accordance with its spontaneous probability; and Risk 3 may arise from both its spontaneous probability and the transition probabilities from Risk 5 and Risk 7.

Finally, the weighted network is analyzed through simulation. A large number of iterations are conducted for each scenario of simulation, where the occurrence of every risk during the run is recorded. The observed probability or frequency of each risk in the simulation may change or not. This new value due to the risk propagation behavior is called 𝑃 ′ . Hence, the output of global risk exposure becomes:

𝑂 ′ = 𝑃 𝑖 ′ 𝑛 𝑖=1 * 𝐼 𝑖 (5)

III. TAILORING IMPORTANCE MEASURES TO THE PROJECT RISK

NETWORK

This section introduces an approach to combining the refined risk assessment (considering risk network instead of individual risks) and the importance measure techniques. We tailor several risk importance measures to the context of project management. The value of the proposed risk IMs can be obtained by the introduced risk network model using simulation techniques. Various IMs can provide project manager with diverse insights on risks and risk interactions with respect to risk management.

A. Risk Importance Measures

Several risk importance measures are proposed based on the refined risk values using the risk network model:

Risk Achievement Value:

𝑅𝐴𝑉 𝑖 = 𝑂 ′ 𝑅 𝑖 = 1 -𝑂 𝑏𝑎𝑠𝑒 (6)
Risk Reduction Value:

𝑅𝑅𝑉 𝑖 = 𝑂 𝑏𝑎𝑠𝑒 -𝑂 ′ 𝑅 𝑖 = 0 (7)
Marginal Risk Value:

𝑀𝑅𝑉 𝑖 = 𝑂 ′ 𝑅 𝑖 = 1 -𝑂 ′ 𝑅 𝑖 = 0 ( 8
)
Here O is the defined objective function for PRM, namely the overall risk exposure; O(base) is the estimated value without carrying out any risk management activities; 𝑂 ′ 𝑅 𝑖 = 1 indicates the achieved risk exposure under the condition that the spontaneous probability of Risk i is set to 1 while the other risks remain the original status (i.e., with the estimated spontaneous probability to occur); 𝑂 ′ 𝑅 𝑖 = 0 indicates the value of O given that the spontaneous probability of Risk i is set to 0 (others remain).

The RAV i specifies the maximum potential increase from the estimated global risk exposure if Risk i is out of control and determined to occur with 100% probability. The RRV i represents the maximum potential decrease of the global risk exposure by investing to directly mitigate Risk i. The MRV i is irrelevant to the estimated value of the spontaneousprobability of R i . It thus reflects the importance of R i with respect to its structural position in the risk network. In other words, it denotes the range of influence of R i on the overall objective function and follows:

𝑀𝑅𝑉 𝑖 = 𝑅𝐴𝑉 𝑖 + 𝑅𝑅𝑉 𝑖 (9)
We divide these measures by O (base) to express their fractional contribution to the overall risk exposure. The ratios are also easier to use as IMs for risk ranking. The formulas are as follows:

Risk Achievement Importance: [START_REF] Salado | Assessing the Impacts of Uncertainty Propagation to System Requirements by Evaluating Requirement Connectivity[END_REF] Risk Reduction Importance:

𝑅𝐴𝐼 𝑖 = 𝑂 ′ 𝑅 𝑖 =1 -𝑂 𝑏𝑎𝑠𝑒 𝑂 𝑏𝑎𝑠𝑒
𝑅𝑅𝐼 𝑖 = 𝑂 𝑏𝑎𝑠𝑒 -𝑂 ′ 𝑅 𝑖 =0
𝑂 𝑏𝑎𝑠𝑒 [START_REF] Baccarini | The concept of project complexitya review[END_REF] Marginal Risk Importance:

𝑀𝑅𝐼 𝑖 = 𝑂 ′ 𝑅 𝑖 =1 -𝑂 ′ 𝑅 𝑖 =0 𝑂 𝑏𝑎𝑠𝑒 (12)

B. Risk Interactions Importance Measures

Besides individual risks, the importance of risk interactions can also be measured in terms of its influence on the global risk

exposure:

Risk Interaction Achievement Importance:

𝑅𝐴𝐼 𝑖→𝑗 𝐼 = 𝑂 ′ 𝑒 𝑖→𝑗 =1 -𝑂 𝑏𝑎𝑠𝑒 𝑂 𝑏𝑎𝑠𝑒 (13)
Risk Interaction Reduction Importance :

𝑅𝑅𝐼 𝑖→𝑗 𝐼 = 𝑂 𝑏𝑎𝑠𝑒 -𝑂 ′ 𝑒 𝑖→𝑗 =0 𝑂 𝑏𝑎𝑠𝑒 (14)
Marginal Risk Interaction Importance:

𝑀𝑅𝐼 𝑖→𝑗 𝐼 = 𝑂 ′ 𝑒 𝑖→𝑗 =1 -𝑂 ′ 𝑒 𝑖→𝑗 =0 𝑂 𝑏𝑎𝑠𝑒 (15)
In a similar sense, 𝑒 𝑖→𝑗 = 1 and 𝑒 𝑖→𝑗 = 0 translate into respectively enhancing the corresponding edge of risk interaction from R i to R j in the network (set its transition probability TP ji to 1) or blocking it (set the transition probability TP ji to 0). The RAI I , RRI I , and MRI I for risk interaction edges have similar implications to the counterpart measures for risk nodes in the network.

IV. APPLICATION TO A REAL ENGINEERING PROJECT

In this Section, we illustrate the application of the proposed approach to a real large engineering project, aimed at building the infrastructure and associated systems of the future tramway in a medium-sized city in Europe. The project includes the construction and implementation of tramway, equipment, and civil work, with 10 years duration and hundreds of millions € budget. The leading company is a designer and manufacturer of trains, which recently extended its scope by proposing -turnkey‖ projects, including not only the trains, but also the complete infrastructure around the trains.

A. Identifying Risks and Building the Risk Network

A project risk list is provided by the project manager and the expert team, containing 56 identified risks at the main level, with their name, domain information, as shown in Table III in the Appendix. Basic characteristics of risks have been assessed by the project manager and associated experts, including qualitative probability and impact scales, as well as criticality measure (product of probability and impact).

Qualitative scales of probability are then converted into quantitative metrics through Eq. ( 1), setting parameters α = 5, β = 8by experience. We adoptimpact scales that existed in the organization to estimate the impact gravity of project risks. Becauserisk impacts are in essence multi-dimensional (namely in terms of schedule, cost, quality, etc.), hence hard to create comparable quantitative scales, in this case study we only multiply the qualitative scales by a constant for quantification. Besides, the importance measures in terms of ratio are not sensitive to the value scales of impact.

Identification of the risk interdependencies (by the steps described in Section 2.2) allows defining the structure of the project risk network, shown in Fig. 2. The network is comprised of 56 risks and 95 risk interactions, with only 5 isolated risks (R8, R11, R15, R23 and R34). For each risk, experts were asked to provide information about the potential causes and effects (to explore the row and the column in the RR matrix corresponding to the considered risk). In the end, the aggregation of local cause-effect relationship identifications makes it possible to display the global risk network. This enables a final meeting to be organized in which interviewees can propose new nodes and connections in the risk graph.

It is mandatory to assess the strength of risk interactions. To do this, two approaches may be considered. The first one is to evaluate them directly using expert judgment, for instance on a 10-level Likert scale. A second solution is to use pair wise comparisons, for instance using the Analytic Hierarchy Process [START_REF] Saaty | The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation[END_REF], as previously introduced in [START_REF] Vidal | Developing an original framework to assess project complexity thanks to the use of the AHP[END_REF].Thanks to the high expertise of interviewees, no AHP-based evaluation was performed in this case, although we would recommend it in most cases.

Furthermore, some bias may be included in the assessment of interactions since it appears that interactions are often conceived in terms of impact and not in terms of precedence. Moreover, this step requires the participation of several experts involved in the project since it necessitates a very wide overview of the project elements and stakes. Great attention should thus be paid to that point when analyzing the results.

One may wonder when is the most appropriate moment to perform this risk network modelling process and the related analysis. Early modelling enables discussions between people who would not have necessarily been in relation due to the project organization to be facilitated. However, information may be neither available nor reliable at the very beginning of the project. The decision about the analysis schedule is thus a balance between doing it early enough and having reliable enough information. In this case, the project had already been launched with eight risk review meetings before the beginning of the study.

B. Results and Analysis

The relevant values of O 𝑏𝑎𝑠𝑒 , 𝑂 ′ 𝑅 𝑖 = 1 and 𝑂 ′ 𝑅 𝑖 = 0 for each risk R i are calculated by simulating the risk network model. In this work, we use the software ARENA ® for simulation. The number of simulation iterations are increased gradually from 1000, 2000, …, to 20 000, and the occurrence of each risk during the run is recorded. The results are then found stable after 14 000 iterations. We adopted 20 000 iterations in this case study forcomputational convenience. The simulation cost is not a limiting factor as the total running time using ARENA on a normal PC is less than 10 minutes. The results shown in Fig. 3 visually illustrate the contribution, namely the varying range of influence, of each risk to the output of global risk exposure.

The values of the proposed risk IMs are then consolidated and compared with the classical risk criticality analysis. We . But in addition, some new risks have been highlighted as key factors by the proposed approach, which are supplementary results and provide the manager with new insights on the risk network. Specifically, R7 (Traffic signaling, priority at intersections), R10 (Travel Time performance) and R2 (Liquidated damages on intermediate milestone and delay of Progress Payment Threshold) are ranked as the top-three risks with marginal influence on the global risk exposure. R10 has a very high RAI but not identified as important risk in terms of its RRI value. It means that although the decrease of risk exposureis relatively low by carrying out mitigation actions on R10, the potential damage/loss could be significant if R10 happens. Thus, the project manager should not propose preventive actions on R10, but keep on monitoring it during the project progress. On the other hand, R7 and R2 also have high RRI values. Hence, it would be worthy to plan risk mitigation actions on them, although the feasibility of actions should be considered.

Because the edges of risk interactions and the nodes of risks in the network are basically different in nature and may be suitable to adopt different kinds of mitigation actions, from the practical point of view, we separately measure and rank them. In this case study, we only consider the Marginal Risk Interaction Importance MRI I for risk interactions. The top-ten critical risk interactions are listed in Table II. These important interactions together with the identified key risks, i.e., the most important factors that affect the output O of risk exposure, are highlighted in Fig. 2. It is obvious that most of the important risk interactions are related to R2 (Liquidated damages on intermediate milestone and delay of Progress Payment Threshold). The reasons are not only that R2 itself has high importance, and it also plays a hub role in the risk network (see Fig. 2). Many other risks will directly or indirectly lead to R2; and R2 has intense relationship with R43 (Return profit decrease) and R55 (Available cash flow decrease) which are also critical risks with high consequences. The project manager should pay attention to the key risk interactions and make efforts to mitigate the risk propagation through them, e.g., by enhancing communication between corresponding actors or preparing alternative solutions to share the risks.

C. Discussion

The risk list used in this research has been directly provided by the manager and corresponded to the list currently used in the project. The formulation and semantics of risks are themselves an issue, and not only in this case. However, the requirement of the manager was to identify the influence of interdependencies between risks on the behavior of the project. The aim was to analyze the gap with the current representation of this behavior, based on classical project risk management techniques and on the classical assumption of independence. Behind the formulation of the risk, there is always an event corresponding to the dissatisfaction of a stakeholder or of a parameter of the project/product. For instance, R5 -Traction/breaking function‖ corresponds to the risk of unsatisfying performance of the traction/breaking function for the client. Similarly, the following risks mean -unsatisfying travel time performance‖, -unsatisfying harmonics level‖, even if the word unsatisfying was not present. This imperfect formulation of risks is a potential source of misinterpretation, not only on risks but also on risk interactions. This is why the work of risk interaction identification and assessment had to be made with several experts and several meetings to be sure to avoid possible ambiguity.

The managers knew that risks are potentially interdependent, but did not have the tools and metrics to study the influence of interdependencies on the global behavior of the project. It means that this complimentary information was immediately judged as useful, since it permitted to:

confirm several intuitions (the manager thinks that this risk may be considered with a higher priority level than its direct criticality assessment); -provide some new findings, e.g., about risks with a significant difference between classical and non-classical indicators and rankings. The reliability of probability estimates in the context of projects is rarely good enough, because of the unicity of projects in terms of at least one of its dimensions (objectives, context, resources, geographical location, etc.) This lack of similar experience does not permit to propose a statistically sound estimation of a probability (or an estimation model). The principle is then to transform these numerical inputs into numerical outputs, where gaps and orders of magnitude are analyzed, not precise values. Knowing that we have this lack of reliability in estimates, or error margin, we suppose that gaps between outputs should be high enough to be significant. The ranking swaps for instance are interesting to analyze, particularly if they involve putting a low-level risk at a high position in importance measure-based rankings, like R10 for instance. On the opposite, R2 and R37 are always very close to each other, and the fact that in one ranking R2 is behind R37 (or not) may not mean that one should be prioritized. It is only when differences are significant, like R43 which was initially the most critical risk, and which appears only one time in the new rankings of RAI, RRI and MRI respectively. The existence of differing rankings according to various indices is not a conflict, since it gives different information and different reasons for proposing a tailored response action for each risk. For instance, a critical risk with less influence on the network can be treated with classical mitigation or avoidance of its direct consequences. A risk with a low criticality but a high importance on the network may be treated with a preventive action to cut propagation chain at the source. This means that the budget invested to prevent this source risk may be cancelled for other risks which are downstream in the reaction chain and initially considered as more critical. We argue that it is usually preferable to act on upstream events of a chain rather than on downstream events, since the latter generally have higher amounts at stake and more expensive treatment actions.

V. CONCLUSIONSAND PERSPECTIVE

This paper presents an original method for project risk analysis and prioritization using importance measures. Risk interactions are accounted for with the purpose of modeling the complexity of interdependent project risks. It is an initial step to develop dependency-driven risk assessment methods.We have showed how the proposed approach can be applied to a real case study of a large engineering project and illustrated that it serves as a powerful complement to the classical project risk analysis.

Various importance measures have been proposed with different purposes for risk analysis and prioritization. For example, the Marginal Risk Importance (MRI) indicates the potential influence range of a risk on the global risk exposure; the Risk Achievement Importance (RAI) provides information to identify the risks which may be dangerous if occurs, even if they are estimated at a low level of exposure; Risk Reduction Importance (RRI) helps to identify the risks which are the most effective targets for planning mitigation actions. The classical criticality analysis of risks can only evaluate the risks that affect the global project performance in a local point of view without taking into account risk interactions. Using the importance measures permits to identify the key factors (i.e., either spontaneous reasons or particular risk interactions) in the risk network that most affect the global performance in terms of risk exposure. For supporting subsequent decisions with respect to risk response planning, this approach enables manager to combine classical risk mitigation strategies (avoidance, mitigation, transfer and acceptance) with new ones which help facing the complexity of risk network: protecting risks due to their position in the network, changing the actors who own risks and/or the organization to adapt it to the current complexity, and reducing the complexity by acting for instance on important risk interactions instead of risks only.

The current study still has a number of limitations. For example, it only considers impact of dependencies on risk materialization; however, the related changes in probability of occurrence and severity of consequenceswill be included. Besides, the risk impacts/consequences are in essence multi-dimensional, namely in terms of schedule, cost, quality, etc., and thus with difficulties to create comparable scales. This approach employs a DSM-based method with pair-wise comparisons to model risk interactions. However, it may fail in accounting for some types of dependencies, for example, a risk may materialize (or change its properties) only as a function of two other risks occurring [START_REF] Salado | The concept of order of conflict in requirements engineering[END_REF].This kind of scenarios should be noticed and future work will be proposed to model them.

Moreover, risk clustering and importance measures of the risk groups will be studied, in order totake into account the synergy or co-effects of risks with respect to mitigation actions planning.Some probabilistic-based risk methods like Monte Carlo analysis will also be considered and compared. Finally, in current study, we assume that the risk network structure do not change during the project development. As perspective, the dynamic situations would be involved in the model.

APPENDIX

Nomenclature

Fig. 1 .

 1 Fig. 1. Matrix-based representation of project risk network (adapted from[START_REF] Fang | A simulation-based risk network model for decision support in project risk management[END_REF]).

Fig. 2 .

 2 Fig. 2. Tramway engineering project risk network (with highlighted important risks and interactions)

Fig. 3 .

 3 Fig. 3. Illustrating the contribution of each risk on the global risk exposure

TABLE I

 I

	highlight the prioritized top-ten risks according to different
	indicators in
			TOP-TEN RISKS ACCORDING TO VARIOUS IMS
	Ranking	Classical risk criticality measure	RAI	Importance Measures of Risks RRI	MRI
		Risk ID	Value	Risk ID Value Risk ID Value Risk ID Value
	1	R43	3.048	R10	0.238	R37	0.079	R7	0.258
	2	R37	2.667	R7	0.190	R16	0.071	R10	0.238
	3	R55	2.667	R6	0.164	R7	0.068	R2	0.213
	4	R2	2.465	R2	0.147	R2	0.066	R37	0.207
	5	R3	1.905	R44	0.147	R18	0.063	R16	0.186
	6	R40	1.541	R37	0.128	R12	0.056	R18	0.164
	7	R12	1.524	R4	0.118	R43	0.048	R6	0.164
	8	R18	1.524	R16	0.115	R55	0.046	R12	0.148
	9	R29	1.524	R27	0.113	R41	0.044	R44	0.147
	10	R7	1.314	R24	0.113	R19	0.038	R41	0.142

Table I .

 I It is found that the rankings using the proposed risk IMs are quite different from the classical risk analysis results. For example, the most critical risk identified by classical analysis was R43 (Return profit decrease), which is closely related to the financial performance of the project.

	However, after taking into account the risk propagation
	behavior, some other risks occupy the pole or premier critical
	position according to different IMs. Another example is R3
	(Vehicle storage in another city), which was regarded
	important, but has fallen out of the top-ten list according to the
	proposed IMs. Overall, several of the critical risks identified by
	classical risk criticality analysis have been confirmed by the
	proposed IMs, e.g., R37 (Risk of partial rejection of our request
	for extension of time), R2 (Liquidated damages on intermediate
	milestone and delay of Progress Payment Threshold), R12
	(Operating certificate delay) and R18 (Civil Work delay &
	continuity)

TABLE II TOP

 II -TEN RISK INTERACTIONS ACCORDING TO MRI I

	Rank	Risk Interaction ID	MRI I
	1	R2->R43	0.220
	2	R2->R55	0.212
	3	R55->R43	0.150
	4	R18->R2	0.139
	5	R12->R2	0.129
	6	R52->R2	0.108
	7	R9->R2	0.096
	8	R48->R2	0.093
	9	R14->R2	0.089
	10	R37->R2	0.089