T S Van Kortryk
email: vankortryk@gmail.com

The Doctoral Students of Richard Feynman

come

Introduction

An ordinary genius is an ordinary fellow ... There is no mystery as to how his mind works. ... It is different with the magicians ... Even after we understand what they have done, the process by which they have done it is completely dark. They seldom, if ever, have students ... Richard Feynman is a magician of the highest caliber.

-Mark Kac [START_REF] Kac | Enigmas of Chance: An Autobiography[END_REF] As the centennial of Richard Feynman's birth approaches, I attempt in this letter to dispel a minor myth about him. The myth is embodied in the words of Mark Kac that I have emphasized above, and in the following statement attributed to one of Feynman's students, Philip Platzman [START_REF] Mehra | The Beat of a Different Drum: The Life and Science of Richard Feynman[END_REF]: "The reason why Feynman did not have many students was because he was very difficult with them, because he didn't really worry about students. ... He had a few students, but not many."

Thus a prevailing belief in the scientific community seems to be [START_REF] Curtright | [END_REF] Feynman had very few doctoral students who completed theses under his supervision. It may be surprising to most people to learn this is not true.

The number of Feynman's doctoral students is actually 31 ± 3, with the uncertainty intended to take into account some unavailable documents as well as possible subjectivity on my part [START_REF][END_REF]. The lineup of students who completed their PhD research under Feynman's discerning gaze began with Michel Baranger, Laurie Brown, and Giovanni Lomanitz at Cornell in 1951, and concluded with Ted Barnes and Thomas Curtright at Caltech in 1977. In addition, two years after Feynman's death he was listed as a co-advisor in Sandip Trivedi's thesis, completed in 1990.

While 31 is not an extremely large number of doctoral students to have mentored during a lifetime as an academic (e.g. compared to the 70+ dissertations supervised by Julian Schwinger during five decades), nonetheless, 31 does amount on average to three PhDs for every four years of Feynman's time as a professor. Moreover, were it not for illness during the last several years of his career, Feynman might have supervised several more students. Among Feynman's doctoral students the most recognized physicist is undoubtedly George Zweig, who was also mentored by Murray Gell-Mann, and who graduated from Caltech in 1964. Soon thereafter Zweig had a major impact on elementary particle physics through his independent invention of the "quark model" of hadrons. However, in my opinion, the research of Feynman's other students has also had significant impact and continues to influence several areas of physics.

The Students

There are PhDs and then there are Feynman PhDs. -Richard Sherman [START_REF] Mehra | The Beat of a Different Drum: The Life and Science of Richard Feynman[END_REF] From theses and PhD dissertation examination documents wherein it was either explicitly stated or otherwise clear that Feynman was the advisor * or co-advisor * * , I find the 31 doctoral students listed here, the first three at Cornell, the others at Caltech: Lorin Vant-Hull (1967) Verification of long range quantum phase coherence ... William Press (1973) Applications of black-hole perturbation techniques Robert Wang (1976) A Study of Some Two-Dimensional Field Theory Models Don Page (1976) Accretion into and emission from black holes Stephen Wolfram (1980) Some topics in theoretical high-energy physics All of these were Caltech students. Originally I included Platzman in this second list. But upon looking at other documents I became convinced that his thesis was effectively co-supervised by Feynman to the extent that he belonged in the first list. (Indeed, as I document later, Platzman's personal listing in the Mathematics Genealogy Project states that Feynman was a co-advisor.) Similar remarks apply for Robert Carlitz [START_REF] Carlitz | 's thesis was not available from the Caltech library when I originally compiled a list of Feynman[END_REF] and Finn Ravndal [START_REF] Ravndal | was also Steven Frautschi[END_REF]. If so, that would justify my head count of 31 "Feynman PhDs" to be a lower bound.

Finally, I find several less compelling cases where Feynman was only a member of the dissertation examination committee at Caltech and was not particularly influential for the research, so far as I can tell. I suspect there are many more such cases that I have not found, since on this point documentation is quite often incomplete and all committee members are not listed. For the last two cases given above, I cannot access the theses to see if Feynman was acknowledged for significant influence.

Sources

At the time of this writing, wikipedia lists only six students to have officially received PhDs with Feynman as the advisor, in alphabetical order: James M. Bardeen, Laurie Mark Brown, Thomas Curtright, Al Hibbs, Giovanni Rossi Lomanitz, and George Zweig. However, this list is obviously far from complete, as documented by the Math Genealogy Project (MGP) and by the Caltech library archives.

According to the MGP, also at the time of this writing, there were at least thirteen other doctoral degrees completed under Feynman's supervision in addition to those listed in wikipedia -one more at Cornell and twelve more at Caltech. In particular, Philip Platzman is in the MGP list but not in wikipedia. I suppose that is because he requested MGP to classify him as a doctoral student of Feynman. Platzman's personal listing in MGP supports my supposition. By way of comparison, and as a measure of the completeness of their database, Schwinger has only 22 of his students listed by the MGP.

In any case, the mother lode of information about Feynman's doctoral students can be found at the Caltech library. A direct search of their online database produces a list of 25 PhD theses where Feynman is described as the advisor or co-advisor, at the time of this writing. By way of comparison, a direct search for Gell-Mann as advisor turns up 16 theses in the Caltech library database. Among these, Hilton, Levine, and Zweig are shown to be co-advised by Feynman and Gell-Mann.

Beyond these publicly accessible sources, the largest amount of documentation that is available to me concerns Thomas Curtright, who has provided this succinctly amusing excerpt from his thesis examination committee papers [START_REF] Curtright | [END_REF]:

  For example: Lipes, Richard Gwin (1969) I. Application of multi-Regge theory ... II. High energy model .... Hill, Christopher Thaddeus (1977) Higgs scalars and the nonleptonic weak interactions. Dally, William J. (1986) A VLSI architecture for concurrent data structures. (restricted) Wawrzynek, John (1987) VLSI concurrent computation for music synthesis. (restricted)

Michel Baranger * * (1951) Relativistic Corrections to the Lamb Shift

Acknowledgements: I thank Professor Curtright for suggesting that there could very well be a widespread misunderstanding about the extent of Feynman's mentoring of doctoral students. Finally, I thank Cosmas Zachos for his comments on various drafts of this manuscript.

Laurie Brown * (1951) Radiative corrections to the Klein-Nishina formula Giovanni Lomanitz * (1951) Second order effects in the electron-electron interaction Albert Hibbs * (1955) The growth of water waves due to the action of the wind William Karzas * * (1955) The effects of atomic electrons on nuclear radiation Koichi Mano * (1955) The self-energy of the scalar nucleon Gerald Speisman * (1955) The neutron-proton mass difference Truman Woodruff * * (1955) On the orthogonalized plane wave method for calculating ... Michael Cohen * (1956) The energy spectrum of the excitations in liquid helium Samuel Berman * (1959) Radiative corrections to muon and neutron decay Frank Vernon * (1959) The theory of a general quantum system interacting ... dissipative system Willard Wells * (1959) Quantum theory of coupled systems having application to masers Henry Hilton * * (1960) Comparison of the beta-spectra of boron 12 and nitrogen 12 Carl Iddings * (1960) Nuclear size corrections to the hyperfine structure of hydrogen Philip Platzman * * (1960) Meson theoretical origins of the non-static two nucleon potential Marvin Chester * * (1961) Some experimental and theoretical observations on ... EMF Elisha Huggins * (1962) Quantum mechanics of the interaction of gravity ... Harold Yura * (1962) The quantum electrodynamics of a medium Michael Levine * * (1963) Neutrino processes of significance in stars George Zweig * * (1964) Two topics in elementary particle physics ... James Bardeen * * (1965) Stability and dynamics of spherically symmetric masses ... Howard Kabakow * * (1969) A perturbation procedure for nonlinear oscillations ... Robert Carlitz * * (1970) Elimination of parity doubled states from Regge amplitudes [5] Mark Kislinger * * (1970) Elimination of parity doublets in Regge amplitudes Finn Ravndal * * (1971) A relativistic quark model with harmonic dynamics [6] Richard Sherman * (1971) Surface impedance theory for superconductors in ... magnetic fields Arturo Cisneros * * (1973) I. Baryon-Antibaryon phase transition ... II. ... the Parton Model Steven Kauffmann * (1973) Ortho-positronium annihilation ... first order radiative corrections Frank (Ted) Barnes * * (1977) Quarks, gluons, bags, and hadrons Thomas Curtright * (1977) Stability and Supersymmetry Sandip Trivedi * * (1990) Topics in quantum gravity

From documents where Feynman was not described as an advisor or co-advisor, but was a member of the PhD examination committee although not the committee chairman, and/or was acknowledged in the thesis for moderate influence and general advice, I find in addition: Fredrik Zachariasen (1956) Photodisintegration of the deuteron Paul Craig (1959) Observations of perfect potential flow and critical velocities in superfluid ... James Mercereau (1959) Diffraction of Thermal Waves in Liquid Helium II Kenneth Wilson (1961) An investigation of the Low ... and the Chew-Mandelstam equations John Andelin (1966) Superfluid drag in helium II Karvel Thornber (1966) I. Electronic Processes ... II. Polaron Motion ...

Summary

From looking at many theses and papers by Caltech students, my overall impression is simply this: Feynman played a major role through his mentoring and supervision of doctoral students. He exerted tremendous influence on graduate student research conducted at Caltech during his four decades there -perhaps even more than his widely perceived influence on Caltech undergraduate studies. I conclude that it is not true Richard Feynman "had a few students, but not many."


El-Hachemi Guerrout
email: e_guerrout@esi.dz

Ramdane Mahiou
email: r_mahiou@esi.dz

Samy Ait-Aoudia
email: s_ait_aoudia@esi.dz

Hidden Markov Random Fields and Particle Swarm Combination for Brain Magnetic Resonance Image Segmentation

Keywords: Brain image segmentation, Hidden Markov Random Field, Swarm Particles Optimization, Dice coefficient

The interpretation of brain images is a crucial task in the practitioners' diagnosis process. Segmentation is one of key operations to provide a decision support to physicians. There are several methods to perform segmentation. We use Hidden Markov Random Fields (HMRF) for modelling the segmentation problem. This elegant model leads to an optimization problem. Particles Swarm Optimization (PSO) method is used to achieve brain magnetic resonance image segmentation. Setting the parameters of the HMRF-PSO method is a task in itself. We conduct a study for the choice of parameters that give a good segmentation. The segmentation quality is evaluated on ground-truth images, using the Dice coefficient also called Kappa index. The results show a superiority of the HMRF-PSO method, compared to methods such as Classical MRF and MRF using variants of ACO (Ant Colony Optimization).

Introduction

With the overwhelming number of medical images, the manual analysis and interpretation of images from different imaging modalities (Radiography, MRI -Magnetic Resonance Imaging, CT -Computed tomography, etc) became a tedious task. This fact underlines the necessity of automatic image analysis, through several operations including segmentation.

Hidden Markov Random Field (HMRF) provides an elegant way to model the segmentation problem. Since the seminal paper of Geman and Geman [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF], Markov Random Fields (MRF) models for image segmentation have been extensively investigated [START_REF] Backes | Medical image retrieval and analysis by Markov random fields and multi-scale fractal dimension[END_REF][START_REF] Guerrout | Hidden Markov Random Fields and Swarm Particles: A Winning Combination in Image Segmentation[END_REF][START_REF] Yousefi | Brain tissue segmentation in MR images based on a hybrid of MRF and social algorithms[END_REF][START_REF] Zhang | Hidden Markov random field model based brain MR image segmentation using clonal selection algorithm and Markov chain Monte Carlo method[END_REF]. The segmentation process consists in finding the hidden information namely the segmented image by observing the data from the original image. We seek the segmented image, according to the MAP (Maximum A Posteriori) criterion [START_REF] Wyatt | MAP MRF joint segmentation and registration of medical images[END_REF]. MAP estimation leads to the minimization of energy function [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]. This problem is computationally intractable. Therefore, optimization techniques are used to compute a solution.

Particle Swarm Optimization (PSO) has emerged as one of the best optimization techniques. This new class of metaheuristics was proposed in 1995 by Eberhart and Kennedy [START_REF] Eberhart | A new optimizer using particle swarm theory[END_REF]. This technique was extensively studied by many researches [START_REF] Kennedy | Particle swarm optimization[END_REF][START_REF] Shi | Empirical study of particle swarm optimization[END_REF]. The selection of PSO parameters, in the algorithm simulation, is a problem in itself [START_REF] Clerc | The particle swarmexplosion, stability, and convergence in a multidimensional complex space[END_REF][START_REF] Eberhart | A new optimizer using particle swarm theory[END_REF][START_REF] Eberhart | Comparing inertia weights and constriction factors in particle swarm optimization[END_REF][START_REF] Guerrout | Hidden Markov Random Fields and Swarm Particles: A Winning Combination in Image Segmentation[END_REF]. A bad choice of parameters can lead to a chaotic behaviour of the optimization algorithm. In this paper we investigate parameters setting and performance of Hidden Markov Random Field (HMRF) and Particle Swarm Optimization (PSO) combination named HMRF-PSO [START_REF] Ait-Aoudia | Medical Image Segmentation Using Particle Swarm Optimization[END_REF][START_REF] Guerrout | Hidden Markov Random Fields and Swarm Particles: A Winning Combination in Image Segmentation[END_REF] in brain MR (Magnetic Resonance) images segmentation. In this specific case, segmentation consists in partitioning the brain image into different characteristic parts that are gray matter, white matter and cerebrospinal fluid. The segmentation evaluation is conducted on ground-truth images from Brainweb1 and IBSR2 databases. The Dice coefficient is used to assess the quality of the segmentation. The HMRF-PSO method is compared to methods using Markov Random Field and variants of Ant Colony Optimization [START_REF] Yousefi | Brain tissue segmentation in MR images based on a hybrid of MRF and social algorithms[END_REF]. The achieved results are promising and show a clear superiority of the HMRF-PSO method. This paper is organized as follows. An overview of previous work is given in section 2. In section 3, we give The Hidden Markov Random Field model principles in the context of image segmentation. The Particle Swarm Optimization and HMRF combination is explained in section 4. Experimental results on medical samples are given in section 5. Section 6 concludes the paper.

Previous work

Brain MRI images segmentation has attracted a particular focus in medical imaging. The importance of this modality has favoured the abundance of research on automatic extraction of image characteristics resulting from medical examinations. The segmentation techniques can be classified in four broad categories: Threshold-based techniques, Regionbased techniques, Classification techniques and Model-based techniques [START_REF] Gordillo | State of the art survey on MRI brain tumor segmentation[END_REF].

Threshold-based techniques [START_REF] Natarajan | Tumor detection using threshold operation in MRI brain images[END_REF][START_REF] Shanthi | Skull stripping and automatic segmentation of brain MRI using seed growth and threshold techniques[END_REF] use image histogram and find one or more intensity thresholds to identify the different classes of the image. In the case of foreground/background separation, one threshold is identified. If the image contains n distinctive classes (n-1) thresholds are necessary. The threshold-based techniques are very noise sensitive. Region based techniques explore the pixels of the image and assemble them in non overlapping regions according to a criterion of homogeneity. In this context, several authors used region growing [START_REF] Roura | MARGA: multispectral adaptive region growing algorithm for brain extraction on axial MRI[END_REF] or watersheds to perform segmentation [START_REF] Grau | Improved watershed transform for medical image segmentation using prior information[END_REF][START_REF] Salman | Image Segmentation Based on Watershed and Edge Detection Techniques[END_REF]. In classification techniques, pixels are grouped based on the some properties of these pixels (grey levels, texture or colour. The groups formed are called clusters. Cmeans based techniques [START_REF] Ahmed | A modified fuzzy c-means algorithm for bias field estimation and segmentation of MRI data[END_REF][START_REF] Zhang | A novel kernelized fuzzy cmeans algorithm with application in medical image segmentation[END_REF] and Markov Random Fields are part of the classification-based methods and are widely used for brain segmentation. In model-based segmentation, as in deformable models and level sets, a model is built for a specific anatomic structure by incorporating a priori information concerning this structure (shape, location, and orientation) [START_REF] Ho | Level-set evolution with region competition: automatic 3-D segmentation of brain tumors[END_REF][START_REF] Mcinerney | Deformable models in medical image analysis: a survey[END_REF]. The presence of noise in the acquired images can severely degrade the segmentation results and thus makes the process of segmentation useless. Denoising is generally performed prior to the effective segmentation [START_REF] Bhateja | A reconstruction based measure for assessment of mammogram edge-maps[END_REF][START_REF] Bhateja | A non-local means filtering algorithm for restoration of rician distributed MRI[END_REF][START_REF] Gupta | A noise robust edge detector for color images using Hilbert Transform[END_REF][START_REF] Makni | Introducing spatial neighbourhood in Evidential C-Means for segmentation of multi-source images: application to prostate multiparametric MRI[END_REF][START_REF] Alankrita | Computer aided detection of brain tumor in magnetic resonance images[END_REF][START_REF] Srivastava | Combination of wavelet transform and morphological filtering for enhancement of magnetic resonance images[END_REF][START_REF] Xiao | Patch-based label fusion segmentation of brainstem structures with dual-contrast MRI for Parkinson's disease[END_REF].

Hidden Markov Random Field (HMRF)

An image is formed of a finite set S of sites corresponding to the pixels, The segmented image,

  1 , , , , sM x x x x    is seen as the realization of another Markov Random Field,   1 , , , , sM X X X X   
, defined on the same lattice S, takes its values in the discrete space ={1,2,…,K}. K represents the number of classes or homogeneous regions in the image. The configuration set is .

The Hidden Markov Random Filed provides an elegant way to model the segmentation problem by using the MAP (Maximum A Posteriori) estimator. This latter consists in finding a realization x of X by observing the data of the realization y, representing the image to segment (The figure 1 shows an example for K=4).

y: Observed Image x: Hidden Image The aim of the MAP estimator is to maximize the probability P(X=x|Y=y) which is equivalent in this context to minimize the function (x,y)

 [2,15]. 2 s 2 s, C (y -)² (x,y) ln( ) (1-2 ( , ) 2 s s s x x s t s S t x xx T                       (1)
Where  is a constant, T is a control parameter called temperature,

 is a Kronecker's delta,   1 , , , , jK        and   1 ,
    1 ,y 0...255 ( ,y)= K K j j f if else             (2)           2 2 2 , ln 1 2 ( y) , ,= 2 j s j st c s t j j j sS y xx T f                      
          1 , , , , i i ij iK m t m t m t m t   
and a velocity

    1 , , , , i i ij iK v t v v v   
at the time t.

Each particle i at the time t compute its own segmented

image           1 , , , , i i is iM x t x t x t x t    using its position   i mtwhere   is x t j  if the nearest mean to s y is   ij mt .
Each particle i at the time t measure its position   i mt

which equal to     , i m t y  . Let           ' ' ' ' 1 , , , , i i ij iK m t m t m t m t   
the best position visited by the particle i till the time t (we called it the local best), so:

                      ' ' ' ' , 1 1 1, , 1 , i i i i i i i y m t y m t if m t mt m t if m t y m t y                (3) Let           1 , , , , jK t t t t       
the best position visited by all the particles till the time t (we called it the global best), so:

          ' ' ' 1 ,..., ,..., i NP t m t m t m t  
Where NP is the number of particles

              ' '' 1 ,, arg min ,..., i NP mt t m t m t yy     (4)
The velocity  

                            ' 11 22 1* ** ** ij ij ij ij ij j ij ij ij j j ij v t w v t L t G t L t c r m t m t G t c r t m t t t               ( 5
)
Where w is called the inertia weight, 1 c and 2 c are the acceleration constants;

 

      11 i i i m t m t v t     (6)
When the maximal number of iterations is reached, we take the global best as the solution  :  

Experimental Results

Parameters Setting

Evaluating the quality of the segmentation using the Dice Coefficient [START_REF] Dice | Measures of the amount of ecologic association between species[END_REF] can only be made where the a priori segmentation is known. The matching between the class

HMRF parameters

Choosing the appropriate parameters for HMRF-PSO combination method is a delicate task. A bad choice can lead to poor results. We will focus in this section on setting the parameters of HMRF process.

PSO parameters

To select the correct settings for PSO algorithm, we have performed a statistical analysis on brain images segmentation differentiating between Grey Matter (GM), White Matter (WM) and Cerebro-Spinal Fluid (CSF) classes. An overview of the results is given below.

The number of iterations is a parameter that has a very important role in convergence and terminates the optimization process. Figure 6 shows the influence of the change in the number of iterations on the segmentation quality. Other process parameters have been set at: c1=0.7, c2=0.9, w=0.7, vmax=5, swarm_size =30 and =1. Figure 7 shows the swarm velocity influence on the quality of the segmentation. Other process parameters have been set at: c1=0.4, c2=0.4, w=0.4, iteration_number=100, swarm_size =40 and =1.

Vmax speed must be limited to avoid the algorithm performance degradation. Beyond a certain speed, the segmentation quality deteriorates. The size of the swarm plays a very important role in the optimization process. This parameter affects robustness and quality of the result. Figure 8 shows the influence of the change of the swarm size on the quality segmentation. Other process parameters have been set at: c1=0.6, c2=0.7, w=0.6, vmax=5, iteration_number=100 and =1.

Comparative study

To assess the HMRF-PSO combination method, we made a comparative study with four segmentation algorithms operating on brain images that are Classical MRF, MRF-ACO (Ant Colony Optimization), MRF-ACO-Gossiping [START_REF] Yousefi | Brain tissue segmentation in MR images based on a hybrid of MRF and social algorithms[END_REF] and LGMM [START_REF] Liu | Image segmentation using a local GMM in a variational framework[END_REF].

To perform a meaningful comparison, we use the brain medical images with their ground truth from IBSR database and Brainweb database with Modality=T1 and Slice thickness=1mm. The comparison will be based on the Dice coefficient. Brainweb [START_REF] Cocosco | Brainweb: Online interface to a 3D MRI simulated brain database[END_REF] The results clearly show the prevalence of the HMRF-PSO method compared with other methods. The HMRF-PSO combination globally outperforms

LGMM method [START_REF] Liu | Image segmentation using a local GMM in a variational framework[END_REF].

Conclusion

In this paper, we have presented a method referred to as HMRF-PSO that combines Hidden Markov Random Fields and Particle Swarm Optimization to perform segmentation. A statistical study was also carried out to set the parameters of the method. The tests conducted have focused on the brain images from the largely used databases, Brainweb and IBSR.

The HMRF-PSO combination outperforms other combination methods tested that are: Classical MRF, MRF-ACO-Gossiping and MRF-ACO methods. Therefore, the proposed method has the potential to be used in computer aided medical diagnosis Systems. Nonetheless, the proposed method has to be tested on images coming from other modalities. A comparative study with other segmentation methods must also be conducted to confirm the method supremacy.

On the other hand, direct search techniques, specifically Nelder-Mead and Torczon methods, are currently under investigation to solve the optimisation problem.

.

  sites. The sites in S are related by neighbourhood system V(S).The image to segment into K classes or the observed image, the Hidden Markov Random Field model as a realization of a family of random variables defined on S, Each random variables {Y s } sS takes its values in the space  obs ={0..255}. The configuration set is  obs .

Figure 1 .

 1 Figure 1. Observed and hidden image.

jS

  contains all the sites s such that the nearest mean to particle i has a position (a solution)

1 ivt

 1  of the particle i at the time t+1 is influenced by its local best

  true positive; FP is the false positive and FN is the false negative.

Figure 2 .

 2 Figure 2. TP, FP and FN.

Figure 3

 3 Figure3shows an IRM scan and its corresponding segmentations in four classes with =1, T 0 =4 and varying the parameter .

Figure 3 .

 3 Figure 3. Segmentation varying .

Figure 4 Figure 4 .

 44 Figure4shows an image and its corresponding segmentations in two classes with =1, =0.98 and varying the parameter T0.

Figure 5 Figure 5 .

 55 Figure5shows an image and its corresponding segmentations in two classes with =0.98, T 0 =4 varying the parameter .

Figure 6 .

 6 Figure 6. Iteration number variation.

Figure 7 .

 7 Figure 7. Maximum velocity variation.

Figure 8 .

 8 Figure 8. Particle number variation.

  , , ,

												jK  
	are respectively the means and the standards deviation
	of the K classes in the segmented image x .
	Computing	the	exact	segmentation
				  		of the image y is impossible
	but		we	can	seek	an	approximation
	  x x 	  sM x x	of the exact segmentation
	x	*		* 1 ( ,..., ,..., ) * * sM x x x	using optimization techniques.
	Our way to look for an approximation  x is to seek an
	approximation     j     	K	of	*     * * 1 ( ,..., ,... ) * j K 
	where
	The segmented image x is calculated after
	computation of  by classifying s y into the nearest
	mean Our goal becomes looking for *  instead * x such that
		*	 arg min ( ,y)  K     	where:

* arg min (x,y) x x 1 ( ,..., ,... ) 1 ,..., ,... ) * j  is the mean of the class j in * x and  j  is the mean of the class j in  x . j  of  (i.e. s xj  if the nearest mean of s y is j  ).

Table 1 .

 1 Parameters of methods used.

	images are simulated MRI volumes for
	normal brain from McGill University. These
	simulations are based on an anatomical model of
	normal brain. In this database, an image can be selected
	by setting modality, slice thickness, noise and intensity
	non-uniformity.
	The Internet Brain Segmentation Repository (IBSR)
	provides manually-guided expert segmented brain data.
	This repository is made available to the scientific
	community by the Neuroimaging Informatics Tools and
	Resources Clearinghouse (NITRC) [20]. Its aim is to
	encourage the evaluation and development of
	segmentation methods.
	The parameters setting for the four methods used in the
	study are summarizes in table 1.
	Method	N°	Parameters
	Classical MRF	1	T: Temperature=4
			T: Temperature=4 , a: Pheromone info.
	MRF-ACO	2	Influence=1, b: Heuristic info. Influence=1, q: Evaporation rate=0.1, w: Pheromone decay
			coefficient=0.1
			T: Temperature=4 , a: Pheromone info.
			Influence=1, b: Heuristic info. Influence=1, q:
	MRF-ACO-Gossiping	3	Evaporation rate=0.1, w: Pheromone decay coefficient=0.1, c1: Pheromone reinforcing
			coefficient=10, c2: Pheromone reinforcing
			coefficient=100
			T: Temperature=4 , =0.98, c1=0.6, c2=0.7,
	HMRF-PSO	4	w=0.6, vmax=5, swarm_size =70,
			iteration_number=100, =1

Table 2

 2

	shows the mean values of Dice coefficient for
	Brainweb database. The slices, used form Brainweb
	database are: 85, 88, 90, 95, 97, 100, 104, 106, 110, 121
	and 130.

Table 2 .

 2 The mean values of DC.

	The method	GM WM CSF Mean
	Classical-MRF	0.75 0.72	0.78	0.75
	MRF-ACO	0.72 0.76	0.78	0.75
	MRF-ACO-Gossiping 0.72 0.76	0.78	0.75
	HMRF-PSO	0.95 0.98	0.93	0.95
	The table 3 shows the mean values of Dice coefficient
	for IBSR database. The slices, used from 1-24(IBSR
	database), are: 18, 20, 24, 26, 30, 32 and 34.

Table 3 .

 3 The mean values of DC.

	The method	GM WM CSF Mean
	Classical-MRF	0.76 0.82	0.24	0.61
	MRF-ACO	0.77 0.82	0.25	0.61
	MRF-ACO-Gossiping 0.77 0.82	0.25	0.61
	HMRF-PSO	0.83 0.85	0.31	0.66

Table 4

 4

shows the comparison between HMRF-PSO with The parameters: c1=0.45, c2=0.8, w=0.45 and LGMM method

[START_REF] Liu | Image segmentation using a local GMM in a variational framework[END_REF]

. We have used the Brainweb databases with different noise levels. The slices used are 85, 88, 90, 95, 97, 100, 104, 106 and 110. The first column (N,I) gives the noise and the intensity nonuniformity.

Table 4 .

 4 The mean values of DC.

		LGMM		HMRF-PSO
	(N,I)	GM WM	CSF	Mean GM WM CSF Mean
	(0%,0%)	0.69 0.66	0.75	0.70	0.95 0.98	0.95	0.96
	(3%,20%) 0.90 0.94	0.891 0.91	0.94 0.96	0.94	0.95
	(5%,20%) 0.91 0.95	0.88	0.91	0.91 0.95	0.92	0.93
	(7%,20%) 0.90 0.95 0.87	0.91	0.88 0.93	0.89	0.90
	(9%,20%) 0.19 0.74	0.73	0.56	0.81 0.90	0.78	0.83

http://www.bic.mni.mcgill.ca/brainweb/

http://www.nitrc.org/projects/ibsr/


P Abivin

I Henaut

C Chaudemanche

J F Argillier

F Chinesta

M Moan

Dispersed Systems in Heavy Crude Oils

Systèmes dispersés dans les bruts lourds -La diminution des ressources de pétrole conventionnel conduit les producteurs à orienter la recherche vers l'exploitation de bruts difficiles. Parmi ceux-ci, les bruts lourds, caractérisés par leur forte viscosité, constituent des réserves de pétrole abondantes. Des procédés particuliers tels que la production froide ou l'injection de vapeur permettent de récupérer ces bruts malgré leur forte viscosité. Lors de la production froide, l'huile peut avoir l'aspect d'une mousse que l'on nomme « foamy oil » : sous l'effet de la dépressurisation, certains éléments légers comme le méthane ou l'éthane peuvent passer à l'état gazeux sous forme de bulles. L'injection de vapeur implique, quant à elle, une production importante d'eau émulsionnée avec le brut. Le but de cet article est d'étudier l'influence de ces phases dispersées (gaz ou eau) sur les propriétés d'écoulement du brut dans les conditions de production (pression, température). À l'aide d'un procédé novateur, des mousses de brut lourd et des émulsions ont été reconstituées et leurs propriétés d'écoulement étudiées à l'aide d'un rhéomètre équipé d'une cellule sous pression. La première partie de l'article concerne la présence de bulles dans le brut lourd. Nous montrons qu'une forte déformation des bulles peut mener à une diminution significative de la viscosité de la mousse. En revanche, lorsque les bulles sont sphériques, elles contribuent à une augmentation de la viscosité du mélange. Dans la deuxième partie, les émulsions eau-dans-huile sont étudiées. De la même façon, sous une vitesse de cisaillement suffisamment élevée pour déformer les gouttes, on observe une diminution de la viscosité apparente de l'émulsion. La dernière partie décrit quelques essais qui combinent l'influence de la présence de bulles et de gouttes d'eau sur les propriétés d'écoulement des bruts lourds, se rapprochant des conditions réelles de production. Nous étudions notamment le rôle de la présence de gouttes d'eau sur le comportement moussant du brut. Finalement, ce travail expérimental montre que la viscosité des fluides de gisements dépend fortement des conditions de production et qu'il est essentiel d'en étudier le comportement rhéologique en fonction de la température, de la pression et de la présence de phases dispersées.

Abstract -Dispersed Systems in Heavy Crude Oils -Due to very large reserves, heavy crude oils represent an important challenge for the oil industry. Their production has to handle very viscous hydrocarbons and involves the use of advanced methods such as cold production or steam injection. Cold production is faced with the foamy oil phenomenon: while extracted, a heavy crude oil is submitted to a depletion that can induce the formation of dispersed gas bubbles and makes it appear as a foam. With steam injection, water is invariably produced and emulsions may be formed. These two situations show how heavy crude oils are concerned with dispersed systems. The aim of the article is to evaluate the impact of this dispersed phase, either bubbles or droplets, on the flow properties of heavy crude oils. To do so, measurements have been carried out with a controlled stress rheometer under pressure and temperature. Live samples were obtained by recombining a stock tank oil with methane inside the pressure cell of the rheometer. The first part of the article is dedicated to the foamy oil phenomenon. It

INTRODUCTION

Because of very large reserves, heavy crude oils represent a challenge for the future. Indeed, the oil companies still have significant difficulties to produce these very viscous oils. Their rheological behavior appears as a key issue and has already been addressed [START_REF] Coustet-Pierre | Caractérisation du comportement rhéologique des bruts lourds en vue de l'optimisation de leur transport[END_REF]. These results on pure heavy oils are of primary importance in the surface pipelines but in wells the influence of the gas and water that are co-produced should also be taken into account. Actually, the flow properties of the resulting multiphase mixture are much more complex than the ones of a single dead crude oil.

The pressure inside a crude oil reservoir results in a thermodynamic equilibrium between the gas cap, the lightest hydrocarbons (methane, ethane...) dissolved in the oil and the liquid crude oil. The amount of dissolved gas in this so-called "live oil" is proportional to the system pressure and obeys Henry's law: [G] = H × P, where H is Henry's constant, depending on both fluids, P the pressure and [G] the amount of dissolved gas. During its extraction and its transport from the reservoir to the wellhead, the live oil is depressurized. The equilibrium given by Henry's law is then broken, the live oil is supersaturated. To restore the equilibrium, the system tends to get rid of the excess of dissolved gas. So the depletion usually leads to the nucleation and growth of bubbles within the oil, what modifies its flow properties. In the case of light oils, bubbles coalesce very quickly and a slug flow can appear out of the well [START_REF] Taitel | Stability of severe slugging[END_REF][START_REF] Henriot | Multiphase production control: application to slug flow[END_REF]. On the opposite, in the case of heavy oils, bubbles remain dispersed and flow within the oil, what gives to the produced oil the aspect of a "chocolate mousse". This phenomenon, called "foamy oil phenomenon" [START_REF] Smith | Fluid flow and sand production in heavy oil reservoirs under solution-gas drive[END_REF][START_REF] Maini | Significance of foamy oil behaviour in primary production of heavy oils[END_REF], appears to cause higher production rates than expected by classical reservoir modelling [START_REF] Smith | Fluid flow and sand production in heavy oil reservoirs under solution-gas drive[END_REF][START_REF] Huerta | Understanding foamy oil mechanisms for heavy oil reservoirs during primary production[END_REF]. Furthermore, favoring the foamy oil behavior is a technique used to enhance heavy oil recovery [START_REF] Weatherhill | Cold heavy oil production at Patos, -Marinza, Albania[END_REF]. To understand it, several theories have been explored. One of them considers that bubbles trapped in the reservoir maintain a high pressure in the porous medium, what enhances the oil recovery [START_REF] Nauroy | Production froide des huiles visqueuses[END_REF]. Other theories suggest that the live oil viscosity is lower with the presence of bubbles, thus improving its transport in the well and the reservoir. For instance, it is proposed that asphaltenes migrate to the surface of bubbles [START_REF] Bauget | Production d'huiles lourdes par dépressurisation : études des interfaces huile-air et modélisation du procédé[END_REF] and the resulted impoverishment of the matrix in asphaltenes decreases the foamy oil viscosity [START_REF] Chaudemanche | Effect of pressure on rheological properties of water-in-crude oil emulsions[END_REF]. However, different experimental studies predict either an increase [START_REF] Bora | Cold production of heavy oils: an experimental investigation of foamy oil flow in porous media[END_REF][START_REF] Fisher | Use of magnetic resonance imaging as a tool for the study of foamy oil behaviour for an extra-heavy crude oil[END_REF][START_REF] Goodarzi | Heavy-oil fluid testing with conventional and novel techniques[END_REF] or a decrease [START_REF] Islam | Mechanics of bubble flow in heavy oil reservoirs[END_REF][START_REF] Albartamani | Experimental studies on "foamy oil" phenomena[END_REF] in the live oil viscosity with the presence of bubbles. The influence of bubbles on the heavy oil flow properties is thus still debated.

In addition to gas, water is invariably produced, leading most of the time to the formation of water-in-oil (W/O) emulsions. The presence of water is particularly encountered with the use of steam injection, process currently utilized for extra heavy oil production. These emulsions are not only stable due to natural surfactants (such as asphaltenes, naphtenic acids and resins) but also very viscous. Actually, their viscosity increases as the water content increases [START_REF] Singh | Flow properties of Alaskan heavy-oil emulsions[END_REF] up to the inversion point, where W/O emulsions are transformed into O/W emulsions. In the field of heavy oil transportation, one way to reach low viscosities is thus to switch from W/O to O/W emulsions. A particular attention has been paid in the literature to the accurate determination of this inversion point [START_REF] Szelag | Rheological properties of emulsions stabilized by acylglycerol emulsifiers modified with dodium carboxylates[END_REF][START_REF] Alboudwarej | Rheology of heavy-oil emulsions[END_REF]. Nevertheless, production rates higher than expected have been noticed [START_REF] Loughead | Lloydminster heavy oil production -why so unusual?[END_REF][START_REF] Huerta | Understanding foamy oil mechanisms for heavy oil reservoirs during primary production[END_REF], with low water-inoil ratio, far from the inversion point.

Thus heavy-oil production process gives rise to a complex dispersed system in which gas bubbles and water droplets flow within the oil, each of them affecting the apparent viscosity of the extracted fluid. The understanding of the flow properties of this multiphase mixture is fundamental to predict the performance of a production plant. So far, not many investigations have been devoted to these particular dispersions. The present work gradually addresses the problem of heavy oil flow properties in real conditions, i.e. with the presence of gas bubbles and water droplets. In the first part, the influence of gas bubbles on the viscosity of heavy oils is analyzed. It is followed in the second part by the study of the demonstrates that contrary to spherical bubbles, deformed bubbles can give rise to significant viscosity reduction, which is governed by the capillary number. The second part deals with water-in-oil emulsions. Again, experimental results show that dispersed droplets increase viscosity unless they get sufficiently elongated by the applied shear rate. The last part of the article is particularly original as it studies some rheological experiments of heavy crude oils in presence of both gas bubbles and water droplets. It investigates to what extent the presence of water influences the bubbly characteristics of live heavy crude oils. The experimental work suggests that for optimum design and operations, it is essential to determine oilfield viscosities i.e. under pressure and temperature on live and changing dispersed systems.

influence of water droplets on heavy-oil rheological behavior under field conditions. Finally, in the third part, some experiments performed on heavy oils containing both gas bubbles and water droplets are presented.

FOAMY OIL PHENOMENON

This section focuses on the foaming behaviour of heavy oils. Its purpose is to assess how the formation and the presence of bubbles can modify heavy oil viscosity. A new experimental approach has been developed to carry out this work. It consists in depleting different live oils in the pressure cell of controlled-stress rheometers. By measuring the viscosity and following the morphology of the created foam, the influence of bubbles on oil viscosity has been established. Particular attention has been paid to the role of shear rate, that in turn controls the shape of the bubbles.

Materials and Methods

The density of the Canadian heavy crude oil used in this study is 0.982 at 25°C, its viscosity is 9 Pa.s at 28.5°C and its asphaltene content is roughly 11 weight % (determined by SARA analysis in heptane). We used two controlled-stress rheometers: Anton Paar Physica MCR 501 and TA Instruments AR2000. Both are fitted with a pressure cell (from 102 to 10 kPa). A schematic diagram of these experimental devices is presented in Figure 1. Magnetic transmission is used between the rheometer motor and the geometry. Pressure is applied to the sample by gas (either nitrogen or methane) and controlled by a pressure regulating valve.

Several geometries can be set: a vane, an helix or a Couettelike cylinder. We equally used the vane and the helix to avoid slipping effects and to improve the oil/gas mixing process. The Couette geometry is used to study the piezodependence of a liquid without dissolving any gas. The calculation of the viscosity is based on the Couette analogy [START_REF] Ait-Kadi | Quantitative analysis of mixer-type rheometers using the Couette analogy[END_REF].

The experiments are divided into two steps. First, the live oil is prepared by introducing gas into the dead oil under pressure and shear (100 s -1 ). The gas slowly dissolves in the oil until the system reaches the equilibrium given by Henry's law. Mixing is very important during the process: the best the mixing, the fastest the system reaches the thermodynamic equilibrium, i.e. saturation. Using the helix and the vane devices, this process is a few hours long and is impossible with a classic Couette geometry. During this operation, the viscosity is continuously measured and drops with the dissolution of gas. The live oil is considered as saturated when the viscosity is stabilized. The gas dissolution conditions were the same for all the samples: 30 bars in methane. The amount of dissolved gas has been measured using PVT cells and approximately corresponds to a GOR (Gas Oil Ratio, ratio between gas volume and oil volume at room pressure and 15°C) of 6. During the second step, the live oil is depressurized from the saturation pressure to atmosphere. Again, the viscosity of the system is continuously measured under a constant shear rate during and after the depletion.

Results on Heavy Crude Oil

Depletions of live heavy crude oils have been performed both in quiescent and dynamic conditions. They lead to drastically different results that are successively described below.

Depletion Under an Oscillatory Shear

The first rheological experiments have been done in quiescent conditions with oscillatory tests, i.e. very low strain (5 Hz, 2%), on the Canadian heavy oil saturated in methane (30 bars). These tests allow to study the viscoelastic behaviour of the foamy oil without any deformation of the bubbles. Indeed, under these conditions, they remain spherical. Figure 2 depicts the evolution of the corresponding complex viscosity after the depletion. We plotted the viscosity ratio which is defined as the ratio between the live oil viscosity and the oil viscosity without dissolved gas. As a result, the equilibrium value is 1, which corresponds to the viscosity ratio when the oil has got rid of all the dissolved gas thermodynamically in excess at room pressure. At t = 0, the viscosity ratio is less than 1 because the dissolved gas, like diluents, reduces the live oil viscosity. After the depressurization, the behavior of the live oil viscosity can be described into three steps (Fig. 2b). First, the viscosity declines drastically due to the pressure drop. This is resulting from the high pressure-dependence of heavy oil viscosity [START_REF] Abivin | Viscosity behavior of foamy oil: experimental study and modeling[END_REF]. Then, the viscosity rises quickly, then slowly to finally tend to the thermodynamics equilibrium. The occurrence of bubbles after the depressurization has been checked using X-Ray experiments (Fig. 2a). The degassing kinetics of different heavy oils has been studied in a previous paper [START_REF] Abivin | Viscosity behavior of foamy oil: experimental study and modeling[END_REF], in which a kinetics model of degassing live heavy oil viscosity is developed. To match this model with the experiments, Eintein's hard-sphere model [START_REF] Einstein | Eine neue Bestimmung der Moleküledimensionen[END_REF][START_REF] Einstein | Berichtigung zu meiner Arbeit: eine neue Bestimmung der Moleküledimensionen[END_REF] was used to describe successfully the impact of bubbles on the viscosity of the foamy oils. This study confirmed that the presence of spherical gas bubbles leads to an increase in apparent foamy oil viscosity.

Depletion Under Shear

Opposite to these oscillatory tests, the same kind of experiments was carried out under different continuous shear rates. Figure 3 depicts the evolution of the Canadian heavy oil viscosity saturated in methane (30 bars) under shear rates of 10, 50 and 100 s -1 . The complex viscosity discussed above is also plotted for comparison. Two observations have to be pointed out. First, it appears that the higher the shear rate, the lower the viscosity ratio. Second, the viscosity seems to stabilize. It should also be noted that the kinetics model used in our previous paper do not match these new experimental results and overestimate the apparent viscosity of the mixture in all cases. As the model was made under the assumption that the bubbles are spherical (use of Eintein's hard sphere model), it seems that this hypothesis is no more valid and that bubbles are probably deformed by shear rate.

The decreasing apparent viscosity of a bubbly material below the viscosity of the material itself has already been observed for viscous fluids like magmas or melt glass and polymers [START_REF] Manga | Viscosity of magmas containing highly deformable bubbles[END_REF][START_REF] Llewellin | The rheology of a bubbly liquid[END_REF][START_REF] Rust | Effects of bubble deformation on the viscosity of dilute suspensions[END_REF][START_REF] Stein | Shear viscosity of rhyolite vapor emulsions at magmatic temperatures by concentric cylinder rheometry[END_REF][START_REF] Bagdassarov | Transient phenomena in vesicular lava flows based on laboratory experiments with analogue materials[END_REF][START_REF] Tuladhar | The development of polymer foam microstructure: experimental observations and matching modelling for polystyrene foams using different blowing agent[END_REF]. These studies have shown that this behaviour is due to the elongation of bubbles under shear. This could explain why the hard sphere model based on the spherical shape cannot describe anymore the viscosity of foamy oils under a high shear rate. As a result, the capillary number, Ca, is often introduced to determine the impact of bubbles on the viscosity of a bubbly liquid. The capillary number is the ratio of the viscous forces, which tends to deform the bubble, to the surface tension that restores its spherical shape. It is written as follows:

(1) where η 0 is the viscosity of the continuous phase (Pa.s), σ the gas/oil interfacial tension (estimated in our case from the measurement of an interface air/Canadian oil: 31 mN/m) and a the droplet radius. For high capillary numbers, bubbles are elongated and the viscosity of the bubbly liquid drops; for low Ca, bubbles remain spherical and contribute to an increase in the material viscosity [START_REF] Rust | Effects of bubble deformation on the viscosity of dilute suspensions[END_REF].

Experiments with a Silicon Oil

The previous experiments have been repeated with a transparent silicon oil (8.5 Pa.s at 28.5°C) instead of using the opaque black heavy crude oil. Figure 4 shows that this oil exhibits the same behaviour as the Canadian heavy oil during depletion.

Two additional equipments have been employed for the mineral oil: a pressurized reactor and a pressurized capillary rheometer. Both are fitted with a window which allows to visualize the foam morphology. In each case, methane is σ γ η a Ca 0 = introduced under pressure (30 bars) until the mineral oil is saturated. Then, the system is depleted either at rest or under shear. Figure 5 presents the results obtained on the reactor. The picture on the left has been taken on the mineral oil 30 minutes after the depletion in quiescent conditions: bubbles are spherical and are not numerous. The picture on the right shows the same system but under shear: bubbles appear elongated and abundant. Under shear, bubbles are maintained much longer in the oil. One can oppose that this is due to our stirring geometry, impeding the bubbles from rising by disturbing the flow lines. In fact, it is not the case because we also observed the stability of elongated bubbles under simple shear.

To illustrate the opposite results obtained in quiescent and shearing conditions, the rising velocity of a bubble in a liquid has been considered. This velocity is given by the following relation [START_REF] Stokes | On the effect of the Internal Friction of Fluidson the motion of Pendulums[END_REF]:

(2) where V is the velocity of the bubble, g the gravitational acceleration, r the bubble radius, ρ L and ρ G respectively the densities of the liquid and the gas phases and η L the viscosity of the liquid matrix. This relation does not fit the case of elongated bubbles as an additional friction term is being required for ellipsoids [START_REF] Perrin | Mouvement Brownien d'un ellipsoïde. I. Dispersion diélectrique pour des molécules ellipsoïdales[END_REF][START_REF] Perrin | Mouvement Brownien d'un ellipsoïde. II. Rotation libre et dépolarisation des fluorescences, Translation et diffusion de molécules ellipsoïdales[END_REF]. Thus the rising velocity appears to be lower when a bubble is elongated horizontally. This is a first explanation to the stability of bubbles in a flow. Experiments on the silicon oil have also been conducted using the Multipass Rheometer (MPR). The MPR is a pressurized capillary rheometer fitted with a window through which the sample can be visualized during the experiments. It has been developed in the Chemical Engineering Department of the University of Cambridge, UK [START_REF] Mackley | The multipass rheometer[END_REF]. The sample under study is driven back and forth through a capillary mid-section by hydraulic pistons. This arrangement allows control of the experimental shear rate or frequency, hydrostatic pressure and temperature. The viscosity is calculated from the pressure drop and the geometrical characteristics of the capillary tube. In addition, the use of specially designed capillaries allows optical birefringence, X-ray measurements or direct visualization of the sample under test. Our results are presented in Figure 6.

The pressure drop measured for the oil without any bubble is compared to the one obtained in the same conditions for the bubbly oil. It can be seen that the pressure drop is lower in the case of foams at high flow rates. Pictures taken during the experiments also show that bubbles are more and more elongated as the flow rate increases. Bubbles are CO 2 added and mixed in the MPR. Thus the lower pressure drop measured in the case of foams is not due to the presence of dissolved gas in the matrix. These data support the assumption that the presence of elongated bubbles can lead to a decrease in the fluid apparent viscosity.

Modelling of Bubbles Deformation

To understand the effect of gas bubbles, and more specially their size, shape and orientation, on the average viscosity, we have computed the homogenized shear modulus of a model material in solid mechanics. The analogy between shear modulus and viscosity under low deformation is used here. Indeed, under low displacements, non linear terms of the Green-Lagrange strain tensor can be neglected and we obtain a linearized deformation tensor, which is similar to the one used in fluid mechanics.

We have considered a periodic representative cell and have applied an affine velocity field on the cell boundary. The averaged shear rates and shear stresses have then been computed, leading to an homogenized shear viscosity. For this calculation the finite element method was applied on the two configurations shown in Figure 7. As expected from our experimental results, the higher the bubble elongation in the flow direction, the lower the homogenized shear viscosity. In these simulations, we have ignored the effects of surface tension and we recognize that the analyzed configurations do not correspond exactly to the experimental observation, but in any case they allow to identify the evolution tendency and to confort, at least qualitatively, the rheological measurement results.

It has been shown in this section that the viscosity of a foamy oil highly depends on the shear conditions. The bubbles can lead to an increase in the viscosity if remaining spherical, while they can induce a drop in the apparent viscosity if elongated under sufficient shear rates. Thus the capillary number is an important characteristic of the studied system as it allows to predict the shape of the bubbles. In this work, the capillary number has been evaluated considering an average bubble diameter of 1 mm, according to our visualizations. It is comprised between 2 and 20 for shear rates comprised between 10 and 100 s -1 . Under the oscillatory tests, the capillary number can be evaluated at 0.01 at the maximum of deformation. These values are in accordance with [START_REF] Rust | Effects of bubble deformation on the viscosity of dilute suspensions[END_REF], who found a critical capillary number around 0.7 above which the bubbles starts to elongate.

Rising and Decreasing Viscosity Regimes

The two flow regimes that have been previously described can be easily demonstrated by a particular experiment. It consists in depleting the system slowly (1 bar by stage every minute) instead of instantaneously. During all the experiment, the shear rate is set at 100 s -1 . The viscosity ratio is plotted against the pressure in Figure 8. Its evolution can be described into three steps. First, an increase can be seen from 30 to 15 bars. This is due to the impoverishment of the matrix in dissolved gas and to the nucleation and growth of bubbles within the oil. At this stage, bubbles are small and remain spherical, contributing to an increase in the apparent viscosity. At P = 15 bars, the viscosity starts to decline. The pressure still decreasing, the bubbles grow by diffusion of the dissolved gas towards the bubbles, by expansion and by coalescence. Since the bubbles are growing, they get deformable and begin to elongate. The flow regime is changing from a classical suspension model to a sandwichtype material. The viscosity is decreasing, despite the impoverishment of the oil in dissolved gas. At 7 bars, the system seems to have reached equilibrium. The apparent viscosity of the system is "stabilized" at a much lower value than the oil itself. So this figure depicts the two flow regimes described above: a regime where the viscosity rises with the presence of spherical bubbles and one where the viscosity drops, due to the elongation of the bubbles under shear.

WATER-IN-OIL EMULSIONS

In this part, the Canadian heavy crude oil behaviour has been investigated regarding the effect of added water content, droplet size, shear rate and temperature. The effect of pressure has also been studied (Chaudemanche et al.) but is out of scope of this paper.

To form water-in-oil emulsions, a 65 mL flask filled with heavy crude oil is placed in a water bath and heated at 40°C during 30 min. Demineralised water is added, until the water/oil volume ratio desired is reached. The water volume percentage used (φ) varies from 0 to 30%. A constant rotational speed is then applied with an Ultra-Turrax T25 to emulsify the water. Three different levels of mixing have been tested: 13 000, 19 000 and 25 000 rpm. Emulsions prepared with the highest one are the most stable. They have been kept at ambient temperature and have been further characterized by microscopy, calorimetry and rheological analysis.

Microscopic Observation

To determine the mean droplet size and the macroscopic structure associated to each water volume percentage, a sample of each emulsion has been analyzed under an optical microscope and a cryogenic scanning electron microscope.

Photomicrographs have been taken and the mean droplet size has been estimated. Figure 9 shows the micrographs of water-in-oil emulsions for φ = 5, 10, 20 and 30%. These micrographs depict the structure generated by the addition of water inside the heavy crude oil. Water droplets are clearly identified on both optical and electron microscopy. The mean droplet size is estimated as a function of the water volume percentage. The diameter is roughly similar whatever the water cut at ∼1 µm. However, the optical micrograph for a 30/70 W/O emulsion shows bigger droplets with a mean diameter of approximately 3 µm. Thus, an increase in water volume percentage seems to correspond with a slight increase in the mean droplet size. We assume that the increase in the mean droplet size is due to a lower available amount of natural surfactants. Actually, in all the experiments, the global volume is constant, meaning that when the water volume percentage increases, the oil content and its surfactant molecule content decreases accordingly. The main mechanism at the origin of bigger droplets, like coalescence, is favoured when asphaltene content is low. Indeed, it is well known that asphaltenes stabilize the water/oil interfaces and prevent coalescence [START_REF] Sjöblom | Our current understanding of water-in-crude oil emulsions: Recent characterization techniques and high pressure performance[END_REF][START_REF] Poteau | Influence of pH on stability and dynamic properties of asphaltenes and other amphiphilic molecules at the oil-water interface[END_REF].

These pictures also show that water droplets may form aggregates. This structure should be revealed by rheological analysis and could greatly influence the global behaviour of the emulsions under shear.

Emulsion Characterisation by DSC

Many techniques can be used to characterize emulsions, but most of them are not appropriate to dark and concentrated W/O emulsions. In order to avoid the dilution of these emulsions, which may create perturbation on the interactions between droplets, the microcalorimetry technique [START_REF] Dalmazzone | Microcalorimetry, in Encyclopedic Handbook of Emulsion Technology[END_REF]) can be advantageously used. Differential Scanning Calorimetry is based on the measurement of heat exchanges between a sample and a reference as a function of time and temperature. The peak area of the recorded signal is related to the heat flow generated or absorbed by the sample. In the present work, each sample has been cooled from ambient temperature to -100°C, maintained 2 min at -100°C and heated to 20°C at a constant scanning rate of 5°C/min. The water droplets, embedded inside the heavy crude oil, have thus crystallised during cooling and then have re-melted during heating. The crystallization temperature is correlated to the droplet size, whereas re-heating of the sample allows the determination of the water content.

The DSC thermograms of W/O emulsions, for φ = 5, 10, 20 and 30%, are presented in Figure 10. Results indicate that the samples analyzed are effectively W/O emulsions. The thermograms of each sample present one major crystallization peak at roughly -40°C, which is typical of a fine emulsion [START_REF] Dalmazzone | Microcalorimetry, in Encyclopedic Handbook of Emulsion Technology[END_REF]. Practically no difference is observed on the crystallization temperatures as a function of the water volume percentage. The determination of the melting energy of the initial water sample allows the calculation of the water content of each emulsified sample from its corresponding melting energy (see Table 1). The experimental water content calculated is in accordance with the theoretical one.

Steady State Flow Curves

As already mentioned, the understanding of the rheological behaviour of emulsions is of importance, particularly for their transportation. It is a key point to determine viscosity as a function of the dispersed phase volume percentage. Through the rheological analysis, we should be able to describe the structure associated with the increase of water volume percentage. In Figure 11, the viscosity measured at 40°C versus shear rate is plotted for each emulsion and compared to the heavy crude oil viscosity. Viscosity vs. shear rate curves show a Newtonian behaviour for low shear rates. The Newtonian viscosity measured increases as the water volume percentage increases. Whatever the emulsion, viscosity decrease for shear rate higher than 100 s -1 is typical of a shear thinning behaviour. In the present work, shear viscosity curves have been fitted with a Williamson model:

(3) where η 0 is the Newtonian viscosity of the emulsion (Pa.s), k the consistency (s) and n the rate index. The fitting

n k ) ( 1 0 γ η η + =
parameters are given in Table 2. The k parameter is equal to the inverse of a shear rate. This parameter increases with the water volume percentage, and indicates that the shear thinning behaviour occurs at lower shear rates. This tendency is consistent with microscopic observations, where the size and numbers of droplet aggregates increase with the water volume percentage. These results are in accordance with measurements in the literature [START_REF] Pal | Effect of droplet size on the rheology of emulsions[END_REF][START_REF] Pal | Scaling of relative viscosity of emulsions[END_REF] where it is found that the droplet size may have a large impact on the viscosity, particularly when going to small droplet sizes. [START_REF] Pal | Effect of droplet size on the rheology of emulsions[END_REF][START_REF] Pal | Scaling of relative viscosity of emulsions[END_REF] argued a non-Newtonian behaviour such as shear thinning appeared to be more pronounced for fine than for coarse emulsions.

Different hypothesis can be formulated to explain the shear thinning behaviour. It may be due to shear-induced disruption of droplet aggregates and deformability of droplets [START_REF] Saiki | Effects of droplet deformability on emulsion rheology[END_REF] or to the droplet break-up occurring for high hydrodynamic stress. To resolve this, the modelling of the rheological behaviour can be fruitful.

Modelling of the Flow Behaviour

A large number of viscosity equations and correlations relating the emulsion viscosity to the dispersed phase volume percentage have been proposed in the literature. They all give expressions for the relative emulsion viscosity, η r , i.e. the ratio of the emulsion viscosity (η) to the pure continuous phase (η c ):

(4)

An overview of emulsion viscosity models and correlations was analyzed by [START_REF] Pal | Novel viscosity equations for two immiscible liquids[END_REF] a few of them are mentioned here.

For very dilute emulsions, the viscosity is given by the following equation [START_REF] Taylor | The viscosity of a fluid containing small drops of another fluid[END_REF]:

(5) where η d is the viscosity of the dispersed phase (Pa.s).

In the limit of η d /η c →∞, Equation (3) reduces to the well-known Einstein equation [START_REF] Einstein | Eine neue Bestimmung der Moleküledimensionen[END_REF][START_REF] Einstein | Berichtigung zu meiner Arbeit: eine neue Bestimmung der Moleküledimensionen[END_REF]:

(6)
The Taylor equation cannot be applied when hydrodynamic interactions between the droplets become significant. In the emulsion literature, it is a common practice to apply suspension viscosity equations to interpret the viscosity/concentration behaviour [START_REF] Saiki | Effects of droplet deformability on emulsion rheology[END_REF]. Among the most widely used equations are: -The Mooney equation [START_REF] Mooney | The viscosity of a concentrated suspension of spherical particles[END_REF], which is an expression for the relative viscosity of dispersions of monodisperse spheres.

(

-The Krieger and Dougherty equation [START_REF] Krieger | A mechanism for non-Newtonian flow in suspensions of rigid spheres[END_REF], which was developed for concentrated suspensions.

(8)

In these last equations, φ m is the maximum packing concentration of particles. At φ m , the viscosity of the suspension becomes infinite. The Krieger and Dougherty equation describes many suspension systems and has also been applied to emulsion systems [START_REF] Tadros | Fundamentals principles of emulsion rheology and their applications[END_REF]. Values expected concerning φ m for suspensions are in the range: 0.64 (random close packing) to 0.74 (hexagonal close packing), but for emulsions larger φ m were explained as being due to droplet polydispersity and deformation [START_REF] Manoj | Characterization of a depletion-flocculated polydisperse emulsion II. Steady state rheological investigation[END_REF]. However, one serious drawback of applying these equations to emulsions is that these equations do not take into account the effect of viscosity ratio

m m r φ φ φ η 5 . 2 ) / 1 ( - - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = m r φ φ φ η / 1 5 . 2 exp φ η 5 . 2 1 + = r φ η η η η η η η ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + = = d c d c c r 5 . 2 1 c r η η η =
(K) between heavy oil and dispersed phase. In the case of bubbles, this parameter was not introduced because gas is considered to be inviscid.

The relative experimental viscosities, at four different shear rates (1, 10, 100 and 500 s -1 ), plotted against the water volume percentage and fitted with the previous correlations are shown in Figure 12.

For these emulsions, the rheological behavior at low shear rates and low volume percentage can be approximated by the Einstein's equation for hard spheres, confirming previous works [START_REF] Dickinson | Rheology of emulsions -The relationship to structure stability[END_REF]. At higher φ, the Krieger and Dougherty model, with φ m closed to 0.64, fits with a good accuracy our experimental relative viscosities: water droplets can be considered as spherical. However, at higher shear rates (above 100 s -1 ) experimental relative viscosities can not be modelled by any correlations, as the models overestimate the emulsion viscosity. These results are in accordance with those found on water droplets deformation in alkyd resin [START_REF] Thompson | The effect of droplet extension on the rheology of emulsions of water in alkyd resin[END_REF]. The authors established that the viscosity reduction at high shear rates was due to an extensive distortion of the water droplets. To clarify this behaviour, it appears necessary to take into account the effect induced by shear stress on the droplet shape by the introduction of the capillary number as seen in the previous section for bubbles.

Droplet Deformation

The capillary number (Equation ( 1)) can also be used to determine the deformability of a droplet under shear rate. The degree of deformation D can be described as a scalar combination of the lengths of the major (L) and minor (B) axes of Relative viscosities vs. water volume percentage at γ ⋅ = 1, 10, 100 and 500 s -1 , fitted with usual correlations.

an elliptical droplet [START_REF] Taylor | The viscosity of a fluid containing small drops of another fluid[END_REF]: D = (L-B)/(L+B) and that for small deformations, D is given by: (9)

where K is the viscosity ratio. Subsequent workers have extended the analysis applied by Taylor to transient flows [START_REF] Cox | The deformation of a drop in a general timedependent fluid flow[END_REF], and extended the validity by performing the calculation to higher order in droplet deformation [START_REF] Frankel | The constitutive equation for a dilute emulsion[END_REF]. The following equation can thus estimate the level of droplet deformation [START_REF] Torza | Particule motions in sheared suspensions XXVII. Transient and steady deformation and burst of liquid drops[END_REF]:

(10)

Figure 13 shows the viscosity ratio between the emulsion viscosity and the Newtonian viscosity plotted against the droplet deformation at 40°C. All curves, whatever the emulsion considered, can be superimposed. The beginning of the shear thinning behaviour can be thus described by the same parameter for each system noted D lim . This analysis was also carried out at 20, 30, 50 and 60°C and no strong variation of D lim was noticed.

As the volume percentage rises, there is an increased possibility for droplets to experience deformation due to greater hydrodynamic forces and droplet collisions caused by shear. Deformation may allow droplets to slide on each other, leading to an emulsion less resistant to the applied shear, therefore less viscous, than the suspension at the same volume percentage.

A viscosity decrease with increasing disperse phase concentration is a well-known phenomenon for polymer blends. Viscosity reduction by more than that predicted on the basis of linear mixing has been attributed to slip between layers of polymer [START_REF] Zhao | Rheology of immiscible polymer blends -interfacial slip[END_REF].

Experimentally, the critical capillary number (Ca crit ) necessary for break-up depends strongly on the viscosity ratio [START_REF] Grace | Dispersion phenomena in high viscosity immiscible fluid systems and application to static mixers as dispersion devices in such systems[END_REF]. If the viscosity ratio is very small, extreme deformation of the droplet is possible without inducing break-up in the steady state. The critical capillary number in simple shear is given by (Hinch and Acrivos, 1980): (11) In the present work, according to Equation ( 11), Ca crit is equal to approximately 15.8. The capillary number calculated from D lim , is Ca lim ∼ 0.6. Thus, we argue that droplets do not break up, but deform into stretched droplets of elliptical cross section (see Fig. 14).

According to this schematic representation, a relaxation time should exist to allow droplets to recover their initial shape. To verify this hypothesis, a 30/70 water-in-oil emulsion has been submitted to an increasing shear rate from 0.1 to 500 s -1 followed by a decreasing shear rate from 500 to 0.1 s -1 . This experiment is performed at 20, 40 and 60°C. The results are given in Figure 15. An hysteresis is observed

3 / 2 054 . 0 - = K Ca crit 2 2 ) 19 ( ) / 20 ( ) 1 ( 4 ) 16 19 ( 5 K C a K K B L B L D + + + = + - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = 16 16 16 19 K K C a D
whatever the temperature and is more emphasised as the temperature decreases. At low temperatures, the heavy crude oil viscosity is high and shear stresses transmitted to water droplets are promoted. Thus the relaxation time is higher when temperature falls, corresponding to highly stretched droplets. When the shear rate reaches 0.1 s -1 , the final viscosity is approximately the same as the initial viscosity. This result is characteristic of a thixotropic behaviour.

In this second part, stable W/O emulsions have been obtained with a mean droplet diameter ∼1µm. By using DSC, we were able to confirm the water content and the nature of each emulsion prepared. The rheological analysis in the steady state proved that for low shear rates water droplets increase the emulsion viscosity. On the contrary, at high shear rates, the viscosities obtained were lower than those predicted by suspension and emulsion models, indicating the necessity to take into account the droplet deformation under shear. As in the case of deformed bubbles, elongated water droplets can lead to a strong decrease in viscosity of the fluid in which they are dispersed. Viscosity ratio versus Deformation (according to Equation 9) for φ = 5, 10, 20 and 30%.

Disruption Orientation

Schematic representation of the droplet disruption and orientation under shear.

GAS BUBBLES AND WATER DROPLETS DISPERSED IN HEAVY CRUDE OILS

In this part, several rheological experiments have been performed on the heavy crude oil containing both gas bubbles and water droplets. The objective is to show how flow properties of real multiphase system can be complex. As in the first section of this paper, bubbles have been created by depletion of a live emulsion. Again, the experiments are divided into two steps: the gas dissolution under shear and pressure followed by the appearance of bubbles provoked by depletion. The experiments have been carried out using both instantaneous and slow depletions from 30 bars to atmospheric pressure.

Methane Dissolution

The methane dissolution in the heavy crude oil and the emulsions is ensured by a helix geometry at 30 bars and T = 28.5°C, under a shear of 100 s -1 . The viscosity is measured as a function of time and we consider that the dissolution is achieved when the viscosity levels off, like in the first section of this paper. The dissolution of methane in water droplets is considered to be negligible [START_REF] Berthezene | Methane solubility in synthetic oil-based drilling muds[END_REF]. Thus the global solubility of methane is related to the oil volume percentage. To verify this hypothesis, we have fitted the viscosity of saturated emulsions with the wellknown Einstein model [START_REF] Einstein | Berichtigung zu meiner Arbeit: eine neue Bestimmung der Moleküledimensionen[END_REF]. As seen in Figure 12, it is quite suitable to predict emulsion viscosity at 100 s -1 . The theoretical viscosities calculated with the Einstein model correspond to the experimental ones when the oil is saturated, thus confirming that the difference in viscosity is only due to the presence of spherical water droplets and not because methane is dissolved in water.

Multiphase Dispersed Systems

The rheological behaviour of a multiphase dispersed system, namely gas bubbles + water droplets embedded inside a heavy crude oil has been compared to the one of a system containing only bubbles. Figure 16 presents the relative viscosity η r (i.e. system viscosity/emulsion viscosity at φ = 0, 10, 20 or 30% without bubbles at atmospheric pressure) during the depletion process under a continuous shear rate of 100 s -1 . The experimental curves show that just like in the case of heavy oil, the foam obtained in emulsions has a relative viscosity stabilized at a value lower than 1. The bubbly emulsion is therefore less viscous than the original emulsion. Again, this phenomenon is attributed to the elongation of the gas bubbles, favouring the liquid flow. The key point to highlight is that this phenomenon is influenced by water droplets. In fact, as in these conditions emulsions rich in water are more viscous, they deform more easily gas bubbles and enhance the viscosity drop.

In Figure 17, the evolution of the emulsion viscosity is plotted during a slow depletion at 1 bar/min. The behaviour observed is totally different compared to an instantaneous depletion. We can observe in the first moment, a viscosity increase whatever the water volume percentage when the pressure has reached 27 bars. When the pressure is equal to 8bars, the viscosity drops for each sample analyzed. This behaviour is characteristic of the presence of deformed gas bubbles. It seems that at this level of pressure, gas bubbles are big enough (due to diffusion or coalescence) to allow their deformation under a constant shear rate. This enables to decrease the global viscosity of the multiphase system. Again, the decrease seems to be influenced by water droplets.

Spherical water droplets increase the viscosity of heavy oils and facilitate bubbles deformation, which in turn decreases viscosity. The case of elongated water droplets should be further studied. Unlike spherical bubbles, deformed ones lead to a decrease in viscosity, which does not promote bubble deformation. These remarks illustrate how rheological properties of mixed dispersed systems can be complex. It is particularly the case of bubbly live oil emulsions.

CONCLUSIONS

In this paper, it is demonstrated that the flow properties of heavy oils are highly modified by the presence of gas bubbles and water droplets. Naturally present within the oil during the production process, these dispersed phases can lead to either a decrease or an increase in heavy oil apparent viscosity, depending on the shear rates applied to the sample during its transport in the well or in the reservoir. Since bubbles and droplets are deformable, two flow regimes can be encountered. Under low shear rates, the suspended phases remain spherical. The material behaves like a classical dispersion and can be modelled in the limit of low volume percentages by a classical hard sphere model. In this case, the apparent viscosity of the produced oil is higher than the oil itself. Above a sufficient shear rate, the suspended phases are elongated in the flow line direction and the material behaves differently. We have shown that the viscosity is then lower than the crude oil itself. The deformability can thus lead to a decrease in the apparent suspension viscosity, what is of primary importance to predict the production rates of a well. It depends mainly on the viscosity ratio and the capillary number. That is why this phenomenon is fundamental in the case of heavy oils. Taking into account these qualitative results, a perspective would be to model the heavy oil viscosity in real conditions, like it is transported in the well. Evolution of the viscosity vs. time and pressure (depletion rate = 1 bar/min) under a continuous shear rate of 100 s -1 for φ = 0, 10, 20 and 30%.

  Figure 1Scheme of the rheometer pressure cell.

Figure 2

 2 Figure 2 Evolution of the viscosity ratio of a Canadian heavy oil saturated in methane (30 bars) and depressurized from 30 bars to room pressure instantaneously. a) X-Ray picture taken 30 seconds after the depressurization. b) Zoom on the first minutes.

  ratio of a silicon oil saturated in methane and depressurized instantaneously from 30 bars to room pressure at different shear rates.

Figure 5

 5 Figure 5Pictures taken 30 minutes after the saturated silicon oil was depressurized instantaneously. Comparison between a depletion at rest (left) and under shear rate (right).

  Figure 6Pressure drop measured for the silicon oil and the silicon oil with bubbles (foam) in the MPR.

  Figure 7Modelling of the shear modulus of a material in the case of spherical holes and elongated holes.

  Figure 9 Observation of W/O emulsions under optical a) and scanning electron b) microscopes.

  Figure 10Thermograms of W/O emulsions for φ = 5, 10, 20 and 30%.

  Figure13

  Figure 16Evolution of the relative viscosity vs. time and pressure (depletion rate = 15 bars/s) under a continuous shear rate of 100 s -1 for φ = 0, 10, 20 and 30%.

TABLE 1 Crystallization

 1

		temperatures, melting/crystallization energies and
	experimental water content for φ = 5, 10, 20, 30 and 100%
		Crystallization	Melting	Crystallization Experimental
	Sample	temperature	energy	energy	water content
		T onset (°C)	(J/g)	(J/g)	(%)
	Distilled water	-18.8	329.2	294.3	100
	5/95	-40.0	16.4	11.6	4.9
	10/90	-39.8	35.9	27.5	10.9
	20/80	-39.6	69.5	48.7	21.1
	30/70	-39.6	91.7	67.5	27.9