L'épuisement des réserves de pétrole et l'augmentation des besoins énergétiques des forces démographiques croissantes, ainsi que le coût énorme de l'extraction du pétrole, nécessitent d'optimiser les méthodes de récupération du pétrole. L'une de ces méthodes dans l'injection de polymère avec des solutions de polymère viscoélastique est une technique de récupération d'huile améliorée chimique (EOR). Il pourrait en résulter une récupération d'huile plus importante que celle prédite à partir d'une analyse du nombre capillaire (le nombre capillaire est le rapport des forces visqueuses avec la tension interfaciale des fluides dans les contacts). Cette récupération d'huile supplémentaire est due aux effets élastiques des solutions de polymère entraînant un déplacement de l'huile hors des pores où elle était piégée, ainsi qu'une extraction améliorée ressentie par les poches d'huile en raison de forces normales ou la stabilisation des filets d'huile contre la rupture. Ces mécanismes peuvent être liés à des effets élastiques linéaires ou non linéaires. Cependant, les effets linéaires et viscoélastiques des solutions de polymère dépendent du poids moléculaire des chaînes de polymère. Par conséquent, l'efficacité de la récupération d'huile par injection de polymère est fortement limitée par la dégradation mécanique et chimique des polymères. La dégradation du polymère entraîne une perte irréversible de viscosité, due à la diminution du poids moléculaire des chaînes polymères. Lorsqu'il est injecté dans le réservoir, dans le cas d'une injection radiale à partir d'un puits, il est connu que les premiers centimètres parcourus à travers la roche sont le domaine crucial où le polymère subit une dégradation.

Dans notre travail, nous parvenons à prédire la dégradation des polymères lorsqu'ils sont injectés à travers des dispositifs de contraction et un milieu poreux. Pour cela, nous utilisons successivement différentes techniques qui ont permis finalement de construire des modèles quantitatifs. Dans un premier temps, nous utilisons des géométries de contraction hyperbolique uniques. Différents solvants et solution de polymère avec différentes propriétés rhéologiques sont injectés à différents flux à travers cette géométrie de contraction hyperbolique et la chute de pression entre l'entrée et la sortie est mesurée. La chute de pression relative de la solution de polymère par rapport à celle du solvant présente un maximum en fonction du flux. Le maximum de cette pression relative résulte de la compétition entre l'extension et la scission des chaînes polymères. Les lignes de courant de l'écoulement de polymère sont visualisées, afin de mesurer la vitesse de déformation. En analysant des passages successifs sur ce dispositif de contraction unique, nous avons développé un modèle universel qui prédit quantitativement la scission. Nous avons trouvé ce modèle indépendant des propriétés physiques de la solution, mais dépendant du Hencky strain imposé par la géométrie.

Lorsque le modèle a été étendu à plusieurs contractions en série, nous avons observé que la déformation vraie imposée à la solution de polymère est différente de la déformation théoriquement imposée par celle de la géométrie. Grâce à la dépendance de notre modèle avec la souche, la scission des polymères pour plusieurs contractions en série peut être prédite.

RÉSUMÉ

Dans une étape suivante, le modèle prédisant la scission des dispositifs de contraction unique a été étendu aux milieux poreux. Nous montrons que ce modèle est applicable aux milieux poreux. Par conséquent, les dégradations et la chute de pression entre l'entrée et la sortie pour l'écoulement d'une solution de polymère à travers un milieu poreux de différentes longueurs peuvent être prédites. L'expérience révèle un très bon accord avec la prédiction du modèle. De plus, nous montrons que la dégradation est une fonction croissante de la longueur. Nous prédisons et observons également une longueur critique sur laquelle aucune dégradation significative n'a été observée. La longueur critique pour atteindre une saturation apparente dans la dégradation est une fonction décroissante du flux. De même, la valeur des plateaux de dégradation augmente avec le flux.

Enfin, le modèle a été étendu à l'écoulement radial -pour imiter les conditions du réservoir. Nous montrons comme précédemment que la dégradation est une fonction croissante de la distance parcourue dans le réservoir jusqu'à ce qu'elle atteigne une valeur de plateau. Enfin, en conditions radiales, nous montrons que la dégradation se produit pendant les premiers cm de la distance parcourue dans le milieu poreux.

Cette thèse comprend 8 chapitres et s'organise comme suivant :

Dans le chapitre 1, nous discutons brièvement des problèmes et des approches implémentées correspondantes dans le manuscrit actuel.

Des solutions de polymères de différentes propriétés rhéologiques sont préparées en modifiant le poids moléculaire, la concentration, la viscosité du solvant et la qualité du solvant. Des polymères de poids moléculaire élevé avec une polydispersité élevée sont utilisés. Les propriétés rhéologiques sont caractérisées à la fois par des conditions de cisaillement et d'écoulement d'extension. Des dispositifs expérimentaux pour les géométries de contraction microfluidique et le milieu poreux sont décrits. Les techniques expérimentales, y compris les expériences d'injection simple et double -qui permettent de sonder la scission -sont expliquées. Une technique de visualisation de l'écoulement a été utilisée dans toutes les expériences d'injection unique pour estimer le véritable taux de déformation.

Dans le chapitre 4, les champs d'écoulement dans les contractions sont discutés en détail. Des solutions de polymère avec différentes propriétés rhéologiques (modifiées par les paramètres chimiques physiques comme le poids moléculaire, la concentration, la viscosité du solvant et la qualité du solvant) sont injectées par le biais de géométries de contraction hyperbolique uniques avec différents « Hencky strain ». Différents régimes d'écoulement ont été observés, à savoir un écoulement de type newtonien, des tourbillons d'angle symétriques stables, une croissance de tourbillon d'angle asymétrique stable, des tourbillons d'angle symétriques dépendant du temps, des tourbillons d'angle asymétriques dépendant du temps, des tourbillons d'angle singuliers stables et des instabilités viscoélastiques. Nous mesurons la longueur des tourbillons dans tous ces différents régimes d'écoulement.

Lorsque des polymères en solvant éthanol / eau sont injectés à travers une géométrie avec un Hencky strain de 3.2 , nous n'avons observé aucune activité de vortex. Cette activité de vortex réduite pourrait être due à la diminution des contributions des contraintes d'extension par rapport aux contraintes de cisaillement. La taille du vortex n'est pas corrélée avec le nombre de Weissenberg. En faisant varier la déformation de Hencky nous trouvons que la taille de vortex adimensionnée augmente logarithmiquement avec le nombre de Deborah. De plus, l'activité du vortex diminue avec la diminution de la hauteur du canal. La taille du vortex adimensionnée varie comme la racine de l'épaisseur de la cellule. Un modèle simple proposé a une dépendance à la loi de puissance plus élevée que les observations expérimentales, ce qui pourrait résulter des effets de modifications des enchevêtrements lorsque les chaines de polymères sont étirées.

Lorsque les différents régimes d'écoulement sont tracés dans l'espace Wi -Re, comme cela est fait dans la littérature, les différents régimes d'écoulement évoluent le long de la pente de ce tracé qui est le nombre élastique. Le type de régime d'écoulement et la transition d'un régime à un autre ne peut pas être prédit à partir du nombre élastique, qui est le rapport des effets élastiques (caractérisés par le nombre de Weissenberg) et des effets inertiels (déterminés à partir du nombre de Reynold).

La présence de tourbillons induit des fluctuations de pression. Cependant, la chute de pression (due au début de l'épaississement extensionnel ou de la scission des polymères) n'est pas influencée par la présence de ces tourbillons ou la transition d'un régime d'écoulement à un autre régime d'écoulement. La formation de tourbillons modifie les champs de vitesse locaux et donc le taux d'extension. Nous utiliserons la visualisation pour calculer un taux de déformation extensionnelle plus précis.

Dans le chapitre 5, des solutions de polymères non enchevêtrés semi-dilués avec différentes propriétés rhéologiques sont injectées à différents débits à travers une seule géométrie de contraction hyperbolique avec une Hencky strain de 3,2 et la chute de pression est mesurée entre l'entrée et la sortie du canal. La chute de pression relative de la solution de polymère par rapport à celle du solvant présente un maximum en fonction du flux. Le maximum résulte de la compétition entre l'extension et la scission des chaînes polymères. Le rapport de pression maximum varie avec la masse moléculaire MW et la concentration c comme en Mw 1±0.2 c 0.7 ±0. 3 tandis que le débit à la pression maximale varie comme Mw -1.8±0.2 c -0.7±0. 3 . Nous montrons que le rapport de chute de pression maximum, qui révèle la compétition entre la viscosité extensionnelle et la scission de la chaîne, se produit pour un nombre de Weissenberg donné.

Nous étudions ensuite comment le débit au rapport de pression maximum augmente pour des passages successifs à travers la même contraction, en raison de différents degrés de dégradation à chaque étape. Cela nous permet de quantifier la scission de la chaîne en fonction du débit. Nous établissons un modèle universel de scission qui est indépendant de la qualité du solvant et du poids moléculaire, pour un Hencky strain donné. Nous montrons que le modèle universel peut prédire le comportement de scission après des passages successifs dans plusieurs expériences d'injection. Le modèle de scission développé se compose d'exposant de scission qui définit l'ampleur de la transition de l'absence de scission à la saturation en scission. L'exposant de scission est indépendant de tout paramètre physique de la solution de polymère mais dépend du Hencky strain. Cette évolution robuste de la scission peut être appliquée dans d'autres applications où les polymères sont injectés plusieurs fois, comme la lubrification dans les moteurs, l'injection par jets et pendant la récupération d'huile ; ainsi que dans les milieux poreux, qui pourraient être représentés par des contractions multiples successives.

Dans le chapitre 6, nous faisons varier le « Hencky strain » pour étudier l'influence du temps de séjour sur la scission du polymère. Pour cela, des solutions polymères aux propriétés rhéologiques différentes sont injectées à travers des géométries avec différentes déformations Hencky allant de 1.4 à 4 unités. Nous avons observé que le véritable taux de déformation au maximum du rapport de perte de charge se produit pour des valeurs plus petites lorsque le Hencky strain augmente. Le taux de déformation au maximum dans la courbe du rapport de chute de pression est inversement proportionnel au temps relaxation de la solution de polymère, et le rapport maximal de chute de pression se produit pour un nombre de Weissenberg donné pour une géométrie avec une Hencky strain donnée. La dépendance exponentielle du temps de séjour requis pour provoquer une scission sous l'application d'une force constante suggère que le processus de scission est un processus activé par l'énergie ; cependant, la présente étude ne peut pas expliquer la dépendance au poids moléculaire pour la force appliquée.

La scission est étudiée avec des doubles passages sur des géométries avec différentes déformations de Hencky. La même loi de scission est valide, mais l'exposant de scission du modèle dépend du Hencky strain. Pour des Hencky strain croissants, nous observons que l'exposant de scission diminue et que la quantité de scission augmente en raison du temps de séjour plus élevé.

De plus, nous avons étudié les passages simples et doubles sur des géométries avec plusieurs contractions en série. Le Hencky strain imposée par la géométrie est constante dans les contractions successives. À partir d'expériences à passage unique, nous observons que la quantité de scission augmente avec l'augmentation du nombre de contractions. Cependant, à partir des expériences à double passage, où le premier passage est à travers une géométrie avec un nombre de contractions donné et le second passage à travers une seule contraction, a suggéré une scission réduite que celle attendue du modèle de scission que nous avons développé. Des expériences de visualisation ont révélé que le véritable Hencky strain dans les contractions successives est réduite en raison de la formation des tourbillons. Avec le vrai Hencky strain calculée à partir des lignes d'écoulement observées en contraction successive et compte tenu de la dépendance de le Hencky strain pour l'exposant de scission, nous avons pu prédire la quantité de scission dans des contractions successives.

Pour les injections radiales en géométrie cylindrique, le flux baisse inversement avec le rayon total parcouru et de sorte que la dégradation est plus faible pour le même flux d'entrée en surface que dans les injections linéaires.

En augmentant le rayon du puits d'injecteur, les tendances de dégradation et le rayon critique passent des conditions d'injection radiales aux conditions d'injection linéaire. L'approche des conditions d'injection linéaire se produit lorsque la courbure des conditions radiales est négligeable sur le rayon de saturation.

Dans des conditions d'injection radiales, le rayon de saturation est de 5 à 10 cm. Une conclusion importante à tirer de cette étude est que des pores propres sont une nécessité pour diminuer la dégradation et ainsi augmenter l'efficacité de la récupération du pétrole. Les différents paramètres du modèle dépendent à la fois de la nature du coeur (comme les impuretés, la taille des pores, les longueurs des pores) et de la solution de polymère (par exemple le poids moléculaire). Le blocage des résultats principaux diminuerait la perméabilité. Une diminution de la perméabilité peut entraîner une diminution de l'exposant de scission a, augmentant ainsi la dégradation qui se traduit par une chute rapide des propriétés viscoélastiques à une courte distance du point d'injection.

Au chapitre 8, les conclusions importantes de notre étude sont mentionnées ainsi que les perspectives de la présente étude.

MOTS CLÉS

Viscoélasticité, viscosité extensionnelle, rhéofluidifiant, CaBER, contraction hyperbolique, scission de polymère, dégradation mécanique du polymère, tourbillons, tourbillons de coin, tracé Wi -Re, temps de séjour, milieux poreux, déformation Hencky, écoulements d'extension, HPAM, récupération améliorée de l'huile, injections linéaires, Injections radiales, Plusieurs contractions en série, Mécanisme activé par l'énergie pour la scission, Numéro de Weissenberg, numéro de Deborah.

The depleting number of oil reserves and the increased energy needs of the growing population forces, as well as the huge cost of oil extraction, requires optimizing the methods for oil recovery. One of these methods in polymer flooding with viscoelastic polymer solutions is a chemical Enhanced Oil Recovery technique (EOR). It could result in oil recovery larger than the one predicted from a capillary number analysis (Capillary number is the ratio of the viscous forces with the interfacial tension of the fluids in contacts). This additional oil recovery is due to elastic effects of the polymer solutions resulting in oil displacement from the deadend pores, enhanced pulling experienced by the oil blobs due to normal forces or stabilization of oil threads against snap off. These mechanisms can be related to either linear or non-linear elastic effects. However, the linear and viscoelastic effects of polymer solutions are dependent on the molecular weight of the polymer chains. Therefore, the efficiency of oil recovery by polymer flooding is greatly limited by mechanical and chemical degradation of polymers. The degradation of polymer leads to an irreversible loss of viscosity, due to decreased molecular weights of the polymer chains. When injected in the reservoir, in the case of a radial injection from a well, it is known the early few centimeters travelled through the rock are the crucial domain where polymer undergoes degradation.

In our work, we succeed in predicting the degradation of polymers when injected through contraction devices and porous medium. For that purpose, we successively use various techniques that have allowed at the end to build quantitative models. In a first step, we use single hyperbolic contraction geometries. Different solvents and polymer solution with different rheological properties are injected at different flux through this hyperbolic contraction geometry and the pressure drop between entrance and exit is measured. The relative pressure drop of the polymer solution to that of the solvent exhibits a maximum as a function of flux. The maximum results from the competition between the extensional and the scission of polymer chains. The streamlines of polymer flow are also recorded, in order to measure the strain rate.

From multiple passages on this single contraction device we developed a universal model that predicts quantitatively the scission. We found this model to be independent on the physical properties of solution, but dependent on the Hencky strain imposed by the geometry.

When the model was extrapolated to several contractions in series, we observed that the true strain imposed on the polymer solution is different from the strain imposed by that of the geometry. Thanks to the dependency of our model with the strain, the quantity of scission of polymers for several contractions in series can be predicted.

In a next step, the model predicting scission from the single contraction devices was extended to porous media. We show that this model is applicable for the porous media. Therefore, the degradations and the pressure drop between the entrance and exit for flow of a polymer ABSTRACT solution through porous medium of different lengths can be predicted. The experiment reveals a very good agreement with the prediction from the model. Furthermore, we show that the degradation is an increasing function of length. We predict and observe also a critical length over which no further significant degradation was observed. The critical length for attaining an apparent saturation in degradation is decreasing function of the flux. Similarly, the value of the degradation plateaus increases with flux.

Lastly the model was extended to the radial flow -to mimic the reservoir conditions. We show as previously that the degradation is an increasing function of distance travelled in the reservoir until it reaches a plateau value. Lastly, in radial conditions we show that the degradation occurs for the first few cms of the distance travelled in the porous medium.

KEYWORDS

Viscoelasticity, extensional viscosity, shear thinning, CaBER, hyperbolic contraction, polymer scission, polymer mechanical degradation, vortices, corner vortices, Wi -Re plot, residence time, porous media, Hencky strain, Extensional flows, HPAM, enhanced oil recovery, Linear injections, Radial injections, Several contractions in series, energy activated mechanism for scission, Weissenberg number, Deborah number. Table 2. 2.3: 𝑊𝑢 is the entrance width of the geometry, 𝑊𝑐 is the width at the throat of the contraction (Refer Figure 3.8), h is the height of the channel, 𝑡𝑟 is the relaxation time of the polymer chain, 𝑡𝑟𝑒𝑠 is the residence time, 𝐷ℎ is the hydraulic diameter given by 𝐷ℎ = 2ℎ𝑊𝑐(ℎ + 𝑊𝑐), 𝐿𝑣𝑠 is the length of the vortices, The geometrical constant 𝐷𝑔𝑒𝑜 = (𝑊𝑐 -𝑊𝑢)𝐿𝑐𝑊𝑢 and G is the modulus. . 2.4: Literature review on vortices in different axisymmetric contraction geometries. 30 Table 2.5: Molecular weight dependence for critical strain rate of onset of degradation found by different researchers and their conditions of degradation. The results summarized above are in fast transient extensional flows where devices consist of an orifice or a contraction. The critical strain rate for onset of degradation scaling with molecular weight as 𝜀𝑓 ∝ 𝑀𝑤 -𝑏. In dilute steady state scission, due to the infinite residence time the polymer chains completely unravels and the value of exponent would be 𝑏 = 2 [START_REF] Odell | Degradation of polymer solutions in extensional flows[END_REF][START_REF] Hunkeler | Polymer solutions under elongational flow: 1. Birefringence characterization of transient and stagnation point elongational flows[END_REF] 3.4: Nomenclature to specify the type of polymer solutions for a given experiment. Here n is the shear thinning index, tcm the relaxation time from the Carreau model, tca is the relaxation time from the CaBER equipment, c* is the critical (overlap) concentration that separates the dilute and semi-dilute regimes and is calculated as the inverse of intrinsic viscosity as in Eq (3.4). σ is the surface tension of the polymer solution which is similar to the surface tension of the solvent used, given the small concentration of polymer. The salinity of all the solutions is 0.6% (w/w) salt composition as described in section 3.1. ......................... 44 Table 3.5: List of geometries used in the current study and their dimensions ....................... 47 Table 3.6: Dimensions of geometries with different Hencky strain. Wu is the entrance width , Wc is the contraction width, Lc is the contraction length h is the channel height, 𝜀𝐻 is the Hencky strain, and α is the aspect ratio. 5.1: Dimensions of the hyperbolic contraction geometry used in this chapter.* is mentioned to differentiate the geometries with same Hencky strain (refer Table 3.5). ........ 82 Table 5.2: Nomenclature for polymer solutions used. Here n is the shear thinning index, and tcm is the terminal relaxation time from the Carreau model. The applicability of the empirical Cox-Merz rule (linking frequency in the linear viscoelastic regime to the shear rate in nonlinear regime) supports the use of terminal relaxation time obtained from the shear viscosity data in extensional regime 115 . c* is the critical (overlap) concentration that separates the dilute and semi-dilute regimes and is calculated as the inverse of intrinsic viscosity. ........... 83 5.9: Illustration of the scission at flow rates above Q max . Consider a first passage for Q1>Q1 max (Fig. 14). In the second passage a maximum occurs for flow rates Q2 max ≅ Q1, as the polymers that stretch for flow rates Q<Q1 (pink shaded portion) have already been degraded and do not contribute for extension while the non-degraded polymer during the first passage at Q1 would only degrade for Q2>Q1. . 5.14: Red stars show the evolution of multiple passages as a function of the 1 st pass for the data of Figure 5.12. In each case the ordinate is the ratio of Q max for successive injections (1 to 2, 2 to 3, etc.) to Q max of the first injection, and the abscissa is the ratio of Q max of the final injection to Q max of the first injection. The solid lines are calculated using (5. 6.3: Hencky strain as a function of critical strain for scission 𝜀𝑇𝑚𝑎𝑥 Each point corresponds to the true strain rate at the maximum of the pressure drop ratio curve R(Q), when injected with 11Mda-0.08/40SG polymer solutions through a geometry with given Hencky strain 𝜀𝐻 of Table 3.6. The strain rate at the maximum 𝜀𝑇𝑚𝑎𝑥 is calculated from the length of the vortices Lvs measured in the viscoelastic instability regime (7 th flow regime of Figure 4.3) for 𝜀𝐻 = 3.9 𝑎𝑛𝑑 3.2; Stable symmetric corner vortex regime (2 nd flow regime of 6.9: Evolution of maximum pressure drop ratio curve from successive injections of 15MDa-0.08/40SG polymer solution. Here, Q1 is the first injection flow rate, Q1 max is the flow rate at the maximum in pressure drop ratio for the non-degraded material, and Q2 max is the flow rate at the maximum in pressure drop ratio of second passage. The solid line shows a fit to Eq. (5.7) for different scission exponent b which depends on the Hencky strain of geometry. Open symbols correspond to the geometries with different geometrical dimensions than that of the solid symbols (Refer Table 3.

List of Tables

List of Figures

Chapter I -Introduction

The depleting number of oil reserves and the increased energy needs of the growing population force the need for efficient methods for oil recovery. In practice, oil recovery is done in 3 different processes namely (Figure 1.1),

• Primary process -the oil is recovered due to resulting pressure difference from the reservoir and the atmospheric pressure. Only 5-10% of the total oil present in the reservoir is recovered in this process

• Secondary process-Water is injected into the reservoir to create the pressure and thereby push the oil in the reservoir. 35-40% of total oil is recovered. Injection of water leads to Rayleigh-Taylor instabilities due to relative low viscosity and the poor interfacial tension of the injected water with the oil in the reservoir.

• Tertiary process-also called enhanced oil recovery (EOR). It consists of 2 different techniques namely, Thermal (injection of steam) and chemical (viscosifiers like polymers are added to the injected solution). Polymer flooding with viscoelastic polymers is a chemical EOR technique which could result in more oil recovery than that predicted from a capillary number analysis (Capillary number is the ratio of the viscous forces to the interfacial tension of fluids in contacts). This additional oil recovery is due to the elastic effects of the polymer solutions resulting in oil displacement from the dead-end pores (enhanced pulling experienced by the oil blobs due to the normal forces) or stabilization of oil threads against snap off. These mechanisms can be related to either linear or non-linear elastic effects. However, the linear and viscoelastic effects of polymer solutions depend on the molecular weight of the polymer chains. Therefore, the efficiency of oil recovery by polymer flooding is greatly limited by mechanical and chemical degradation of polymers. Degradation leads to an irreversible loss of viscosity due to decreased molecular weights of the polymer chains. When injected in the reservoir, in the case of a radial injection from a well, it is known that the early few centimeters travelled through the rock are the crucial domain where polymer undergoes degradation.

Several studies have addressed the scission by measuring the onset of scission either on laboratory scale, either in single contractions devices or porous medium, which is a 3D complex network of contractions and expansions. They proposed empirical approaches to predict scission in these devices. However, some of these empirical approaches have tried unsuccessfully bridging the gap between single contraction devices and porous medium.

In this thesis we ask following questions:

• Is it possible to quantify scission using an empirical approach either to predict the scission in contraction devices or the porous medium?

• If so, is it possible to apply the empirical approach of a single contraction device to contraction device with several contractions and further to a porous medium with a complex network of contractions and expansions?

• Finally, if a successful approach can be developed, can this be extended to reservoir injecting conditions to understand different parameters improving the recovery process?

To answer to these questions, we use the following approach.

In the first step we use a single hyperbolic contraction geometry. Different solvents and polymer solutions with different rheological properties are injected through this hyperbolic contraction geometry and the pressure drop between entrance and exit is measured. The ratio of pressure drop of the polymer solution to that of the solvent results in a maximum as a function of flux. The maximum occurs as a result of competition between the extensional and then scission of polymer chains. The streamlines of polymer flow are also recorded from the optical visualization experiments, in order to measure the true strain rate. From multiple passages on this single contraction device we developed a universal model that predicts quantitatively the scission. We found this model to be independent of the physical properties of solution, but dependent on the Hencky strain imposed by the geometry. When the model was extrapolated to several contractions in series, we observed that the true strain imposed on the polymer solution is different from the strain imposed by smooth laminar flow in geometry, due to the formation of the vortices. Thanks to the dependency of the universal model on the Hencky strain, the quantity of scission of polymers for several contractions in series can be predicted. Furthermore, we found this model can be adapted to porous media.

In brief, the scission of polymers is first addressed in single contraction devices with different geometrical parameters, generalized to several contractions in series and finally to a porous media. Lastly, the model is extended to reservoir conditions, to understand the life of polymers while travelling in an oil well.

This thesis consists of 8 chapters and is organized as follows:

In chapter 1, we briefly discuss the problems and the corresponding implemented approaches in the current manuscript.

In chapter 2, firstly few generalities of the different materials are discussed briefly. Then the onset of coil-stretch transition and the extensional viscosity phenomena as discussed in the literature are detailed. A review on formation of upstream vortices, parameters influencing their onset, type of the flow regime, different transitions and their dimensions are mentioned.

Chapter In chapter 5, different solvents and semi-dilute unentangled polymer solutions with different rheological properties are injected at different flow rates through a single hyperbolic contraction device and the pressure drop is measured between entrance and the exit of the channel. The relative pressure drop of the polymer solution with that of the solvent exhibits a maximum when observed as a function of flow rate (pressure drop ratio curve). This maximum can be thought as the results of the relative competition between the extension and scission of the polymer chains. Thanks to the flow field measurements, we were able to calculate the true extensional strain rates. We observe that the strain rate at the maximum is inversely related to the relaxation time and therefore occurs for a given Weissenberg number for all polymer solutions used (molecular weight, concentration, solvent viscosity, solvent quality). From the double passages we have observed that polymer undergoes scission when injected through hyperbolic contraction devices and we propose a model to predict the scission. This model is robust, independent of physical chemical parameters and is extendable for multiple passages.

In chapter 6, polymer solutions with different rheological properties are injected through geometries with different Hencky strains. Similar inverse relation as earlier holds between the strain rate at the maximum of pressure drop ratio curve and the relaxation time of the polymer solution. Double passage experiments in geometries with different Hencky strain resulted in the similar universal model, but with different parameters. In continuation, with single and double passages on contractions in series, we observed that the true Hencky strain imposed on the polymer chain is different from the Hencky strain corresponding to the geometry. It was possible to calculate the scission in successive contractions from the models, to predict scission and its dependency on the Hencky strain.

In chapter 7, we extend the model to predict the scission of polymers developed on a single contraction microfluidic device to porous medium. The model was found to be applicable to porous media and we were able to measure the degradation and the pressure drop between the entrances of exit of the porous medium of different lengths. Degradation was observed to increase with the length of the porous medium until a critical value, over which no further significant degradation was observed. In addition, when the model was extrapolated to the reservoir conditions (radial flows), we observed the degradation was important in the first few cms of the distance travelled in the reservoir.

In chapter 8, the important conclusions of our study are mentioned along with the perspectives of the current study.

Chapter II -Literature Review

Generalities on rheology

Rheology is word of Greek origin derived from "Rheo" means flow and "logy" means study, study of the deformation and flow of matter is termed as Rheology. What is a fluid? The simple definition for this question is, everything which flows. This rule out solids from being fluids. It is worth to cite here one of the well-known prophesies as quoted by Deborah "Mountains flowed before the Lord", which means mountains do flow, but during an infinite time scale which is only possible for the lord and not for human beings whose life span is about several decades. This led to the introduction of fundamental dimensionless Deborah number De which is defined as the ratio of relaxation time and the time of deformation. The relaxation time depends upon the internal structure of the system.

For instance, mantle made up of rocks can be considered as solids at time scales of human life but can be modelled equivalent to liquids by considering the convective motion in the mantle which are responsible for continental drift which is hardly noticeable for human beings since the continental plates move at a few centimetres per year. Another live example is the pitch drop experiment, started in 1927 in the university of Queensland where a piece of pitch, which appears to be solid at human time scales, flows under its own weight and drips at a rate of one drop during several years. Its viscosity is of order 10 8 Pa.s.

For De<<1, is a viscous liquid because the deformation time is greater than the relaxation dynamics of the system. For De >>1 the system has sufficient time to completely relax within the time scale of observation and therefore the system behaves like a elastic solid. For De=1 the relaxation time is comparable to the observation time, so the system behaves as a viscoelastic material.

In this section, we present constitutive models that described the state of the system in these 3 regimes of Deborah number.

Cauchy equation

In a continuum theory, a fluid flow can be well explained by the Cauchy's equation which includes conservation of mass and incompressibility implying 𝛁. 𝐮 = 𝟎, (2.1) and conservation of momentum is expressed by

𝛒 ( 𝛛𝐮 𝛛𝐭 + (𝐮. 𝛁)) 𝐮 = 𝛁. 𝛕 + 𝛒𝐠, (2.2)
where u(r, t) is the local velocity, r and t are the position and time respectively. ρ is the density of fluid, 𝛕(r, t) is the local stress tensor, g represents gravity.

Stress tensor

To completely specify stress (force per unit area) at any point in the fluid, one needs nine components since each component of the stress must be defined not only by the direction in which it acts but also the orientation of the surface upon which it is acting. The first index i indicates the direction of stress, and the second index j represents the orientation of the surface upon which it is acting. Therefore, the i th component of the force acting on a surface whose outward normal points in the j th direction is τij. For classical materials, the stress tensor is symmetric i.e τij= τji. Moreover, for incompressible fluids, the stress tensor τ can be also written as

𝛕 = -𝐏𝐈 + 𝛔, (2.3)
Where P is the pressure and σ is the extra stress tensor.

Strain rate tensor

The strain rate tensor D is a physical quantity which describes the rate of change of the deformations of a material in the surroundings of position r at time t (it can also be defined as the derivative of the strain tensor with respect to time),

𝐃 = 𝟏 𝟐 (𝛁𝐮 + 𝛁𝐮 𝐭 ). (2.4)
There are basically 2 types of flows; namely shear and extensional flows (2.9)

The extensional viscosity is defined as the ratio of the first normal stress to the strain rate, 𝛈 𝐞𝐱𝐭 = 𝛔 𝟏𝟏 -𝛔 𝟐𝟐 𝛆̇.

(2.10)

Elastic solids

An elastic solid can be described by Hooke's Law. Hooke's Law states that the applied stress σ is proportional to the corresponding strain γ = δ L ⁄ , where δ is the relative displacement of a body of length L. The proportionality constant between stress and strain is defined as the modulus of elasticity, G, 𝛔 = 𝐆𝛄.

(2.11) The tensorial version is similar. Let us assume that the body is labelled by its coordinates in space x. After deformation, each point originally at x will move by a vector u. Let xi and ui be the components of x and u in direction i.

The shear strain is

γ ij = 1 2 ( ∂u i ∂x j + ∂u j ∂x i
) and the stress tensor is σ ij . The proportionality constant between σ ij and γ ij for i ≠ j is the shear modulus, G. Under uniaxial loading in direction 1, the extensional stress is also proportional to the extensional strain , and the proportionality constant is given by the Young's modulus E. Both the shear modulus G and the elastic modulus E define the stiffness of the material are related by 𝐄 = 𝟐𝐆(𝟏 + 𝛎).

(2.12)

Here the Poisson ratio ν is the ratio of transversal strain to axial strain, under uniaxial loading.

A material which obeys Hooke's Law, instantly responds to the applied stress and also recovers it original shape completely upon removal of the applied stress.

Strain

The elongational strain has a particular expression which is the Hencky strain ε H . This is the integrated strain over the whole deformation, is given by

𝛆 𝐇 = ∫ 𝐝𝛆 = ∫ 𝛅𝐥 𝐥 𝐋 𝐟 𝐋 𝐢 = 𝐋𝐧 ( 𝐋 𝐟 𝐋 𝐢 ), (2.13)
where L f is the final dimension and L f is the initial dimension. The Hencky strain is considered the total strain from the complete history of imposed deformations.

Viscous fluids

Ideal viscous fluids can be described by the Newtonian law. For Newtonian fluids like water, glycerol, ethanol and their mixtures, the viscosity η is constant for any shear rate, and the extra stress tensor σ and the velocity field are related by the simple constitutive relation, 𝛔 = 𝟐𝛈𝐃 (2.14) In simple shear flow the first and second normal stress are 0, i.e. N1 = N2 = 0, and the viscosity is given by ratio of shear stress to the shear rate as in Eq. ( (2.17) For a Newtonian solvent the stress tensor σ s is given by Eq.(2.14). Several models have been proposed for calculating the contribution of polymer to the stress tensor σ P . There are two regimes of viscoelasticity, namely linear and the non-linear regime. Some of the constitutive equations proposed in these 2 regimes are mentioned in Table 2.1.

Linear viscoelasticity

The linear viscoelastic regime occurs at small deformations. The Maxwell model is the simplest viscoelastic model

Maxwell model

James Clerk Maxwell in 1867 proposed a simplest analytical description of viscoelastic liquids by analogy with a purely viscous damper and a purely elastic spring connected in series. Under applied shear stress, the total stresses σ T and the total strains ε T are defined as follows For the elastic spring the l stress σ e and the strain γ e are proportional with a proportional constant G M 𝛔 𝐞 = 𝐆 𝐌 𝛄 𝐞 .

( and the total strain is given by the sum of strains from the individual components 𝛄 = 𝛄 𝐞 +𝛄 𝐯 .

(2.22)

The ratio of viscosity from the viscous damper to the modulus of elasticity from the elastic spring is the relaxation time t r of the damper,

𝐭 𝐫 = 𝛈 𝐌 𝐆 𝐌 (2.23)
For a step strain of magnitude γ imposed on the system at time t=0, for purely elastic solids, the stress instantaneously spikes up to the equilibrium stress Gγ from the Hooke's law and remains at this value until the applied stress is removed . ) .

(2.28)

This leads for the relaxation modulus to the relation

𝐆(𝐭) = 𝛔(𝐭) 𝛄 = 𝐆 𝐌 𝐞 ( -𝐭 𝐭 𝐫
) .

(2.29)

In practice, polymer fluids exhibit multiple relaxation modes, so the stress is often described by a combination of several Maxwell components.

Note that this Maxwell model applies only for small strains. Every material has a region of linear response at sufficiently smaller strains that the stress relaxation modulus being independent of the imposed strain.

Non-linear viscoelasticity

The regime for which the stress is non-linearly dependent on the strain or strain rate is called the non-linear visco elastic regime. This non-Newtonian regime occurs for large deformations.

For non-Newtonian regime, the viscosity depends on the rate of deformation Based on the type of response, the non-Newtonian regime, can be further classified into two regimes, namely shear thinning and extensional thickening (extensional) regimes.

Shear thinning

In simple shear flows, viscoelastic materials have a Newtonian plateau in viscosity at low shear rates where the viscosity is independent of the shear rate. Over a critical shear rate the viscosity start to decrease as a function of shear rate, which is called as the shear thinning regime (shown in Figure 2.2). The drop-in viscosity continues until it attains a second plateau at very high shear rates, often referred as a lower Newtonian plateau.

Extensional thickening

Besides shear forces, a fluid can experience extension such as in accelerated flows through contraction devices. The resistance to extension is called extensional or elongational viscosity. As depicted in Figure 2.2, the viscosity increases with the extensional rate which is called as the extensional thickening effect. Furthermore, the extensional viscosity depends on the type of extension e.g. uniaxial, equibiaxial or planar extension.

Model Constitutive equations Additional comments

Oldroyd-B t r σ p ∇ + σ p = 2η P E X ∇ = ∂X ∂t + (u. ∇)X -(∇u)X -X(∇u) T
Model doesn't predict the nonlinear regime or shear thinning regime. It predicts infinite extensional viscosity for finite strain rate.

FENE σ p = 𝐻𝑄 1 -< 𝑄 2 > 𝑄 0 2 ⁄
H is the spring constant; 𝑄 connection vector, 𝑄 0 2 is the maximum stretching in chain FENE model predicts well the shear stresses and the finite extensibility parameter cuts the infinitely growing extensional viscosity for a finite strain rate unlike the Oldroyd-B model

FENE-P σ p = G(fA -δ) t r A ∇ + fA = δ f = 1 + 3 b (1 + trace(σ p ) 3G )
FENE-P is a bead spring model. It is not able to show the hysteresis effects that polymers have, while the FENE model can.

Geisekus

HPAM polymer solutions flows through planar contractions and porous media

The objective of this thesis is to study the scission of HPAM polymer solutions flowing through planar contraction devices and porous media. In this section we discuss briefly the literature on rheological properties of polymer solution, the onset of extension phenomenon, flow instabilities observed during contraction flows, scission of polymers and finally the extension and scission of polymers in porous medium.

Rheology of polymer solution

Linear polymer-chains are macromolecules with several successive monomer units. Polymer chains in solution general take a random coil (or random walk) confirmation in solutions. Rheological properties such as viscosity and relaxation time are dependent on the effective size of these polymer coils, which is dependent on several parameters.

Polymer stiffness

HPAM polymer is made up of acrylic and amide monomer units. Two consecutive monomers of partially hydrolysed poly acrylamide are connected with a C-C link and therefore have respective preferred configurations. However, over a certain length the influence of a given monomer on the other is no more applicable, such a length is called as the Kuhn length denoted by b (or a random walk step size). A polymer coil can be thus simply assumed to be made of many such Kuhn units. The lower the number of monomers in a given Kuhn unit (i.e. Kuhn length b being smaller), higher is the flexibility of the polymer chain.

Chemical composition of monomers.

For HPAM polymers, acrylic monomers can hydrolyse in acrylate monomers which are negatively charged. HPAM are characterized by their degree of hydrolysis.

Solvent Quality

For polymer solutions, both monomer-solvent (m-s) and monomer-monomer (m-m) interactions are important and must be considered. Excluded volume is the volume occupied by a molecule of the polymer cannot be occupied by the other molecule due to repulsion of electronic clouds. For a θ-solvent the excluded volume interactions are zero (i.e. no attractive or repulsive interactions) [START_REF] Hiemenz | Polymer chemistry[END_REF][3][4] . For good solvent the excluded volume interactions are non-zero and are positive (repulsive), therefore polymer coils are swollen. For bad solvents the excluded volume parameter is non-zero and negative (attractive) therefore polymer coils are collapsed. Charles W. Manke et al. [START_REF] Kong | Effect of solvent quality on the conformation and relaxation of polymers via dissipative particle dynamics[END_REF] performed dissipative particle dynamics technique using the form which was earlier suggested by Hoogerbrugge and Koelman [START_REF] Hoogerbrugge | Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics[END_REF][START_REF] Koelman | Dynamic simulations of hard-sphere suspensions under steady shear[END_REF] . From the gyration radii of the polymer, for single linear chains in solvents, they have shown the transition from expanded confirmation to a collapsed state when the solvent quality (adjusting the amplitude of repulsive interactions) is changed from good to poor solvents.

Polymer concentration

Based on the relative amount of the polymer chains in a given volume of solution, different concentration regimes include dilute, semi-dilute unentangled, semi-dilute entangled and concentrated. In dilute solutions the polymer chains are relatively separated from one another, and with increase in concentration the polymer coils touch one another. The critical overlap concentration is defined by c*. For c<c* polymer solutions are dilute and for c>c* polymer solutions are in the semi-dilute regime [START_REF] Castelain | A study of the viscosity of cellulose derivatives in aqueous solutions[END_REF] . With increasing concentration the parts of polymer coils begins to entangle, defined by the crossover concentration ce and further increase in concentration would lead to a concentrated regime 4 .

Salinity level

Charged polymer (polyelectrolytes) confirmations are greatly dependent on the presence of salts in the solution. Salt screens the charge on the monomers and the polymer chains assume a coil confirmation. In case of absence of these salt ions the charges on the monomers are not screened and due to the repulsive forces, the polymer chains can assume an extended confirmation [START_REF] Michael Rubinstein | Scaling theory of polyelectrolyte solutions[END_REF] .

The molecular weight and concentration scalings for the coil size, relaxation time and the viscosity of neutral polymer solutions with different solvent qualities in different concentration regimes are given in Table 2.2.

Dilute Semi-dilute unentangled Semi-dilute entangled Fractal exponent (ν) Good solvent 0.588(~3/5) Θ-solvent 0.5 (1/2) Good solvent 0.588(~3/5) Θ-solvent 0.5 (1/2) Good solvent 0.588(~3/5) Θ-solvent 0.5 (1/2) Radius R R ≈ b ( v b 3 ) 2ν-1 N ν R ≈ ξ ( N g ) 1 2 ⁄ Where ξ = b ( v b 3 ) 2ν-1 ν ⁄ g ν R ≈ a√ N N e ≈ b√N a ≈ b√N e And g = ( b 3 v ) 3(2ν-1) (3ν-1) ⁄ ϕ -1 (3ν-1) ⁄ R ∝ N 0.58 R ∝ N 0.5 R ∝ N 0.5 ϕ -1 (3ν-1) ⁄ R ∝ N 0.5 ϕ -1 (3ν-1) ⁄ R ∝ N 0.5 R ∝ N 0.5
Relaxation time Zimm relaxation time

t Zimm ≈ η s R 3 kT ≈ η s R 3 ( v b 3 ) 6ν-1 N 3ν kT t Rouse ≈ η s ξ 3 kT ( N g ) 2 ≈ η s b 3 N 2 (φ) 2-3ν 3ν-1 ⁄ kT t reptation = ( ξ b ) 3 ( N e g ) 2 ( N N e ) 3 t Zimm ∝ N 1.76 t Zimm ∝ N 1.5 t Rouse ∝ φ 0.31 N 2 t Rouse ∝ φN 2 t reptation ∝ φ 1.6 N 3 t reptation ∝ φ 2.3 N 3 Specific Viscosity η sp ≈ η s φN 3ν-1 η sp ≈ η s Nφ 1 3ν-1 ⁄ η sp ≈ N 3 N e 2 φ 3 (3ν-1) ⁄ η sp ∝ φN 0.78 η sp ∝ φN 0.5 η sp ∝ Nφ 1.3 η sp ∝ Nφ 2 η sp ∝ N 3 φ 3.9 η sp ∝ N 3 φ 4.7
Table 2.2: Radius of the coil R, relaxation time tr and specific viscosity scalings with Molecular weight and concentration in the three concentration regimes (dilute, semi-dilute non entangles and semi-dilute entangled) for a neutral polymer in good and θ-solvent.

Adapted from Colby (2010) 4 .

Shear flows

Shear flows occur in geometries with no velocity gradient in the flow direction. Shear flow is a superposition of extension and rotation, in which a polymer chain stretches along the extensional direction but rotates so that the coil then contracts. This leads to weak stretching 10, 11 . The phenomenon of shear thinning in polymer solutions has been well studied, and originates from the orientation and disentanglement of chains under flow Doi and Frederick [START_REF] Doi | The theory of polymer dynamics[END_REF] . The Carreau-Yasuda model, is a phenomenological model which adequately describes the shear-thinning behavior of many polymer solutions,

𝛈 𝐩 (𝛄̇) = 𝛈 ∞ + (𝛈 𝐩𝟎 -𝛈 ∞ ) (𝟏+(𝒕 𝐜𝐦 𝛄̇) 𝟐 ) (𝟏-𝐧) 𝟐 , (2.30)
where η ∞ is the viscosity at infinite shear rate (assumed here to be the one of the solvents ~ηs ), η p0 the zero-shear viscosity, γ̇ shear rate, and n < 1 is the shear thinning index.

Extensional flows-onset of extensional viscosity

In pure extensional flow the chains stretch and eventually completely uncoil. At macroscopic scales this leads to a large increase of the extensional viscosity [7]. De Gennes (1974) predicted that polymer coils in dilute solution under extensional flow will experience a sudden coil-stretch transition at a critical strain rate εĊ which is inversely proportional to the longest relaxation time of the polymer coil [START_REF] De Gennes | Coil-stretch transition of dilute flexible polymers under ultrahigh velocity gradients[END_REF] .

By increasing concentration the critical strain rate εĊ decreases, as inferred from capillary extensional rheology experiments [START_REF] Clasen | How dilute are dilute solutions in extensional flows?[END_REF] and from Brownian dynamics simulations with hydrodynamic interactions [START_REF] Stoltz | Concentration dependence of shear and extensional rheology of polymer solutions: Brownian dynamics simulations[END_REF] . Indeed, for both dilute solutions and semi-dilute solutions, the strain rate for the onset of stretching was found to decrease with concentration [START_REF] Haas | Characterization of dilute polyacrylamide and polystyrene solutions by means of porous media flow[END_REF][START_REF] Keller | The extensibility of macromolecules in solution; a new focus for macromolecular science[END_REF][START_REF] Georgelos | Apparent thickening behavior of dilute polystyrene solutions in extensional flows[END_REF] . This was attributed to interactions between polymers during stretching and suggested an ultra-dilute regime where there would be no entanglements or interactions under stretching [START_REF] Clasen | How dilute are dilute solutions in extensional flows?[END_REF][START_REF] Chow | Entanglements in polymer solutions under elongational flow: a combined study of chain stretching, flow velocimetry and elongational viscosity[END_REF] .

The extensional viscosity at high extension rates is primarily due to the resistance offered by the stretching of polymers. The transient extensional viscosity depends on both strain rate and strain. Gupta et al. measured the extensional stress growth as a function of strain in dilute and semi-dilute polystyrene solutions using a filament stretching device [START_REF] Gupta | Extensional viscosity of dilute polystyrene solutions: Effect of concentration and molecular weight[END_REF] . They found that, while linear shear stress relaxation follows Zimm relaxation due to hydrodynamics, extensional stress growth is better-described by Rouse dynamics. The Zimm-Rouse transformation occurred at a Hencky strain ε H ≈ 2 . In this paper we work in the semi-dilute unentangled regime with a geometry whose Hencky strain is greater than 2, where the Rouse dynamics are applicable.

Extension-dominated flows present complex molecular configurations than that the can be qualitatively understood by a single dumbbell model. Chu et al., showed that the complex configurations of individual DNA molecules under extension includes single dumbbells, halfdumbbells, kinked chains and the folded molecules [START_REF] Perkins | Single polymer dynamics in an elongational flow[END_REF] .

Extensional flows are studied in cross slot flow devices [START_REF] Odell | Degradation of polymer solutions in extensional flows[END_REF] , four-roll mill, contraction devices (both axisymmetric and planar), and filament stretching devices. Our study involves flow of polymer solution through hyperbolic planar contraction devices and in porous media. Flows in such devices are mixed flows with both shear and extension flows experienced in different regions of the flow cell. Flow of polymer solutions in these contraction devices are often complex due to the flow instabilities which occur at the upstream of the contraction. In mixed flows [START_REF] Doyle | Relaxation of dilute polymer solutions following extensional flow[END_REF] (extensional and shear dominated flows) the dissipated energy can be observed in macroscopic measurements such as the pressure. In the next section we discuss these flow instabilities in detail.

Vortices onset and their growth

For non-Newtonian solutions flowing through contractions, where extensional forces are dominant can develop vortices at the entrance of the contraction. The study of formation of vortices is important due to their applications in wide variety of industrial contexts, such as injection molding of polymer melts, melt spinning and film blowing.

The flow of a viscoelastic fluid through contraction-expansion is a complex flow with strong shear forces close to the wall and extension along the centreline at the upstream and expansion/extension in orthogonal to the flow direction at the downstream. Lip vortices, Moffatt eddies, stable axisymmetric corner vortices, axis asymmetric corner vortices, time-fluctuating corner vortices, viscoelastic turbulence/time-dependent unstable flows have been reported in the axisymmetric contraction and expansion geometries with sharp and rounded corners 23-26 27-33 , hyperbolic contractions-sudden expansion geometry , stagnation flows and serpentine geometries [START_REF] Galindo-Rosales | Viscoelastic instabilities in micro-scale flows[END_REF] . Table 2.4

The onset and the growth in each of these flow regimes depends on the type of geometry, contraction ratio β, aspect ratio α, contraction length Lc, height h, polymer flexibility, Deborah or Weissenberg number, and Reynolds number, all of which are defined below. A brief review of studies on formation of vortices is mentioned in Table 2.4. Here we discuss the briefly the important findings in the literature that are relevant to our study. In most of the cases, at very low flow rates and for corresponding dimensionless numbers (Wi, De, Re) a Newtonian like flow is reported. Evans and Walter et al. studied entry flow problems for Boger and shear thinning fluids in planar and square-square contraction geometries in contraction ratios β ranging from 4 < β <80. A Boger fluid has no vortices for any contraction ratio and the flow is always Newtonian like [START_REF] Evans | Flow characteristics associated with abrupt changes in geometry in the case of highly elastic liquids[END_REF][START_REF] Evans | Further remarks on the lip-vortex mechanism of vortex enhancement in planar-contraction flows[END_REF] . For a shear thinning fluid they reported lip vortices at low Weissenberg number; Lip vortices are vortices which occur at the mouth of contraction and that do not touch the side walls of the geometry. Rothstein and Mckinley et al. [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] , in their studies on axisymmetric contraction geometries with contraction ratios 2 < β <8, observed lip vortices for β <2, but did not observe any lip vortices for β = 4 and 8. However Lanzaro et al. [START_REF] Lanzaro | Effects of contraction ratio on non-linear dynamics of semidilute, highly polydisperse PAAm solutions in microfluidics[END_REF] , in their studies of polyacrylamide polymer solutions with 3.3 <c/c*<16.6 in abrupt planar contraction geometries for 4 < β <16, reported no lip vortices for β = 4 and lip vortices for β = 8 and 16. 2D numerical simulation in planar contraction flow studies made by Alves et al. with the PTT (Phan-Thien Tanner) model describing a shear thinning fluids for contraction ratios ranging 4 < β <100, found lip vortices for β>10 and but not for lower contraction ratios [START_REF] Alves | On the effect of contraction ratio in viscoelastic flow through abrupt contractions[END_REF] . 3D-numerical simulations for 8:1 planar contraction geometries with varying depth (height of the channel) made by Omowunmi et al. suggested aspect ratio α is another key parameter. They reported that lip vortices were prevalent for geometries with lower aspect ratio [START_REF] Omowunmi | Modelling the three-dimensional flow of a semi-dilute polymer solution in microfluidics-on the effect of aspect ratio[END_REF] . Jonathan and McKinley et al. suggested a dimensionless normal stress ratio parameter ℵ , which is the ratio of the normal stresses resulting along the walls due to shear flow to the extensional stresses along the central line of the extensional flow. They reported the lip vortices only for ℵ>0.055±0.005 [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] . In summary, the occurrence of lip vortices depends on the type of the polymer solution and geometrical parameters. However, the geometrical dependences reported in the literature are not consistent. In our study the aspect ratio α is in the range of 0.18<α<0.16 and the contraction ratio β are in the range of 4<α<50, we do not find the formation of lip vortices in the experimental range. We found that the most relevant parameter was the dimensionless normal stress ratio parameter ℵ proposed by Jonathan and McKinley et al. [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] . With increasing Weissenberg number (flow rate) corner vortices are reported to set in. Vortices which occur at the corners and grow along the walls of the geometry are called corner vortices. Evans and Walter et al., for shear thinning fluids in planar and square-square contraction geometries in contraction ratios β ranging from 4 < β <80, reported corner vortex growth whose dimensions are strongly dependent on the Weissenberg number [START_REF] Evans | Flow characteristics associated with abrupt changes in geometry in the case of highly elastic liquids[END_REF][START_REF] Evans | Further remarks on the lip-vortex mechanism of vortex enhancement in planar-contraction flows[END_REF] . The vortex size is reported to be larger in the axisymmetric contractions than in the planar contractions. This is because the total Hencky strain accumulated in uniaxial extension in axisymmetric is higher than in planar extension in planar contractions [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] . However Galindo et al. [START_REF] Galindo-Rosales | Viscoelastic instabilities in micro-scale flows[END_REF] found that, the Weissenberg number was not very good parameter for predicting the dimensionless vortex size χ vs , for injecting different concentrations of PAA aqueous solutions through hyperbolic planar contraction geometries. Cloitre et al. studies on rigid fiber suspensions flow through a conical axisymmetric contraction. They suggested that vortices result from the mechanical balance between the extensional stresses in the flow direction and the shear forces in the radial direction. Such that the dimensionless vortex size χ vs is proportional to square root of the ratio of extensional to shear viscosity [START_REF] Mongruel | Axisymmetric orifice flow for measuring the elongational viscosity of semi-rigid polymer solutions[END_REF] . Evans and Walter et al. reported that, stable upstream vortices become unstable at moderate Weissenberg and Reynolds numbers in both axisymmetric and planar contractions geometries as reported by 35, 36 . Chiba et al. studied shear thinning aqueous polyacrylamide solutions flowing through circular contraction devices with contraction ratio β=2. They reported other elastic instabilities like asymmetric helical flow patterns, azimuthally-varying elastic corner vortices, and 'buckling' flow structures [START_REF] Chiba | Instabilities in a circular entry flow of dilute polymer solutions[END_REF] .

Wi-Re space

Flow regimes are often characterized by Elastic number, defined as the ratio of the Weissenberg number (characterizes the elasticity of the polymer) and the Reynolds number (characterizes the inertial effects). Rodd et al. studied aqueous PEO polymer solutions in planar contraction devices. They suggested that different flow regimes evolve along a given elastic number, which is the slope of Wi-Re plot (see Figure 2.3). In their studies the critical Weissenberg number for transition from one flow regime to another flow regime occurs for: higher Weissenberg number for higher elastic number, lower Weissenberg number for moderate elastic number and slightly higher Weissenberg number for lower elastic number (the critical Weissenberg number curve attains a minimum for varying elastic number). They have also explained that the formation of vortices and other viscoelastic effects develop due to interplay of elasticity and fluid inertia [START_REF] Rodd | The inertioelastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries[END_REF][START_REF] Rodd | The importance of downstream events in microfluidic viscoelastic entry flows: Consequences of increasing the constriction length[END_REF] . This kind of plot was also implemented by Hidema et al. [START_REF] Hidema | Effects of flexibility and entanglement of sodium hyaluronate in solutions on the entry flow in micro abrupt contraction-expansion channels[END_REF] for sodium hyaluronate (hyaluronic acid sodium salt, Na-HA) solutions in planar abrupt contraction devices, and Lanzaro et al. for aqueous polyacrylamide solutions in abrupt contraction geometries. However, our experimental finding contradicts such a dependence of different flow regimes on elastic number. Measurements of the pressure drop for polymer solutions flowing through abrupt axisymmetric contraction-expansions showed a substantial increase in pressure drop compared to that of a Newtonian fluid with same viscosity and flow rates. The occurrences of these vortices/visco-elastic instabilities to have no relevance to this additional pressure drop [START_REF] Rothstein | Extensional flow of a polystyrene Boger fluid through a 4: 1: 4 axisymmetric contraction/expansion[END_REF] .In other words the increase in pressure drop is due to the stretching of polymers and

Polymer scission

As the chain uncoils and extends, hydrodynamic drag between solvent and polymer can lead to scission of polymers for extension rates exceeding a critical rate 𝜺̇𝒇 when the extensional force approaches or exceeds that required to break a C-C bond (typically F≈nN per chain) [START_REF] Müller | Degradation of semidilute polymer solutions in elongational flows[END_REF] . More precisely, scission can be explained from thermal activation assisted by the increase in tension, which decreases the activation energy barrier required for breaking the C-C bond [START_REF] Odell | Degradation of polymer solutions in extensional flows[END_REF] .

Scission in extensional flow leads to huge viscosity losses [START_REF] Philippe | Flow-induced polymer degradation probed by a high throughput microfluidic set-up[END_REF] , and both stretching and scission depend on flow history and on the precise geometry of extensional flow [START_REF] Hunkeler | Polymer solutions under elongational flow: 1. Birefringence characterization of transient and stagnation point elongational flows[END_REF][START_REF] Hunkeler | Polymer solutions under elongational flow: 2. An evaluation of models of polymer dynamics for transient and stagnation point flows[END_REF] . In stagnation devices (e.g. cross slot or four mill devices) the residence time is considered infinite for the polymers trapped at the stagnation point. The critical strain rate 𝜺̇𝒇 for scission in this case scales with molecular weight as 𝛆̇𝐟 ∝ 1/M w 2 10, 54 . This scaling results from concentrating the viscous drag on an extended string of 𝑁 beads at the center of the chain, where the maximum chain tension 𝜎 scales as as σ ∝ 𝜂 𝑠 𝜀Ṅ [START_REF] Hiemenz | Polymer chemistry[END_REF] where 𝜂 𝑠 is the solvent viscosity [START_REF] Odell | Flow-induced chain fracture of isolated linear macromolecules in solution[END_REF] .

In contrast, for transient extensional flows there are several observed scalings for the critical strain rate 𝜀̇𝑓 ∝ 𝑀 𝑤 -𝑏 , depending on molecular weight, concentration, solvent quality, and whether the flow is laminar or turbulent (Table 1). The weak molecular weight dependence in turbulent flow (𝑏 < 1.3) is believed to be due to partial stretching. An exponent 𝑏 = 2 would apply for a fully stretched polymer chain, but turbulent flow frustrates this and reduces full extension relative to laminar flow. However, as shown by Table 1, there is not yet a distinct agreement on this topic. Breakage in transient flow has been modelled by several authors [START_REF] Vanapalli | Scission-induced bounds on maximum polymer drag reduction in turbulent flow[END_REF] . In Ryskin's "yo-yo model " of surface energy accumulation, polymer chains under extension stretch first at the centre [START_REF] Ryskin | Calculation of the effect of polymer additive in a converging flow[END_REF] . The remaining coiled portions at the extremes experience high viscous drag that is balanced by an increased tension at the midpoint. Larson and Magda 57 studied a bead-spring model and found maximum extension at the centre of the chain, which is supported by the hairpin model of Nguyen and Kaush [START_REF] Nguyen | Chain extension and degradation in convergent flow[END_REF] . Brett suggested that flow-induced scission leads to a narrower molecular weight distribution, with the decrease in polydispersity due to chain scission from pass to pass suggesting a precise facture pattern [START_REF] Buchholz | Flow-induced chain scission as a physical route to narrowly distributed, high molar mass polymers[END_REF] . By contrast, Muller suggested that for highly concentrated polymer solutions, scission is more randomized due to entanglements [START_REF] Müller | Degradation of semidilute polymer solutions in elongational flows[END_REF] . In dilute solutions tension is exerted by solvent-monomer friction, whereas in semi-dilute solutions stress is also transmitted through entanglements. In the latter case the stress per chain which effects degradation is controlled by the entanglement density 𝑐/𝑐 * 60 , where c is the polymer concentration and 𝑐 * is the overlap concentration. In Table 2.6 we mention the different conditions of polymer scission and the reported scission mechanism. To summarize, there is not yet any clear consensus of the precise mechanism of scission in polymer chains under flow.

The quality of solvent is also an important parameter in chain scission. An increase in solvent viscosity decreases the critical strain rate for the onset of stretching [START_REF] Durst | Flows of dilute hydrolyzed polyacrylamide solutions in porous media under various solvent conditions[END_REF] . Odell and Keller proposed a weak dependence for onset of degradation with solvent viscosity, 𝜀ḟ ∝ 𝜂 𝑠 -0.25 62 .

Indeed, degradation is more dependent on solvent quality than on solvent viscosity. In good solvents, polymers are more swollen than in bad solvent, and thus more easily stretched by flow gradients. A stretched polymer in poor solvents resemble a pearl necklace (with the 'pearls' comprising small coiled sections of polymers) [START_REF] De Gennes | Kinetics of collapse for a flexible coil[END_REF][START_REF] Montesi | Collapse of a semiflexible polymer in poor solvent[END_REF] . The extensional viscosity at a given extensional rate is observed to be lower in bad solvents than in good solvents [START_REF] Georgelos | Apparent thickening behavior of dilute polystyrene solutions in extensional flows[END_REF] . Moussa et al. studied the influence of solvent quality on degradation in turbulent flow, in which the motion between two counter rotating eddies involves extensional deformations [START_REF] Durst | Laminar and turbulent flows of dilute polymer solutions: a physical model[END_REF] . They found that the onset for degradation in theta solvents occurred at a lower Reynolds number than in good solvents [START_REF] Moussa | Effect of solvent on polymer degradation in turbulent flow[END_REF] . Similarly, Odell et al. studied polystyrene degradation in cross slot devices in toluene (good solvent) and decalin (theta solvent), finding a lower onset fracture strain rate in the second case. However such comparison is delicate since both relaxation time and c/c* are different in the two systems [START_REF] Philippe | Flow-induced polymer degradation probed by a high throughput microfluidic set-up[END_REF] .

Polydispersity influences the critical strain rate for the onset of stretching [START_REF] Keller | The extensibility of macromolecules in solution; a new focus for macromolecular science[END_REF] . Increasing the high molecular weight tail at a given average molecular weight should increase the extensional viscosity [START_REF] Palangetic | Dispersity and spinnability: Why highly polydisperse polymer solutions are desirable for electrospinning[END_REF][START_REF] Veerabhadrappa | Polymer screening criteria for EOR application-a rheological characterization approach[END_REF] . Polydispersity should have a similar influence on degradation (a distribution that enhances the higher molecular weight tail is more likely to undergo scission), although there have been few systematic studies.

As reviewed by Odell et al. [START_REF] Odell | Degradation of polymer solutions in extensional flows[END_REF] , from the calculations of molecular shape and the coil diameter of the polymer chain in simple shear flow by Cerf et al. [START_REF] Cerf | Recherches théoriques et expérimentales sur l'effet Maxwell des solutions de macromolécules déformables[END_REF] and Lumley et al. [START_REF] Lumley | On the solution of equations describing small scale deformation[END_REF] , Odell argued that shear flows are restricted from extension and thereby any scission. In addition, they believed that this degradation in laminar shear flows are arising due to presence of turbulence which can be poorly described by a Reynolds number (inertia) for the case of viscoelastic polymer systems.

For applications such as oil recovery, solutions of flexible polymers are injected into a porous medium, which comprises pores and necks that resemble contraction-expansion geometries which will be discussed in detail in next section. Several authors attempted to study scission in porous media by performing experiments on multiple contractions, where they observed that the most significant decrease in the molecular weight distribution occurs during the first contraction and depends on the strain and strain rate of the contraction [START_REF] Clay | Molecular degradation of concentrated polystyrene solutions in a fast transient extensional flow[END_REF] . So, the literature on polymer scission under extensional flow indicates that there exist multiple parameters influencing scission. The scission of polymer limits their usage in application like oil recovery. In the next section we discuss the extension and scission of polymers for flows in Porous medium.

Porous medium

Polymer flooding is a promising EOR technique, where polymer solutions are injected into the reservoir to recover the oil. Several researchers both from field and laboratory studies, have shown improved oil recovery [START_REF] Veerabhadrappa | Polymer screening criteria for EOR application-a rheological characterization approach[END_REF][START_REF] Huh | Residual oil saturation from polymer floods: laboratory measurements and theoretical interpretation[END_REF][START_REF] Levitt | Polymer flooding of heavy oil under adverse mobility conditions[END_REF][START_REF] Selby | Potential of non-thermal methods for heavy oil recovery[END_REF][START_REF] Wang | Viscous-elastic polymer can increase microscale displacement efficiency in cores[END_REF] . This improved oil recovery is discussed as:

The microscopic i.e. the oil recovery from trapped oil dead pores and

The macroscopic enhancement, like delayed formation of oil ganglia, from breaking of the coil chain due to the viscoelasticity nature.

Flows in porous media are complex mixed flows (shear and extension) due to multiple contractions and expansions. However, in porous media the coil stretch transition is a quite controversy topic. In early works of several researchers, the onset of extensional behaviour of polymer solutions was found to occur at a critical Deborah number [START_REF] Haas | Characterization of dilute polyacrylamide and polystyrene solutions by means of porous media flow[END_REF][START_REF] Durst | Laminar and turbulent flows of dilute polymer solutions: a physical model[END_REF][START_REF] Marshall | Flow of viscoelastic fluids through porous media[END_REF][82][START_REF] Hirasaki | Analysis of factors influencing mobility and adsorption in the flow of polymer solution through porous media[END_REF][START_REF] Southwick | Molecular degradation, injectivity, and elastic properties of polymer solutions[END_REF][START_REF] Haas | EFFECTS OF DILUTE POLYMER SOLUTIONS ON POROUS MEDIA FLOWS[END_REF][START_REF] Chauveteau | The onset of dilatant behaviour in non-inertial flow of dilute polymer solutions through channels with varying cross-sections[END_REF] . The extensional behaviour was modelled using a FENE dumbbell model to predict extensional viscosity [START_REF] Bird | Rheology and kinetic theory of polymeric liquids[END_REF] , and also explain the apparent extensional viscosities measured from screen viscometers [START_REF] Lim | The interpretation of screen-factor measurements[END_REF] . Different critical Deborah number for the onset of the extensional behaviour were observed, for De=0.5 in 65 De=0.1 in [START_REF] Marshall | Flow of viscoelastic fluids through porous media[END_REF] and De=10 in [START_REF] Chauveteau | Molecular interpretation of several different properties of flow of coiled polymer solutions through porous media in oil recovery conditions[END_REF] .This discrepancy in the Deborah was explained by Heemskerk et al. due to the difficulty in calculating the stretch rate in porous medium [START_REF] Heemskerk | Quantification of viscoelastic effects of polyacrylamide solutions[END_REF] . To model the rheological behaviour in porous medium, Ghoniem et al. studied onset of viscoelastic behaviour of partially hydrolysed polyacrylamide in successive contractions and enlargements of Plexiglass, and multiple passages on packed column of sand. They proposed a quantity SG (suggested group), for which the onset of viscoelastic behaviour is found in the Plexiglas holds true for the onset in the pack sand column. They also found the onset occurs for a fixed suggested group SG for whatever the porous core, polymer and the solvent conditions. In their case SG was the product of the stretch rate and the Deborah number [START_REF] Ghoniem | Extensional flow of polymer solutions through porous media[END_REF] In contrast Rodriguez et al. attributed the onset of thickening, was due to the formation of a transient polymer network. Howe et al. [START_REF] Howe | Flow of concentrated viscoelastic polymer solutions in porous media: effect of MW and concentration on elastic turbulence onset in various geometries[END_REF] proposed that the onset of extensional thickening is due to elastic turbulence. This was supported by observations of elastic turbulence and flow thickening in sinuous pipe flows [START_REF] Burghelea | Elastic turbulence in von Karman swirling flow between two disks[END_REF] . Lastly, James et al. reported that the normal stresses in extension are found to relate to the normal stresses developed in shear measurements [START_REF] James | N1 stresses in extensional flows[END_REF] .

Shear thickening in porous media is measured by a resistance factor or mobility reduction which is the ratio of the pressure drop due to injection of polymer solution at a given flow rate to the pressure drop for flow of solvent at the same flow rate [START_REF] Southwick | Molecular degradation, injectivity, and elastic properties of polymer solutions[END_REF] . Polymers in extended state develop huge extensional stresses which there by reduces the mobility of the solution.

Culter et al. observed degradation of viscoelastic polymers from the huge extensional stresses developed from the entrance effects of the capillaries [START_REF] Culter | Entrance effects on capillary degradation of dilute polystyrene solutions[END_REF] . Maerker et al. observed that the mechanical degradation of polymer solutions injected in porous media is due to the extensional stresses. They characterized degradation the loss in screen factor after the injection through porous medium. For a screen factor measurement, a solvent of fixed volume is allowed to pass through a five 100µm-mesh screens in series and time taken is noted, next a polymer of the same volume is passed through the mesh screens and the time taken is noted. The screen factor is the ratio of time taken by the polymer solution to the time taken by the solvent. They found that the degradation occurred for a strain rate above a critical strain rate and is an increasing function of strain rate. Maerker also found that screen factor loss increased as the ionic strength of the solution increased upon adding NaCI to HPAA solutions [START_REF] Maerker | Shear degradation of partially hydrolyzed polyacrylamide solutions[END_REF] .

Haas and Kulicke et al. observed that the resistance factor or mobility reduction decreased significantly in the second passage for a pre-degraded polymer solution. The pre-degraded polymer solution is collected from a first passage through a porous medium at a flow rate greater than critical value. The onset of shear thickening shifted to high flow rates in second passage for the polymer solutions pre-degraded at flow rate above this critical value for degradation [START_REF] Haas | Characterization of dilute polyacrylamide and polystyrene solutions by means of porous media flow[END_REF] . Farinato et al. studied mechanical degradation when injecting a high molecular weight poly (acrylamide-co-Na acrylate) in 1M NaCl through a porous medium. They observed the resistance factor or mobility reduction to decrease in successive passages [START_REF] Farinato | Polymer degradation in porous media flow[END_REF] . Ghoniem et al. studied the loss of resistance factor in multiple passages, observed a decrease in the pressure drop resistance factor with the number of passages. They correlated the shift in pressure drop resistance factor with quantity SG, (SG is the product of the extension rate and the Deborah number) [START_REF] Ghoniem | Extensional flow of polymer solutions through porous media[END_REF] . Muller et al. injected PEO solutions through porous media for several times and observed degradation only for Reynolds number in shear thickening regime [START_REF] Moreno | Flow-induced degradation of hydrolyzed polyacrylamide in porous media[END_REF] . The resistance factor or the mobility reduction decrease with the number of passes until it reached an asymptotic value. They have also reported that, when the degraded solution from the several passages is injected for final time through porous medium (passage number where it asymptotes), the onset of extensional thickening in the final injection was observed to occur at that Reynolds number where the polymer solution have been degraded during the previous injections. Similar results were reported by Sàez et al., This suggests that not all the polymers are degraded during the first passage [START_REF] Saez | Flow of monodisperse polystyrene solutions through porous media[END_REF] . During each passage a small quantity of polymers are degraded, which is the reason for decrease in the resistance factor from pass to pass. The resistance factor reaches an asymptotic value after several passes, where all the polymers capable of degradation are been degraded.

Degradation depends on the rheological properties of the polymer solutions,(molecular weight , concentration, solvent quality, solvent viscosity, as described elsewhere 100 ), and the nature of porous medium.

M.c.Laren et al. proposed that degradation increases with increasing the particle size. They quantified degradation by measuring the pressure drop in porous media of different lengths 101 , Muller et al. also found higher degradation for a higher particle size [START_REF] Moreno | Flow-induced degradation of hydrolyzed polyacrylamide in porous media[END_REF] . However, Kaser and Keller et al. 101 observed that the degradation increases with decreasing particle size 102 . Seright et al. 103 observed that the degradation is related to the particle diameter DP as u/dp 2 , where u is the Darcy velocity. studied degradation of HPAM through a porous medium of length 6mm and suggested that degradation reaches an asymptotic value for this length. They also suggested that degradation could reach an asymptotic value for radial injection. The quantity of degradation is lower under radial conditions than that observed in linear injection 106 . They quantified degradation from the loss in shear viscosities after injecting through a porous medium, measured on a Couette flow device.

In summary the origin of the onset of extensional thickening behaviour remains a question need to be understood. The degradation of polymer is primarily due to the extensional stresses. Furthermore, it is confirmed that no degradation occurs in pure shear flows. Literature provides wide range of dependence on the length of the porous medium for degradation. Also there exists a gap between the studies of linear injection from the radial injection.

Chapter III -Materials and methods

Polymer solution

Partially-hydrolyzed polyacrylamide (HPAM) with 30% hydrolysis (Figure 3.1) was obtained as powder from the Flopaam series (SNF Floerger, Andrezieux, France). The polymers were synthesized by the supplier using radical polymerization and are highly polydisperse, with Mw/Mn~4-5. Oligomeric polyacrylamide chains (PAM) were also obtained from SNF. The molecular weights obtained are summarized in Table 3.1. A 0.6% (w/w) salt aqueous solution is prepared with the salt composition (Table 3.2). Stock aqueous solutions 90% (w/w) of glycerol and 90% (w/w) ethanol were prepared by adjusting the salt composition to 0.6%. A PAM solution with an oligomeric concentration of 5% is prepared in 0.6% salt solution. The PAM solution behaves as a Newtonian solvent in our experimental range of flow rates.

Initial mother solutions with HPAM polymer concentration, 0.5 % (w/w) were prepared in a 0.6 % (w/w) salt solution, by adding slowly the polymer and simultaneously stirring with the help of a mechanical stirrer at a speed of 500rpm for duration of 2 hours. The mother solution thus prepared is stored for one day at room temperature. The mother solution is then diluted to the desired polymer concentrations using different Newtonian solvents. The viscosity of the solvent is modified by adding glycerol or PAM solution (M 𝑤 = 500kDa) or ethanol to water. All the polymer solutions prepared in the current work have a fixed salt composition of 0.6 % (w/w). Homogenous mixing is ensured by gently stirring the solutions overnight with a magnetic stirrer. Diluted solutions are then filtered using 5µm nylon screen (we verified that the nylon screen does not degrade the polymer).

The question may arise whether to consider the polyelectrolyte effects, i.e, the charge interaction between polymers. We follow the analysis proposed by Dobrynin et al. [START_REF] Michael Rubinstein | Scaling theory of polyelectrolyte solutions[END_REF] .The criteria for crossover from polyelectrolyte to non-polyelectrolyte effects is determined by the concentration of salts 𝐶 𝑠 and the polymer charge concentration 𝐶 𝑝 = 𝑐/𝐴 (where 𝑐 is the total monomer concentration and 𝐴 is the number of monomers per effective charge). If 𝐶 𝑠 > 𝑐/𝐴, then we do not expect polyelectrolyte effects since solvent ions can screen all of the polymer charges. For our systems the salt concentration 𝐶 𝑠 = 215mM. The polymer concentrations used (~0.08%w/w) results in polymer monomer concentration of order 𝑐~11mM (0.8/71). For polymers with 30% hydrolysis two charges are separated by 2 monomers (𝐴 = 3), leading to a polymer charge concentration 𝐶 𝑝 = 3.7mM, which is very low compared to the salt concentration. This confirms that the salt ions are in excess and our solutions are free from polyelectrolyte effects.

Shear characterization

The viscosities 𝜂 𝑠 of Newtonian solvents and 𝜂 𝑝 of the polymer solutions are measured using a ProRheo low shear LS300 rheometer. Viscosities are measured as a function of shear rate in the range 0.1-100 s -1 . The high sensitivity of this equipment delivers accurate values even at a shear stress of 20µPa. In Figure 2 we show a typical flow curve of polymer solution (viscosity as a function of shear rate) and the solvent used to prepare the solution. The viscosity of a Newtonian solution is constant with shear rate. The viscosity of a polymer solution has a Newtonian plateau at low shear rate followed by a shear thinning regime at a critical shear rate which is related to the relaxation time of the polymers.

Solvent-quality and Concentration domain

As explained in the introduction, the quality of the solvent and the concentration regime both affect chain scission. The relative increase of viscosity due to added polymer is expressed by the specific viscosity η sp ,

𝛈 𝐬𝐩 = 𝛈 𝐩 -𝛈 𝐬 𝛈 𝐬 . (3.1)
The reduced viscosity is the ratio of specific viscosity and polymer concentration c,

𝛈 𝐫 = 𝛈 𝐬𝐩 𝐜 . (3.2)
With no polyelectrolyte effects the specific viscosity depends on concentration according to the Huggins equation, The intrinsic viscosities (Table 3.3) depends on the solvent and obey [𝜂] 0𝑆𝐺 > [𝜂] 40𝑆𝐺 > [𝜂] 15𝑆𝐸 . The Fox-Flory equation predicts that the intrinsic viscosity is proportional to the hydrodynamic volume occupied by the polymer [𝜂] ∝ 𝑉 ℎ ∝ 𝑅 𝑔 3 [35] .In good solvents, monomer-monomer repulsion swells the polymer coils, while in poorer solvents a decrease of coil size can be achieved due to a decrease in monomer-monomer repulsion (glycerol) or by disfavoring the monomer-solvent interaction (ethanol). From the intrinsic viscosity values, we conclude that the three solvent are good solvent for HPAM. The case of water is known 107 . Glycerol reduces the solvent quality -as compared to water -in a manner similar to a salt, which screens polymer charges and leads to a decrease in the polymer coil size [START_REF] Dupas | Mechanical degradation onset of polyethylene oxide used as a hydrosoluble model polymer for enhanced oil recovery[END_REF]108,109 . Ethanol reduces the solvent quality 61 even more. poor solvents) may better describe the semi-dilute unentangled regime 111 , which we study here. We observe a concentration scaling η sp ∝ c 1.4±0.2 consistent with the expected good solvent semi-dilute unentangled regime.

The transition from a dilute regime to semi-dilute regime occurs for a critical concentration c*. The critical concentration is inversely related to the intrinsic viscosity given as

𝐜 * = 𝟏 [𝛈] , (3.4)

Relaxation time

Relaxation time of the polymer chain is defined as the time taken by the polymer chain to diffuse through a distance equal to its chain length.

The relaxation time of the polymer solutions is determined by two different techniques: the first from simple shear and the second from extensional measurements. The first technique utilizes shear thinning measurement through the Carreau model (t cm ) identified as the terminal relaxation time. Terminal relaxation time can be defined as the average relaxation time of the polymer chain system with a distribution of chain lengths, not all the polymer chains (especially the longer chain lengths) relax within the limit of the terminal relaxation time. Terminal relaxation time is related to the weight average molecular weight (M w ) and presence of minute quantities of long polymers does not influence its precise values.

The second technique utilizes the capillary break extensional rheometer -CaBER (t ca ) identified as the longest relaxation time. Within the limit of the longest relaxation time, all the polymer chains with a distribution of chain lengths have completed the relaxation process. Unlike the terminal relaxation time the longest relaxation time is greatly influenced by the presence of the long polymer chains and it is therefore related to the higher moments of molecular weight such asM z , M z+1 etc…

Shear thinning and Carreau model

The Carreau model is a phenomenological model which describes the shear thinning of polymer solution. It depends on a characteristic time that -thanks to the Cox and Merz rule -can be identified with the terminal relaxation time. We extract the relaxation time t cm of polymer solutions by fitting our experimental viscosity data to,

𝛈 𝐩 (𝛄̇) = 𝛈 ∞ + (𝛈 𝐩𝟎 -𝛈 ∞ ) (𝟏+(𝒕 𝐜𝐦 𝛄̇) 𝟐 ) (𝟏-𝐧) 𝟐 , (3.5)
Where η ∞ is the viscosity at infinite shear rate (assumed here to be the one of the solvent ~ηs ), η p0 is the zero shear viscosity, γ̇ is the shear rate and n < 1 is the shear thinning index.

The rheological data of different polymer solutions are presented in Table 3.4. The relaxation time t ca of polymer solutions in semi-dilute unentangled polymer solutions obeys scaling of Rouse relaxation time (t r ), which follows t r ∝ c -0.31 𝑀 2 112 .

Filament thinning dynamics of capillary extension break rheometer-CaBER

Capillary thinning experiments were conducted with a HAAKE CaBER 1 as shown in The first regime is ill-defined and is still dominated by gravitational sagging. In the second regime the diameter of filament diameter is still higher, and the strain rates are low, therefore resulting surface stresses are balanced by Newtonian viscous forces. The third regime, the strain rates are high, but the viscosity decreases over time due to alignment of the polymers. The forth regime is when the elastic forces originating from the extension of the polymers balance the surface stresses. The filament diameter at the midpoint D mid (t) decays exponentially with time and can be modeled using upper convected Maxwell model (UCM model) is given as

𝐃 𝐦𝐢𝐝 (𝐭) = 𝐃 𝟎 ( 𝐆 * 𝐃 𝟎 𝟐 * 𝛔 ) 𝟏 𝟑 𝐞 ( -𝐭 𝟑 * 𝐭 𝐜𝐚 ⁄ ) , ( 3.6)
Where D 0 = initial diameter, G is the elastic modulus, σ is the surface tension in dyne/cm, t ca is the relaxation time.

Relaxation time t ca can be obtained by fitting the UCM model to the experimental data of filament diameter decay of this forth regime, on a semi-log plot as shown in Figure 3.6. The extensional dominate regime occurs before the filament breaks down. Table 3.4: Nomenclature to specify the type of polymer solutions for a given experiment. Here n is the shear thinning index, tcm the relaxation time from the Carreau model, tca is the relaxation time from the CaBER equipment, c* is the critical (overlap) concentration that separates the dilute and semi-dilute regimes and is calculated as the inverse of intrinsic viscosity as in Eq (3.4). σ is the surface tension of the polymer solution which is similar to the surface tension of the solvent used, given the small concentration of polymer. The salinity of all the solutions is 0.6% (w/w) salt composition as described in section 3.1.

Furthermore, it is possible to calculate the extensional viscosity of the solution using the filament diameter decay as shown in Figure 3.6. By assuming that the surface stresses (σ/D mid (t)) are balanced by the resulting extensional stresses, one can compute the extensional viscosity η ext , as given Eq.(3.7).

𝛈 𝐞𝐱𝐭 = 𝛔 𝐃 𝐦𝐢𝐝 (𝐭)𝛆̇ , ( 3.7)
where the extensional strain rate ε̇ is obtained from the rate of filament diameter decay as

𝜺̇= -𝟐 𝑫 𝒎𝒊𝒅 (𝒕) 𝒅𝑫 𝒎𝒊𝒅 (𝒕) 𝒅𝒕 , (3.8)
and the extensional viscosity is given by

𝛈 𝐞𝐱𝐭 = - 𝛔 𝟐 𝐝𝐃 𝐦𝐢𝐝 (𝐭) 𝐝𝐭 . ( 3.9)
The Hencky strain ε H of the midpoint of the filament is given by

𝛆 𝐇 = ∫ 𝛆̇𝐝𝐭.
(3.10) By substituting the strain rate 𝜀̇ from Eq.(3.8) we have

𝛆 𝐇 𝐠𝐞𝐨 = -𝟐 ∫ 𝐝𝐃 𝐦𝐢𝐝 (𝐭) 𝐃 𝐦𝐢𝐝 (𝐭) 𝐭 𝟎 = 𝟐𝐥𝐧 ( 𝐃 𝟎 𝐃 𝐦𝐢𝐝 (𝐭) ), (3.11)
Where initial diameter 𝐷 0 is equivalent to that of the plate diameter Dcp=6mm. However as described by Clasen, the more appropriate strain accumulated by the polymer must be calculated from the onset of extensional hardening regime in the filament thing dynamics.

The onset of extensional hardening regime occurs for ε̇≅ 𝐃 𝐦𝐢𝐝 (𝐭) ).

(3.12)

The Hencky strain of the geometry as in Eq. (3.11) and the appropriate Hencky strain calculated from the Eq. (3.12) as a function of time are plotted in Figure 3.7.

Figure 3.7: Strain rate calculated from Eq.(3.8)as a function of Hencky strain of the geometry ε H geo of CaBER as in Eq. (3.11) and the appropriate Hencky strain ε H calculated from the Eq. (3.12.

Strain rate, Hencky strain and residence time in the contraction cell

Flow through a contraction geometry results in strong extensional flow. A hyperbolic contraction geometry was chosen to obtain a constant average rate of stretching ℇ ̇zz along the length of contraction zone Lc (3.9). Table 3.5 summarizes all the geometries and their dimensions, used for the current study. We follow the protocol of Nghe et al. in fabricating our microfluidic devices 51 :

1. Geometries are fabricated on a silicon wafer in a negative pattern (fabricated by the company microlLIQUID©, Gipuzkoa -Spain)

2. This geometry is transferred from a silicon wafer to a polydimethylsiloxane (PDMS) model.

3. We transferred the geometry from PDMS to microfluidic device using Norland Optical Adhesive 81 ("NOA81") obtained from Thorlabs©, New Jersey-US. NOA81 is a single component liquid adhesive that cures in seconds to a tough, hard polymer when exposed to ultraviolet light. ).

Geometry

(3.15)

For a hyperbolic profile, the width decreases as one passes along the contraction length.

𝑊(𝑧) ∝ 1 𝑧+𝑧 𝑜 , following 𝐖(𝐳) = 𝐀 𝐳+𝐳 𝐨 , ( 3.16)
where 𝐴 is a constant characteristic of the geometry, For a contraction starting at z=z0 with a width 𝑊(0) =Wu and at distance z= Lc, , 𝑊(𝐿 𝑐 ) = 𝑊 𝑐 , A is given by the relation 𝐀 = 𝐖 𝐮 𝐳 𝐨 , (3.17) Therefore, the width is given by The constant z0 is the width at the contraction throat and this gives the relation:

𝐖(𝐳) = 𝐖 𝐮 𝐳 𝐨 𝐳+𝐳 𝐨 . ( 3
𝐳 𝐨 = 𝐖 𝐜 𝐋 𝐜 𝐖 𝐮 -𝐖 𝐜 . (3.19)
From the previous relation, we can derive the strain rate in the hypothesis that the flow lines follow the geometry -which as will be shown later, is not always the case.

𝛆̅ = 𝐐 𝐋 𝐜 𝐡 ( 𝟏 𝐖 𝐜 - 𝟏 𝐖 𝐮
).

(3.20)

The total accumulated (Hencky) strain undergone by the polymer at the throat of contraction is given by

𝛆 𝐇 = ∫ 𝛆̅(𝐳)𝐝𝛕 𝐫𝐞𝐬 . ( 3

Optical visualization

Polymer solutions in contractions are well known to exhibit vortices above a critical flow rate [START_REF] Boger | Viscoelastic flows through contractions[END_REF][START_REF] White | Review of the entry flow problem: experimental and numerical[END_REF][START_REF] Boger | Experimental removal of the re-entrant corner singularity in tubular entry flows[END_REF][START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] . A typical situation is shown in Figure 3.8, where the red dotted ovals represent the zone of vortices. Polymer solutions for fluorescence imaging are prepared by mixing polystyrene spherical beads (Thermofischer) of 10µm diameter at a concentration of 3.6*10 6 beads/ml or 1µm diameter with a particle concentration of 10 10 beads/ml. The beads are loaded with a fluorescent dye with spectra in the visible range (absorption at a wavelength of 490 nm and emission at 520nm). Fluorescence imaging is carried out using a Nikon microscope with broadband illumination. Filters inbuilt into the microscope select the range of illumination.

Flow visualization is performed using a 2X-4X objective. A high-speed camera is placed in-line to record the flow at a rate of 50 frames per sec with a high exposure time of 60-80s.Nikon microscope is equipped with a platform moving in X-Y direction.

The selected configurations of camera and the objective limits the field of observation in one single image corresponds to 6 mm x 6 mm. Therefore, the complete geometry is imaged by moving the platform. Images obtained at different locations at different times are stitched using the channel dimension to obtain one complete image of the flow lines (Figure 3.9). The time between images obtained is about few milliseconds. Several such complete images are obtained at different time periods and are superposed in the Z plane based on their standard deviation.

Single contraction devices

Single injection experiments

In the single injection experiments (Figure 3.10) pressure sensors (ELVESYS©, Paris, France) are placed close to the entrance and exit. We have checked that the total pressure drop from all the connecting tubes -except the contraction itself -contributes less than 5-10% of the total pressure measured. Thus, because the pressure sensors are connected very near to the entrance and the exit, we assume that the entire measured pressure drop originates from the contraction. The pressure sensors have ranges of either 0-7*10 5 Pa (+10 3 Pa) or 0-2*10 5 Pa (+0.3 *10 2 Pa). The choice of range limits the sensitivity and we choose the pressure sensor for each experiment.

We perform single injection experiments with contraction devices of different geometries.

For this the solvent is injected at different flow rates and the pressure drop ΔP s (Q) between the entrance and exit is measured. In the next step the polymer solution is injected at different flow rates and the pressure drop ΔP p (Q) is measured.

Figure 3.10: Single injection device including the injection system, the tubing, the micromodel, the collecting bottle and pressure sensors at entrance and exit.

Model Geometry

In the first step we use a model hyperbolic contraction geometry (Figure 3.8) with Hencky strain 3.2*, units with the geometrical dimensions as given in Table 3.5.

The solvent is injected through the contraction at different flow rates 𝑄 to measure the pressure difference 𝛥𝑃 𝑠 (𝑄). The pressure drop due to solvent increases linearly with flow rate (Figure 3.11) 𝛥𝑃 𝑠 (𝑄) = 𝑅 𝑠 𝑄, where the resistance factor 𝑅 𝑆 depends on the solvent viscosity and the geometry. The Reynolds number is estimated as

Re (Q, 𝛈 𝐬 ) = 𝛒𝐮 ̅(𝐳)𝐝 𝐡 𝛈 𝐬 , ( 3.27)
where

d h = 4A(z) 2(h+W(z))
, u ̅(z)is the average velocity calculated as in Equation (3.14), and ρ is the solvent density. On this geometry we work with different solvents at (Reynolds number at the contraction) Re<1000. Deviation from linear dependence of pressure drop on flow rate are observed for a Reynolds number Re=400, which results from the flow field deformation due to inertia at the exit of contraction similar to shown in Figure 3.15 (similar observations were made by [START_REF] Galindo-Rosales | Viscoelastic instabilities in micro-scale flows[END_REF] ). In our study, we do not consider these deviations and we use the extrapolated linear behavior of pressure drop due to solvent even for Re> 400. For polymer solutions in this geometry we keep Re < 225, and there are no inertial effects.

Next we measure the pressure difference 𝛥𝑃 𝑝 (𝑄) for the polymer solutions in Table 3. .

(3.28)

A typical plot of pressure drop ratio R(Q) has a maximum, as shown in Figure 3.12, at values of pressure and flow rate respectively R max and Q max .

Figure 3.12: Pressure drop ratio curve of a given polymer solution when injected through hyperbolic contraction geometry with Hencky strain 3.2*(refer Table 3.5). The maximum occurs of at pressure ratio 𝑅 𝑚𝑎𝑥 and flow rate 𝑄 max .

Influence of Hencky strain

To study the importance of Hencky strain on scission, the Hencky strains of the contraction devices are varied by changing the entrance width Wu and the contraction width Wc. The contraction length Lc and the height of the channel h are kept constant for this purpose. The geometrical dimensions are given in Table 3.6.

40/60 glycerol/water solvent with viscosity 𝜂 𝑠 ≅ 3𝑐𝑃, is injected at different flow rates through geometries with different Hencky strain. The pressure drop of the solvent is plotted as a function of flow rate Q in the Figure 3.13. The pressure drop increases linearly with the flow rate in all the geometries. Due to inertial effects, slight deviations from the linearity observed in geometries ε H = 3.9 for Re=222 and for ε H = 3.2 at Re=312 (see Figure 3.13 b), as observed earlier section 3.7.1.1.The reason for this deviation is earlier explained in section 3.7.1.1 and is due to inertia. However, we again ignore these deviations and extrapolate the linear behavior of solvents for higher Reynolds number.

Double passage experiments

Figure 3.16 illustrates a double passage experiment. We first measure the pressure drop ratio curve as explained in the previous section. The blue solid line in the first graph of Figure 3.16 is the first injection pressure drop ratio curve having a maximum at Q1 max . In step 1 we collect the outlet the solution at a given flow rate Q1. In step 2 the collected Q1 outlet solution is injected for a second time at flow rate Q2 to measure the second pressure drop ratio curve, as shown by the orange dotted line with a maximum at Q2 max . We observe that this maximum depends on the initial flow rate of collection Q1 and Q1 max .

A given multiple injection curve is denoted as R (QN; QN-1) where QN is the flow rate at the N th passage and QN-1 is the flow rate at which the polymer was degraded during the N-1 st passage. Figure 9 illustrates multiple passage experiments were performed for successive injections and collections. In this experiment a pressure drop curve R(QN; QN-1) depends on the entire injection history (Q1, Q2, … QN-1). The experimental procedure is similar to the double injection experiments (Figure 9).

Multiple passage experiment

Multiple injection experiments were performed for successive injections and collections. In this experiment a pressure drop curve PR (QN; QN-1) depends on the entire injection history (Q1, Q2, … QN-1). The experimental procedure is similar to the double passage experiments (Figure 3.17)

Figure 3.17: Multiple injection experiments. The blue solid curve is the first injection pressure drop ratio curve R(Q1) with maximum at Q1 max . The orange dotted curve is the second injection R (Q2; Q1) with maximum at Q2 max ; the green solid curve is the third injection pressure drop R (Q3; Q2) with maximum at Q3 max , and the black dotted line is the fourth injection pressure drop ratio curve R (Q4; Q3) with maximum at Q4 max .

In the later sections we demonstrate a universal model using single hyperbolic contraction geometries which predict the scission in successive passages. We apply our findings to multiple contractions in series, before applying this model to a real porous medium, which is generally a 3D complex network of porous network. To quantify the scission in successive contractions, we perform a double passage experiments with few modifications from that of the procedure mentioned in section 3.7.2. A nondegraded polymer solution is injected at a constant flow rate Q1 = 83.33µL/S through geometries with different number of contractions in series and the solution at the outlet is collected. The collected solutions are then injected at flow rates Q2 through a single contraction and the pressure drop ratio curves R (Q2, Q1, Nc) where Q1 is the flow rate at which the polymer solution is degraded by injecting through a geometry of NC contractions in series during first passage and Q2 is the flow rate for a second injection through the geometry with a only one contraction (Figure 3.20).

Contractions in series

Figure 3.20: Schematic illustration of the experimental procedure of a double passage experiments for contractions in series. In the first step 15MDa-0.08/40SG polymer solution is injected through geometries with different number of contractions in series and the solution is collected at the exit Q1(Nc). Here Q1=83.33µL/s . The collected solution is injected for the second time through geometry with only one contraction Q2 (Nc=1). The pressure drop ratio curve R (Q2, Q1, Nc) of a second passage is plotted for these solutions collected in different geometries

Porous medium

Porous medium is a material with voids in a solid skeletal matrix. It is a complex 3D network of complex contractions and expansions. The rock porosity generally depends upon the nature of the rock material. A porous medium is often characterized by its porosity, pore size, permeability, tortuosity and other parameters. The solid matrix and the pore network are often continuous, but closed voids are inaccessible and therefore an effective porosity is of our interest.

. Porous cores

Cores of lengths, namely 1.1, 2.17, 3.07 and 7.78 mm, and diameter 12 mm were cut from sintered ceramic slabs supplied by Hilton Instruments Ltd. For simplicity, all the lengths of the porous medium are rounded to the nearest integer in the rest of the paper. The porous media were initially cleaned using ultra-sonication and then dried using CO2 at a pressure of 2 bar. This process was repeated for at least 3 times to ensure the porous media were free from any displaceable impurities. The porosity was measured using a pycnometer, Accupyc II 1340 from Micrometrics. The porosity of all the cores was about 40% (±5%). Permeability to water was determined by injecting brine at ambient temperature at different flow rates. A pycnometer uses the gas displacement method to measure volume of the placed sample.

Nitrogen is used as displacement medium. The porous medium is placed in the sample chamber (Figure 3.21) and the nitrogen is let into the pycnometer. In the first step, valve 1 opens and gas enters the reference chamber with a volume of Vr. Once the system equilibrates and the pressure Pr is measured. In the second step valve 2 opens and the gas enters the sample chamber (whose empty volume is Vs) with the porous medium. Pressure Ps is measured after the system is equilibrated.

The volume of solid material of the porous rock sample Vpm is calculated as

𝐕 𝐩𝐦 = 𝐕 𝐬 + 𝐕 𝐫 𝟏 - 𝐏 𝐫 𝐏 𝐬 (3.29)
The total volume of the porous medium VT is calculated as

𝐕 𝐓 = 𝛑𝐝 𝟐 𝐋 𝟒 (3.30)
The non-solid volume or void volume of the porous medium Vp is calculated as 𝐕 𝐩 = 𝐕 𝐓 -𝐕 𝐩𝐦 (3.31) Porosity φ is the ratio of the volume of the non-solid volume to the total volume of the porous medium.

𝛗(%) = 𝐕 𝐩 𝐕 𝐓 * 𝟏𝟎𝟎 (3.32)
The porosity φ(%) of the porous medium of different lengths is about 40% (±3%);

Polymer solution

Partially hydrolyzed polyacrylamide, HPAM-FLOOPAAM 3630, is obtained from SNF, France. The solvent, water with a salinity of 0.6% is prepared with the salt composition in Table 3.2. The salt water is filtered using a 0.2μm filtrate screen, since the average pore diameter is of order 5-10µm therefore an extremely impurity-free solution is essential. Mother solution of polymer with molecular weight 19MDa and a polymer concentration of 0.5%(w/w) in a 0.6% salt water is prepared as described in section 2.1. The mother solution is then diluted to a final concentration of 0.08%(w/w) and filtered using 5μm nylon screen(19MDa-0.08/0SG).

The polymer viscosity is measured using the Low shear viscometer LS-300 as mentioned in section 2.2. The polymer viscosity at zero shear rate ηp0= 18 cP.

Experimental set up

The assembly used to study the pressure drop as a function of the flow rate is shown schematically in Figure 3.22. It comprised a TELEDYNE ISCO 1000D syringe pump. A polyetheretherketone cell embeds the porous medium. Three ROSEMOUNT 3051S differential pressure sensors set on different ranges (0-60*10 2 Pa, 0-600 *10 2 Pa and 0-20 *10 2 Pa) which measures the pressure difference between the entrance and exit. To ensure the tightness of the cell, the porous medium is positioned between two O-seal rings (black).

A by-pass tubing is used to eliminate air bubbles.

Solvent Injection-Permeability

The flow of a Newtonian solvent through a capillary follows the Poiseuille's law, where the pressure gradient over the capillary of length L, is directly proportional to the viscosity and is inversely proportional to the surface area. The flow of a solvent through a porous medium follows Darcy's Law, where the pressure gradient over the porous medium of length L, is directly proportional to the viscosity of the solvent and inversely proportional to permeability and the surface area as in Eq. (3.33). Where k w is the initial permeability of solvent, water measured from Darcy law of Eq. (3.33), S is the surface area, η s is the solvent viscosity, L is the length of the porous medium. From the slope of Figure 3.24, we can calculate the permeability of the solvent k w . We obtain the permeability to be 2±0.25 Darcy (1 Darcy≅10 -12 m 2 ) respectively for the 1, 2, 3 and 8 mmlong cores

∆𝐏 𝐬 = 𝐋𝛈 𝐬 𝐤 𝐰 𝐉 (3.

Polymer injection

Next the polymer solution 19MDa-0.08/0SG is injected through a given porous medium of length L at different fluxes J. The pressure drop ∆P p (J, L).due to injection of polymer is measured between the entrance and exit.

The mobility reduction Rm is defined as the relative resistance to the flow offered by the injection of a solution (for instance polymer solution) at a given flow rate to the resistance to the flow offered by injecting a reference solution at the same flow rate. Here the reference solution is the solvent, salt water. Hence, 𝐑 𝐦 (𝐉, 𝐋) = ∆𝐏 𝐩𝐨𝐥𝐲𝐦𝐞𝐫 (𝐉, 𝐋) ∆𝐏 𝐒𝐨𝐥𝐯𝐞𝐧𝐭 (𝐉, 𝐋) (3.34)

Although this equation is similar to Eq.(3.28) for microfluidic contraction devices, the relative resistance in porous medium depends on both the viscosities and the permeability of the polymer solution and the solvent water, unlike the dependence on viscosity in microfluidic geometries. Mathematically,

𝐑 𝐦 = 𝛈 𝐩 𝛈 𝐬 𝐤 𝐬 𝐤 𝐩 (3.35)
Where η p apparent viscosity of the polymer, η s is the viscosity of the solvent, k s permeability of the solvent and k p is the permeability of the polymer.

The permeability of polymer solution is altered due to adsorption of polymer chains on the walls of porous rock, resulting in a decreased pore size. On contrary the adsorption of polymers in microfluidic geometries has negligible effect on the permeability.

A typical mobility reduction is plotted as a function of flux J in the Figure 3.25: mobility reduction as a function of low rate for a porous medium of length L=3mm, for a 19MDa-0.08/0SG polymer solution.

Degradation

Degradation is often used in the engineering field to demonstrate the deterioration of the physical properties. Polymers scission in porous media is often called degradation.

The degradation is evaluated by the ratio of the viscosity of the solution after injecting through a porous medium to that of the viscosity of the solution before injecting. A drop in the viscosity arises from the scission or breaking of polymer chains.

The zero-shear viscosity of a polymer solution is measured using the low shear viscometer as η p initial . The polymer solution is then injected through a given porous medium of length L at a given flow rate Q. The solution is collected at the exit of the porous medium and its zeroshear viscosity is measured as η p final . Degradation is calculated as 𝐃𝐞𝐠𝐫𝐚𝐝𝐚𝐭𝐢𝐨𝐧 (%) = 𝛈 𝐩 𝐢𝐧𝐢𝐭𝐢𝐚𝐥 -𝛈 𝐩 𝐟𝐢𝐧𝐚𝐥 𝛈 𝐩 𝐢𝐧𝐢𝐭𝐢𝐚𝐥 -𝛈 𝐬 * 𝟏𝟎𝟎 (3.36)

Double passage

Double passage experiments were done on a porous medium of length Lo=1mm. A 19MDa-0.08/0SG polymer solution is injected through a porous medium of length Lo at a given flux J1 and the mobility reduction 𝑅 𝑚 (J 1 , 𝐿 𝑜 ) is calculated as in Eq. (3.34). The polymer solution is collected at the outlet of the first injection at flux J1 and is injected for a second time through the same porous medium (Lo=1mm). The mobility reduction R m (J 2 , J 1 ) is plotted as a function of flux J. Similar experiments were performed with the same solution on porous medium of different lengths.

Chapter IV -vortices

Introduction

In this chapter we discuss the different upstream vortex regimes observed. A brief literature review on entry flow problems was given in section 2. In section 4.6, a short discussion on the influence of vortices on the pressure drop is given.

Before going into details, we recall the dimensionless numbers from Table 2

Influence of rheological properties and flow rate on vortices

In this section polymer solutions with different rheological properties are injected through a hyperbolic contraction [(5-0.2)/34]0.2 (with a Hencky strain ε H = 3.2*). The rheological properties are varied by choosing different molecular weights, solvent viscosity and solvent quality. In Table 4.1 the polymer solutions used in the current study and their corresponding rheological properties are shown. As discussed in section 3.3, adding glycerol to water increases the viscosity, but decreases the solvent quality. The addition of ethanol in water does not modify the solvent viscosity but decreases the solvent quality.

We observe three different flow regimes upstream using the geometry [(5-0.2)/34]0.2. At low flow rates no vortices were observed thus laminar, and flow like that of a Newtonian solvent. However small disturbances (fluctuations) in streamlines, but without vortices, were observed, as shown in Figure 4.1 for Wi = 1.51. We ignore these disturbances and consider them like the situation with no vortices. Stable corner vortex growth regime sets in at Wi ̴ 2, as shown in the same figure. The corner vortex increases in size for increasing flow rate until the vortices becomes chaotic. The transition from one flow regime to another (laminar→vortices →viscoelastic instabilities/chaotic) does not occur for a fixed Weissenberg number or Reynolds number. This will be discussed in detail in section 4.5.

Vortices formed at the mouth of contraction and which do not touch the sidewalls are called lip vortices. Lip vortices are obvious in the case of abrupt contractions but were not observed in the current geometry.

From the literature it appears that lip vortices occur for polymer solutions with high shear thinning index and contradictory experimental observations, for the influence of contraction ratio β (refer section 2.2.4). For geometry [(5-0.2)/34]0.2 the contraction ratio β=25 and the shear thinning index n as reported in Table 4.1 are quite low, since we stay in semi-dilute unentangled regime. A low shear thinning index could explain the lack no lip vortices in our case. The most relevant parameter to predict the occurrence of lip vortices was proposed by Jonathan and McKinley et al., who have suggested a dimensionless normal stress ratio parameter ℵ (Eq.(4.1)), which is the ratio of the normal stresses along the walls due to shear flow to the extensional stresses along the central line of the extensional flow. Lip vortices only occur for ℵ>0.055±0.005 [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] . It holds true in our case with their proposed equation derived from the Oldroyd B-model, for which we find the Trouton ratio Tr(ε H ) is of several thousands.

ℵ = 𝐒(𝛄̇) 𝐓𝐫(𝛆 𝐇 ) = 𝟐 ( 𝛈 𝐩 𝛈 𝐬 ⁄ ) 𝟑 + ( ( 𝛈 𝐩 𝛈 𝐬 ⁄ ) (𝟐𝐃𝐞 -𝟏) ⁄ ) 𝐞𝐱𝐩[𝟐𝛆 𝐇 (𝟏 -(𝟐𝐃𝐞) -𝟏 )] (4.1)
Here Tr(ε H ) is the Trouton ratio evaluated at the total Hencky strain accumulated along the centerline of the contraction-expansion and S(γ̇) the shear-rate-dependent stress ratio. The laminar regime, stable corner vortex regime and the chaotic regime were observed in almost all the solutions of Table 4 In order to understand the effects in more detail, in the next section we vary the Hencky strain of the geometry.

Influence of Hencky strain

In this section, the Hencky strain of the geometry is varied by changing the contraction width and or entrance width of the geometry. While doing so, we keep the contraction length L c (reduced from that of the geometry [(5-0.2)/34]0.2) and the height h constant (refer to Table 3.6). We use four different molecular weights in glycerol/water. We do not observe any lip vortices in any experiments, so we believe that the dimensionless normal stress ratio parameter ℵ parameter proposed by Jonathan et al. to be the most relevant to predict their occurrence [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] .

4) Stable asymmetric corner vortex growth (green). 5) Time-dependent corner vortex growth (orange). 6) Stable-singular corner vortex (yellow) .7) Chaotic/Viscoelastic instabilities (red).

We observe seven typical flow regimes (Refer to Figure 4.3):

1. Laminar regime with no vortices: The flow is laminar similar to a newtonian fluid with without any formation of upstream vortices 2. Stable symmetric corner vortices: two stationary upstream axisymetric (in the direction of flux) corner vortices (Unlesss mentioned the number of vortices are two in all the cases).

3. Time-dependent symmetric corner vortices: Two upstream corner vortices of almost the same size that fluctuate in time.

4.Stable asymmetric corner vortices: two stationary upstream corner vortices whose dimensions are not equal.

5. Time-dependent asymmetric corner vortices: two upstream corner vortices whose respective average dimensions are not equal, fluctuating in time.

6.Stable singular corner vortices: only one stationary corner vortice observed upstream of the contraction. We observe this case only for Wi=7.2 and Re=0.5, For the polymer solution 15MDa-0.08/40SG in geometry [(2-0.5)/2.6]0.2 7. Viscoelastic instabilities: Chaotic flow with no identifiable vortices can be identified. We also attribute to this domain a fluctuating singular corner vortices.

Not all of the seven flow regimes are observed in each experiment. The type of flow regimes , transitions between regimes and their dimensions were unique for each experiment differ as shown in detail below. In Table 4.2 we report the rheological properties of the polymer solutions, the geometrical dimensions used in this section and their corresponding flow regimes.

To study the influence of the Hencky strain on the growth of the vortices we measure the length of vortices L vs to calculate the dimensionless vortex size χ vs (Eq (2.31)). For this we do not limit ourself to the stable symmetric corner vortices (as in section 4.2) but also measure the dimensions of other flow regimes shown in Figure 4.3. Note that the images shown are superposition of multiple images obtained at diferent times. Therefore the values considered for analysis are obtained from the individual images (of about 15-20 images).Previously researchers have measured the corner vortices sizes , havenot explicitly limited to this stable corner vortices. For example, a study of PEO solutions in water through a 16:1 planar contraction, observed many flow regimes (Rodd et al. [START_REF] Rodd | The inertioelastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries[END_REF] ): assymetric/bistable singular corner vortices, symmeteric stable corner vortices, divergent flows, vortex growth+divergent flows, lip vortices etc…The plot of dimensionless vortex size χ vs as a function of the Weissenberg number consist data from for all the above mentioned regimes .

We define the vortex regime for a deviation in the flow line by an angle greater than or equal to 25° , where we measure the vortex length.

At first, an 11Mda-0.08/40SG polymer solution is stdied in geometries with different Hencky strains and the lengths of the vortices Lvs are measured. The dimensionless vortex size χ vs does not collapse on to a single curve when plotted as a function of Deborah number (Figure 4.4).

Sousa et al. observed that the growth of corner vortices is well captured by the Deborah number, for polyacrylamide solution in 40/60 glycerol/water solvent in square-square axisymetric contractions [START_REF] Sousa | Effect of the contraction ratio upon viscoelastic fluid flow in three-dimensional square-square contractions[END_REF] . Similar results were observed by Rotheinstein et al. [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] . Note that Deborah number is the ratio of Weissenberg number and the Hencky strain, while the Hencky strain is the logarithamic modified contraction ratio as in Eq.(3.22).

In addition, Hidema et al. studied corner vortices in geometries with different contraction ratios β and aspect ratio α. They found that the dimensionaless vortex size seems to collapse on to a single curve when plotted as a function of Weissenberg number normalized by the contraction ratio β for a given aspect ratio α. Furthermore, the vortex dimensions was larger in channels with a higher aspect ratio [START_REF] Hidema | Effects of flexibility and entanglement of sodium hyaluronate in solutions on the entry flow in micro abrupt contraction-expansion channels[END_REF] . For vortices in cylindrical or square geometries the vortices were larger than in the planar contraction geometries [START_REF] Rodd | The inertioelastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries[END_REF][START_REF] Rodd | Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries[END_REF] .

In our case the aspect ratios (

) are not same, so we use a modified dimensionless vortex size ξ vs given as, In this case (Figure 4.5), all the curves (single polymer solution in geometries with different Hencky strain) collapses on to nearly a single curve when plotted as a function of Deborah number. The modified dimensionless vortex size ξ vs , scales roughly as ξ vs ≈ ln De, these findings are in coherent with the observations of other researchers [START_REF] Rodd | The inertioelastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries[END_REF][START_REF] Evans | Flow characteristics associated with abrupt changes in geometry in the case of highly elastic liquids[END_REF][START_REF] Evans | Further remarks on the lip-vortex mechanism of vortex enhancement in planar-contraction flows[END_REF][START_REF] Rothstein | Extensional flow of a polystyrene Boger fluid through a 4: 1: 4 axisymmetric contraction/expansion[END_REF] . We found similar results for different molecular weights 3.6.

Note that the Deborah number is the ratio of the Weissenberg number and the Hencky strain (assuming a constant strain rate along the hyperbolic profile), so the Deborah number takes into account the contraction ratio. We considered the height of the channel in calculating the modified dimensionless vortex size to accommodate varying aspect ratio. Such a collapse suggests that height of the channel is an important parameter in predicting the vortex growth.

Cloitre et al. studied with semi-rigid Xanthum polymer solutions in a conical axisymmetric contraction, and suggested that the vortices arise from the mechanical balance between the extensional and shear stresses [START_REF] Mongruel | Axisymmetric orifice flow for measuring the elongational viscosity of semi-rigid polymer solutions[END_REF] . Further they observed that the dimensionless vortex size χvs and the pressure drop ΔP (measured between entrance and exit) are a function of the Trouton ratio 𝐓𝐫. The dimensionless vortex size found to depend on the Trouton ratio according to

𝛘 𝐯𝐬 = (𝐓𝐫) 𝟏 𝟐 ⁄ (4.3)
Furthermore, the pressure drop as calculated to be

∆𝐏 = 𝛈 𝐬𝐡𝐞𝐚𝐫 𝐐 𝐜 𝟑 (𝐓𝐫) 𝟏 𝟐 ⁄ (4.4)
Where η ext is the extensional viscosity and η shear is the shear viscosity, Q is the flux, c is dependent on the geometry and has units of length.

Extending their model for vortices in axisymmetric conical contractions to the vortices in planar contractions, we suggest that the dimensionless vortex size depends on the height of the channel. Therefore, in the next section the channel height is varied.

Influence of height

A 15Mda-0.08/40SG solution is injected through geometries with same Hencky strain ε H = 3.9 but varying heights. In all the cases, singular time-dependent corner vortices regime was observed (i.e. viscoelastic instability). The dimensionless vortex size is plotted as a function of Weissenberg number in Figure 4.6. The dimensionless vortex size decreases with decrease in height. In axisymmetric contraction geometry, the sizes of the vortices are larger than in planar contraction geometries for the same contraction ratio. One argument for smaller vortices in planar contractions is that the total Hencky strain imposed is different between uniaxial and planar kinematics This was found in numerical studies using Phan-Thien/Tanner (PTT) PTT model [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] Cloitre et al. suggested for axisymmetric contraction geometries, the vortices result from the mechanical balance between the extensional stress component in the radial direction (or flow direction) and the shear stress component from the other directions. We extend this to the planar contraction geometries, to balance the extensional stress component in the Z direction (flow direction) with the shear stress arising from the Y direction (channel height h). For this following assumptions are made, 1. Flow is uniform between the boundaries of the vortices 2. The Extensional viscosity is constant between vortices,

3.

The normal stresses at the boundary of the vortices are small and are therefore neglected.

Therefore, the mechanical stress balance equation is:

𝛛𝛔 𝐳𝐳 𝛛𝐳 + 𝛛𝛔 𝐲𝐳 𝛛𝐲 = 𝟎. (4.5)
We assume a shear stress:

𝛔 𝐲𝐳 = 𝛈(𝛄̇)𝛄̇, (4.6)
with a viscosity dependence on shear rate given by a Carreau law as

𝛈(𝛄̇) = 𝐤(𝛄̇) 𝐧-𝟏 . (4.7)
Where the exponent is between n=1 (no shear thinning) and n<1 shear thinning. For the stress to be an increasing function of the shear rate, we assume n>0. The shear rate may be written as:

𝛄̇= 𝛛𝐮 𝐲 𝛛𝐲 . (4.8)
Therefore, the stress gradient can be given as , where ξ is the characteristic length scale of the problem, and a is the velocity at the upstream of the vortices. The extensional stress is given by Comparing the two previous expressions gives:

𝛔 𝐳𝐳 = 𝛈 𝐞𝐱𝐭 𝛅𝐮 𝐳 𝛅𝐳 . ( 4
𝐡 -(𝐧+𝟏) ( 𝐳 𝛏 ) ψ𝐧 ∝ 𝐳 𝛃-𝟐 𝛏 ψ (4.17)
The exponent of z must be the same, which leads to: < 1 for n<1. For a 15MDa-0.08/40SG solution with n=0.8, Hence the dimensionless vortex size is related to height as χ vs ∝ h 0.9 . Experimentally we observe dependence to be χ vs ∝ h 0.5 . We suspect and therefore question whether this deviation in height dependence result from the increased interactions between polymer chains under extension as reported in literature of section 2.2.3. Also, the vortices dimensions are measured in time-dependent flows (7 th flow regime), where local velocity fields are complicated to measure and non-steady flow probably violates the power law dependence.

𝛙 = -𝟐 𝐧 -𝟏 (4.18) So that ξ ∝ h
Shear thinning may help to explain the dependence of the vortices with the thickness.

Obviously more experiments are necessary with stable corner vortices on both top and bottom at the upstream of contraction, in order to verify the above models. From the above results we see that although the Deborah number or Weissenberg number could eventually describe the vortex size χvs, it is also essential to be aware of the type of flow regime and the dynamics. The velocity field in the region of vortices is greatly dependent on the type of vortices, and therefore the extensional rate.

In the next section we plot the typical Wi-Re often used to classify the flow regimes based on the Elastic number Eq.(2.36).

Flow regimes on Wi-Re plot

Rodd et al. [START_REF] Rodd | Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries[END_REF] studies for PEO aqueous solutions through planar contraction devices studied the transition from one flow regime to another using a Wi-Re plot. In their studies the critical Weissenberg number curve for transition from one flow regime to another flow regime presents a minimum for varying elastic number. They have also explained that the formation of vortices and other viscoelastic effects develop due to interplay of elasticity and fluid inertia.

To understand the role of elasticity and the inertial effects in our experiments, in Figure 4.8 we show the different regimes for all the polymer solutions as a function of Weissenberg number and the Reynolds number, so called Wi-Re space. The slope for each of the Wi-Re data sets corresponds to a particular elastic number El ≡ Wi

Re

. For a given geometry changing the relaxation time of the polymer solution -likewise the Weissenberg number and the viscosity of the polymer solution-likewise the Reynold's number, gives an access to a range of elastic numbers. Elastic number can also be modified by changing the Hencky strain of the geometries which simultaneously modify Weissenberg number and Reynolds number. In the current study we work with elastic numbers ranging from 0.009<El<53. The seven flow regimes mentioned in Figure 4.3 of section 4.3 are observed on this Wi-Re space. The occurrence of these flow regimes and the transition between flow regimes is not correlated with the Weissenberg number or the Reynolds number. For example, for elastic number El= 0.02 the stable corner vortices appear at Wi=1.3 and Re =66, but at Elastic number El=0.025 no vortices were observed. In addition, for elastic number El=7.8, the 15Mda-0.08/40SG solution when injected through geometry with Hencky strain ε H = 1.8 , had no vortices for Wi<2.8, a stable singular corner vortex sets in for Wi=5.6, which finally transforms into timedependent asymmetric corner vortices for Wi >11. On the other hand, a 15MDa-0.08/40SG when injected through geometry with Hencky strain ε H = 3.9 has only a viscoelastic instability regime, for Wi>3.5.

For the elastic number El≈1.8, the 11Mda-0.08/40SG solution at Hencky strain 𝜀 𝐻 = 1.8 have stable symmetric corner vortices (Wi<20) and time dependent symmetric corner vortices (Wi>30.5, Re>16). While for the similar elastic number 11Mda-0.08/40SG polymer solution when injected through geometry with Hencky strain 𝜀 𝐻 = 1.4 have stable symmetric corner vortices (Wi=3.6), stable asymmetric corner vortices (7.3<Wi<27) and time-dependent asymmetric corner vortices (Wi>34.7). Also, 19MDa-0.08/40SG when injected through geometry with Hencky strain 𝜀 𝐻 = 3.2*, resulted in Newtonian flow like regime (Wi<1.57), stable corner vortices (2.4<Wi<5.5) and viscoelastic instabilities (Wi>6.3) for the similar elastic number (El≈1.7).

These results are quite different from the observations reported in literature. To best of our knowledge previous studies used mostly one or two polymer solutions in different geometries, or a single geometry with different polymer solutions. Therefore, we conclude that the Wi-Re space (or elastic number) does not discriminate different flow regimes.

Although, the vortices of non-Newtonian fluids through contraction geometries have been extensively studied, there are very few systematic studies on occurrences of vortices in hyperbolic planar contraction geometries with varying geometries and polymer solution properties.

Influence of vortices on the pressure drop measured

In all of these experiments the pressure drop between the entrance and exit was measured.

The fluctuations in the pressure were observed to increase in presence of the vortices. For Newtonian like laminar flows the fluctuations were absent, while for stable corner vortices the fluctuations were small. For time-dependent and viscoelastic unstable flows the pressure fluctuations were higher. A larger vortex activity increases the intensity of pressure fluctuations. The occurrences of vortices, the type of flow regime or the transition between flow regimes, has no direct relevance to the changes in the average pressure drop data, despite the increase in pressure fluctuations. Similar observation were reported by McKinley et al. [START_REF] Rothstein | Extensional flow of a polystyrene Boger fluid through a 4: 1: 4 axisymmetric contraction/expansion[END_REF] . The formation of vortices modifies the local velocity fields and thereby the extensional rates, which will be accounted for the next chapter. 4.2.

Conclusion

Viscoelastic fluids injected through a contraction easily develop upstream vortices. In this chapter the occurrence of vortices was studied in detail with different geometries and polymer solutions with different rheological properties.

We observed seven flow regimes: Laminar flow, stable symmetric corner vortices, stable asymmetric corner vortices, time-dependent symmetric corner vortices, time-dependent asymmetric corner vortices, stable singular corner vortices and viscoelastic instabilities.

For Hencky strain ε H = 3.2 * , we measured vortex length L vs . For polymers in an ethanol/water solvent, we observed no vortex activity (L vs =0). This reduced vortex activity might be due to the decreased of extensional stresses contributions relative to the shear stresses. The dimensionless vortex size χ vs was poorly determined by the Weissenberg number.

Upon varying the Hencky strain ε H , we find the modified dimensionless vortex size ξ vs (where

ξ vs = L vs W u
) is increasing with Deborah number as a logarithmic function. Furthermore, the vortex activity decreases with decreasing the channel height. The dimensionless vortex size χ vs scales as χ vs ∝ √h. A shear thinning model may describe this scaling, but a simple model has a higher power law dependence than the experimental observations, which could be due to the effects of polymer entanglements in the extended state.

When the different flow regimes are plotted in Wi-Re space, as done in the literature, the different flow regimes evolve along the slope of this plot which is the Elastic number. The type of flow regime and the transition from one regime to another cannot be predicted from the Elastic number which is the ratio of elastic effects (characterized by the Weissenberg number) and the inertial effects (determined from the Reynold's number).

The presence of vortices induces fluctuations in the pressure. However, the pressure drop (due to onset of extensional thickening or scission of polymers) is not influenced by the occurrences of these vortices or transition from one flow regime to another flow regime. Similar observations were made by Rothstein & McKinley et al. [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF] . The formation of vortices does modify the local velocity fields and thereby the extensional rate. We will use this to calculate a more accurate extensional strain rate in the next chapter.

Chapter IV -Model geometry to study scission

Introduction

We study polymer scission under extensional flow by measuring the pressure drop through a hyperbolic contraction as a function of mass flow rate. In the first step a solvent is injected through the contraction geometry and the pressure drop is measured from the absolute pressure recorded by the pressure sensors at the entrance and exit. The pressure drop of the solvent increases linearly with the flow rate, While the pressure drop of a polymer increases non-linearly with the flow rate. The ratio of the pressure drop of the polymer solution to that of the pure solvent exhibits a maximum for a given flow rate, revealing a crossover between chain stretching and scission. The aim of this chapter is to discuss the behavior around the maximum pressure ratio in a single contraction.

In most polymer solutions the maximum pressure ratio in a single contraction is extremely difficult or impossible to reach with a reasonable pressure. This kind of maximum was observed by Georgelos et al. [START_REF] Georgelos | Apparent thickening behavior of dilute polystyrene solutions in extensional flows[END_REF] , which they attributed to viscoelastic turbulence. In order to observe the influence of scission on this cross-over the onset of viscoelastic turbulence must be delayed or its influence should be weak compared to the effects of degradation. To do so we choose very high molecular weight HPAM (described in detail below), which has a few limitations. Firstly, these polymers are synthesized by radical polymerization, which yields fairly high polydispersity. Secondly, it is challenging to obtain reliable chromatography studies of the molecular weight distribution on such high molecular weights, because of the difficulty to get clean light scattering signal, and because of the risk of chain scission during the measurement itself. For this reason, we do not report explicit measurements of the change of molecular weight during scission; rather, we infer scission from pressure measurements after repeated passages through contraction flows.

This chapter consists of 5 sections.

In section 5.2 polymer solutions with varying molecular weight and concentration in a 40/60 glycerol water are injected through a given hyperbolic contraction geometry. The dependence the maximum of the pressure drop ratio curve a molecular weight and concentration is studied.

In section 5.3 , we discuss how vortices modify the extensional flow regime and we propose a modification due to the vortex to calculate the true strain rate.

In section 5.4, the true strain rates calculated at the maximum are compared with the relaxation times tcm measured from a Carreau model. Here we have not use the relaxation times tca measured from the CaBER since we are not sure if the CaBER measurements have any induced scission during filament thinning dynamics. The correlation of true strain rate with relaxation time suggests that the cross over (maximum) occurs for a given Weissenberg number.

In section 5.5, to understand and quantify scission, a double passage is performed on a given geometry from the solution collected during the first passage at a given flow rate. We find a universal relation for predicting the scission by injecting different polymer solutions in a given geometry, based on the measurement of the maximum pressure ratio as a function of flow rate through the first passage. Surprisingly, this relation is independent of polymer concentration, molecular weight, solvent quality and solvent viscosity.

Lastly in section 5.6 , a multiple passage is performed to verify if the universal model developed from the double passages can be iterated to successive passages. We observe the universal model holds for successive passages.

Influence of Molecular weight and Concentration

In this chapter we work with a single hyperbolic contraction device [(5-0.2)/34]0.2 as given in Table 5.1

Table 5.1: Dimensions of the hyperbolic contraction geometry used in this chapter.* is mentioned to differentiate the geometries with same Hencky strain (refer Table 3.5).

The aim of this study is to quantify degradation during polymer elongation. Unfortunately, as mentioned before, we have not been able to perform actual measurements of the molecular weight distribution, to directly detect scission after passing through a contraction. For a given polymer solution, the mechanical resistance to polymer stretching is reflected in the increase of the pressure drop ratio (Figure 5.1). As the flow rate (and thus extension rate) increases the polymer chains stretch, which increases the pressure drop ratio. Stretched polymer coils are susceptible to scission, which then decreases the pressure drop ratio. The competition between elongation and scission leads to the maximum pressure drop ratio R max at a flow rate Q max . Figure 5.1 shows the pressure drop ratio for different molecular weights. For smaller molecular weights the pressure drop ratio decreases and the maximum pressure drop ratios occurs at higher flow rates, as expected. We scale each pressure drop curve by the maxima R max and Q max (Figure 5.2). The superposition is effective for Q> Q max but fails for Q< Q max . We attribute the failing at low flow rate to the different strain rate dependence of extensional and shear viscosities with molar mass, and the fact that low flow rates are dominated by simple shear. The good superposition at large flow rates indicate that degradation has the same dependence on flow rate as does the extensional viscosity, which

Viscoelastic instabilities

Viscoelastic instabilities easily occur for polymer solutions flowing through contraction devices, often via vortices forming at the entrance (Figure 5.5). Rothstein and McKinley [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF][START_REF] Rothstein | Extensional flow of a polystyrene Boger fluid through a 4: 1: 4 axisymmetric contraction/expansion[END_REF] showed that the onset of vortices occurs at a critical Deborah number, defined as the ratio of the polymer relaxation time 𝑡 𝑟 to the residence time 𝑡 𝑟𝑒𝑠 :

𝐃𝐞 = 𝐭 𝐫 𝐭 𝐫𝐞𝐬
(5.3) They observed three different domains of vortices: lip vortices, corner vortices and unstable vortices. As discussed in chapter 4 we do not find lip vortices but we find corner vortices with a well-defined length and circulation, similar to those observed in Rothstein and McKinley [START_REF] Rothstein | The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop[END_REF][START_REF] Rothstein | Extensional flow of a polystyrene Boger fluid through a 4: 1: 4 axisymmetric contraction/expansion[END_REF] . Cloitre et al., suggested that these vortices result from the balance between extensional and shear forces [START_REF] Mongruel | Axisymmetric orifice flow for measuring the elongational viscosity of semi-rigid polymer solutions[END_REF] . Hinch suggested that vortices could influence the complete stretching of polymer and/or possibly hold the stretch of polymers for longer times than that possible in laminar flows of the same geometry 118 . We find that the vortex length 𝐿 𝑣𝑠 increases with flow rate until they become unstable or chaotic. We suggest that this is likely due to the 'natural' vortex size exceeding the size permitted by the contraction device.

HPAM solutions in water-glycerol mixtures form vortices with length L vs increasing as a function of flow rate Q. On the other hand, HPAM solutions in the ethanol-water solvent which has poorer solvent quality (Sec. 3.3), do not form vortices prior to chain scission. We speculate that the poorer solvent quality of ethanol-water leads to a smaller ratio of extensional thickening relative to the shear viscosity, which suppresses vortex formation in accordance with the Cloitre argument.

The occurrence of vortices increases the effective contraction length and promotes earlier stretching of the polymer chains (Figure 5.5). Hence the strain rate experienced by the solution is lower than that specified by the hyperbolic geometry. To estimate the true strain rate, we increase the contraction length L c in Eq.(3.20) by the vortex length L vs to obtain a modified "true" extension rate

𝛆̇𝐓 = 𝐐 (𝐋 𝐜 +𝐋 𝐯𝐬 )𝐡 ( 𝟏 𝐖 𝐜 - 𝟏 𝐖 𝐮
).

(5.4)

In addition, we have checked that even in the presence of vortices, the stream lines follow nearly hyperbolic trajectories (but the question of whether the boundary conditions of zero velocity strictly hold at a bounding hyperbolic surface (Figure 5.5) is still not answered). Hence expression Eq.( 5.4), which assumes that the average extensional strain rate is constant in the contraction, is a good approximation. In all the data presented here, and for which we quote a strain rate (Figure 5.6 and Table 5.2) according to Eq.( 5.4), we restrict ourselves to situations for we observe symmetric corner vortices. However, we observe viscoelastic turbulence/instabilities and/or unsteady or asymmetric vortices in some high flow rate situations.

In section 5.4, we use the true extensional rate 𝜀Ṫ.

Flow conditions at the pressure drop ratio maximum

The extensional strain rate in a hyperbolic geometry is calculated using Eq. (5.4). Although the pressure maximum is not correlated with vortex formation, a vortex will reduce the extension rate due to earlier stretching of the chains at the (upstream) beginning of the vortex.

The pressure maximum is associated with chain scission, and the attainment of either a critical tension necessary to rupture a C-C bond (of order pN/chain), or a chain tension sufficiently close to the rupture tension for long enough time to allow thermal activation of rupture [START_REF] Odell | Degradation of polymer solutions in extensional flows[END_REF] . A critical tension will be set by a critical stress and its corresponding extension rate, while activation will depend on the residence time at high chain tension. Hence the pressure maximum could be determined by either a critical Weissenberg number (to exceed the rupture tension) The pressure maximum is not a direct indicator of the tension during chain scission, since the pressure drop ratio combines shear flow and extensional flow as explained by Cloitre et al. [START_REF] Mongruel | Axisymmetric orifice flow for measuring the elongational viscosity of semi-rigid polymer solutions[END_REF] .

In Figure 5.6 we show the true extension rate (proportional to Wi) as a function of the relaxation time. Since in this chapter we use only one geometry and thus one Hencky strain (𝜀 𝐻 = 3.2 * ), the Weissenberg and Deborah numbers give us similar understanding. In order to differentiate whether chain rupture occurs instantaneously during transient stretching (controlled by the Weissenberg number), rather than gradually as in an activated process (controlled by the Deborah number) one needs further controlled experiments at different Hencky strains.

In Figure 5.7, the normalized pressure ratios superpose quite well for Wi≥16.25±2, but poorly for Wi<16.25±2.The probable reason for this poor superposition is the different dependences of extensional and shear viscosities on molecular weight, as explained in section 5.2.

The expected crossover between extension and degradation occurs at around Wi=16. To confirm that this crossover is due to degradation and to detect scission below and above this crossover, we next perform double injection experiments.

Double passage -evidence for scission

In earlier sections vortex lengths were measured to evaluate the true extension rate, and situations with complex flows (elastic turbulence, asymmetric and oscillating vortices, etc) were omitted. In this section, we parameterize the flows in terms of flow rates, rather than correcting to determine the true apparent extensional strain rate. Unfortunately, the data of the vortices are not available for second passages as these experiments are time consuming, However we will show below that this difference is surprisingly unimportant)

We next show that double injection allows us to follow polymer scission in detail. We first inject a solution with a specific flow rate Q1 and collect the output of the solution. We then inject this collected output a second time at flow rate Q2 to measure the pressure drop ratio curve R (Q2; Q1) (Figure 5.8).

The maximum of the second-passage pressure drop ratio curve R (Q2 max ; Q1) shifts to higher flow rate and lower pressure drop ratio in comparison to the first-passage pressure drop ratio curve R (Q1). When the collection flow rate Q1 is increased, the maximum pressure ratio R max of the second passage shifts to lower values and Q 2 max shifts to higher values. This is consistent with the finding (Eqs. (5.1) and (5.2)), that for decreasing molecular weight the maximum of the pressure drop ratio curve shifts to lower values of R max and higher Q max . This confirms the presence of scission, so that the maximum of the re-injected curve provides information about polymer degradation, with the help of scalings in Eqs.(5.1) and (5.2).

For the native solution 19 MDa-0.08/40SG-[(5-0.2)/34]0.2 we find Q1 max =20.8 µL/s (Figure 5.8).

For solutions pre-degraded (collected) at flow rates Q1>>Q1 max (i.e. Q1=83.3 µL/s; 125 µL/s; 167 µL/s) the maximum of the pressure occurs at a flow rate nearly equal to the degradation flow rate, i.e. Q2 max ≅ Q1. This suggests that all polymers capable of degrading at extension rate corresponding to Q1 have been degraded as illustrated in Figure 5.9. For polymer solutions pre-degraded at flow rates Q1<<Q1 max (i.e. Q1=8.3 µL/s; 4.2 µL/s ;), we find Q2 max ≅ Q1 max , suggesting weak degradation. Similar experiments were done using different molecular weights and solvents (Table 5.

2).

The relative position of Q2 max versus Q1 and Q1 max suggests the master curve shown in Figure 5.10. Here we plot the maximum of the second pressure drop ratio curve Q2 normalized by the maximum of its respective non-degraded solution (i.e. Q2 max /Q1 max ), as a function the normalized pre-degrading flow rate Q1/Q1 max . In this representation, the two asymptotic regimes described above can been observed. For flow rate of the first injection below the maximum pressure flow rate, i.e. for Q1/ Q1 max <<1, we see that Q2 max / Q1 max tends toward 1, indicating an absence of polymer scission. For flow rate above the maximum pressure flow rate, i.e. Q1/ Q1 max >> 1, we find

Q 2 max /Q 1 max ≈ Q 1 /Q 1 max indicating a saturation scission regime, i.e. Q2 max = Q1.
Moreover, for the present contraction all the data collapse onto a master curve, for any molecular weight, for all concentrations (above but close to c*) and for all solvent qualities. This is one major result of this thesis which indicates a general relation for flux ratio of the flow rate at the maximum pressure rate for the first and second passages, for a given contraction geometry, independent of solvent quality, concentration and molecular weight.

This relation can be described by

𝐐 𝟐 max 𝐐 𝟏 max = 𝐟 ( 𝐐 𝟏 𝐐 𝟏 max ) = [𝟏 + ( 𝐐 𝟏 𝐐 𝟏 max ) 𝐛 ] 𝟏 𝐛 . (5.7)
Here exponent b corresponds to the inverse of the sharpness of the crossover regime. When b is larger a sharper cross over regime is observed. Since this exponent determines the transition from no scission to scission dominated regime, we therefore call this exponent the scission exponent.

Figure 5.9: Illustration of the scission at flow rates above Q max . Consider a first passage for Q1>Q1 max (Fig. 14). In the second passage a maximum occurs for flow rates Q2 max ≅ Q1, as the polymers that stretch for flow rates Q<Q1 (pink shaded portion) have already been degraded and do not contribute for extension while the non-degraded polymer during the first passage at Q1 would only degrade for Q2>Q1. A similar master curve can be obtained using the strain rate instead of the flow rate. It superposes precisely to the one in flow rate as seen in Figure 5.10. This is obvious in the case of polymer solution 19MDa-0.08/15SE (polymer in ethanol), which doesn't exhibit any vortices in the experimental range of flow rates, so that the true strain rate and flow rate are exactly proportional. However, it's also valid in the case of the polymer solution 11MDa-0.08/40SG which exhibits vortices (Figure 5.10), open symbols.

To obtain the true strain rate master curve requires measuring the length of the upstream vortex. In certain cases, the vortex boundaries were difficult to define and therefore to omit such difficulties we plot the master curve with flow rate instead of true strain rate.

From the previous scalings between flow rate maximum and molecular weight (Eq.( 5.2)) we can derive the relation between the molecular weights of initial polymers and subsequent passages. The molecular weight M of the solution is characterized by Q max , which for a nondegraded polymer solution is given by

𝐐 𝟏 𝐦𝐚𝐱 = 𝐤𝐌 𝐢𝐧 -𝟐 , (5.8
) where the constant k is specific for a given polymer solution and contraction geometry. We will assume that this relation also holds for polymers that have been degraded. Hence,

𝐐 𝟐 𝐦𝐚𝐱 = 𝐤𝐌 𝐨𝐮𝐭 -𝟐 , (5.9
) where M out is the molecular weight after the 1 st injection at a given flow rate Q.

Upon substituting Eqs.(5.8) and (5.9) into Eq. (5.7) we find the relation between inlet and outlet molecular weights after one passage through a given contraction:

𝐌 𝐨𝐮𝐭 𝐚𝐩𝐩 = 𝐌 𝐢𝐧 [𝟏 + ( 𝐌 𝐢𝐧 𝟐 𝐐 𝐤 ) 𝐚 ] - 𝟏 𝟐𝐚 . (5.10)
The outlet molecular weight calculated here is an apparent molecular weight, since the polymers that degrade are at the high molecular weight part of the distribution so that the injection and outlet molecular weight distributions are likely to have different functional forms.

We plot the apparent outlet molecular weight of Eq.(5.10) with respect to the injected molecular weight (Figure 5.11). In the limit of vanishing flow rate, where one expects no degradation, the outlet molecular weight is same as that of the injected molecular weight (linear red line). For a given flow rate the degradation starts to deviate from this linear behavior at a critical molecular weight specific for that flow rate, and then degradation saturates, suggesting that any polymer having molecular weight above this critical value will undergo degradation.

Multiple passages

We next generalize this approach to many passages. Figure 5.12 summarizes the results for a fluid subjected to four successive injections. In each case we choose

Q 1 = Q 2 = Q 3 = 33.3µL/s > Q 1 max .
After each injection the maximum shifts to lower pressure values and higher flow rates, which suggest additional degradation from pass to pass. However, the amount of degradation is not the same from pass to pass.

The maximum of the native solution 19 MDa-0.08/40SG-[(5-0.2)/34]0.2 is at Q1 max =20.8 µL/s. The flow rate chosen for the first collection is Q1=33.3 µL /s > Q1 max , which thus induces significant degradation. The flow rates chosen for subsequent collections were the same as in the first passage, which in each case is smaller than the maximum for that passage (i.e. .12 shows the shifts of the maxima with degradation during successive injections. The history of degradation can be traced by the evolution of maxima from the multiple injection curves. In Figure 5.13 we plot the ratio of the maximum of (N+1) st pass to the maximum of the N th pass i.e. QN+1 max / QN max , as a function of the normalized flow rate at which it was degraded (collected) during N th pass, QN+1 max / QN max as red stars.

Q 2 < Q 2 max , Q 3 ≪ Q 3 max ).
Figure 5.13: Evolution of the maximum for the data of Figure 5.12 of a given passage with respect to the maximum of the preceding passage QN max /QN-1 max , for passages 1, 2, 3, and 4.

The model shows a fit to Eq. (5.7) with scission exponent a=2.5.

The points from multiple injections follow the previously obtained trend of for two successive passes shown in Figure 15 (based on Eq. (5.7) with scission exponent 𝑎 = 2.5). Therefore this relation is capable of predicting the location of the maximum for a pre-degraded polymer solution based on the conditions of degradation (Q1, Q2, Q3……. QN).

Based on this correspondence, we suggest using the entire injection history, in conjunction with the characteristic degradation curve from the first passage, to describe the evolution through multiple passages. The relation between the second and the first passage is given by Eq. (5.7). Similarly, the relation between the third and second passages should be given by

𝐐 𝟑 𝐦𝐚𝐱 𝐐 𝟐 𝐦𝐚𝐱 = 𝐟 ( 𝐐 𝟐 𝐐 𝟐 𝐦𝐚𝐱 ) = [𝟏 + ( 𝐐 𝟐 𝐐 𝟐 𝐦𝐚𝐱 ) 𝐚 ] 𝟏 𝐚 . (5.11)
Combining Eqs.(5.7) and (5.11) yields

𝐐 𝟑 max 𝐐 𝟏 max = 𝐟 ( 𝐐 𝟐 𝐐 𝟏 𝐦𝐚𝐱 * 𝐟( 𝐐 𝟏 𝐐 𝟏 𝐦𝐚𝐱 ) ) = [𝟏 + ( 𝐐 𝟏 𝐐 𝟏 max ) 𝐚 + ( 𝐐 𝟐 𝐐 𝟏 max ) 𝐚 ] 𝟏 𝐚 .
(5.12)

This can be easily generalized to N passages:

𝐐 𝐍 max 𝐐 𝟏 max = [𝟏 + ∑ ( 𝐐 𝐢 𝐐 𝟏 max ) 𝐚 𝐍-𝟏 𝐢=𝟏 ] 𝟏 𝐚 .
(5.13)

In the case of N passages at the same flow rate 𝑄 𝑖 = 𝑄 1 one finds

𝐐 𝐍 max 𝐐 𝟏 max = [𝟏 + (𝐍 -𝟏) ( 𝐐 𝟏 𝐐 𝟏 max ) 𝐚 ] 𝟏 𝐚 .
(5.14) Figure 5.14: Red stars show the evolution of multiple passages as a function of the 1 st pass for the data of Figure 5.12. In each case the ordinate is the ratio of Q max for successive injections (1 to 2, 2 to 3, etc.) to Q max of the first injection, and the abscissa is the ratio of Q max of the final injection to Q max of the first injection. The solid lines are calculated using (5.14) for N=2, 3, 4 with scission exponent a=2.5.

Figure 5.14 shows that, with respect to the maximum of the non-degraded solution, we observe an upward shift of the maximum flow rate Q max , which corresponds to more scission and lower molecular weight after each pass. Figure 5.14 shows the iteration of Eq.(5.14) for N = 2,3,4, which agrees well with the measured maxima.

Conclusions

Semi-dilute unentangled high molecular polymer solutions were used to study chain scission in a given model contraction. Using a single microfluidic device with a Hencky strain of 3.2, we show that the pressure drop ratio of the polymer solution relative to pure solvent has a maximum as a function of flow rate, which originates from the scission of the polymer chains.

The maximum pressure ratio scales with molecular weight and concentration as R max ∝Mw 1±0.2 c 0.7 ±0.3 while the flow rate at the maximum scales like Q max ∝Mw -1.8±0.2 c -0.7±0.3 . These scaling are similar to the findings of Nghe et al. obtained at the onset of degradation 116 .

We show that the maximum pressure ratio, which reveals the competition between extensional viscosity and chain scission, occurs for a given Deborah or Weissenberg number, although experiments for different Hencky strain are required in order to determine whether such a simple condition applies. We study how the flow rate at maximum pressure ratio increases for successive passages through the same contraction, due to different degrees of degradation at each stage. This allows us to quantify the chain scission as a function of the flow rate. We demonstrate a universal pattern of scission that is independent of solvent quality and molecular weight, for a given Hencky strain. We show that the universal pattern can predict the degradation behavior after successive passes in multiple injection experiments. This robust evolution of scission can be applied in other applications where polymers are injected multiple times, such as lubrication in engines, injection through jets, and during oil recovery; as well as in porous media, which could be represented as successive multiple contractions. In the next section we vary the Hencky strain to study the influence of residence time on polymer scission.

Chapter VI -Hencky strain effect on critical Weissenberg number for scission and the scission exponent.

Introduction

In section 5.4 we concluded that the strain rate at the maximum pressure drop ratio corresponds to the crossover between stretching and scission. It was also shown that the maximum in a pressure drop ratio occurs for a given Weissenberg number suggesting that polymer undergoes scission instantaneously during transient stretching. However, if the polymer scission occurs gradually as in an activated process then it would be controlled by the Deborah number. So, in this chapter, to study the effect of Deborah number, we vary the Hencky strain. Deborah number is the ratio of the residence time and the relaxation time. In this chapter we demonstrate the role of Hencky strain in precisely predicting scission. We then analyze the case of multiple contractions in series, in which the true Hencky strain in the successive contractions is different from the Hencky strain imposed by the geometry.

This chapter is basically divided into 2 main subsections. First, we discuss the effects of Hencky strain in single hyperbolic contraction geometries, and secondly, we discuss scission of polymers in multiple contractions in series. Subsection 6.2 is further divided into 3 sections.

In section 6.2.1 single injections on geometries with different Hencky strains are done for a given polymer solution. The resulting true strain rates at the maximum are studied as a function of Hencky strain.

In section 6.2.2, polymer solutions with different molecular weights are injected through geometries with different Hencky strains to determine the Weissenberg number at the maximum for different Hencky strains.

In section 6.2.5 double injections are performed on geometries with different Hencky strain with a given polymer solution. We propose a scission model like the one described in chapter 5, to predict the scission in successive passages. We will show how the model varies with Hencky strain. With these understandings, we next predict the scission for several contractions in series.

In section 6.3, several contractions of similar Hencky strain are used (section 3.8.). The solution is collected at the outlet of first passage through geometries with different number of contractions in series. The collected solutions are injected for a second time through single contraction geometry; surprisingly the amount of scission is lower than expected. The flow visualization data confirmed that in presence of vortices the true Hencky strain in the successive contractions is smaller than the Hencky strain imposed by that geometry. We were able to predict the degradation in the successive contractions by considering this true Hencky strain.

6.2 Hencky strain effect of critical strain rate for scission

Single injection experiments: effect of geometry

In this section the Hencky strain of the geometry is varied by changing the entrance width W u and the contraction width W c , as shown in Table 3.6, reproduced below.

Table 3.6: Dimensions of geometries with different Hencky strain. Wu is the entrance width , Wc is the contraction width, Lc is the contraction length h is the channel height, 𝜀 𝐻 is the Hencky strain, and α is the aspect ratio.

The Pressure drop ratio R presents a maximum when plotted as a function of the flow rate. Decreasing Hencky strain shifts the curves towards, higher flow rates except for Hencky strain 3.2, the reason for this behavior could be due to the mixed flow (shear and extension) in the contraction geometries and are not in the scope of this thesis. However, the true strain rate at the maximum ε̇T max (considering the vortices) shifts to lower values for increasing Hencky strain ε H , shown in later sections.

When the pressure drop ratio R is normalized both by the maximum in the pressure drop ratio and the corresponding flow rate, all the curves superpose as shown in Figure 6. 3.6).

To understand the scission of polymers in geometries with varying Hencky strain, the true strain rate 𝜀̇𝑇 𝑚𝑎𝑥 is calculated according to Eq.(5.4)

ε̇T max = Q max (L c +L vs )h ( 1 W c - 1 W u )
Earlier we argued that the true strain rate at the maximum ε̇T max is the critical strain rate for crossover between stretching and scission.

Eq.(5.4) is accurate if the boundaries set by the vortices are similar to a quasi-stationary hyperbolic contraction profile (so as to have a constant strain rate along the contraction profile set by the vortices), which we assume to be the case. In Figure 5.6 of section 5.4, the resultant vortices shape resembles a hyperbolic contraction profile. The length was measured for stable symmetric corner vortices and to some extent time dependent symmetric corner vortices. Moreover, in section 5.4, we omitted the regime of viscoelastic instabilities. Therefore, such an assumption of constant strain rate is reasonably good. But for experiments done in this chapter, different flow regimes were observed. For Hencky strains 3.9 and 3.2 for all polymer solution in glycerol/water solvent, the maximum in pressure drop ratio occurred for flow in the viscoelastic instability regime (7 th flow regime of Figure 4.3). For the same polymer solutions in Hencky strains ranging from 2.3 to 1.4 the maximum occurred in stable/time-dependent and symmetric/asymmetric corner vortex regimes (2-5 flow regimes of Figure 4.3). The measurements on the precise local strain rates for these different flow regimes are quite difficult. At this moment whatever the flow regime we assume the strain rate to be constant in between the vortices. We define the vortex regime for a deviation in the flow line by an angle greater than or equal to 25°, where we measure the vortex length and calculate the true strain rate at the maximum ε̇T max from the Eq.(5.4) (Note that this includes all the vortices regimes 2-7, of Figure 4.3).

We plot this strain rate at the maximum ε̇T max as a function of Hencky strain ε H in Figure 6.3. For higher Hencky strain the cross over occurs at lower strain rates ε̇T max .The curve is a boundary between no significant degradation to significant degradation. i.e. strain rates and the Hencky strains in the region below this curve have no significant degradation while strain rates and the Hencky strains above this curve would result in significant degradation. Figure 6.3: Hencky strain as a function of critical strain for scission 𝜀̇𝑇 𝑚𝑎𝑥 Each point corresponds to the true strain rate at the maximum of the pressure drop ratio curve R(Q), when injected with 11Mda-0.08/40SG polymer solutions through a geometry with given Hencky strain 𝜀 𝐻 of Table 3.6. The strain rate at the maximum 𝜀̇𝑇 𝑚𝑎𝑥 is calculated from the length of the vortices Lvs measured in the viscoelastic instability regime (7 th flow regime of Figure 4.3) for 𝜀 𝐻 = 3.9 𝑎𝑛𝑑 3.2; Stable symmetric corner vortex regime (2 nd flow regime of Figure 4.3) for 𝜀 𝐻 = 2.3 𝑎𝑛𝑑 1.8; Stable asymmetric corner vortex regime (4 th flow regime of Figure 4.3) for 𝜀 𝐻 = 1.4.

Molecular weight Influence

In the next step, we study the influence of molecular weight in each of geometries with different Hencky strain.

Polymers with different molecular weight are injected at different flow rates through geometry with a Hencky strain 𝜀 𝐻 = 2.3, and the pressure drop ratio is reported as a function of flow rate as in Figure 6.4. We normalized the pressure drop ratio with maximum in pressure drop ratio and the corresponding flow rate as in Figure 6.5. Similar observations as in section 5.2 were made, where a decrease in molecular weight, shifts the maximum towards lower pressure drop ratio and higher flow rates. In summary, from single injection of a give polymer solution in different Hencky strains (section 6.2.1), for higher Hencky strain we observed smaller true strain rate at the maximum. In each of the geometries with different Hencky strain, we observe the similar molecular weight dependence (section 6.2.2) as observed for Hencky strain ε H =3.2* (section 5.2). To calculate the Weissenberg number at the maximum of the pressure drop ratio in different Hencky strains, we multiply the relaxation time with the true strain rates at the maximum in the corresponding geometries. In chapter 5 the relaxation time t cm is obtained by fitting a Carreau model with the non-linear shear viscosity data obtained from the Couette flow device. But it could be more relevant to use the extensional relaxation time like the one obtained from the filament thinning dynamics of a CaBER experiments. But even before that it is essential for us to evaluate if CaBER results suffer from scission of polymers during filament thinning dynamics which we address in the next section.

Does degradation occur in filament thinning dynamics of CaBER experiment?

We first ask whether if there is any polymer scission in the CaBER measurements? If not, then the CaBER will give a more reliable measure of the Rouse time than using the Carreau model. The Carreau model applies the empirical Cox-Merz rule, which is only approximately valid. While the CaBER gives direct access to the longest relaxation time of a polymer solution (Rouse time).

The experimental procedure and analysis of the CaBER measurements are explained in section 3.4.2. Figure 6.6 shows the strain rates as function of Hencky strain 𝜀 𝐻 measured from filament thinning dynamics of the polymer solution 11MDa-0.08/40SG in the CaBER experiment, as well as strain rate at the maximum as a function of Hencky strain 𝜀 𝐻 measured from the pressure drop ratio curves for geometries with different Hencky strain.

The strain rates from CaBER are smaller, or in the regime of no significant degradation, to the critical strain rates at the maximum measured from the microfluidics. This assures that there is negligible degradation in the filament thinning dynamics of the CABER experiment. Figure 6.6: Strain rates as function of Hencky strain 𝜀 𝐻 measured from filament thinning dynamics of the CaBER experiment, and the critical strain rate at the maximum(cross over) 𝜀̇𝑇 𝑚𝑎𝑥 as a function of Hencky strain 𝜀 𝐻 measured from the pressure drop ratio curve for injection of a 11MDa-0.08/40SG through microfluidic geometries.

Weissenberg number in different Hencky strains

The critical strain rate for scission ε̇T max measured from different molecular weights is inversely related to relaxation time of the polymer solution measured from CABER t ca as in Figure 6.7. Therefore, for a geometry with a given Hencky strain ε H scission occurs for a given Weissenberg number, Wi max . Figure 6.8 shows that for higher Hencky strain scission occurs at a lower Weissenberg number. We thus generalized a result of the previous chapter to various Hencky strain. The maximum of pressure ratio occurs for a Weissenberg number that depends on the Hencky strain of the cell. It would be interesting to understand how this variation can give us information on the scission mechanism but this is out of the scope of the thesis.

Double injections

From the single injection experiments it is evident that scission cannot be solely determined from the Weissenberg number, but also depends on the Hencky strain ε H the Hencky strain, and thus residence time or the Deborah number. Next we perform double passages on geometries with different Hencky strains to observe the transition into a degradation regime as done in section 5.5.

A polymer solution is pre-degraded during a first passage at flow rate Q1 and then reinjected for a second time at different flow rates for the second pressure drop ratio curve R (Q2, Q1). The second pressure drop ratio curve is shifted to lower values of pressure drop ratio, because of the scission in the first passage.

In Figure 6.9 we plot the flow rate at the maximum in the pressure drop ratio of second passage 𝐐 𝟐 max , as a function of flow rate at which the solution is pre-degraded during the first passage 𝐐 𝟏 . Both the quantities are normalized by the flow rate at the maximum in the pressure drop ratio of the first passage 𝐐 𝟏 max .

The scission evolution in each Hencky strain can be modeled using the similar mathematical expression described in Eq.(5.7) and is reported below. The scission model consists of scission exponent b which describes the broadness of transition into degradation regime. (5.7)

The experimental data of scission evolution in each Hencky strain is described by Eq.(5.7), for different values of scission exponent b (represented as solid lines in Figure 6.9).

To further validate the model, similar experiments of double passages are done with the 19MDa-0.08/15SE through a geometry with Hencky strain ε H = 3.9 . The experimental points correlating the flow rate at the maximum in pressure drop ratio of second passage to the flow rate at which the solutions were pre-degraded during the first passage (both normalized by the flow rate at the maximum in the pressure drop ratio of the first passage) lie on the same plot as for 15MDa-0.08/40SG polymer solution with Hencky strain ε H = 3.9 . Figure 6.9: Evolution of maximum pressure drop ratio curve from successive injections of 15MDa-0.08/40SG polymer solution. Here, Q1 is the first injection flow rate, Q1 max is the flow rate at the maximum in pressure drop ratio for the non-degraded material, and Q2 max is the flow rate at the maximum in pressure drop ratio of second passage. The solid line shows a fit to Eq. (5.7) for different scission exponent b which depends on the Hencky strain of geometry. Open symbols correspond to the geometries with different geometrical dimensions than that of the solid symbols (Refer Table 3.5).

We obtain the same trends for different Hencky strains as observed in section 5.5, but the broadness of the transitions, and thereby scission exponent b are Hencky strain 𝜀 𝐻 dependent.

For higher Hencky strains the scission exponent b is smaller (Figure 6.10), which indicates a broader crossover to the scission regime. In other words, scission occurs even for lower flow rates (Q < Q max ). Similarly the scission exponent b is larger for smaller Hencky strains, which indicates the crossover is sharper and therefore that scission is more deterministic (i.e. negligible scission for Q < Q max and scission for Q > Q max ).

The scission exponent b is linearly related to Hencky strain and we observe the phenomenological relation 𝐛 ≅ -𝟏. 𝟏𝛆 𝐇 + 𝟔 (6.1) Figure 6.10: Scission exponent b as a function of Hencky strain, from the experimental data of Figure 6.9 .

However it is likely that the scission exponent should not become negative for very high Hencky strain.

Discussion

For single injection experiments the total pressure drop includes both the shear and the extensional contributions. Upon varying the Hencky strain both the shear and the extensional contributions are modified. The precise quantitative contributions are not in scope of the current study as we do not have the data on the local velocity fields. The normalized pressure drop ratio and the flow rate as in Figure 6.2, shows a failure of superposition for lower flux.

Because the Newtonian viscosity is a linearly related with the molecular weight of the polymer Mw, like the maximum pressure drop ratio. The absence of superposition at low flow rates is likely due to the details of the shear thinning and extension behaviors in these geometries. The maximum in pressure drop ratio occurs for low strain rates for higher Hencky strains and vice versa.

The critical residence time at the maximum t res max can be calculated simply from the definitions of residence time, strain rate and Hencky strain as explained in section 3.5:

𝐭 𝐫𝐞𝐬 𝐦𝐚𝐱 = 𝛆 𝐇 𝛆̇𝐓 𝐦𝐚𝐱 (6.2)
This equation applies if the strain rate is constant (in time) between the vortices. In other words, the boundaries set by the vortices must resemble a quasi-stationary hyperbolic contraction profile (which we assume).

The critical residence time t res max at the maximum is decreasing exponentially with the critical strain rate ε̇T max at the maximum as shown in Figure 6.11, according to The maximum force F b max , the C-C bond can hold before breaking is given from the derivative of the Eq. (6.5):

𝐅 𝐛 𝐦𝐚𝐱 = 𝐔 𝟎 𝟐𝐝 at a separation length l = l e + dln2
The Arrhenius rate (𝑲 𝟎 ) as given by Odell et al. [START_REF] Odell | Flow-induced chain fracture of isolated linear macromolecules in solution[END_REF] is given in

𝐊 𝟎 = 𝛖𝐞 𝐔 𝟎 [-𝐏+𝐝 𝐥𝐧( 𝟏+𝐏 𝟏-𝐩 )+⋯/k B 𝐓] (6.6)
Where k B T is the thermal energy. However, most researchers neglect the logarithmic term in the exponent From the first term of Taylor series expansion of the exponent, we obtain

𝐊 𝟎 = 𝐂 𝟏 𝐞 𝐅 𝐛 𝐝𝐤 𝐁 𝐓 (6.7)
When straight polymer chain is placed in the elongation flow, the chain stretches in the direction of Flux. The breaking of chain occurs at the center and can be understood by applying the Stoke's law. An analogy of such straight chain can be made to the bead rod model, where beads of radius a are connected by frictionless rods of length 𝐫 𝐛 . In elongational flow filed the completely stretched out molecule resembles to the one shown in Figure 6.12. The whole polymer chains move with the fluid, there exists a relative velocity between the individual beads (except at the center, which moves along the fluid) and the surrounding fluid of varying magnitude and direction. The magnitude of the relative velocity increases as one approach to the chain ends. Due to the friction between the beads and solvent, results the tension on the chain with the maximum stress being accumulated at the center. When this stress accumulated at the center exceeds the strength of the string would result in a cleavage. The frictional forces F i on an i th bead can be calculated from Stoke's law as, 𝐅 𝐢 = 𝟔𝛑𝛈 𝟎 𝐚𝐯 𝐫𝐢 s (6.8) where s is the shielding factor 119 of one bead on another for extensional strain rates experienced, v ri is the relative velocity of bead with respect to solvent.

If 𝐫 𝐛 is the separation between two adjacent beads, the relative velocity of an ith bead is given as 𝐯 𝐫𝐢 = 𝛆̇𝐢𝐫 𝐛 ( ε̇ is the extensional strain rate). So F i is given as 𝐅 𝐢 = 𝟔𝛑𝛈 𝟎 𝐚𝛆̇𝐢𝐫 𝐛 s (6.9)

The force at the center of the chain 𝐅 𝐜 , would be the sum of forces 𝐅 𝐢 all acting on all the beads of the chain. 𝐅 𝐛 = 𝛈 𝟎 𝐥𝐫 𝐛 𝛆̇𝐍 𝐱 (6.11)

In dilute solutions for a perfectly stretched chain x=2 [START_REF] Odell | Degradation of polymer solutions in extensional flows[END_REF][START_REF] Nguyen | Mechano-Chemical Degradation of Polymer Solution in Capillary Flow: Laminar and Turbulent Regime[END_REF][START_REF] Odell | Flow-induced chain fracture of isolated linear macromolecules in solution[END_REF][START_REF] Nguyen | Chain extension and degradation in convergent flow[END_REF] .

In dilute solutions for a partially stretched chain x<2 [START_REF] Nguyen | Chain extension and degradation in convergent flow[END_REF]120,121 Figure 6.12: Illustration of completely stretched polymer analogy with bead-rod model. The length of the arrow represents the intensity of relative solvent velocity, which increases when approached to the ends. The central bead c is the center of gravity of the chain which moves with a average velocity of the surrounding fluid.

To determine the exponent x in our case, by comparing Eq.( 6.3) with Eqs.(6.7) and (6.11) we get 𝐲 ∝ 𝐅 𝐛 𝐝 ∝ 𝛈 𝟎 𝐍 𝐱 𝐝 Figure 6.13: Exponent y of Eq.( 6.3) obtained from the experimental data in Figure 6.11 as a function of Molecular weight of the polymers.

We plot the exponent y as a function of molecular weight in Figure 6.13. The plot suggests a molecular weight dependence with an exponent x ≅ 2.5 greater, than the exponent predicted for a completely stretched state x=2.

The potential explanation for this deviation could be the following: For a single polymer chains in dilute solution, the solvent viscosity is the appropriate term for calculating the force, which leads to the exponent 1. For semi-dilute solutions the effective viscosity could be molecular weight dependent due to entanglements and/or by dynamic interactions.

2. The broadness of potential energy curve (there by parameter d) could be dependent on the molecular weight, since increase in the monomers would result in more compliance under stress, which would increase the effective width (or broadness) of potential energy curve. At this point we suggest detailed theoretical and numerical studies could help us better understand the mechanism of scission, which is not in the scope of the current study.

To conclude, our experimental findings suggest that scission mechanism to be an energy activated process.

We provide the Rouse calculations for critical tension for causing scission in Annex II. The proposed model cannot explain the critical Weissenberg number for scission and the scission exponent b.

With the current experimental understanding, we next perform single and double injection experiments on geometries with several contractions in series (with same Hencky strains in all the contractions separated by a fixed distance), to approach a model in porous medium.

Multiple contractions in series 6.4.1 Single Injection

In this section multiple contractions are placed in series (Figure 6.14) with a separation of 5mm between successive contractions. The Hencky strain for all contractions is ε H =2.3. where R h is the hydraulic resistance and depends on the geometrical parameters.

The 19MDa-0.08/40SG polymer solution is injected into multiple contraction and their pressure drop ∆P p (Q, N c ) is noted. Pressure drop ratio R(Q, N c ) is calculated as, 𝐑(𝐐, 𝐍 𝐜 ) = ∆𝐏 𝐩 (𝐐, 𝐍 𝐜 ) ∆𝐏 𝐬 (𝐐, 𝐍 𝐜 ) , (6.13) where N c is the number of contractions.

Pressure drop ratio R(Q, N c ) is plotted in Figure 6.15. For higher contraction number N c , the pressure drop ratio R(Q, N c ) occurs for higher pressure drop ratio and smaller flow rates. The total length of microfluidic device is same for all the geometries. Even for one contraction, the polymer contribution to pressure drop ratio is relatively higher than the solvent. With increase in number of such contractions in series, results in higher pressure drop ratio. In successive contractions the scission occurs during each contraction event, however the amount of degradation is smaller in the successive contractions, and therefore the flow rate at maximum in pressure drop ratio Q max shifts to lower values with increasing number of contractions in series N c . An analogy can be made with a porous medium (refer Chapetr 7), for increasing the number of contractions N c (for a fixed total length of the geometry) is equivalent to decreasing the permeability of a porous medium. For a porous medium when the permeability of the porous medium is decreased the pressure drop ratio (and thereby the maximum) occurs for higher pressure drop ratio and smaller flow rates. In order to predict scission in the successive contractions, we in the next step perform a double passage experiments as explained in section 3.7.2.

Double injection on contraction in series

In order to understand the polymer scission during successive contractions and to apply the degradation model developed in chapters 5 and 6, double injections (section 3.7. The pressure drop ratio R(Q 2 , Q 1 , N c ) is plotted as a function of flow rate in Figure 6. 16. The pressure drop ratio curves R(Q 2 , Q 1 , N c ) shifts to lower pressure drop ratio and higher flow rates, suggesting higher degradation with increasing the number of contraction N c during the first passage. Q 2 max (Q 1 , N c ), shifts to higher values as the number of contractions is increased, consistent with increasing degradation for more contractions. We next try to predict the maxima using the models developed in section 6.2.

Degradation on single contraction with Hencky strain 2.3 units

In order to predict the polymer scission we use the model proposed in Eq.(5.7), the scission model consists of scission exponent b which depends on Hencky strain as discussed in section 6.2.5. For scission through a single contraction with Hencky strain ε H = 2.3 units, the scission exponent b = 3.25.

In this experiment we conducted a first passage at constant flow rate 𝐐 𝟏 through a geometry with 𝐍 𝐜 contractions followed by a second passage at flow rate Q 2 through a single contraction. Therefore, the scission model should be applied for the N c +1 th passage to calculate the maximum of second passage; It is interesting to note lesser amount of scission for successive contractions. What can explain this deviation? In chapter 5, we showed that vortices modify the local strain rate, but for double passages, polymer solutions with different vortex activity, still collapsed onto a single master curve given by Eq.(5.4). From section 6.2.5, we concluded that the scission is lower for lower Hencky strain. Hence, we suggest that scission is due to reduced effective Hencky strain in successive contractions, which we study in the next section. The formation of vortices in single contractions was discussed in section 4. Corner vortices grow in size until they reach the length of the cell, after which they buckle. Similar corner vortices are observed in successive contractions. When a polymer enters from a width of 5mm and passes through a contraction of 0.5mm the polymer becomes stretched. At the exit the polymer re-enters the width of 5mm where the stretched polymer relaxes and attains the original coiled stated. In other words, at low flow rates when the length of the unperturbed vortices Lvs is lesser than the separation distance LG between successive contractions, the width accessible by polymer solution remains same as that of the cell. Therefore, the amount of strain undergone by the polymer in each contraction should be equal to that imposed by the geometry. Therefore to predict the Q 2 max (Q 1 , N c ), Eq.( 6.15) can be used with the scission exponent b corresponding to the effective Hencky strain ε H eff (N c ) in a given contraction. We first predict the scission in each contraction of geometry with multiple contractions in series and then predict the maximum for the second passage through a single contraction geometry. 2. The polymer which undergo scission in the first passage does not have the similar Q 2 max as predicted by the Eq.(5.7). Since the predicted Q 2 max is only applicable in a geometry of same Hencky strain as in the first injection (Figure 6.24).

Figure 6.24: The first contraction, the Hencky strain remains the same as that of the cell with a single contraction.

The first issue, the scission exponent b depends on Hencky strain according to Eq.(6.12) 𝐛 = -𝟏. 𝟏𝟒 * 𝛆 𝐇 + 𝟔.

Q max corresponds to an equivalent molecular weight (apparent Mw) as in section 5.5, according to,

Here 𝑖 = 1,2,3 … 𝑁 𝑐

The polymer exiting the 𝑖 𝑡ℎ contraction with molecular weight 𝑀 𝑜𝑢𝑡 (𝑄 1 , 𝑖) is the inlet molecular weight for the (𝑖 + 1) 𝑡ℎ contraction, 𝐌 𝐨𝐮𝐭 (𝐐 𝟏 , 𝐢) = 𝐌 𝐢𝐧 (𝐐 𝟏 , 𝐢 + 𝟏) (6.21) This process is iterated from i=1 to 𝑁 𝒄 and the final M out (𝑄 1 , 𝑁 𝒄 ) is calculated. Polymer with M out (𝑄 1 , 𝑁 𝒄 ) collected at the outlet of the first passage.

In the last step, this collected solution is injected through a single contraction with ε H =2.3 units. For a single contraction we always observe that the true Hencky strain is that imposed by the geometry so we can simply use Eqs. (6.16) and (6.17

Conclusion

Scission of semi-dilute unentangled polymers is studied in geometries with Hencky strains ranging from ε H =1.4 to ε H =4 units. The true strain rate at the maximum of the pressure drop ratio R occurs at smaller values for larger Hencky strains. The scission process seems to be an energy activated process; however, the current study cannot explain the molecular weight dependence of the applied force. The scission is studied with double passages on geometries with different Hencky strains. The same scission law is valid, but the scission exponent b of the model depends on the Hencky strain. The broadness of transition from no scission to scission dominated regime increases with increasing Hencky strain.

Scission is then studied in geometries with several contractions in series. From single passage experiments we observe that the amount of scission increases with an increasing in number of contractions. However double passages experiments, where the first passage is through a geometry Nc number of contractions and the second passage is through a single contraction, suggested a reduced scission than expected from the scission model we developed. Visualization experiments revealed that the true Hencky strain in successive contractions is reduced due to the formation of the vortices. With the true Hencky strain calculated from the observed flow lines in successive contraction, and considering the dependence on the Hencky strain, we were able to predict the amount of scission in successive contractions. In the next section we extend our understanding to a porous media, which is a complex network of contractions and expansions with 3D connectivity.

Chapter VII -Porous medium

Introduction

The objective of this chapter is to establish a predictive law for the degradation of polymer solutions flowing through porous media, using porous cores of different lengths. For this purpose, we first measure the mobility reduction R m for injecting a polymer solution through the porous media. When plotted as a function of the flux J, typical mobility reduction curves have maxima R m max at flux J max .

In a second series of experiments (double passage), pre-degraded solutions are prepared by collecting flow outlet through the core of length L 0 =1 mm (first pass). This pre-degraded solution is then re-injected through the same core (second pass), and the second mobility reduction curve is measured. From the analysis of the mobility reduction curves from the two passages, a multiple passage model (MP model) is proposed to predict the mobility reduction curve through a core having a length NL 0 , by assuming that a pass through a core of length NL 0 is equivalent to N successive passes through a core of length L 0 .

The model is then used to predict the degradation in a core of length L. Degradation shall increase with core length, reaching an apparent steady state value after a critical length L c (i.e. there is negligible scission for L > L c ). This critical length decreases with increasing flux, and the amount of degradation at apparent steady state increases with the flux. By applying the model to 2D radial injection, we observe that degradation reaches a steady state value at a critical radius r c from the injector, which shifts to higher values for increasing injector well radius R w . As expected, at large injector well radius R w , the critical radius r c tends towards the critical length L c for 1D injection (linear flows), in the limit r c ≪ R w .

To the best of our knowledge, this is the first study in which: Degradation is measured in short cores, Degradation kinetics (degradation versus core length) is measured experimentally and predicted from a mathematical model, Mobility reduction is predicted as a function of length. The link is made between linear (1D) and radial (2D) injection.

This chapter is organized in 7 sections as follows, In section 7.2, we present the results of single injection experiments on porous media of different lengths. The mobility reduction as a function of flux, and the degradation inferred from the loss in the viscosity measurements (from a Couette flow device-based viscometer, at low shear rates) are discussed.

In section 7.3, we perform double passage experiments on porous medium of length L 0 =1mm. I.e. a solution collected at outlet of the first injection, is injected for a second time.

We get the similar trends of scission of polymers to those observed in single contraction devices of sections 5.5 and 6.2.5.

In section 7.4, with the help of the Galindo Cross model and the scission trends, we propose a multiple passage model (MP model) to predict the mobility reduction curve of porous media of any length, based on the mobility reduction curve measured from the shorter cores.

In section 7.5, we successfully compare the predicted mobility reduction curve results with the experiments.

In section 7.6, we compare degradation predictions from the MP model with the degradation from the experiments, inferred from the loss in the viscosity of polymer solution before and after injection through the porous media. We show that the trends of degradation are captured reasonably with this model.

In section 7.7, the application of the MP model to industrial context is discussed. We apply the MP model to both 1D flows (linear injections, constant flux) and 2D radial flows (flux decreases with the radius). We show that the degradation attains a plateau in each case after critical length (or radius). The degradation at the plateau and the length (or radius) to attain this plateau occurs for higher values in 1D flows in comparison to the 2d radial flows (for the same entrance flux at wellbore surface). We demonstrate degradation in 2D radial flows is most important during the first few centimetres. The probability of scission is higher, close to the injector due higher fluxes. In addition, blocking of pores with any impurities decreases the permeability of the rock, which may promote the degradation. Therefore, for better efficiencies we highlight the fact for a need of clean practices of the polymer flooding to prevent polymer undergoing scission in the first few centimetres.

Lastly in section 7.8, we discuss the important conclusions of the experiments on the porous medium and suggest the further points of improvements that could be done in establishing more reliable models.

Single injection experiments on porous medium of different length

The single injection experiments are explained in section 3.9.5. Briefly, the solvent is first injected at different fluxes through a given porous medium of length L and the pressure drop ΔP s , between entrance and the exit is measured. Then the polymer solution is injected and the pressure drop ΔP P is measured. The mobility reduction R m is defined as the ratio of pressure drop of polymer to that of the pressure drop of the solvent as given in, Eq. ( .

The polymer viscosities are measured in Couette flow devices for low shear rates. Degradation is quantified by the relative loss in viscosity as in, Degradation is expressed in %.

In Figure 7.1, the mobility reduction is plotted as function of flux J for porous media of different lengths L. The maximum in the mobility reduction curve originates from the competition between the polymer stretching and the polymer scission. For each core length the mobility reduction curve has a maximum R m max at a flux J max . The shear thinning behavior classically observed in polymer propagation studies on long cores occurs at lower fluxes (typically of few cms). On small porous cores of few mms (as in our case) at such small flux, the pressure drop is below the sensitivity limits of our pressure sensors. The mobility reduction curves are normalized in Figure 7.2. The superposition is quite good for both J < J max and J > J max , except for longest core(8 mm) for which the superposition fails for J > J max .

The corresponding degradation curves are plotted in Figure 7.3. For a given flux, degradation increases with core length. However, for the 8 mm-long core, the degradation seems to be lower than expected. The shift of J max to lower fluxes for longer cores is consistent with the higher degradation in longer cores, as inferred from chapters 5 and 6. The influence of length on degradation will be discussed in detail in the later sections.

Double passages

A second injection is performed on the same core to measure the effect of pre-degradation on the mobility reduction. This additional information enables us to build a model for predicting the mobility reduction and the degradation during successive injections.

The polymer solution was degraded at different levels in core of length L 0 =1 mm by changing flux J 1 . For each solution collected during the first passage at J 1 , the mobility reduction curve R m (J 2 , J 1 ) was measured by re-injecting the solution at flux J 2 through the same core. The mobility reduction curves for J 1 = 0.46, 1.01, 1.85 and 3.7 m.h -1 and reinjection are plotted in Figure 7.4.

When the flux J 1 is increased, the maximum of the mobility reduction R m max (J 2 , J A similar superposition was performed by Jouenne et al. 123 for polymer solutions with different molecular weights, concentrations and salinities injected through a short sintered ceramic core with permeability of water, K w = 4 Darcy. They found that variations in concentration or pre-degradation level have no significant influence on the width of the normalized mobility reduction curves. For commercial polymers of different molecular weights, they also observed a weak change of the width. They attributed this difference in width to the high polydispersity of the commercial polymers. In conclusion, the width of mobility reduction curve is weakly or not effected with the physicochemical properties of the solution.

Furthermore, we assume that this superposition holds for not only the first and second pass but also holds for any number of successive passages.

Modeling the mobility reduction curve

In continuation to the shear thinning regime, which is well described by Carreau model, Galindo Rosales et al. 124 proposed a mathematical equation to model the shear thickening and the degradation regime. This model involves several fitting parameters to describe the apparent viscosity or the equivalent mobility reduction as a function of flux or shear rate.

R m (𝑄) = { R m max + R m c -R m max 1 + (τ 2 ( Q -Q c Q max -Q ) 𝑄) n 2 if 0 < J ̂< 1 1 1 + (τ 3 Q -Q max ) n 3 if J ̂≥ 1
, where τ 2 is a constant which characterizes the shear thickening regime, τ 3 is a constant which characterizes the degradation regime. The fitting exponents n 2 and n 3 are used to differentiate the dependency of shear thickening and degradation regimes with flux. Q c is the flow rate of onset of extension and R m c is the mobility reduction at the onset of extension.

Though the physics behind these fitting parameters is mostly phenomenological, the intuitive model they propose is an efficient method to predict the apparent viscosity. Ideally the fitting parameters can be related to the physics of the polymer micro configurations.

We slightly modify the model proposed by Galindo Rosales et al. 124

where J ̂= J J max . The normalized mobility reduction curves in Figure 7.5 are fitted by the modified Cross model (black line) of Eq.(7.1) with the following fitting parameters: n 2 =1.45, n 3 =1.25, τ 2 =3, τ 3 =0.0075.

Prediction of the mobility reduction curve

We aim to predict the mobility reduction for injection of a polymer through a core of length L = NL 0 from N successive passages on porous core of length L 0 .

To predict the mobility reduction curve R m (J N , L 0 = 1mm) in a N th passage through the porous core of length L 0 = 1mm, it is essential to predict the maxima (Rm i max , J i max ) of i th passage through the porous core of length L 0 = 1mm. during each pass for i = 1,2,3 … N. Note that the flux J i , is constant in 𝑁 passages.

Reinjected solutions

As seen in Figure 7.4, when solutions, (pre-degraded during a first pass at flux J 1 ) are reinjected for a second pass at flux J 2 the maximum in mobility reduction curve is shifted from J 1 max to J 2 max . This shift, expressed as the ratio J 2 max / J 1 max , and is plotted in Figure 7.6 as a function of the ratio J 1 / J 1 max . The experimental trend can be described by a

𝐉 𝟐 𝐦𝐚𝐱 𝐉 𝟏 𝐦𝐚𝐱 = 𝐟 𝟐 ( 𝐉 𝟏 𝐉 𝟏 𝐦𝐚𝐱 ) = [𝟏 + ( 𝐉 𝟏 𝐉 𝟏 𝐦𝐚𝐱 ) 𝐚 ] 𝟏 𝐚 , (7.2)
as for single contractions. The best fit for scission exponent a =1.85, which corresponds to the sharpness of the crossover. In other words, the regime in which J 2 max / J 1 max deviates from 1 but is still not proportional to J 1 / J 1 max .

The first injection of the non-degraded solution has maximum in the mobility reduction R m max (0, J 1 ) , at flux J 1 max , while the reinjected solutions have maxima R m max (J 2 , J 1 ) at flux J 2 max , which depends on J 1 . Next, we establish a relation between the maximum in mobility reduction and their corresponding flux. In Figure 7.7, The maximum mobility reduction R m max is plotted as a function J max (from the 2 passes). The maximum mobility reduction scales as R m max ∝ (J max ) -0.5±0.1 and is given as: From the knowledge of these two parameters, the whole mobility reduction of a solution re-injected through a 1mm-long core (it is possible for any length) can be predicted by using Eq.(7.1) as in,

𝐑 𝐦 𝐦𝐚𝐱 = 𝐀(𝐉 𝐦𝐚𝐱 ) -𝟎.𝟓 (

Multiple passage model (MP-model) to predict the Mobility reduction curve

It is shown in the section 5.6, by injecting successively a polymer solution in a hyperbolic contraction, the scission model of Eq.(5.7) is valid for N successive injection. A third pass related to the second pass is equivalent to the second pass related to the first pass and similarly a N th pass related to the (N -1) th pass. Given that the normalized mobility reduction curves during the first and the second passes are superimposable (Figure 7.5), we assume that Eqs. (7.2) and (7.3) are also valid for N passes in a more complex porous medium.

With this assumption, we attempt to predict the mobility reduction curves for a porous medium of length L = NL 0 (where L 0 =1 mm) by extending Eq. ( 7.2) we get:

where the function 𝑓 𝑁 is the function for N th passage described in Eq.(7.2). Note that the flux same in N passages.

J N max J 1 max for one iteration (N=2 passages, J 2 max J 1 max
)is given as By using Eq.( 7.3), the maximum of the mobility reduction for the N th pass through porous medium of length L 0 =1 mm is: R m max (J N , L 0 ) for one iteration (N=2 passages, R m max (J 2 , L 0 ) ) is given as At last using the Eq. (7.1) describing the unique shape of the normalized mobility reduction curve, the whole mobility reduction curve (at any flux J, note here the flux is same in N passages) can be predicted by:

Where J ̂N = J N J N max
The mobility reduction of N th passage R m (J N , L 0 ) for one iteration (N=2 passages, R m (J

Where J ̂2 = J 2 J 2 max = J 2 J 1 max [1+( J 1 J 1 max ) a ] 1 a
and fitting parameters: n 2 =1.45, n 3 =1.25, τ 2 =3, τ 3 =0.0075.

With the above equations as inputs, the pressure contribution of a porous medium of length L = NL 0 is equal to the sum of the pressure drops measured during N successive injections at a flux J through a porous medium of length L 0 . This assumption will be valid only if the entrance pressure drop is negligible. This is true in our case, as tested with 4 passages (data not shown here). The mobility reduction on the porous medium of length NL 0 is expressed as:

𝐑 𝐦 (𝐉 𝟏 , 𝐍𝐋 𝐎 ) = ∑ 𝐑 𝐦 (𝐉 𝐢 , 𝐋 𝟎 ) 𝐍 𝐢=𝟏 𝐍 (7.12)
Where the R m (J i , L 0 ) are given by Eqs. (7.2) and (7.12). The R m (J i , L 0 ) for one iteration (N=2 passages, R m (J , (7.13) where

J ̂𝟏 = J 1 J 1 max and J ̂2 = J 2 J 2 max = J 2 J 1 max [1+( J 1 J 1 max ) a ] 1 a
.

Therefore when injected at a flux J on porous medium of length L = NL 0 the mobility reduction is simply obtained from the average of the entire history of mobility reduction in N passages at a flux through the porous medium of length L 0 as in Eq.(7.12).

The predicted mobility reduction curve (MP R m (J 1 , L = NL 0 )) for N =1 to 8 (L 0 = 1 mm) are compared with the experimentally obtained mobility reduction curves R m (J 1 , L) (for L =1 mm, 2 mm, 3 mm, 4mm, 6 mm, 8 mm) in Figure 7.8. The predicted mobility reductions fit quite well with the experimentally obtained results. This is the major result of this chapter These findings were true for any L 0 and L 0 can be extrapolated to even very small lengths of porous medium where the experiments are quite complicated to perform on such fine porous medium (e.g. 0.1mm or even of the order of a single contraction in the porous medium). The mobility reduction gives access to the pressure losses in porous medium of given length when injected by a polymer solution at a flux J. In the next section we use the MP model to predict the degradation in porous medium of given length L.

Modeling polymer degradation

The experimental degradation (Eq.(3.36)) is calculated using the viscosities measured in simple shear for polymer solutions before and after injection through a porous medium at a given flux J.

The degradation calculated by Eq.(3.36) during the first passage Deg [J, L o ], is compared with the ratio of maximum of the mobility reductions R m max (J 2 , J 1 ) and R m max (0, J 1 ) at a flux J (This will be the flux J 1 at which the solution is collected after the first injection) 𝐃𝐞𝐠 [𝐉, 𝐋 𝐨 ] = 𝟓𝟕(𝐫 𝟐 ) 𝟐 -𝟏𝟓𝟕(𝐫 𝟐 ) + 𝟏𝟎𝟎,

where

r 2 = R m max (J 2 , J 1 ) R m max ( J 1 )
. Note that the Deg [J, L o ] = 0% for r 2 = 1 and Deg [J 1 , NL o ] = 100% for r 2 = 0. However, r 2 is never going to be zero in reality, even after the polymer is broken infinitely r 2 would be a non-zero number. This suggests the degradation is always less than 100%. Similarly degradation in a porous medium of length L = NL 0 , is calculated from (N + 1) th passage on the porous medium of length L 0 =1 mm(since N passages on porous medium of length L 0 is equivalent to one passage through a core of length L = NL 0 for a given flux, as in section 7.5) and as given, where r N+1 = R m max (J N+1 , J 1 ) R m max (0, J 1, )

.

The degradation Deg [J 1 , NL o ] for one injection on porous core of length NL o , single iteration (N=2 passages, Deg [J 1 , 2L o ] ) is given as,

The degradation calculated using Eq.(7.15) is compared with the experimental degradation; the predicted values deviate slightly. Note that degradation is calculated from The shear viscosity measured in Couette flow device; the shear viscosity is well-determined by the weight-averaged molecular weight M w , while the extensional stress that induces scission may be better determined by higher moments of the molecular weight such as M Z or M Z+1 . Nonetheless, the degradation trends are reasonably captured with this model. In the next section we extrapolate the MP-model to higher lengths both in linear and 2D radial injection conditions.

𝐃𝐞𝐠 [𝐉 𝟏 , 𝐍𝐋

𝐨 ] = 𝟓𝟕(𝐫 𝐍+𝟏 ) 𝟐 -𝟏𝟓𝟕(𝐫 𝐍+𝟏 ) + 𝟏𝟎𝟎, (7.15

Application in an industrial context

The influence of length on linear injection is a long-standing problem and, as mentioned in section 2.2.6, the loss in screen factor or equivalently degradation is related to length as Here, we use the MP model and find that degradation increases with increasing in length under linear injection conditions. However, degradation is most important for the first 30-40cms, after which a limiting value of degradation is reached.

For 2D radial injection conditions we find that degradation increases sharply during the first few cms and saturates beyond some particular length. The amount degraded under radial injection is lower for the same entrance flux than for linear injections. In addition, when the radius of the injector well is increased the degradation trends shift from those of the radial injecting conditions ( For linear injection (Figure 7.11) the flux is 1D and constant throughout the porous media, so all J i are equal (Let's say J). We extrapolate the MP-model to longer 1D porous media to understand the influence of length and flux on degradation. Degradation from a single passage on porous media of length L = NL o at a given flux J, is equivalent to degradation from N passages on porous medium of length L o , at the flux J. For calculating degradation we use Eqs.(7.6), (7.8) and (7.15). The combined equation is given as Eq. (7.6) is a power law function which approaches 1 only when the flux approaches zero (i.e. J = 0). However, for flux lower than the critical flux J c for the onset of shear thickening (i.e. J<J c ) polymers are not stretched, so should not undergo degradation. Odell et al. [START_REF] Odell | Degradation of polymer solutions in extensional flows[END_REF] suggested that polymers do not degrade in shear flow. Although shear thinning and transition to flow thickening is not observed clearly in our experiments, therefore we slightly modify Eq. (7.6) for N+1 passages at constant flux J 1 as, We suppose that there is a critical flux ratio 𝐉 𝐉 𝐍-𝟏 max = 0.05 below which there is no degradation. Here f N (0.05) = 1.006266, so the correction to the previous model is weak and difficult to measure. But this new expression allows the degradation to plateau at some point. The modified equation for calculating degradation on a porous medium of length l at a flux J is given as,

,(7.17)

Here J N max is obtained from f(

J 1 J N-1 max ) , J N-1 max is obtained from f( J 1
J N-2 max ) ……. and J 2 max is obtained from f( J 1 J 1 max ). In Figure 7.12, degradation calculated using Eqs.(7.21) is plotted as a function of length for different fluxes J. Degradation increases sharply for the first few mms, followed by a weaker increase with length. The degradation attains a quasi-saturate (or apparent steady state) degree of degradation for a given flux at critical length L c . We define saturation conditions when the degradation gradient is lower than 0.005%/mm. This finding contradicts the power law dependence proposed by [START_REF] Maerker | Shear degradation of partially hydrolyzed polyacrylamide solutions[END_REF]104,105 , explained in section 2.2.6. In Figure 7.13, we plot critical length L c as a function of flux; for a higher (lower) flux the quasi-saturation in degradation occurs at smaller (longer) lengths. The amount of degradation occurring at this critical length L c increases with increasing flux.

In the next step we extrapolate the MP model to radial injections (2D flux), to understand how the degradation evolves as a function of radius of the reservoir. We find that degradation is less pronounced in radial injections than in linear injection, and depends on the radius of the reservoir. Radial injections are close to the flow conditions on the reservoir field. To observe the degradation in radial injection we assume an injector well of height 12mm and radius R w . The entrance flux J (at the well bore surface) in 2D radial injections are adjusted to have the same flux as that in the linear injecting conditions. The flux is no longer constant at the successive radii. In the successive radii the flux J r , decreases inversely with the total radius R tot (= r + R w ) and is given as, 𝐉 𝐫 = 𝐉𝐑 𝐰 𝟐𝛑(𝐫 + 𝐑 𝐰 ) (7.20)

Radial injection

Here r is the distance traversed in the reservoir and R w is the injector well radius. Therefore, for successive passages (or successive radii) 𝑓 𝑁 is calculated as Note that we iterate on a length scales of porous core L 0 = 1.1mm, and N = r+L 0 L 0 . The flux at the maximum in N th pass is related to N-1 th pass, N-1 th pass is related to N-2 th pass and so on.

𝐉
We calculate degradation using Eq.(7.21), (7.8) and (7.15) in radial conditions for the same entrance flux J at the surface, as in linear conditions. The combined equation is given as, Note that the flux at the maximum (J N max ) in N th pass is related to N-1 th pass, N-1 th pass is related to N-2 th pass and so on.

In Figure 7.15 degradation calculated from Eq. (7.22) is plotted as function of radius r, traversed in the reservoir, for two different Injector well radii. For a smaller injector well radius R w , the flow conditions are purely radial (flux drops as an inverse function of total radius R tot (= r + R w ) as shown in Eq. (7.20). The amount of degradation is lower under radial condition in comparison to amount of degradation under linear conditions for the same injected flux. The degradation increases in the first few cms and attains a perfect saturation plateau; above which one observe no further degradation. The degradation is larger for a higher injector well radius for the same flux. For a larger injector well radius R w , the increase in degradation is sharper in comparison to the increase in degradation for a smaller injector radius R w . In Figure 7.16 we plot the critical radius r c to obtain degradation saturation under radial conditions as a function of injector well radius R w for 2 different fluxes. For increasing injector well radius the critical radius r c increases and asymptotes towards the linear injecting conditions. The degradation is plotted as a function of injector well radius in Figure 7.17, for different injection fluxes. The amount of degradation at saturation or critical radius r c in radial conditions (symbols) is smaller and increases with the increasing injector well radius. Finally, the degradation asymptotes towards the linear injecting conditions (solid lines). On the reservoir field the injector well, radii are of 30mm-200mm, which are thus dictated by radial injecting conditions (the flux drops inversely with the total radius of the reservoir traversed). Degradation is lower under radial conditions than that in linear injecting condition for the same entrance flux at the surface. For radial conditions degradation saturates within the first 5-10 cm, so it is important to implement clean practices for better efficiency of the polymer flooding. Notably, a high permeability at the entrance of the wellbore must be maintained, since the exponent a (see Eq. (7.2) and Figure 7.6), which controls degradation, may depend on the pore geometry. A lower permeability may result in lower exponent a and hence higher degradation. In Figure 7.18 we plot the degradation as a function of radius r, traversed in the reservoir, for varying scission exponent a (we believe that a lower permeability would result in lower scission exponent-thereby greater degradation. Likewise, higher permeability would result in higher scission exponent-thereby lower degradation). Decreasing the scission exponent a (lower permeability) would result in sharper increase in degradation in first few cm of porous medium. The critical radius r c , where the degradation saturates, occurs at smaller values for lower scission exponent a (lower permeability). The degradation at the critical radius r c is higher for lower scission exponent a (lower permeability). At this stage we discuss the limitations and requirements for controlled experiments to further explore the proposed MP model. In the current study the crossover between shear thinning at low shear rates and extensional thickening at higher rates are poorly captured. Due to the smaller thickness of the porous medium, measurements at low flow rates were limited by the sensitivity of the pressure sensors. Consequently, the critical flux J c assumed in the current study provides a good fit, but accurate values of J c in successive passages would give more reliable results.

Furthermore, a more appropriate measure of degradation could be developed instead of the shear viscosity, which is not as sensitive to the high molecular weight fraction responsible for scission and extensional hardening.

Conclusion

In this work we have studied degradation in porous cores of very short lengths (1-8 mm). We define the mobility reduction R m as the ratio of the pressure drops induced due to a polymer solution to that due to the solvent. The mobility reduction curve has a maximum at a given flux, which can be used to identify scission and model flow through composite media. We proposed a multi-pass "MP-model" to predict the mobility reduction curve of a porous medium of length NL o , as equivalent to performing N successive passages on a porous medium of length L o . This model was validated against porous media of different lengths L = NL o . The scission exponent 𝑎 in the double passages MP-model depends on the permeability; decreasing the permeability may decrease the exponent and increase the degradation.

For 1D linear injections the degradation is observed to increase with length and saturate at a critical length L c . The critical length decreases, and the degradation increases at this critical length with increasing flux. For 2D radial injections the flux drops inversely with the total radius traversed and so that the degradation is lower for the same entrance flux J at the surface as in the 1D injections. Upon increasing the injector well radius R w the trends of degradation and critical radius r c cross over from radial to linear injecting conditions. The approach to linear injection conditions occurs for r c R w ⁄ ≪ 1, for which the curvature of the radial conditions is negligible over the saturation radius.

In radial injecting conditions the saturation radius r c is 5-10cm. One important insight to be gained from this study is that clean pores are a necessity in order to decrease degradation and thereby increasing the efficiency of oil recovery. The various parameters of the model are dependent both on nature of the core (such as impurities, pore size, pore lengths) and of the polymer solution (e.g. molecular weight). Blocking of the core results would decrease the permeability. A decrease in permeability may result in lower scission exponent a, thereby increased degradation which results in a rapid drop of viscoelastic properties from a short distance from the injection point.

Chapter VIII -Conclusion and perspectives

Conclusions

The efficiency of oil recovery with polymer flooding is greatly limited by the mechanical degradation of polymers. In our work, we succeed in predicting the degradation of polymers.

For that purpose, we successively use various techniques that have allowed eventually building a quantitative model.

Firstly, single hyperbolic contraction geometry was developed using a microfluidics. In order to predict the polymer scission, we performed double passages of polymer solution through geometry. We succeed to develop a model that predicts scission independently of the rheological properties of the polymer solution. The parameter of the model depends on the Hencky strain of the geometry. From this knowledge, scission was tested on several contractions in series. The true Hencky strain imposed on the polymer chains, as revealed by the flow field observations, was observed to be smaller than that of the Hencky strain imposed by the geometry. We calculated the scission in successive contractions in series from the dependence of model to the true Hencky strain.

In the next step, the model was extended to porous media of different lengths. We show that this model can predict the degradation and the pressure drop between the entrance and exit for flow of a polymer solution through porous media of different lengths. The predictions agree well with experiments. Furthermore, we show that the degradation increases as a function of length. We predict and observe a critical length beyond which no further significant degradation was observed. The critical length for attaining an apparent saturation in degradation is decreasing function of the flux. Similarly, the amount of degradation plateaus at this critical length, with a value increasing with flux.

Lastly, the model was extended to radial flow, to mimic the actual reservoir conditions. We show, as previously, that the degradation is an increasing function of the distance travelled in the reservoir until it reaches a plateau value. For radial conditions, the degradation occurs for the first few cms of the distance travelled in the porous medium. Blocking of pores with impurities decreases the permeability of the reservoir, which results in higher degradation. Therefore, we highlight the importance of clean practices for better efficiency of oil recovery process with the polymer flooding.

Perspectives

We did not determine the precise nature of scission in our work. Such as how scission depends on the chain tension, chain length and flow. To answer to it, it would be interesting to compare the molar mass distributions of degraded samples with that of non-degraded samples in various geometry. This would allow us to better understand the scission mechanism and thereby implementing the molecular models explaining the scission of polymers.

Secondly, we have observed numerous complex velocity fields in the various geometries, but we did not measure them. This will be needed to better understand the extensional and shear contributions from the flow of polymer solution in contraction devices. The curve has a maximum for 𝑅′(Q pd max ) = 0, and the width of the curve is calculated from the second derivative of the above equation at the maximum, i.e. Width = R′′(Q pd max ).

In Fig iv) the width of the pressure drop ratio curves is plotted as a function of weight fraction of high molecular weight polymer in the binary solution. While the width of pressure drop ratio curve changes with the weight fractions of high and low molecular weight polymers in the solution. The polydispersity seems to have no significant influence on the width of the pressure drop ratio curve R′′(Q pd max ). We conclude that the critical Weissenberg number measured at the maximum of the pressure drop ratio for injection of a given polymer solution through geometries with different Hencky strain does not occur for a critical energy criteria. Note that the polydispersity is not considered in the current model and the calculations are for a single polymer chain. More realistic models are needed by considering many polymer chains (as in semi-dilute solutions) and the effects of polydispersity.

1 :

 1 Models, constitutive equations proposed for behaviour of non-Newtonian polymer solution. ................................................................................................................................... 21 Table 2.2: Radius of the coil R, relaxation time tr and specific viscosity scalings with Molecular weight and concentration in the three concentration regimes (dilute, semi-dilute non entangles and semi-dilute entangled) for a neutral polymer in good and θ-solvent. Adapted from Colby (2010) 4 . ................................................................................................................. 24 Table

  ......................................................................................... 26 Table

  . .................................................... 33 Table 2.6: literature review on the effect of the molecular weight distribution on the flow induced scission. ...................................................................................................................... 34 Table 3.1: Molecular weights of native polymers of HPAM. In all cases the polydispersity is 𝑀𝑤/𝑀𝑛~4 -5. ...................................................................................................................... 37 Table 3.2: Salt composition for 0.6% (w/w) salt solution. Salts are obtained from Sigma Aldrich .................................................................................................................................................. 38 Table 3.3: Intrinsic viscosity and for 19MDa polymer in three different solvents. ................. 40 Table

  ................................................................................. 52 Table 4.1: Nomenclature for polymer solutions used. Here 𝑛 is the shear thinning index, and tca is the relaxation time measured from the CaBER. c* is the critical (overlap) concentration that separates the dilute and semi-dilute regimes and is calculated as the inverse of the intrinsic viscosity. ..................................................................................................................... 66 Table 4.2: The seven flow regimes (as in Figure 4.3) observed for different polymer solutions injected through different geometries with different Hencky strains 𝜀𝐻. The Symbols identify the flow regimes for the corresponding Weissenberg corresponding Reynolds numbers and match the data plotted in Figure 4.8. ...................................................................................... 79 Table

Figure 1 .

 1 Figure 1.1: Three stages of oil recover. Namely, primary, secondary and tertiary process 1 . 11 Figure 2.1: Schematic representation of uniaxial deformation shrinking in Y and Z directions and stretching in the X-direction, equi-biaxial deformation where shrinking takes place in the Z direction and expands in X & Y directions, planar extensions where the material shrinks in the Y direction and expands in the Z direction. ....................................................................... 17Figure 2.2: (left) viscosity as a function of shear rate, depicting shear thinning behavior in shear flows. (Right) viscosity as a function of strain rate, depicting extensional thickening behavior in extensional flows. ................................................................................................. 22 Figure 2.3: Different flow regimes on Weissenberg number (Wi) -Reynolds number (Re) plot from Rodd et al. 43 .................................................................................................................... 28 Figure 2.4: Different contraction geometries: axisymmetric and square-square contraction geometries (3D, uniaxial extension is experienced), planar abrupt and hyperbolic contraction devices (Planar extension). ...................................................................................................... 30 Figure 3.1: Partially hydrolyzed polyacrylamide (HPAM) ........................................................ 37 Figure 3.2: Viscosity of a polymer solution and solvent as a function of shear rate obtained using the low shear rheometer, Solid line: Carreau model fit to obtain the polymer relaxation time, 𝑡𝑐𝑚. ................................................................................................................................ 39 Figure 3.3: Specific viscosity as a function of concentration for molecular weight 19MDa in three different solvents. Scaling 1 denotes the dilute regime, while scaling 1.3 denotes the semi-dilute unentangled regime. Inset: reduced viscosity as a function of concentration. .. 40 Figure 3.4: The HAAKE CaBER extensional rheometer. Schematic illustration of the stages of filament evolution in a CaBER experiment. ............................................................................. 42 Figure 3.5: Diameter evolution as a function of time t for a polymer solution 19MDa-0.08%-40SG. ........................................................................................................................................ 43 Figure 3.6: Filament diameter 𝐷𝑚𝑖𝑑, as a function of time for 19MDa-0.08-40/60 glycerol/water, polymer solution. Red solid line is upper convected Maxwell model as mentioned in Eq.(3.6). ............................................................................................................. 44 Figure 3.7: Strain rate calculated from Eq.(3.8)as a function of Hencky strain of the geometry εH geo of CaBER as in Eq. (3.11) and the appropriate Hencky strain εH calculated from the Eq. (3.12. ........................................................................................................................................ 46 Figure 3.8: Coordinates and dimensions of the hyperbolic geometry. The red dotted zone shows corner vortices, the black lines denote the boundaries of the cell, and the blue solid line is the boundary of the flow of polymer solution. ............................................................. 46 Figure 3.9: Fluorescent images obtained from the Nikon microscope. A single image is obtained from images captured at different locations (red dotted squares are of dimension 6 mm x 6 mm, seven such images are stitched to get one complete image). 15 such images at different time periods (∆t≈ 𝑚𝑠) are superimposed to obtain the final images. .................... 49 Figure 3.10: Single injection device including the injection system, the tubing, the micromodel, the collecting bottle and pressure sensors at entrance and exit. ............................... 50 Figure 3.11: Typical pressure drop 𝛥𝑃𝑝(𝑄) for a polymer solution and solvent, 𝑃𝑠(𝑄) when injected through hyperbolic contraction geometry with Hencky strain 3.2*(refer Table 3.5). The maximum Reynolds numbers are 𝑅𝑒𝑠150, 𝜂𝑠 = 250, 𝑅𝑒𝑝150, 𝜂𝑝 = 16. ..................... 51 Figure 3.12: Pressure drop ratio curve of a given polymer solution when injected through hyperbolic contraction geometry with Hencky strain 3.2*(refer Table 3.5). The maximum occurs of at pressure ratio 𝑅𝑚𝑎𝑥 and flow rate 𝑄max. .......................................................... 51

1 : 11 Figure 2 . 1 :

 11121 Figure 1.1: Three stages of oil recover. Namely, primary, secondary and tertiary process 1 . 11 Figure 2.1: Schematic representation of uniaxial deformation shrinking in Y and Z directions and stretching in the X-direction, equi-biaxial deformation where shrinking takes place in the Z direction and expands in X & Y directions, planar extensions where the material shrinks in the Y direction and expands in the Z direction. ....................................................................... 17Figure 2.2: (left) viscosity as a function of shear rate, depicting shear thinning behavior in shear flows. (Right) viscosity as a function of strain rate, depicting extensional thickening behavior in extensional flows. ................................................................................................. 22 Figure 2.3: Different flow regimes on Weissenberg number (Wi) -Reynolds number (Re) plot from Rodd et al. 43 .................................................................................................................... 28 Figure 2.4: Different contraction geometries: axisymmetric and square-square contraction geometries (3D, uniaxial extension is experienced), planar abrupt and hyperbolic contraction devices (Planar extension). ...................................................................................................... 30 Figure 3.1: Partially hydrolyzed polyacrylamide (HPAM) ........................................................ 37 Figure 3.2: Viscosity of a polymer solution and solvent as a function of shear rate obtained using the low shear rheometer, Solid line: Carreau model fit to obtain the polymer relaxation time, 𝑡𝑐𝑚. ................................................................................................................................ 39 Figure 3.3: Specific viscosity as a function of concentration for molecular weight 19MDa in three different solvents. Scaling 1 denotes the dilute regime, while scaling 1.3 denotes the semi-dilute unentangled regime. Inset: reduced viscosity as a function of concentration. .. 40 Figure 3.4: The HAAKE CaBER extensional rheometer. Schematic illustration of the stages of filament evolution in a CaBER experiment. ............................................................................. 42 Figure 3.5: Diameter evolution as a function of time t for a polymer solution 19MDa-0.08%-40SG. ........................................................................................................................................ 43 Figure 3.6: Filament diameter 𝐷𝑚𝑖𝑑, as a function of time for 19MDa-0.08-40/60 glycerol/water, polymer solution. Red solid line is upper convected Maxwell model as mentioned in Eq.(3.6). ............................................................................................................. 44 Figure 3.7: Strain rate calculated from Eq.(3.8)as a function of Hencky strain of the geometry εH geo of CaBER as in Eq. (3.11) and the appropriate Hencky strain εH calculated from the Eq. (3.12. ........................................................................................................................................ 46 Figure 3.8: Coordinates and dimensions of the hyperbolic geometry. The red dotted zone shows corner vortices, the black lines denote the boundaries of the cell, and the blue solid line is the boundary of the flow of polymer solution. ............................................................. 46 Figure 3.9: Fluorescent images obtained from the Nikon microscope. A single image is obtained from images captured at different locations (red dotted squares are of dimension 6 mm x 6 mm, seven such images are stitched to get one complete image). 15 such images at different time periods (∆t≈ 𝑚𝑠) are superimposed to obtain the final images. .................... 49 Figure 3.10: Single injection device including the injection system, the tubing, the micromodel, the collecting bottle and pressure sensors at entrance and exit. ............................... 50 Figure 3.11: Typical pressure drop 𝛥𝑃𝑝(𝑄) for a polymer solution and solvent, 𝑃𝑠(𝑄) when injected through hyperbolic contraction geometry with Hencky strain 3.2*(refer Table 3.5). The maximum Reynolds numbers are 𝑅𝑒𝑠150, 𝜂𝑠 = 250, 𝑅𝑒𝑝150, 𝜂𝑝 = 16. ..................... 51 Figure 3.12: Pressure drop ratio curve of a given polymer solution when injected through hyperbolic contraction geometry with Hencky strain 3.2*(refer Table 3.5). The maximum occurs of at pressure ratio 𝑅𝑚𝑎𝑥 and flow rate 𝑄max. .......................................................... 51

Figure 2 . 2 : 22 Figure 2 . 3 : 28 Figure 2 . 4 : 39 Figure 3 . 3 : 40 Figure 3 . 4 : 42 Figure 3 . 5 : 43 Figure 3 . 6 : 44 Figure 3 . 7 : 46 Figure 3 . 8 : 46 Figure 3 . 9 : 49 Figure 3 . 10 : 50 Figure 3 . 11 : 51 Figure 3 . 13 : 53 Figure 3 . 14 : 53 Figure 3 . 15 : 54 Figure 3 . 16 : 55 Figure 3 . 17 : 56 Figure 3 . 20 : 60 Figure 3 . 23 :

 22222328243933403442354336443746384639493105031151313533145331554316553175632060323 Figure 1.1: Three stages of oil recover. Namely, primary, secondary and tertiary process 1 . 11 Figure 2.1: Schematic representation of uniaxial deformation shrinking in Y and Z directions and stretching in the X-direction, equi-biaxial deformation where shrinking takes place in the Z direction and expands in X & Y directions, planar extensions where the material shrinks in the Y direction and expands in the Z direction. ....................................................................... 17Figure 2.2: (left) viscosity as a function of shear rate, depicting shear thinning behavior in shear flows. (Right) viscosity as a function of strain rate, depicting extensional thickening behavior in extensional flows. ................................................................................................. 22 Figure 2.3: Different flow regimes on Weissenberg number (Wi) -Reynolds number (Re) plot from Rodd et al. 43 .................................................................................................................... 28 Figure 2.4: Different contraction geometries: axisymmetric and square-square contraction geometries (3D, uniaxial extension is experienced), planar abrupt and hyperbolic contraction devices (Planar extension). ...................................................................................................... 30 Figure 3.1: Partially hydrolyzed polyacrylamide (HPAM) ........................................................ 37 Figure 3.2: Viscosity of a polymer solution and solvent as a function of shear rate obtained using the low shear rheometer, Solid line: Carreau model fit to obtain the polymer relaxation time, 𝑡𝑐𝑚. ................................................................................................................................ 39 Figure 3.3: Specific viscosity as a function of concentration for molecular weight 19MDa in three different solvents. Scaling 1 denotes the dilute regime, while scaling 1.3 denotes the semi-dilute unentangled regime. Inset: reduced viscosity as a function of concentration. .. 40 Figure 3.4: The HAAKE CaBER extensional rheometer. Schematic illustration of the stages of filament evolution in a CaBER experiment. ............................................................................. 42 Figure 3.5: Diameter evolution as a function of time t for a polymer solution 19MDa-0.08%-40SG. ........................................................................................................................................ 43 Figure 3.6: Filament diameter 𝐷𝑚𝑖𝑑, as a function of time for 19MDa-0.08-40/60 glycerol/water, polymer solution. Red solid line is upper convected Maxwell model as mentioned in Eq.(3.6). ............................................................................................................. 44 Figure 3.7: Strain rate calculated from Eq.(3.8)as a function of Hencky strain of the geometry εH geo of CaBER as in Eq. (3.11) and the appropriate Hencky strain εH calculated from the Eq. (3.12. ........................................................................................................................................ 46 Figure 3.8: Coordinates and dimensions of the hyperbolic geometry. The red dotted zone shows corner vortices, the black lines denote the boundaries of the cell, and the blue solid line is the boundary of the flow of polymer solution. ............................................................. 46 Figure 3.9: Fluorescent images obtained from the Nikon microscope. A single image is obtained from images captured at different locations (red dotted squares are of dimension 6 mm x 6 mm, seven such images are stitched to get one complete image). 15 such images at different time periods (∆t≈ 𝑚𝑠) are superimposed to obtain the final images. .................... 49 Figure 3.10: Single injection device including the injection system, the tubing, the micromodel, the collecting bottle and pressure sensors at entrance and exit. ............................... 50 Figure 3.11: Typical pressure drop 𝛥𝑃𝑝(𝑄) for a polymer solution and solvent, 𝑃𝑠(𝑄) when injected through hyperbolic contraction geometry with Hencky strain 3.2*(refer Table 3.5). The maximum Reynolds numbers are 𝑅𝑒𝑠150, 𝜂𝑠 = 250, 𝑅𝑒𝑝150, 𝜂𝑝 = 16. ..................... 51 Figure 3.12: Pressure drop ratio curve of a given polymer solution when injected through hyperbolic contraction geometry with Hencky strain 3.2*(refer Table 3.5). The maximum occurs of at pressure ratio 𝑅𝑚𝑎𝑥 and flow rate 𝑄max. .......................................................... 51

4 ) 68 Figure 4 . 4 : 71 Figure 4 . 5 : 71 Figure 4 . 6 : 72 Figure 4 . 7 : 84 Figure 5 . 3 : 85 Figure 5 . 5 : 85 Figure 5 . 6 : 87 Figure 5 . 7 : 87 Figure 5 . 8 :

 4684471457146724784538555855687578758 Stable asymmetric corner vortex growth (green). 5) Time-dependent corner vortex growth (orange). 6) Stable-singular corner vortex (yellow) .7) Chaotic/Viscoelastic instabilities (red). .................................................................................................................................................. Dimensionless vortex size 𝜒𝑣𝑠 (2.31) as a function of Deborah number, for an 11MDa-0.08/40SG solution at different Hencky strains as mentioned in Table 3.6 ............... Modified dimensionless vortex size 𝜉𝑣𝑠 (4.2) as a function of Deborah number, for an 11MDa-0.08/40SG solution at different Hencky strain as mentioned in Table 3.6.......... Dimensionless parameter 𝜒𝑣𝑠 as a function of Weissenberg number, For a 15MDa-0.08/40SG solution in geometries with heights ranging from 0.05mm -0.2mm. ................... Dimensionless parameter 𝜒𝑣𝑠 as calculated in normalized by ℎ as a function of Weissenberg number, when 15MDa-0.08/40SG is injected through geometries with heights ranging from 50µm -200µm..................................................................................................... 75 Figure 4.8: Wi-Re plot for different hyperbolic contraction-expansion geometry and when injected with polymer solutions with different rheological properties. Each line corresponds to a given elastic number El as given in Eq.(2.36). The symbols are explained in Table 4.2. .. 79 Figure 5.1: Pressure drop ratio curve for varying molecular weight (Mw) in 40/60 glycerol/water mixture; the maximum Reynolds numbers are 𝑅𝑒𝑝175, 19𝑀𝐷𝑎 = 19.5, 𝑅𝑒𝑝200, 7𝑀𝐷𝑎 = 38. ............................................................................................................. 83 Figure 5.2: Pressure drop ratios for varying molecular weight (Mw) in a 40/60 glycerol/water mixture and flow rates normalized by their respective maxima. ............................................ Molecular weight dependences of R max and Q max in 40/60 glycerol-water. The straight lines have the slopes indicated. ................................................................................. 84 Figure 5.4: concentration dependences of R max and Q max in 40/60 glycerol-water. The straight lines have the slopes indicated. ............................................................................................... Experimentally observed vortices in a contraction geometry, where Lvs is the length of a vortex. .................................................................................................................... Comparison of maximum extension rate as a function of polymer relaxation time for all samples (with all three solvents). The data are fit by a critical Weissenberg number Wi𝑐 = 16±2. 25. ...................................................................................................................... Pressure drop ratios normalized by their respective maxima as a function of Weissenberg number, Wi. True strain rates are used for calculating Weissenberg number (where the vortex length is taken into consideration, as in Eq. 14) and this plot corresponds to a polymer concentration of 0.08%. ..................................................................................... Pressure drop ratio curves of polymer solutions for a second pass at flow rate Q2 after pre-degradation from a first injection at flow rate Q1. The first injection is shown as symbols along the upper abscissa. .......................................................................................... 89 Figure

90 Figure 5 . 10 : 90 Figure 5 . 11 : 92 Figure 5 . 12 : 92 Figure 5 . 13 :

 90510905119251292513 ...................................................................................... Universal curve for the flow rate Q2 max (respectively strain rate 𝜀𝑇2𝑚𝑎𝑥) of the maximum of pressure drop ratio, obtained in a second passage after a first passage at Q1 (or 𝜀𝑇1). Both are normalized by the flow rate (or strain rate) at the maximum pressure drop ratio of the first passage. Filled and unfilled symbols correspond to flow rate and strain rate respectively. The solid line represents Eq. (5.7) with scission exponent a=2.5. Uncertainties are shown only for the 19MDaa-0.08/40SG sample. Blue triangles are the pressure drop ratio curve as a function of normalized flow rate of the same sample and the blue dashed line is a guide to the eye. ...................................................................................................................... Apparent outlet molecular weight as a function of injected molecular weight at different flow rates Q1, as given by Eq.(5.10). ......................................................................... Pressure drop ratio curves of four passes, pre-degraded during every injection at same flow rate 𝑄1 = 𝑄2 = 𝑄3 = 𝑄 = 33.3 µ𝐿/s. ................................................................. Evolution of the maximum for the data of Figure 5.12 of a given passage with respect to the maximum of the preceding passage QN max /QN-1 max , for passages 1, 2, 3, and 4. The model shows a fit to Eq. (5.7) with scission exponent a=2.5. .......................................... 93 Figure

94 Figure 6 . 1 :

 9461 14) for N=2, 3, 4 with scission exponent a=2.5. ........................................................................................... Pressure drop ratio R as a function of flow rate Q, for a 11Mda-0.08/40SG polymer solution through geometries with different Hencky strain (

Figure 4 . 3 ) 99 Figure 6 . 4 :Figure 6 . 6 :Figure 6 . 7 :

 4399646667 for 𝜀𝐻 = 2.3 𝑎𝑛𝑑 1.8; Stable asymmetric corner vortex regime (4 th flow regime of Figure 4.3) for 𝜀𝐻 = 1.4. ..................................................................................................... Pressure drop ratio curve as function of flow rate Q for polymers with different molecular weight, with polymer concentration of 0.08% (w/w) in 40/60glycerol/water solvent with solvent viscosity 𝜂𝑠 ≅ 3𝑐𝑃, in a [(5-0.5)/2.6]0.2 geometry with Hencky strain 𝜀𝐻 = 2.3. ................................................................................................................................................ 100 Figure 6.5: Normalized pressure drop ratio 𝑅𝑅𝑚𝑎𝑥 as a function of normalized flow rate 𝑄𝑄𝑚𝑎𝑥 for polymer solutions of different molecular weights fixed polymer concentration of 0.08% (w/w) in 40/60glycerol/water solvent with solvent viscosity 𝜂𝑠 ≅ 3𝑐𝑃, in a [(5-0.5)/2.6]0.2 with Hencky strain 𝜀𝐻=2.3. ................................................................................. 100 Strain rates as function of Hencky strain 𝜀𝐻 measured from filament thinning dynamics of the CaBER experiment, and the critical strain rate at the maximum(cross over) 𝜀𝑇𝑚𝑎𝑥 as a function of Hencky strain 𝜀𝐻 measured from the pressure drop ratio curve for injection of a 11MDa-0.08/40SG through microfluidic geometries. ..................................... 101 Strain rate at the maximum in pressure drop ratio 𝜀𝑇𝑚𝑎𝑥 as a function of relaxation time measured from the CaBER 𝑡𝑐𝑎.( Note that there is no degradation during the filament thinning dynamics of CaBER measurements) ......................................................... 102 Figure 6.8: Critical Weissenberg number calculated from tca at maximum of a given pressure drop ratio R as a function of Hencky strain 𝜀𝐻. ..................................................................... 102 Figure

Figure 6 . 26 :Figure 7 . 14 :

 626714 Figure 6.26: Geometrical constant k as a function of Hencky strain obtained from the slopes of Figure 6.26. ........................................................................................................................ 118 Figure 6.27: 𝑄2𝑚𝑎𝑥𝑄1, 𝑁𝑐 calculated from equation (6.15), experimental (green triangle) and true values (red star) calculated from Eq. (6.22) considering the true Hencky strain in the successive contractions. Here, 𝑄1=83.33µL/s. ...................................................................... 119 Figure 7.1: Mobility reduction curves Rm calculated using Eq. (3.34) as a function of flux for injecting a 19MDa-0.08/0SG solution through porous media of different lengths. The pink circles correspond to the maxima (Jmax, Rmmax) of each mobility reduction curve. ........ 122 Figure 7.2: Normalized mobility reduction 𝑅𝑚𝑅𝑚𝑚𝑎𝑥 as a function of normalized flux 𝐽𝐽𝑚𝑎𝑥 . ................................................................................................................................... 123 Figure 7.3: Degradation calculated using Eq.(3.36) as a function of flux J in porous media of length 1, 2, 3 and 8 mm respectively. ................................................................................... 123 Figure 7.4: Mobility reduction curves of 19MDa-0.08/0SG solutions re-injected through the 1mm-long core after a first injection at flux J1 through the same core. J1 is marked along the upper abscissa. ....................................................................................................................... 124 Figure 7.5: Normalized mobility reduction 𝑅𝑚𝑅𝑚𝑚𝑎𝑥 as function of normalized flux 𝐽𝐽𝑚𝑎𝑥 for the first pass and the reinjected solutions on the porous medium of length L0=1 mm. The black solid line is a fit to Eq.(7.1) with fitting parameters 𝑛2=1.45, 𝑛3=1.25, 𝜏2=3, 𝜏3=0.0075. ................................................................................................................................................ 124 Figure 7.6: Ratio of the fluxes J2max / J1max at which mobility reduction is maximum during the first and the second pass on the 1mm-long core as a function the ratio J1/ J1max where J1 is the flux at which the solution was collected in the first pass. Experimental points are the open symbol. The solid line is the best fit of Eq. (7.2) by taking 𝑎 =1.85. ............................. 127 Figure 7.7: Evolution of 𝐑𝐦𝐦𝐚𝐱 as a function of 𝐉𝐦𝐚𝐱 (from the 2 passes on porous core of length 𝐋𝟎 = 𝟏𝐦𝐦 ). .............................................................................................................. 127 Figure 7.8: Predicted mobility reductions MP -RmJ 1 , NL0 (represented by solid lines) and experimental mobility reduction (represented by symbols) on porous medium of lengths L =1 mm, 2 mm, 3 mm, 4 mm, 6 mm, 8 mm as a function of flux. Here N is the number of passages on porous medium of length L0=1.1. .................................................................................... 130 Figure 7.9: Degradation during the first passage on porous medium of length 1 mm, as a function of ratio of maximum of the mobility reduction during the second passage 𝑅𝑚𝑚𝑎𝑥 (𝐽 2, 𝐽 1), to that during the first passage 𝑅𝑚𝑚𝑎𝑥 (0, 𝐽 1).The solid blue line is the fit calculated using Eq.(7.14). ..................................................................................................... 131 Figure 7.10: Degradation as a function of flux J for different lengths of porous media. Symbols corresponds to the experiments and solid lines are calculated using Eq.(7.15) for lengths equal to N*1mm, where N is the number of passages through a porous medium of length L0=1 mm. ................................................................................................................................................ 132 Figure 7.11: Illustration of linear injection experiment on porous medium of length L. Flux J is constant all along the length of the core. The scission zone (where polymer undergoes scission) corresponds to first few cms. .................................................................................. 133 Figure 7.12: Degradation as a function of length for different flux for a flow of polymer solution in 1D linear flows. In the inset the same graph is plotted on a linear scale, which highlights that degradation occurs sharply in the first few mms and then a gradual increase in the degradation. .................................................................................................................... 134 Figure 7.13: Critical length 𝐿𝑐 and degradation at critical length 𝐿𝑐 as a function of surface flux J. 𝐿𝑐 is the critical length at which degradation attains the quasi-saturation in degradation. ................................................................................................................................................ 135

Figure 1 . 1 :

 11 Figure 1.1: Three stages of oil recover. Namely, primary, secondary and tertiary process 1 .

Figure 2 . 1 :

 21 Figure 2.1: Schematic representation of uniaxial deformation shrinking in Y and Z directions and stretching in the X-direction, equi-biaxial deformation where shrinking takes place in the Z direction and expands in X & Y directions, planar extensions where the material shrinks in the Y direction and expands in the Z direction. The extensional stress tensor is given by

Figure 2 . 2 :

 22 Figure 2.2: (left) viscosity as a function of shear rate, depicting shear thinning behavior in shear flows. (Right) viscosity as a function of strain rate, depicting extensional thickening behavior in extensional flows.

3 :

 3 𝑊 𝑢 is the entrance width of the geometry, 𝑊 𝑐 is the width at the throat of the contraction (Refer Figure 3.8), h is the height of the channel, 𝑡 𝑟 is the relaxation time of the polymer chain, 𝑡 𝑟𝑒𝑠 is the residence time, 𝐷 ℎ is the hydraulic diameter given by 𝐷 ℎ = 2ℎ𝑊 𝑐 (ℎ+𝑊 𝑐 ) , 𝐿 𝑣𝑠 is the length of the vortices, The geometrical constant 𝐷 𝑔𝑒𝑜 = (𝑊 𝑐 -𝑊 𝑢 ) 𝐿 𝑐 𝑊 𝑢 and G is the modulus.

Figure 2 . 3 :

 23 Figure 2.3: Different flow regimes on Weissenberg number (Wi) -Reynolds number (Re) plot from Rodd et al.[START_REF] Rodd | Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries[END_REF]

Figure 2 . 4 :

 24 Figure 2.4: Different contraction geometries: axisymmetric and square-square contraction geometries (3D, uniaxial extension is experienced), planar abrupt and hyperbolic contraction devices (Planar extension).Axisymmetric contractions, 3D square-square contractions and planar contractions have been used to study the formation of vortices. The first two types of devices implement uniaxial extension while the planar contractions have a planar extension kinematics. The different flow regimes and their growth dynamics cannot be described only by the Weissenberg, Deborah and Reynolds numbers. Other parameters such as the contraction ratio β, aspect ratio α, polymer flexibility, extension and shear viscosities are also important. The transition from one type of flow regime to other has not resulted in any addition changes pressure drop measurements. In the next section we discuss the polymer scission problem in extensional flows.

3 .

 3 There have been few systematic studies on the influence of the length of the porous medium L D , on degradation in linear injections. Marker et al. proposed for high flow rates of injection, loss of screen factor scales with dimensionless parameter of length as ε(L D ) m , where m = 1 Later, Morris et al. proposed that this exponent m is dependent on the molecular weight of the polymer injected and therefore is equal to m=𝑀 * 10 -7 where M is the molecular weight of the polymer in Daltons 104, 105 . In studies on porous media on length scales of a few cms Marie et al. observed m =0.07 for HPAM polymers and 0.03 for ATBS polymers 105 . Jouenne et al.

Figure 3 . 1 :

 31 Figure 3.1: Partially hydrolyzed polyacrylamide (HPAM) Glycerol and ethanol of analytical grade, oligomeric poly acrylamide (PAM) with M 𝑤 = 500kDa were obtained from Sigma Aldrich.

Figure 3 . 2 :

 32 Figure 3.2: Viscosity of a polymer solution and solvent as a function of shear rate obtained using the low shear rheometer, Solid line: Carreau model fit to obtain the polymer relaxation time, 𝑡 𝑐𝑚 .

Figure 3 . 3 :

 33 Figure 3.3: Specific viscosity as a function of concentration for molecular weight 19MDa in three different solvents. Scaling 1 denotes the dilute regime, while scaling 1.3 denotes the semi-dilute unentangled regime. Inset: reduced viscosity as a function of concentration. 𝛈 𝐫 = [𝛈] + 𝐤 𝐇 [𝛈] 𝟐 𝐜, (3.3) which defines the intrinsic viscosity [𝜂] and the Huggins parameter 𝑘 𝐻 , which originates from polymer-polymer interactions. Figure 3.3 shows the specific and reduced viscosities as a function of concentration for molecular weight 19MDa (Table3.3).

Figure 3 .

 3 Figure 3.3 shows two expected concentration scalings of the specific viscosity with concentration. η sp ∝ cM w 0.7 in the dilute regime where relaxation dynamics best follow the Zimm model 110 , while the Rouse model (η sp ∝ c 1.3 M w 2 for good solvents, η sp ∝ c 2 M w 2 for

Figure 3 . 4 ( 5 ,

 345 Thermo Fisher Scientific, Karlsruhe, Germany) in Clasen laboratory. The capillary extension break rheometer consists of 2 circular plates with a diameter Dcp=6mm.The circular plates are initially separated at a distance of d0 which makes the initial aspect ratio of filament Λ f fluid sample is placed between the circular plates, and care is taken to avoid the sample sagging from the plates. The sample takes the form of a slender cylindrical bridge with a diameter equivalent to the diameter of circular plates. The top plate is then swiftly pulled upward to a final aspect ratio ofΛ f = h f D cp = 1.3 (in a time span of 50ms). A laser micrometerand a high-speed camera are placed in-line to monitor the evolution of filament diameter over time. We have followed the methodology proposed by Clasen et al., to correct the lasermicrometer readout after the filament was broken, this ensures accuracy in filament diameter data measured up to few micrometers.

Figure 3 . 4 :

 34 Figure 3.4: The HAAKE CaBER extensional rheometer. Schematic illustration of the stages of filament evolution in a CaBER experiment.

Figure 3 . 5 :

 35 Figure 3.5: Diameter evolution as a function of time t for a polymer solution 19MDa-0.08%-40SG. The filament diameter thinning dynamics is explained by Clasen et.al. in his studies on CaBER measurements of polystyrene (PDI in diethyl phthalate solvent 113 , which illustrates four distinct regimes of filament thinning.

Figure 3 . 6 :

 36 Figure 3.6: Filament diameter 𝐷 𝑚𝑖𝑑 , as a function of time for 19MDa-0.08-40/60 glycerol/water, polymer solution. Red solid line is upper convected Maxwell model as mentioned in Eq.(3.6). The relaxation times from Carreau model (shear rheology) and CaBER measurements (extensional rheology) of different polymer solutions given in Table 3.4.

2 3 t

 23 ca . The diameter at the onset of extensional hardening regime D onset , would provide more appropriate strain accumulated by the polymer chain. Thus, the appropriate Hencky strain is given as,

Figure 3 . 8 :

 38 Figure 3.8: Coordinates and dimensions of the hyperbolic geometry. The red dotted zone shows corner vortices, the black lines denote the boundaries of the cell, and the blue solid line is the boundary of the flow of polymer solution.

Table 3 . 5 :

 35 List of geometries used in the current study and their dimensionsHere we describe the flow characteristics of the hyperbolic contraction geometry shown in Figure3.8. We consider fluid moving with velocity uz (x, y, z) along the Z direction, with width W(z) along X and height h along Y. For a given geometry the extension rate can be estimated as follows.The imposed uniform volume flux Q in a given geometry, is given by𝐐 = 𝐖(𝐳) 𝐡 𝐮 ̅(𝐳),(3.13)where 𝐮 ̅(𝐳) = ∬ 𝐮 𝐳 (𝐱,𝐲,𝐳) 𝐝𝐱 𝐝𝐲 𝐖(𝐳)𝐡 , (3.14)is the average velocity per area at a given z. The average extensional rate or the equivalent rate of stretching is given by 𝛆̅(

  The residence time τres, i.e. the time during which a given polymer remains in the contraction, is the ratio of the volume V of the contraction to the flow rate Q

Figure 3 . 9 :

 39 Figure 3.9: Fluorescent images obtained from the Nikon microscope. A single image is obtained from images captured at different locations (red dotted squares are of dimension 6 mm x 6 mm, seven such images are stitched to get one complete image). 15 such images at different time periods (∆t≈ 𝑚𝑠) are superimposed to obtain the final images.

Figure 3 . 11 :

 311 Figure 3.11: Typical pressure drop 𝛥𝑃 𝑝 (𝑄) for a polymer solution and solvent, 𝑃 𝑠 (𝑄) when injected through hyperbolic contraction geometry with Hencky strain 3.2*(refer Table 3.5). The maximum Reynolds numbers are 𝑅𝑒 𝑠 (150, 𝜂 𝑠 ) = 250, 𝑅𝑒 𝑝 (150, 𝜂 𝑝 ) = 16.

  4 and calculate the ratio of the pressure drops between polymer solution and pure solvent at the same flow rate, 𝐑(𝐐) = ∆𝐏 𝐏 (𝐐) 𝚫𝐏 𝐬 (𝐐)

Figure 3 . 13 :

 313 Figure 3.13: Pressure drop 𝛥𝑃 𝑠 as a function of a) flow rate, b) Reynold's number, when injected 40/60 Glycerol/water solvent with viscosity 𝜂 𝑠 ≅ 3𝑐𝑃 through geometries with different Hencky strain.Later polymer solution, for instance 11Mda-0.08/40SG is injected at different flow rates through the geometries with varying Hencky strain, and the pressure drop ΔP p (Q) as function of flow rate is plotted in Figure3.14.

Figure 3 . 14 :

 314 Figure 3.14: Pressure drop 𝛥𝑃 𝑝 as a function of flow rate when injected with viscosity 11Mda-0.08/40SG η p ≅ 22cP through geometries with different Hencky strains.

Figure 3 . 15 :

 315 Figure 3.15: Flow lines at the exit of contraction when water is injected through geometry 𝜀 𝐻 = 2.3 at a) Re=25, the flow is laminar. Deviation from the laminar flow are observed at b)Re=53, c) Re=72 d) Re=90.

Figure 3 . 16 :

 316 Figure 3.16: Double injection technique (Steps 1 and 2). The blue solid curve is the first injection pressure drop ratio curve R(Q1) with maximum at Q1 max . The orange dotted curve is the second injection R (Q2; Q1) with maximum at Q2 max .

Figure 3 . 18 :

 318 Figure 3.18: Contractions in series entrance width, Wu = 5mm, contraction width Wc = 0.5mm, contraction length Lc = 2.6mm, expansion length Ld = 1.3mm height h = 0.2mm, distance between successive contractions LG = 5mm. Several contractions are placed in series without changing the complete length of the geometry and the dimensions of the contraction. The distance between any 2 successive contractions is LG =5mm. In the first step the solvent 40/60 glycerol/water is injected at different flow rates in the different geometries. In Figure 3.19 we plot the pressure drop due to solvent as a function of flow rate. The slope of this plot increases by almost a constant value when replacing a rectangular portion of the geometry with an additional contraction.The 19MDa-0.08/40SG polymer solution is injected through the geometries and the pressure drop ratio curves R (Q1, MC) are plotted as a function of flow rate Q. From this data we observed increase in scission with increase in number of contractions in series.

Figure 3 . 19 :

 319 Figure 3.19: Pressure drop of solvent ΔPs as a function of flow rate Q (ml/min) for different geometries with contractions in series.

Figure 3 .

 3 Figure 3.21: (on left) Gas pycnometer from MICROMETRICS. (On right) working principle of Gas pycnometer. We use N2 as the displacement medium. Two absolute pressure sensors are in the pycnometer one at the reference chamber and one at the sample chamber. The volume of reference cell is Vr and the volume of the empty sample chamber is Vs.

Figure 3 . 22 :

 322 Figure 3.22 : Experimental set up comprising ISCO pump, PEEK holder, porous medium, tubing, O-seal rings are placed on the top and bottom surfaces to hold the porous medium firmly, three differential pressure sensors of ranges 0-20*10 5 Pa, 0-0.6*10 5 Pa, 0-0.06*10 5 Pa measure the pressure drop between entrance and exit.

Figure 3 . 23 :

 323 Figure 3.23: Pressure drop of a solvent water ∆𝑃 𝑠 as a function of Q for different lengths of porous media.

Figure 3 . 24 :

 324 Figure 3.24: Slopes Sp for porous media from Figure 3.23 are plotted as a function of their corresponding lengths.

2 . 4 .

 24 The experimental techniques are discussed in section 3.6. The rheological characterization of polymer solutions are discussed in sections 3.2 and 3.4. The relaxation time measured from CaBER tca are used for calculating the Weissenberg number or the Deborah number. This chapter is organized in 5 sections. In section 4.2 polymer solutions are injected through a given hyperbolic geometry (of Hencky strain 3.2) at different flow rates and the flow lines are recorded. The rheological properties are varied by changing the molecular weight, solvent viscosity and the solvent quality. Three different flow regimes were observed: Newtonian flow regime, corner vortices and viscoelastic instabilities. The dimensionless vortex size of the corner vortices is studied as a function of Weissenberg number. In section 4.3 the Hencky strain is varied by changing the geometrical dimensions, including the entrance width Wu and the contraction width Wc. Seven different flow regimes were observed, including the three previous ones, and the dimensionless vortex sizes are studied as a function of Deborah number. In section 4.4, the height of the channels is varied while keeping the other geometrical dimensions constant. The vortex size is studied as a function of height of the channel. In Section 4.5, different flow regimes observed in sections 4.2-4.4 are studied as a function of Weissenberg and the Reynolds number as done in literature.

Figure 4 . 1 :

 41 Figure 4.1: upstream vortices in channel [(5-0.2)/34]0.2 for polymer solution 15MDa-0.08/0SG for different Wi and Reynolds number.Our experimental results, namely the absence of lip vortices, poor collapse of the dimensionless vortex size with Weissenberg number, and no vortices with ethanol solvents suggests that the Weissenberg number is not the only parameter to control the vortex formation and dynamics.

Figure 4 . 2 :

 42 Figure 4.2: Dimensionless vortex size 𝜒 𝑣𝑠 (Eq.(2.31)) as a function of Weissenberg number.

Figure 4 . 3 :

 43 Figure 4.3: The seven flow regimes observed. 1) Laminar with no vortices (black) .2) Stable symmetric corner vortices (blue). 3) Time-dependent symmetric corner vortices (magenta).

Figure 4 . 4 : 6 Figure 4 . 5 :

 44645 Figure 4.4: Dimensionless vortex size 𝜒 𝑣𝑠 (2.31) as a function of Deborah number, for an 11MDa-0.08/40SG solution at different Hencky strains as mentioned in Table3.6

  . Nigen et al. studied Boger fluid in axisymmetric and planar contraction geometry, and found no vortices for (planar contraction) 114 .

Figure 4 . 6 :

 46 Figure 4.6: Dimensionless parameter 𝜒 𝑣𝑠 as a function of Weissenberg number, For a 15MDa-0.08/40SG solution in geometries with heights ranging from 0.05mm -0.2mm. In Figure 4.7 we show that the dimensionless vortex size normalized by the square root of height of the channel collapses onto a single function of Weissenberg number.

  , which ensures that the velocity vanishes at the boundaries, z=±h/2 and let us assume u(z=0)= u max .

  -𝟐+(𝛂-𝟏)(𝐧-𝟏) .

(4. 11 )

 11 However the extensional stress gradient is independent of y i.e

(4. 13 )

 13 Let us assume now that u max = a (

n+1 2 ξ

 2 is proportional to the vortex dimensions L vs (length of the vortices and thereby dimensionless vortex size χ vs 𝛏 ∝ 𝛘 𝐯𝐬 = 𝐋 𝐯𝐬 𝐖 𝐮For Newtonian fluid n=1, we get χ vs ∝ h, But for a shear thinning fluid n<1 we have χ vs ∝

Figure 4 . 7 :

 47 Figure 4.7: Dimensionless parameter 𝜒 𝑣𝑠 as calculated in normalized by √ℎ as a function of Weissenberg number, when 15MDa-0.08/40SG is injected through geometries with heights ranging from 50µm -200µm.

Figure 4 . 3 )

 43 observed for different polymer solutions injected through different geometries with different Hencky strains 𝜀 𝐻 . The Symbols identify the flow regimes for the corresponding Weissenberg corresponding Reynolds numbers and match the data plotted in Figure 4.8.

Figure 4 . 8 :

 48 Figure 4.8: Wi-Re plot for different hyperbolic contraction-expansion geometry and when injected with polymer solutions with different rheological properties. Each line corresponds to a given elastic number El as given in Eq.(2.36). The symbols are explained in Table 4.2.

2 :

 2 Nomenclature for polymer solutions used. Here n is the shear thinning index, and tcm is the terminal relaxation time from the Carreau model. The applicability of the empirical Cox-Merz rule (linking frequency in the linear viscoelastic regime to the shear rate in nonlinear regime) supports the use of terminal relaxation time obtained from the shear viscosity data in extensional regime115 . c* is the critical (overlap) concentration that separates the dilute and semi-dilute regimes and is calculated as the inverse of intrinsic viscosity.

Figure 5 . 1 :

 51 Figure 5.1: Pressure drop ratio curve for varying molecular weight (Mw) in 40/60 glycerol/water mixture; the maximum Reynolds numbers are 𝑅𝑒 𝑝 (175, 19𝑀𝐷𝑎) = 19.5, 𝑅𝑒 𝑝 (200, 7𝑀𝐷𝑎) = 38.

Figure 5 . 2 :

 52 Figure 5.2: Pressure drop ratios for varying molecular weight (Mw) in a 40/60 glycerol/water mixture and flow rates normalized by their respective maxima. The molecular weight and concentration dependence for R max and Q max in 40/60 glycerolwater (Figure 5.3 and Figure 5.4) follow 𝐑 𝐦𝐚𝐱 ∝ 𝐌 𝐰 𝟏±𝟎.𝟐 𝐜 𝟎.𝟕±𝟎.𝟐 , (5.1) 𝐐 𝐦𝐚𝐱 ∝ 𝐌 𝐰 -𝟏.𝟖±𝟎.𝟐 𝐜 -𝟎.𝟕±𝟎.𝟐 . (5.2)

Figure 5 . 3 :

 53 Figure 5.3: Molecular weight dependences of R max and Q max in 40/60 glycerol-water. The straight lines have the slopes indicated.

Figure 5 . 4 :

 54 Figure 5.4: concentration dependences of R max and Q max in 40/60 glycerol-water. The straight lines have the slopes indicated.

Figure 5 . 5 :

 55 Figure 5.5: Experimentally observed vortices in a contraction geometry, where L vs is the length of a vortex.

Figure 5 . 6 :

 56 Figure 5.6: Comparison of maximum extension rate as a function of polymer relaxation time for all samples (with all three solvents). The data are fit by a critical Weissenberg number Wi 𝑐 = 16±2. 25.

Figure 5 . 7 :

 57 Figure 5.7: Pressure drop ratios normalized by their respective maxima as a function of Weissenberg number, Wi. True strain rates are used for calculating Weissenberg number (where the vortex length is taken into consideration, as in Eq. 14) and this plot corresponds to a polymer concentration of 0.08%. 𝐖𝐢 = 𝛆̇𝐭 𝐫 (5.5) or a critical Deborah number (to account for thermal activation during extension) 𝐃𝐞 = 𝐭 𝐫 𝐭 𝐫𝐞𝐬 = 𝛆ε 𝐇 𝐭 𝐫 (5.6)

Figure 5 . 8 :

 58 Figure 5.8: Pressure drop ratio curves of polymer solutions for a second pass at flow rate Q2 after pre-degradation from a first injection at flow rate Q1. The first injection is shown as symbols along the upper abscissa.

Figure 5 . 10 :

 510 Figure 5.10: Universal curve for the flow rate Q2 max (respectively strain rate 𝜀̇𝑇 2 𝑚𝑎𝑥 ) of the maximum of pressure drop ratio, obtained in a second passage after a first passage at Q1 (or 𝜀Ṫ 1 ). Both are normalized by the flow rate (or strain rate) at the maximum pressure drop ratio of the first passage. Filled and unfilled symbols correspond to flow rate and strain rate respectively. The solid line represents Eq. (5.7) with scission exponent a=2.5. Uncertainties are shown only for the 19MDaa-0.08/40SG sample. Blue triangles are the pressure drop ratio curve as a function of normalized flow rate of the same sample and the blue dashed line is a guide to the eye.

Figure 5 . 11 :

 511 Figure 5.11: Apparent outlet molecular weight as a function of injected molecular weight at different flow rates Q1, as given by Eq.(5.10).

Figure 5 . 12 :

 512 Figure 5.12: Pressure drop ratio curves of four passes, pre-degraded during every injection at same flow rate 𝑄 1 = 𝑄 2 = 𝑄 3 = 𝑄 = 33.3 µ𝐿/s.

Figure 5

 5 Figure5.12 shows the shifts of the maxima with degradation during successive injections. The history of degradation can be traced by the evolution of maxima from the multiple injection curves. In Figure5.13 we plot the ratio of the maximum of (N+1) st pass to the maximum of

  2. The superposition fails in the shear dominating regime i.e. Q< Q max , as was observed for the [(5-0.2)/34]0.2 geometry (ε H = 3.2*) when injected with polymers of different molecular weight.

Figure 6 . 1 :

 61 Figure 6.1: Pressure drop ratio R as a function of flow rate Q, for a 11Mda-0.08/40SG polymer solution through geometries with different Hencky strain (Table3.6).

Figure 6 . 2 :

 62 Figure 6.2: Normalized pressure drop ratio 𝑅 𝑅 𝑚𝑎𝑥 as a function of normalized flow rate 𝑄 𝑄 𝑚𝑎𝑥 for a 11Mda-0.08/40SG polymer solution through geometries with different Hencky strain (Table3.6).

Figure 6 . 4 :

 64 Figure 6.4: Pressure drop ratio curve as function of flow rate Q for polymers with different molecular weight, with polymer concentration of 0.08% (w/w) in 40/60glycerol/water solvent with solvent viscosity 𝜂 𝑠 ≅ 3𝑐𝑃, in a [(5-0.5)/2.6]0.2 geometry with Hencky strain 𝜀 𝐻 = 2.3.

Figure 6 . 5 :

 65 Figure 6.5: Normalized pressure drop ratio 𝑅 𝑅 𝑚𝑎𝑥 as a function of normalized flow rate 𝑄 𝑄 𝑚𝑎𝑥 for polymer solutions of different molecular weights fixed polymer concentration of 0.08% (w/w) in 40/60glycerol/water solvent with solvent viscosity 𝜂 𝑠 ≅ 3𝑐𝑃, in a [(5-0.5)/2.6]0.2 with Hencky strain 𝜀 𝐻 =2.3.

Figure 6 . 7 :

 67 Figure 6.7: Strain rate at the maximum in pressure drop ratio 𝜀̇𝑇 𝑚𝑎𝑥 as a function of relaxation time measured from the CaBER 𝑡 𝑐𝑎 .( Note that there is no degradation during the filament thinning dynamics of CaBER measurements)

Figure 6 .

 6 Figure 6.11: critical residence time at the state of scission 𝑡 𝑟𝑒𝑠 𝑚𝑎𝑥 as a function of critical strain rate for scission 𝜀̇𝑇 𝑚𝑎𝑥 , when injected with a polymer solution of molecular weight ranging from 7 MDa to 19 MDa with a polymer concentration of 0.08% (w/w) in 40/60 glycerol/water mixture in geometries with Hencky strains varying from 3.9 to 1.4. Each symbol corresponds to geometry with a given Hencky strain, while the color represents the molecular weight of the polymer. Scission is described as an activated process or thermally activated process (often referred to as TABS model) by many others in past. In brief, to cause scission the c-c bond has to undergo cleavage. The potential energy U as function of separation between the two covalently bonded carbon atoms is well described by a Morse function, by 𝐔 = 𝐔 𝟎 (𝐞 [ -𝟐(𝐥-𝐥 𝐞 ) 𝐝 ] -𝟐𝐞 [ -(𝐥-𝐥 𝐞 ) 𝐝] ),

  the center is regarded as the breaking force of the chain 𝐅 𝐛 .So for dilute polymer chain systems under complete stretching, Force for breaking scales as 𝐅 𝐛 ∝ 𝐍 𝟐 . However, Nguyen and Kaush et al. have proposed a lower scaling for Force for breaking 𝐅 𝐛 number of beads N.

Figure 6 . 14 :

 614 Figure 6.14: Multiple contractions in series, with Hencky strain 𝜀 𝐻 =2.3 in successive contractions. The distance between 2 successive contractions is 𝐿 𝐺 = 5𝑚𝑚.The height of the channel is 200µm. Single injections experiments were explained in section 3.7.1. A solvent is injected at different flow rates and the pressure drop between the entrance and exit of the geometry are measured. The pressure drop increases linearly with flow rate as in Figure 3.19,

Figure 6 . 15 :

 615 Figure 6.15: Pressure drop ratio 𝑅(𝑄, 𝑁 𝑐 ) as a function of Q for geometries with different number of contractions in series 𝑁 𝑐 for 19MDa-0.08/40SG polymer solution.

Figure 6 . 16 :

 616 Figure 6.16: Pressure drop ratio curves 𝑅(𝑄 2 , 𝑄 1 , 𝑁 𝑐 ) of a second injection at flow rate 𝑄 2 through a single contraction, after a first passage at a flow rate 𝑄 1 = 83.33µL/s through a geometry with NC contractions.The maximum of the pressure drop ratio curve is R max (Q 2 , Q 1 , N c ) and the flow rate at this maximum Q 2 max (Q 1 , N c ) are obtained by superposing the pressure drop ratio curves 𝑅(𝑄 2 , 𝑄 1 , 𝑁 𝑐 ) with their corresponding maximum. This ensures the reliability of thus obtained data from curves with ill-defined maxima.

Figure 6 . 17 :

 617 Figure 6.17: Flow rate at the maximum 𝑄 2 𝑚𝑎𝑥 (𝑄 1 = 83.33µ𝐿/𝑠) in pressure drop ratio of second passage through a single contraction, as a function of number of contractions 𝑁 𝑐 during first passage.

6. 4 . 4

 44 Optical visualization in contraction in series during first passageTo better understand the degradation behavior in successive contractions, the flow behavior is visualized, as shown in Figure6.19, Figure6.20 and Figure6.21.

Figure 6 . 19 :

 619 Figure 6.19: Streamline images of 19Mda-0.08/40SG in geometries with 𝑁 𝑐 = 1, 2, 3 contractions for a flow rate Q1=4.16µL/s.

  But upon increasing the flow rate the length of the vortices increases, and when Lvs reaches the LG the width accessible by polymer is reduced due to the buckling effect of the flow lines. This decrease of accessible width is almost similar in all the contraction in series. The decreased accessible width decreases the Hencky strain (ε H eff (N c ) = 𝑙𝑛 ( by the fluid in the successive contractions. This decreased Hencky strain is plotted in Figure6.22 and Figure6.23. We suggest that the lesser amount of degradation is due to the decreased Hencky strain ε H eff (N c ) in the successive contraction.

Figure 6 . 20 :

 620 Figure 6.20: Images of 19Mda-0.08/40SG in geometries with 𝑁 𝑐 =1, 2, 3 and 4 contractions at a flow rate Q1=8.33µL/s(on left), Q1=25µL/s (on right).

Figure 6 . 21 :

 621 Figure 6.21: Images of 19Mda-0.08/40SG in geometries with 𝑁 𝑐 =1, 2, 3 and 4 contractions at a flow rate Q1=42µL/s(on left), Q1=83.3µL/s (on right).

6. 4 . 5 Figure 6 . 22 :

 45622 Figure 6.22: Effective Hencky strain 𝜀 𝐻 as a function of contraction number 𝑁 𝑐 , when a19Mda-0.08/40SG solution is injected through a geometry with 𝑁 𝑐 = 4 contractions at different flow rates.

Figure 6 . 23 :

 623 Figure 6.23: Effective Hencky strain 𝜀 𝐻 as a function of contraction number 𝑁 𝑐 , when a 19Mda-0.08/40SG solution is injected at flow rate Q1=83.33µL/S through geometries with different number of contractions 𝑁 𝑐 . In order to predict the Q max in each contraction for multiple contractions, we need to consider the change in effective Hencky strain, and the change in 1. The scission exponent b of Eq.(5.7) depends upon the Hencky strain.

  ) to get the true Q 2 max (Q 1 , N c ) as in 𝐐 𝟐 𝐦𝐚𝐱 (𝐐 𝟏 , 𝐍 𝐜 ) = 𝐤(𝛆 𝐇 𝐞𝐟𝐟 , 𝐍 𝐜 )(𝐌 𝐨𝐮𝐭 (𝐍)) -𝟐 (6.22)We plot true Q 2 max (Q 1 , N c ) as function of the number of contractions N c , in Figure6.27.

Figure 6 .

 6 Figure 6.25: Q max as a function of molecular weight Mw in single contractions with different Hencky strain.

Figure 6 . 26 :

 626 Figure 6.26: Geometrical constant k as a function of Hencky strain obtained from the slopes of Figure 6.26. The Q 2 max (Q 1 , N c ) calculated from Eq. (6.22) values closely fit the experimentally obtained values, suggesting that the degradation in successive contractions is a function of the effective Hencky strain .The decrease in the accessible width decreases the Hencky strain and therefore the amount of degradation decreases in the successive contractions.

Figure 6 .

 6 Figure 6.27: 𝑄 2 𝑚𝑎𝑥 (𝑄 1 , 𝑁 𝑐 ) calculated from equation (6.15), experimental (green triangle) and true values (red star) calculated from Eq. (6.22) considering the true Hencky strain in the successive contractions. Here, 𝑄 1 =83.33µL/s.

Figure 7 . 1 :Figure 7 . 2 :

 7172 Figure 7.1: Mobility reduction curves Rm calculated using Eq. (3.34) as a function of flux for injecting a 19MDa-0.08/0SG solution through porous media of different lengths. The pink circles correspond to the maxima (J max , R m max ) of each mobility reduction curve.

Figure 7 . 3 :

 73 Figure 7.3: Degradation calculated using Eq.(3.36) as a function of flux J in porous media of length 1, 2, 3 and 8 mm respectively.

Figure 7 . 4 :

 74 Figure 7.4: Mobility reduction curves of 19MDa-0.08/0SG solutions re-injected through the 1mm-long core after a first injection at flux J1 through the same core. J1 is marked along the upper abscissa.

Figure 7 . 5 :

 75 Figure 7.5: Normalized mobility reduction 𝑅 𝑚 𝑅 𝑚 𝑚𝑎𝑥 as function of normalized flux 𝐽 𝐽 𝑚𝑎𝑥 for the first pass and the reinjected solutions on the porous medium of length L0=1 mm. The black solid line is a fit to Eq.(7.1) with fitting parameters 𝑛 2 =1.45, 𝑛 3 =1.25, 𝜏 2 =3, 𝜏 3 =0.0075. In Figure 7.5 the mobility reduction curves are normalized with respect to R m max and J max . The superposition is very good for all pre-degradation levels (or flux J 1 ), unlike the case of planar contraction devices of chapters 5 and 6 (Figure 5.7, Figure 6.2, and Figure 6.5). The possible reason for such a good superposition is that in porous media the flow has a stronger extensional component, compared to simple shear component, due to complex 3D contractions and expansions of the porous network. Conversely in a single contraction the contribution due to simple shear is comparable to that of extension.

  , for this we set the Q c = R m c = 0. The modified Cross model we propose is given in terms of normalized mobility reduction

  parameters are same as earlier: n 2 =1.45, n 3 =1.25, τ 2 =3, τ 3 =0.0075.

Figure 7 . 6 :

 76 Figure 7.6: Ratio of the fluxes J 2 max / J 1 max at which mobility reduction is maximum during the first and the second pass on the 1mm-long core as a function the ratio J 1 / J 1 max where J 1 is the flux at which the solution was collected in the first pass. Experimental points are the open symbol. The solid line is the best fit of Eq. (7.2) by taking 𝑎 =1.85.

Figure 7 . 7 :

 77 Figure 7.7: Evolution of 𝐑 𝐦 𝐦𝐚𝐱 as a function of 𝐉 𝐦𝐚𝐱 (from the 2 passes on porous core of length 𝐋 𝟎 = 𝟏𝐦𝐦 ).

Figure 7 . 8 :

 78 Figure 7.8: Predicted mobility reductions MP -R m (J 1 , NL 0 ) (represented by solid lines) and experimental mobility reduction (represented by symbols) on porous medium of lengths L =1 mm, 2 mm, 3 mm, 4 mm, 6 mm, 8 mm as a function of flux. Here N is the number of passages on porous medium of length L 0 =1.1.

Figure 7 . 9 :

 79 Figure 7.9: Degradation during the first passage on porous medium of length 1 mm, as a function of ratio of maximum of the mobility reduction during the second passage 𝑅 𝑚 𝑚𝑎𝑥 (𝐽 2 , 𝐽 1 ), to that during the first passage 𝑅 𝑚 𝑚𝑎𝑥 (0, 𝐽 1 ).The solid blue line is the fit calculated using Eq.(7.14).

Figure 7 . 10 :

 710 Figure 7.10: Degradation as a function of flux J for different lengths of porous media. Symbols corresponds to the experiments and solid lines are calculated using Eq.(7.15) for lengths equal to N*1mm, where N is the number of passages through a porous medium of length L 0 =1 mm.

1 3,

 1 ε(L D ) m . Several researchers have proposed different values of m, Marker et al. has proposed m = Morris et al. proposed m to be molecular weight dependent as m=𝑀 * 10 -7 , Marie et al. has proposed m =0.07 for HPAM polymers and 0.03 for ATBS polymers 105 .

Figure

7 . 14 )

 714 to those of the linear injecting (Figure7.11) conditions.

7. 7 . 1 Figure 7 . 11 :

 71711 Figure 7.11: Illustration of linear injection experiment on porous medium of length L. Flux J is constant all along the length of the core. The scission zone (where polymer undergoes scission) corresponds to first few cms.

Figure 7 . 12 :

 712 Figure 7.12: Degradation as a function of length for different flux for a flow of polymer solution in 1D linear flows. In the inset the same graph is plotted on a linear scale, which highlights that degradation occurs sharply in the first few mms and then a gradual increase in the degradation.

Figure 7 . 13 :

 713 Figure 7.13: Critical length 𝐿 𝑐 and degradation at critical length 𝐿 𝑐 as a function of surface flux J. 𝐿 𝑐 is the critical length at which degradation attains the quasi-saturation in degradation.

Figure 7 . 14 :

 714 Figure 7.14: Illustration of radial injecting conditions with Injector well radius 𝑅 𝑤 , height 12mm. The flux 𝐽 𝑖 𝑟 follows the Eq.(7.20).

Figure 7 . 15 :

 715 Figure 7.15: Degradation as a function of radius traversed in the reservoir for different entrance fluxes at the surface, for varying injector well radii R w a) R w = 24 mm, b) R w = 1200mm.

Figure 7 . 16 :

 716 Figure 7.16: Critical radius 𝐫 𝐜 as a function of injector well radius 𝐑 𝐰 , with the critical length 𝐋 𝐜 in the linear injection for a) surface flux 𝐉=1.15 m.h -1 .and for b) 𝐉 =3.15 m.h -1 .In Figure7.16 we plot the critical radius r c to obtain degradation saturation under radial conditions as a function of injector well radius R w for 2 different fluxes. For increasing injector well radius the critical radius r c increases and asymptotes towards the linear injecting conditions. The degradation is plotted as a function of injector well radius in Figure7.17, for different injection fluxes. The amount of degradation at saturation or critical radius r c in radial conditions (symbols) is smaller and increases with the increasing injector well radius. Finally, the degradation asymptotes towards the linear injecting conditions (solid lines).

Figure 7 . 17 :

 717 Figure 7.17: Degradation at critical radius r c as a function of injector well radius R w on the same plot the critical length L c in the linear injection (solid lines) for flux J=1.15 m.h -1 and (right) for J=3.15 m.h -1 .

Figure 7 . 18 :

 718 Figure 7.18:Degradation as a function of distance travelled in the porous core for varying scission exponent (or permeability). The injection well radius 𝑅 𝑤 is 24mm, flux at the entrance J is 5.15 m/h. Pink circles highlights the critical radius, over which there would be no further polymer scission, and degradation at the critical radius 𝑟 𝑐 .

Fig i :

 : Fig i: Pressure drop ratio R as a function of flow rate Q, for injection 15MDa-0.08/40SG polymer solution through geometry [(5-0.2)/34]0.2 with Hencky strain 𝜀 𝐻 = 3.2 * .

=12. 5

 5 MDa) during the first passage through contraction geometry [(5-0.2)/34]0.2 with Hencky strain ε H = 3.2 * . Total polymer concentration is 0.08% (w/w). These binary polymer solutions are then injected through the same contraction device. The corresponding pressure drop ratio curves R, are plotted as a function of flow rate Q in Fig ii). The flow rate at which maximum occurs Q pd max , in the pressure drop ratio curve for binary solutions is linearly dependent on the weight fraction of the high molecular weight polymer Fig iii):𝐐 𝐩𝐝 𝐦𝐚𝐱 = 𝐐 𝐀 𝐦𝐚𝐱 𝛗 𝐀 + 𝐐 𝐁 𝐦𝐚𝐱 𝛗 𝐁 I.(ii)Here φ A -is the weight fraction of A φ A . Q A max is the flow rate at the maximum for φ A = 1, and Q B max is the flow rate at the maximum for φ B = 1. The reason for such linear dependence is not in the scope of the current study.

Fig ii:

  Fig ii: Pressure drop ratio R as a function of flow rate Q of binary polymer solutions prepared by mixing polymers of different apparent molecular weights which are collected at flow rates QA=8.33µL/s <Q max (M w app =18.9MDa), QB=83.33µL/s <Q max (M w app =12.5 MDa) in the first passage through contraction geometry [(5-0.2)/34]0.2 with a Hencky strain ε H = 3.2 * . The solid line is the fit to polynomial equation of order 4 (Eq. I.(iii)). A given pressure drop ratio curve can be described a polynomial function of 3 rd and 4 th order as shown by a red solid line of Fig ii).

Fig iii:

  Fig iii: Flow rate at the maximum in the pressure drop ratio curve R(Q) for the data shown in Fig ii.

Fig iv:p 2 , 3 (- 1 +

 231 Fig iv: Width of the pressure drop ratio curve R(Q) for the data shown in Fig ii. Conclusion: The binary polymer solutions are prepared by mixing polymers different apparent molecular weights, by collecting the polymer solution at 2 different flow rates during the first passage. The binary solutions are injected through the same contraction as in their first passage.The flow rate at the maximum in the pressure drop ratio curve, decreases linearly with the weight fraction of high molecular weight polymer. Moreover, the polydispersity did not show any significant influence on the resulting width of the pressure drop ratio curves.

Fig v:

  Fig v: Geometrical function 𝑓 𝑒𝑥𝑡 as a function Weissenberg number for different Hencky strain the solid lines are calculated from the model. The blue circles are geometrical function 𝑓 𝑒𝑥𝑡 calculated from the Eq. II. (xvi) for Weissenberg number at the maximum of the pressure drop ratio when injected 15MDa-0.08/40SG polymer solution through geometries with different Hencky strains. By combining Eqs.II.(xiii) , II.(xv) , and II.(xvi) we get, < 𝐗 𝐩𝐱 𝐗 𝐩𝐱 > +< 𝐗 𝐩𝐲 𝐗 𝐩𝐲 > + < 𝐗 𝐩𝐳 𝐗 𝐩𝐳 > = 𝐍𝐛 𝟐 𝟔𝛑 𝟐 𝐩 𝟐 𝐟 𝐞𝐱𝐭 (𝐖𝐢 𝐩 , 𝛆 𝐇 )

Table 3 .

 3 6). .................................. 98 Figure 6.2: Normalized pressure drop ratio 𝑅𝑅𝑚𝑎𝑥 as a function of normalized flow rate 𝑄𝑄𝑚𝑎𝑥 for a 11Mda-0.08/40SG polymer solution through geometries with different Hencky strain (Table 3.6). ..................................................................................................................... 98 Figure

  ). The injection well radius 𝑅𝑤 is 24mm, flux at the entrance J is 5.15 m/h. Pink circles highlights the critical radius, over which there would be no further polymer scission, and degradation at the critical radius 𝑟𝑐. ..................................... 140

	Symbols
	t r	-Relaxation time of the polymer chain
	t res	-Residence time
	t ca	-Longest relaxation time measured from filament thinning dynamics in CaBER
		experiment
	t cm	-Terminal relaxation time measured from fitting Carreau model for the non-
		linear shear viscosity experimental data
	σ	-Stress (could be shear or extensional)
	τ	-stress tensor
	δ	-Deformation
	ε	-Strain
	u	-velocity
	ε̇	-strain rate
	ε H	-Hencky strain
	Q	-Mass flow rate
	J	-Mass flux (flow rate/ surface area)
	η	-Viscosity in general
	η ext , η shear	-Extensional viscosity andShear viscosity
	η p , η s	-Shear viscosity of polymer and solvent (Newtonian fluid)
	ΔP, ΔP p , ΔP s	-Pressure drop, pressure drop due to injection of polymer and solvent
		respectively
	k B	-Boltzmann constant
	Re	-Reynolds number
	Wi	-Weissenberg number
	De	-Deborah number
	T	-Temperature
	G	-Shear modulus
	E	-Elastic modulus
	γ̇	-Shear rate
	W u , W c	-Entrance width and Contraction width
	L c	-length of the contraction profile
	L vs	-Length of the vortices
	h	-Height of the channel
	χ vs	-Dimensionless vortex size dimension
	ξ vs	-Modified dimensionless vortex size dimension
	n	-shear thinning index
	M w	-Molecular weight of the polymer
	R m	-Mobility reduction
	φ	-Weight fraction
	F b	-Breaking force
	U o	-Disassociation energy

Table 2 .

 2

1: Models, constitutive equations proposed for behaviour of non-Newtonian polymer solution.

  Xanthum solutions. Pressure drop and vortex size, lead to the determination of the elongation viscosity of the solutions, and to a correct estimation of the length of the xanthan molecules.Flow line imaging of, PEO-water solutions injected through planar contraction devices. Stable vortex regions occur for low Wi, In contrast to the previous works made byRodd et al. (2005a) for similar conditions.

	Cloitre et al. Semi-dilute Alves et al. Cylindrical β =24 Xanthum in 75/25 or 85/15 glucose/water solvent of c=0.01-0.05% η p = 22.6-0.43 Pa-S η s = 1.99 -0.23 Pa-S tr = 222.8-0.82 S n= 0.47-0.56 Re<7 square 84.99/14.01 NA Newtonian flow. Inertia to be negligible Re≤1.
	32	β =4	Glycerin/Water.	At low Reynolds numbers, inertial effects are negligible
		𝐿 𝐶 𝑅 2	= 100	90.99/7.51 𝜂 𝑠 = 0.125 Pa. s	and vortices dimension asymptotes to non-zero value.
				Glycerin/Water.
				Rest salt
					𝜂 𝑠 = 0.367
				PA100		0<De <15-vortex length decreases.
					𝜂 𝑝 = 0.487	0<De <54	15<De>54 vortex dimensions increases.
				P300			0<De <46.7	De>54 periodic time-dependent-elastic instability
				𝜂 𝑝 = 0.735
							0<De <15-vortices length decreases
							15<De<46.7 vortices dimensions increases
							De>46.7 periodic time-dependent-elastic instability
	Oliveira et al.	Axisymmetric	Viscoelastic solution	0.1<De<1000	Numerical study
		β =2-100	modeled with PTT	Oldroyd B model for Newtonian solvent.
				model for strain 𝜀 =
				0 -0.5
	Rodd et al.	planar abrupt	PEO (Mw = 2MDa),	3.8 ≤ El ≤ 89,
	(2005 & 2007)	contraction	0.58 ≤ c/c * ≤ 3.5	0<Wi ≤ 548
	28, 29	β =16				0.44<Re<64
		α = 1/8
	Miller and	Same as Rodd et	PEO-aqueous		6.2 ≤ El ≤ 612, 4
	Cooper-White	Square al. (2005a)	PAA in 40/59.9-solutions.	0.5<De<400; ≤	1. With Newtonian solvents, upstream vortices are
	44 (2009) 46	β =2.4-12	Glycerol/water Steady state shear	Wi ≤ 300	observed. The length of the vortices decreases with
				c*=0.06 viscosities and viscous	Reynold's number.
				𝜂 𝑝 = 1.62 Pa. s; and elastic moduli,	2. Lip vortices not observed in experimental range of
				𝜂 𝑠 =0.03 Pa. s; CaBER relaxation	Deborah number but are observed in simulations for
				τ = 32 s. times.		De=0.231.
	Gulati et al.	Planar abrupt	a	DNA	aqueous	diverging stream lines at Used both streak imaging and digital particle image
	(2008) 47	contractions	solution (c/c * = 4)	De>10, for β =2.4 velocimetry (DPIV). Visualized very symmetric and
		β =2				10 3 ≤ El ≤ 10 6	De>20, for β =4 stable corner vortices upstream from the contraction.
		α=0.9				3. Simulations using Finite volume method and for the The centerline velocity profiles are sensitive to Wi.
							1 ≤Wi ≤ 630	polymer PTT model.
							10 -2 ≤γ˙ ≤ 2 * 10 2
							s -1
	Hidema et al.

Lip Vortex growth relates to De. Corner vortex growth relates to De/ β. Kang et al. 45 Planar abrupt contraction 2 wt.% PEO, 1 wt% cellulose aqueous solutions 𝛾̇< 10 7 Microchannel for measurement of viscosities at high shear rates. Bagley correction used to calculate the end effects. Different vortices regimes were characterized using streak images and pressure drop measurements Proposed Wi-Re map to define the occurrences of a flow regime. Vortex dimensions and their stability dependent on Wi and the Elastic number This map illustrates that the critical flow rates vary with both Re and Wi. For lower elastic numbers. The transition between flow regimes is a weak function of El, such that transitions occur at marginally higher Weissenberg numbers. 42 Planar abrupt contractions β =16,8 α=1/8,1/4 NA-HA/PBS NA-HA/Water ~1MDa C=0.3%(w/w) 7 ≤ El ≤ 1240 1 ≤Wi ≤ 10 4 Observed Newtonian like flow regime, lip vortices, stable corner vortices, time-dependent corner vortices and the elastic instability flow regimes. For geometry with same aspect ratio, vortex growth

Table 2 .

 2 4: Literature review on vortices in different axisymmetric contraction geometries.

  10 -2 ≤Re ≤ 10 2 collapsed onto a single curve when plotted as a function of Wi/ β. The flow regimes were studied in Wi-Re space

	Lanzaro et al.	Planar abrupt	PAA aqueous	14.7<El<476.8	Quantitative measurements of velocity fields and
	(2011) 38	contractions	solutions.	1.46 ≤Wi ≤	pressure drops.
		β =4, 8, 16	C=0.1-0.5%(w/w)	131.7	Changing the elastic number El in a given geometry
		α=1/16,1/8.	Shear and filament		changed the stability of observed flow patterns.
			thinning.
	Galindo Rosales	Planar hyperbolic	PAA (Mw = 18 MDa)		For symmetric geometries are Newtonian like,
	Et al. (2011 &	contractions.	Shear and extensional		Steady and symmetric flow regimes were reported for
	2014) 34, 48		characterizations		low Weissenberg numbers. Upon increasing the
					Weissenberg number elastic instabilities sets in.
					Dimensionless vortex size follows the Weissenberg
					number for low values but fails for higher values.

Table 2 .

 2 5: Molecular weight dependence for critical strain rate of onset of degradation found by different researchers and their conditions of degradation. The results summarized above are in fast transient extensional flows where devices consist of an orifice or a contraction. The critical strain rate for onset of degradation scaling with molecular weight as 𝜀̇𝑓 ∝ 𝑀 𝑤 -𝑏 . In dilute steady state scission, due to the infinite residence time the polymer chains completely unravels and the value of exponent would be 𝑏 = 2[START_REF] Odell | Degradation of polymer solutions in extensional flows[END_REF][START_REF] Hunkeler | Polymer solutions under elongational flow: 1. Birefringence characterization of transient and stagnation point elongational flows[END_REF] .

	Polymer	Solvent	Mw range	solvent	Concentration	b	Flow type	Reference
			(MDa)	quality	regime
	PEO	Water	2-7	Good	Dilute	1	contraction	52
	Polystyrene Decalin	2-4	Poor	Dilute	1.3	Turbulent-	52, 53
							contraction
	Polystyrene Dimethylpthalate	2-1	Good	Dilute	0.95	Turbulent-contraction 58
	HPAM	Water (20g/l NaCl)	8-12	Good	Semi-dilute	un 1	Turbulent-	72
					entangled		contraction
	HPAM	Water (20g/l NaCl)	8-18	Good	Semi-dilute	3	Laminar-	72
							contraction
	PEO	Water	1-8	Good	Semi-dilute	1.22	Turbulent-contraction	72
		Water/glycerol (20/80)		Poor		1.42

Table 2 .

 2 6: literature review on the effect of the molecular weight distribution on the flow induced scission.

Table 3 .

 3 1: Molecular weights of native polymers of HPAM. In all cases the polydispersity is 𝑀 𝑤 /𝑀 𝑛 ~4 -5.

	Grade	Mass Molar (MDa)
	3630S	19
	3530S	15
	3430S	11
	3330S	10
	3230S	7
	PAM	0.5
	Salt	Composition in %
	Na2SO4 0.0074
	KCl	0.0111
	CaCl2	0.0745
	MgCl2	0.0561
	NaCl	0.4724
	NaN3	0.02

Table 3 .

 3 2: Salt composition for 0.6% (w/w) salt solution. Salts are obtained from Sigma Aldrich

Table 3 .

 3 3).

	Name	Solvent	[η] (dl/g)
	0SG	Water	0.0056
	40SG	40/60 glycerol/water 0.0034
	15SE	15/85 ethanol/water 0.00225

Table 3 .

 3 3: Intrinsic viscosity and for 19MDa polymer in three different solvents.

Table 3 .

 3 4.

	Name	Mw	c	Solvent	n	tcm	tca	c*	c/c* η s	η p0
		(MDa)	%(w/	Composition		(s)	(s)	%(w/w)		(mPa-s)	(mPa-s)
			w)	%(w/w)
	7MDa-0.08/40SG	7	0.08	40% Glycerol 0.88 0.21 0.03	0.043	1.86 3.18	14
	10MDa-0.08/40SG	10	0.08	40% Glycerol 0.86 0.31 0.045 0.035	2.28 3.18	17.75
	11MDa-0.08/40SG	11	0.08	40% Glycerol 0.85 0.4	0.06	0.032	2.5	3.18	21.35
	15MDa-0.08/40SG	15	0.08	40% Glycerol 0.8	0.86 0.21	0.029	2.76 3.18	32.5
	19MDa-0.08/40SG	19	0.08	40% Glycerol 0.76 1.41 0.27	0.026	2.31 3.18	42.8
	19MDa-0.06/40SG	19	0.06	40% Glycerol 0.86 0.96		0.026	3.85 3.18	20.37
	19MDa-0.1/40SG	19	0.1	40% Glycerol -	-		0.026	3.08 3.18	-
	19MDa-0.08/0SG	19	0.08	Water	0.74 0.63 0.115 0.018	4.44 0.9	18.6
	15MDa-0.08/0SG	15	0.08	Water	0.8	0.43 0.067 0.022	3.63 0.9	13.1
	11MDa-0.08/0SG	11	0.08	Water	0.81 0.18 0.019 0.03	2.67 0.9	7.4
	19MDa-0.08/0,9SP	19	0.08	0,9% PAM	-	-		-	-	3.21	-
	19MDa-0.08/15SE	19	0.08	15% Ethanol 0.84 0.36 0.044 0.045	1.78 1.44	8.8
	15MDa-0.08/15SE	15	0.08	15% Ethanol 0.89 0.22 0.027 0.5	1.6	1.44	6.7
	11MDa-0.08/15SE	11	0.08	15% Ethanol 0.92 0.15 0.014 0.064	1.25 1.44	4.7

  .18)

	[(5-0.2)/34]0.2	5	0.2	34	0.2	3.2*
	[(2.5-0.05)/2.6]0.2	2.5	0.05	2.6	0.2	3.9
	[(1.25-0.0.05)/2.6]0.2 1.25	0.05	2.6	0.2	3.2
	[(5-0.5)/2.6]0.2	5	0.5	2.6	0.2	2.3
	[(3-0.2)/2.6]0.2	3	0.5	2.6	0.2	1.8
	[(2-0.2)/2.6]0.2	2	0.5	2.6	0.2	1.4
	[(2.5-0.05)/2.6]0.1	2.5	0.05	2.6	0.1	3.9*
	[(2.5-0.05)/2.6]0.05	2.5	0.05	2.6	0.05	3.9**
	[(5-0.05)/2.6]0.05	5	0.05	2.6	0.05	4.6*

Table 3 .

 3 6: Dimensions of geometries with different Hencky strain. Wu is the entrance width , Wc is the contraction width, Lc is the contraction length h is the channel height, 𝜀 𝐻 is the Hencky strain, and α is the aspect ratio.

	Geometry	Wu (mm)	Wc (mm) Lc (mm) h (mm)	𝛆 𝐇	α(=WU/h)
	[(2.5-0.05)/2.6]0.2	2.5	0.05	2.6	0.2	3.9	0.02
	[(1.25-0.05)/2.6]0.2	1.25	0.05	2.6	0.2	3.2	0.01
	[(5-0.5)/2.6]0.2	5	0.5	2.6	0.2	2.3	0.04
	[(3-0.5)/2.6]0.2	3	0.5	2.6	0.2	1.8	0.067
	[(2-0.5)/2.6]0.2	2	0.5	2.6	0.2	1.4	0.1

Table 2 . 3 :

 23 .3, required for the current study. 𝑊 𝑢 is the entrance width of the geometry, 𝑊 𝑐 is the width at the throat of the contraction (Refer Figure3.8), h is the height of the channel, 𝑡 𝑟 is the relaxation time of the polymer chain, 𝑡 𝑟𝑒𝑠 is the residence time, 𝐷 ℎ is the hydraulic diameter given by 𝐷 ℎ =

	Equation number Quantity	Symbol Formula
	(2.16)	Reynolds number	𝐑𝐞	𝐑𝐞 =	𝛒𝐮𝐃 𝐡 𝛈
	(2.31)	Dimensionless Vortex size	𝛘 𝐯𝐬	𝛘 𝐯𝐬 =	𝐋 𝐯𝐬 𝐖 𝐮
	(2.32)	Contraction ratio	𝛃	𝛃 =	𝐖 𝐮 𝐖 𝐜	= 𝐞 𝛜 𝐇
	(2.33)	Aspect ratio	𝛂	𝛂 =	𝐡 𝐖 𝐮
	(2.34)	Weissenberg number	Wi	Wi=𝛆̇𝐭 𝐫
	(2.35)	Deborah number	De	𝐃𝐞 =	𝐭 𝐫 𝐭 𝐫𝐞𝐬

Table 4 .

 4 .1, except for polymer solutions in ethanol solvent.It was quite surprising to note the absence of vortices or chaotic behavior with polymer solutions in ethanol for whatever the flow rate. Polymers in water and glycerol-water exhibit a stable corner vortex that grows with Weissenberg number. We measure the length of the stable corner vortices Lvs, for different polymer solutions. In Figure4.2 we plot the dimensionless vortex size χ 1: Nomenclature for polymer solutions used. Here 𝑛 is the shear thinning index, and tca is the relaxation time measured from the CaBER. c* is the critical (overlap) concentration that separates the dilute and semi-dilute regimes and is calculated as the inverse of the intrinsic viscosity.

	Name	Mw	Solvent	tca	c*	c/c*	η s	η p0	Shear thinning
		(MDa)	Composition	(s)	%(w/w		(mPa-	(mPa-s)	index n
			%(w/w)		)		s)
	19MDa-0.08/40SG	19	40% Glycerol	0.27	0.032	2.5	3.18	21.35	0.76
	15MDa-0.08/40SG	15	40% Glycerol	0.21	0.029	2.76	3.18	32.5	0.8
	11MDa-0.08/40SG	11	40% Glycerol	0.07	0.026	2.31	3.18	42.8	0.85
	19MDa-0.08/0SG	19	Water	0.115	0.018	4.44	0.9	18.6	0.74
	15MDa-0.08/0SG	15	Water	0.067	0.022	3.63	0.9	13.1	0.8
	11MDa-0.08/0SG	11	Water	0.019	0.03	2.67	0.9	7.4	0.81
	19MDa-0.08/15SE	19	15% Ethanol	0.044	0.045	1.78	1.44	8.8	0.84
	15MDa-0.08/15SE	15	15% Ethanol	0.027	0.5	1.6	1.44	6.7	0.89
	11MDa-0.08/15SE	11	15% Ethanol	0.014	0.064	1.25	1.44	4.7	0.92

vs (Eq.(2.31)) as a function of Weissenberg number. The dimensionless vortex size χ vs increases as function of Weissenberg number for polymers in water and glycerol-water solvents, while with ethanol the dimensionless vortex size χ vs is always zero. Figure

4

.2, shows that the dimensionless vortex size χ vs does not scale well with Weissenberg number. Similar findings were observed by Galindo Rosales et al., for aqueous dilute polyacrylamide (PAM) in a hyperbolic contraction

[START_REF] Galindo-Rosales | Viscoelastic instabilities in micro-scale flows[END_REF]

. On the other hand, Rodd et al. observed that the dimensionless vortex size χ vs is well predicted by the Weissenberg number, For polyethylene oxide (PEO) solution in a planar abrupt contraction geometry, with contraction ratio β=16 28 .

Table 4 . 2

 42

		Laminar flow	Stable symmetric corner vortices	Time-dependent symmetric corner vortices	Stable asymmetric corner vortices	Time-vortices dependent asymmetric corner	singular corner vortices	Viscoelastic instabilities
				[(5-0.2)/34]0.2 𝛆 𝐇 = 𝟑. 𝟐*
	19-0.08/0SG	0.7<Wi<1.3	1.7<Wi<5.36					6.7<Wi< 13.4
	η p0 = 18.3 cP	2.2<Re< 5.4	5.5<Re<17.7	-	-	-	-	22.16<Re<44.3
	t ca = 0.115 s	'*'	'*'					'*'
	19-0.08/40SG	0.8<Wi< 1.6	2.4<Wi<5.5					6.3<Wi< 23.6
	η p0 = 42.8 cP	0.5<Re< 0.9	1.4<Re<3.2	-	-	-	-	3.7<Re<13.9
	t ca = 0.27 s	'+'	'+'					'+'
	19-0.08/15SE η p0 = 8.8 cP t ca = 0.044 s	0.5<Wi< 10.3 'X' 9.5<Re< 189	-	-	-	-	-	-
	11-0.08/0SG	0.1<Wi<1.1	1.3<Wi<4.4
	η p0 = 7.4 cP	5.5<Re<54.8	65.8<Re<219	-	-	-	-	-
	t ca = 0.019 s
	11-0.08/40SG η p0 = 17.7 cP	0.4<Wi<0.8 2.1<Re< 4.2	1.6<Wi<7.3 8.3<Re<37.5	-	-	-	-	-
	t ca = 0.038 s
	11-0.08/15SE η p0 = 4.7 cP t ca = 0.014 s	0.2<Wi<3.3 18.1<Re<362	-	-	-	-	-	-
			[(2.5-0.05)/2.6]0.2 𝛆 𝐇 = 𝟑. 𝟗
	15-0.08/40SG η p0 = 32.5 cP t ca = 0.21 s	-	-	-	-	-	-	6.6<Wi< 26.4 0.2<Re<0.8

: The seven flow regimes (as in

Table 5 .

 5 make sense because each is related to intra-chain tension. Furthermore, we have studied the influence of polydispersity and found that it has no significant influence on the width of the above pressure drop ratio curve R (detailed in Annex I).

	Geometry	Entrance width	Contraction	Contraction length Lc	Channel	Hencky strain
		Wu (mm)	width Wc (mm)	(mm)	height h (mm)	𝛆 𝐇
	[(5-0.2)/34]0.2 5	0.2	34	0.2	3.2*

  2) were performed. The polymer solution is injected at a flow rate Q 1 , through a geometries with N c contractions and the flow outlet is collected. This solution is injected for a second time at flow rate Q 2 through a geometry with N c = 1, and the pressure drop ∆P p (Q 2 , Q 1 , N c ) is measured. The pressure drop ratio of second injection through the single contraction R(Q 2 , Q 1 , N c ) is calculated, 𝐑(𝐐 𝟐 , 𝐐 𝟏 , 𝐍 𝐜 ) = ∆𝐏 𝐩 (𝐐 𝟐 , 𝐐 𝟏 , 𝐍 𝐜 ) ∆𝐏 𝐬 (𝐐 𝟐 , 𝐍 𝐜 = 𝟏) ,

	(6.14)

  Figure 6.18: predicted 𝑄 𝑚𝑎𝑥 (𝑄 2 , 𝑄 1 ) calculated from equation (6.15) is compared with experimental 𝑄 𝑚𝑎𝑥 (𝑄 2 , 𝑄 1 ) as a function of number of contractions in series 𝑁 𝑐 in the geometry during first passage. Second passage is always on single contraction.

						𝟏
		𝐐 𝟐	𝐦𝐚𝐱 (𝐐 𝟏 , 𝐍 𝐜 ) = 𝐐 𝐍 𝐜 +𝟏 max = [𝟏 + 𝐍 𝐜 (	𝐐 𝟏 max (𝐍 𝐜 =𝟏) 𝐐 𝟏 ) 𝟑.𝟐𝟓	] 𝟑.𝟐𝟓	𝐐 𝟏 max (𝐍 𝐜 = 𝟏) .	(6.15)
	In Figure 6.18 we compare 𝐐 𝟐	𝐦𝐚𝐱 (𝐐 𝟏 , 𝐍 𝐜 ) predicted from Eq.(6.15) with the experimental
	𝐐 𝟐	𝐦𝐚𝐱 (𝐐 𝟏 , 𝐍 𝐜 ) as a function of number of contractions N c . The predicted 𝐐 𝟐	𝐦𝐚𝐱 (𝐐 𝟏 , 𝐍 𝐜 )
	calculated from Eq.(6.15) is higher than the experimentally obtained values suggesting a lower
	amount of scission than the predicted values.

  )

		𝐀 (𝐉 𝟏 max [𝟏 + 𝟐 ( 𝐉 𝟏 𝐉 𝟏 𝐦𝐚𝐱 ) 𝐚	]	𝟏 𝐚	) -𝟎.𝟓	𝟐	𝐀 (𝐉 𝟏 max [𝟏 + 𝟐 ( 𝐉 𝟏 𝐉 𝟏 𝐦𝐚𝐱 ) 𝐚	] 𝟏 𝐚	) -𝟎.𝟓
	𝐃𝐞𝐠 [𝐉 𝟏 , 𝟐𝐋 𝐨 ] = 𝟓𝟕	𝐑 𝐦 𝐦𝐚𝐱 ( 𝐉 𝟏 , 𝐋 𝐨 )				-𝟏𝟓𝟕	𝐑 𝐦 𝐦𝐚𝐱 ( 𝐉 𝟏 , 𝐋 𝐨 )		) + 𝟏𝟎𝟎, (7.16)
	(				)

  𝐃𝐞𝐠 [𝐉 𝟏 , 𝐍𝐋 𝐨 ] = 𝟓𝟕

	𝐀	𝐉 𝐍 max	[𝟏 + ( 𝐉 𝐍 𝐉 𝟏 max ) 𝒇 𝑵 (𝟎. 𝟎𝟓) 𝐚	] 𝟏 𝐚		𝐉 𝐍 max > 𝟎. 𝟎𝟓 𝐉 𝟏	-𝟎.𝟓	𝟐
	(	{	𝟏				𝐉 𝟏 𝐉 𝐍 max < 𝟎. 𝟎𝟓 )
			𝐑 𝐦 𝐦𝐚𝐱 ( 𝐉 𝟏 , 𝐋 𝐨 )
	(							)
	𝐀	𝐉 𝐍 max	[𝟏 + ( 𝐉 𝐍 𝐉 𝟏 max ) 𝒇 𝑵 (𝟎. 𝟎𝟓) 𝐚	]	𝐚 𝟏	𝐉 𝐍 max > 𝟎. 𝟎𝟓 𝐉 𝟏	-𝟎.𝟓
	-𝟏𝟓𝟕	(	{	𝟏 𝐑 𝐦 𝐦𝐚𝐱 ( 𝐉 𝟏 , 𝐋 𝐨 ) 𝐉 𝟏 𝐉 𝐍 max < 𝟎. 𝟎𝟓 )	+ 𝟏𝟎𝟎.

  08/40SG polymer solution through geometry [(5-0.2)/34]0.2 with Hencky strain 𝜀 𝐻 = 3.2 * . Annex table i: Viscosity and Q pd max of binary solutions prepared by mixing polymer solution collected at flow rate QA=8.33µL/s <Q max (M w app =18.9MDa) QB=83.33µL/s >Q max (M w app

	𝐌 𝐰 𝐚𝐩𝐩 =18.9MDa	𝐌 𝐰 𝐚𝐩𝐩 =12.5 MDa	𝛈 𝐩𝟎 (mPa-s)	𝐐 𝐩𝐝 𝐦𝐚𝐱
	𝛗 𝐀	𝛗 𝐁
	1	0	31.9	37.5
	0.9	0.1	29.2	44.16
	0.8	0.2	28.1	50
	0.75	0.25	27.7	52.5
	0.5	0.5	27.1	58.33
	0.25	0.75	23.5	75
	0.2	0.8	22.78	78.9
	0	1	21.5	83.33

Dans le chapitre 2, quelques généralités des différentes solutions de polymères sont discutées brièvement. Ensuite, le début de la transition d'étirements de la pelote polymérique et les phénomènes de viscosité extensionnelle décrits dans la littérature sont détaillés. Une revue de la formation des tourbillons en amont, des paramètres influençant leur apparition, du type de régime d'écoulement, des différentes transitions et de leurs dimensions est discutée. Un bref examen de la littérature sur la scission des polymères sous écoulement extensionnel révèle que la scission des polymères est principalement due aux contraintes d'extension et qu'il existe plusieurs paramètres influençant la scission. De plus, il est confirmé qu'aucune scission ne se produit dans les écoulements de cisaillement simples, car la contribution d'extension nette est négligeable. La scission des polymères en milieu poreux est estimée à partir de la perte de viscosité de la solution de polymère avant et après injection à travers les milieux poreux. Le terme d'ingénierie de « dégradation » quantifie cette perte de viscosité. En littérature, on retrouve plusieurs analogies faites avec de multiples contractions en série ou de multiples passages sur une géométrie donnée, pour aborder l'évolution de la dégradation en fonction de la longueur parcourue dans le milieu poreux. Une large gamme de lois de dépendance à la longueur du milieu poreux pour la dégradation à un taux de déformation (ou débit) donné est proposée. De plus, nous avons observé qu'il existe un écart entre les études d'injection linéaire (conditions de laboratoire) et l'injection radiale (conditions de réservoir).Au chapitre 3, nous discutons des différentes techniques expérimentales pour étudier la scission à la fois dans les géométries de contraction microfluidique et dans le milieu poreux.

Dans le chapitre 7, nous avons étudié la dégradation dans des milieu poreux de courtes longueurs (1-8 mm). Nous mesurons la réduction de mobilité comme le rapport des pertes de charge induites par une solution de polymère à celles dues au solvant. La courbe de réduction de la mobilité a un maximum à un flux donné, qui peut être utilisé pour identifier la scission et modéliser l'écoulement à travers des milieux composites.Nous avons proposé un modèle MP (multi-passes) pour prédire la courbe de réduction de mobilité d'un milieu poreux de longueur NL o , comme équivalent à effectuer N passages successifs sur un milieu poreux de longueur L o . Ce modèle a été validé contre des milieux poreux de différentes longueurs L = NL o . L'exposant de scission du modèle MP dépend de la perméabilité ; la diminution de la perméabilité peut diminuer l'exposant et augmenter la quantité de dégradation.Pour les injections linéaires, la dégradation augmente avec la longueur et sature à une longueur critique. La longueur critique diminue et la dégradation augmente à cette longueur critique avec l'augmentation du flux.

Annex I-Influence of Polydispersity

The Polydispersity index, which measures the heterogeneity in sizes of the polymer is defined to be, 𝐏𝐃𝐈 = 𝐌 𝐖 𝐌 𝐍 .

I.(i)

As discussed in section 3.1, the polymer samples from the SNF supplier are highly polydisperse. In this section our objective is to address the influence of polydispersity on the pressure drop ratio curves. For this we prepare binary solutions, of two different molecular weights. Shear characterization of solutions is performed as in section 3.2 and the rheological properties of the binary solutions are mentioned in Annex table i. Single injection experiments are performed as in section 3.7.1.

Pre-degraded polymers

To trace the signature of polydispersity in the degraded samples, and understand the mechanism of scission, the polymer solutions are degraded during a first pass.

A 15MDa-0.08/40SG polymer solution is injected through the contraction geometry [(5-0.2)/34]0.2 having a Hencky strain ε H = 3.2 * and the pressure drop ratio is measured for different flow rates, in section 3.7.1.The pressure drop ratio curve has a maximum at flow rate Q max = 37.5µL/s . We now choose 2 flow rates QA=8.33µL/s<Q max and QB=83.33µL/s >Q max and collect the solution at the outlet of the first injection respectively. Scission due to contraction decreases the molecular weight of the polymer chains. An apparent molecular weight of the degraded samples can be calculated by from Eq.(5.10) of section 5.5. The polymer solution collected at flow rate QA=8.33µL/s <Q max has an apparent molecular weight of M w app =18.9MDa, while for QB=83.33µL/s <Q max the apparent molecular weight is

II. (xxii)

We define the scission selectivity as the ratio of energy accumulated at the center of the chain to the average energy:

.

II. (xxiii)

When selectivity S=2, The energy is concentrated the chain break at its middle, but if S≈ 1, the energy is distributed homogenously over the chain and scission is random.

In Fig vi ), the selectivity calculated from Eq.II.(xxiii). is plotted as a function of Weissenberg number for geometries with different Hencky strains ε H . For Higher Hencky strain the scission is more probable at the mid chain while for lower Hencky strains ε H , the selectivity is equiprobable along the chain length and therefore the scission could be random.

RÉSUMÉ

L'épuisement des réserves de pétrole et l'augmentation des besoins énergétiques des forces démographiques croissantes, nécessitent d'optimiser les méthodes de récupération du pétrole. L'une de ces méthodes dans l'injection de polymère avec des solutions de polymère viscoélastique est une technique de récupération d'huile améliorée chimique (EOR). L'efficacité de la récupération d'huile par injection de polymère est fortement limitée par la dégradation mécanique et chimique des polymères. Dans notre travail, nous parvenons à prédire la dégradation des polymères lorsqu'ils sont injectés à travers des dispositifs de contraction et un milieu poreux. Dans un premier temps, nous utilisons des géométries de contraction hyperbolique uniques. En analysant des passages successifs sur ce dispositif de contraction unique, nous avons développé un modèle universel qui prédit quantitativement la scission. Nous avons trouvé ce modèle indépendant des propriétés physiques de la solution, mais dépendant du Hencky strain imposé par la géométrie. Lorsque le modèle a été étendu à plusieurs contractions en série, nous avons observé que la déformation vraie imposée à la solution de polymère est différente de la déformation théoriquement imposée par celle de la géométrie. Dans une étape suivante, le modèle prédisant la scission des dispositifs de contraction unique a été étendu aux milieux poreux. Nous montrons que ce modèle est applicable aux milieux poreux et peut même prédire la dégradation (ou la scission) des polymères dans des conditions de réservoir (écoulements radiaux).

MOTS CLÉS

Viscoélasticité, viscosité extensionnelle, contraction hyperbolique, scission de polymère. Abstract

The depleting number of oil reserves and the increased energy needs requires optimizing the methods for oil recovery. One of these methods in polymer flooding with viscoelastic polymer solutions is a chemical Enhanced Oil Recovery technique (EOR). The efficiency of oil recovery by polymer flooding is greatly limited by mechanical and chemical degradation of polymers.

In our work, we succeed in predicting the degradation of polymers when injected through contraction devices and porous medium. In a first step, we use single hyperbolic contraction geometries. From multiple passages on this single contraction device we developed a universal model (independent of the physical properties of solution) that predicts quantitatively the scission. We found this model to be dependent on the Hencky strain of the geometry. When the model was extrapolated to several contractions in series, we observed that the true strain imposed on the polymer solution is different from the strain imposed by that of the geometry. In a next step, the model predicting scission from the single contraction devices was extended to porous media. We show that this model is applicable for the porous media and can even predict the degradation (or scission) of polymers in reservoir conditions (radial flows).

Keywords

Viscoelasticity, extensional viscosity, hyperbolic contraction, polymer scission.