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Experimental study of non-Newtonian displacement flows in vertical eccentric annuli Storey, Stefan 2007

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E X P E R I M E N T A L S T U D Y O F N O N - N E W T O N I A N D I S P L A C E M E N T F L O W S IN V E R T I C A L E C C E N T R I C A N N U L I by STEFAN STOREY BASc. (Engineering Physics) University of British Columbia, 2005 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Mechanical Engineering THE UNIVERSITY OF BRITISH COLUMBIA May 2007 © Stefan Storey, 2007 Abstract The main focus of this work is to experimentally analyze displacement flows in vertical eccen-tric annuli. The experiment models the industrial process of oil well cementing and examines interface dynamics during the process of fluid displacement. Interface dynamics are described phenomenologically in terms of three broad classifications, namely steady, unsteady and static. Experiments were devised to simulate all three interface types and the results are used to analyse the displacement flows in terms of displacement kinematics and also in terms of displacement efficiency. Specifically, three main fluids are are investigated. Xanthan and Carbopol© solu-tions are used to simulate cement and drilling muds, and glycerin is used as an experimental control. The interface dynamics are captured by a simple optical technique and the images post-processed to ascertain interface speed and elongation and displacement efficiency. The results are compared to numerical predictions computed using a 1-D lubrication model. We conclude with a discussion on the significance of results achieved in this work together with relative merits and limitations of the experimental technique ii Table of Gontents Abstract ti Table of Contents " i List of Tables v List of Figures • y i Acknowledgement v " Chapter 1. Introduction 1 1.1 Process description ' 12 Commonly occuring problems in primary cementing 2 1.3 Thesis Motivation. 5 1.4 Review of previous experimental work 5 1.5 Thesis Overview • 1' Chapter 2. Experimental design consideration 12 2.1 Wellbore geometry 12 22 Rheological Considerations 13 22.1 Fluid characteristics 13 222 Selection of laboratory fluids 14 2.3 Modelling flow in eccentric annuli 15 2.3.1 Scaling field conditions 16 Chapter 3. Experimental Set-up 21 3.1 Overview 21 3.1.1 Annulus... 22 3.12 Flow Loop 23 3.1.3 Control and measurement devices 24 3.1.4 Optical Setup 24 32 Analysing interface dynamics 28 32.1 Methodology for finding velocity of Interface 28 33 Experimental fluid rheologies. 29 Chapter 4. Results and Discussion 35 4.1 Procedure: A typical experiment 36 42 Phenomenology 36 42.1 Gap Phenomena 37 422 Wall Effects 40 43 Part I: Overview of experimental results 40 4.4 Part II: Details of individual experiments 47 4.4.1 Newtonian control experiments 47 iii Table of Contents AA2 Herschel-Bulkley displacements 48 4.4.3 Power Law fluids displacing Herschel-Bulkley fluids 50 4.4.4 Double Interface Experiment 51 4.4.5 Effciency of displacement for a static finger 53 4.5 Summary of experiments • 55 Chapter 5. C o n c l u s i o n 57 Chapter 6. Recommendations -59 6.1 Image acquisition System 59 62 Experimental Methodology 59 63 Apparatus 60 6.4 Gap dispersion , 60 Bib l iography 61 A p p e n d i x A 63 A . 1 Mixing methodology for polymer solutions 63 A p p e n d i x B • 68 B. l Images of apparatus 68 A p p e n d i x C 71 C. l Secondary Flow Loop .: 71 C.2 Instrumentation and Control Schematic 72 iv List of Tables A . l Table of dimensional parameter settings 65 A.2 Table of dimensionless parameter settings 67 \ v List of Figures 1.1 Overview of the primary cementing process 2 1.2 Schematic of the primary cementing process 3 1.3 Set-up for numerical models 9 2.1 Displacement classification 16 3.1 Schematic of the eccentricity control 22 3.2 Schematic of main flow loop 23 3.3 Schematic of the optical set •. 25 3.4 Schematic of image set up '. 25 3.5 Sample image from displacement experiment 26 3.6 Schematic for image transformation 27 3.7 Transformed images with edge detection 27 3.8 Combined saturation curves 29 3.9 Rheological properties of xanthan 31 3.10 Yield determination of Carbopol 32 3.11 Rheological properties of Carbopol 32 3.12 Rheology of Carbopol with varying temperature 33 3.13 Rheology of Carbopol before and after pumping 34 4.1 Images of example displacements 38 4.2 Image of propagating thin finger 39 4.3 Images of gap dispersion 41 4.4 Static wall layer 42 4.5 Buoyancy parameter distribution 43 4.6 Output of lubrication model 44 4.7 Gap dispersion plots 45 4.8 Effect of eccentricity 46 4.9 Newtonian control experiments 48 4.10 Herschel-Bulkley displacements 49 4.11 Flow rate and eccentricity trends 50 4.12 Consistency curves 51 4.13 Schematic showing double interface set-up 52 4.14 Three interface experiment 53 4.15 Displacement efficiency at various flow rates. .54 4.16 Amplitude of instability 55 4.17 Comparison of Carbopol and Xanthan displacements 56 B.l Full image of apparatus as built June 2006 69 B. 2 Close-up image of apparatus 70 C l Schematic of secondary flow loop 71 C. 2 Schematic of measurement and control system 73 vi Acknowledgement This work could not have been done without the support of many people. Ian Prigaard and Mark Martinez have been great supervisors, providing guidance and support. I would also like to thank Schlumberger and Trican Oilfield Services for their logistical support during the project, and for their assistance during the 2005 field trip to Alberta. A special thanks to Andreas Putz and Teodor Burgulea for their help in the Complex Fluids Laboratory. This work was completed with the support of an NSERC. I am very grateful to NSERC for providing me with this funding, and continuing to support my research. vii Chapter 1 Introduction 1.1 Process description Primary cementing operations are vital to the process of completing oil and gas wells. After a borehole is drilled, the borehole walls are often lined before any oil and gas can be extracted. Typically, this lining is comprised of a steel pipe, called the casing, and a cement backing. After a section of wellbore has been drilled and the drill string has been removed, all that remains in the borehole is drilling fluid. Before the cementing operation can begin, a steel casing pipe is inserted into the borehole which is filled with drilling fluid. To complete a section of the well, cement must be injected into the gap between the casing wall and the formation bedrock, and the drilling fluid fully displaced. Figure 1.1(a) shows a schematic of cement being injected in the annular gap and Figure 1.1(b) shows a fully cemented casing. Once the cement has set, the sealed steel casing serves to support the walls of the formation rock and to ensure zonal isolation along the length of the well. To achieve the displacement, a sequence of fluids is pumped down the inside of the casing, and upon reaching the bottom of the well the fluids are forced back upwards inside the annular gap. The sequence of fluids varies according to the type of drilling mud, and depth of the well-bore, but typically one or more spacer fluids are pumped first, or a scavenger cement slurry, followed by the cement slurry. The final step is to ensure that no cement remains inside the casing. This is achieved by pumping drilling fluid down the inner pipe to push any remaining cement to the bottom of the wellbore. When the rear edge of the cement reaches the bottom of the well, pumping ceases and the cement is allowed to set. The section of the well is now sealed 1 Chapter 1. Introduction Figure 1.1: Overv iew of the p r imary cementing process, showing the basic steps involved ce-ment ing a wellbore. and dr i l l i ng can recommence. T h e d r i l l s t r ing is reinserted and the bo t tom of the well can be dr i l led to a greater depth. T h e entire process is repeated unt i l the desired number of sections are centered in place and the correct borehole depth and length is at ta ined F i g . 1.2 shows a typ ica l sequence of events. A ful ly completed wel l is composed of many nested casings arranged telescopically w i th each casing over lapping the previous section. A s the well is dr i l led deeper, the casings typical ly have decreasing diameters. We l l inner diameters range f rom 50cm and decline to less than 10cm in the produc ing zone. Cemented sections have lengths that vary according to f ield condi t ions, but are typ ica l ly 200-1000m. 1.2 Commonly occuring problems in primary cementing Ideally, the cement and steel casing wi l l prevent any fluid communicat ion between the fluids in the format ion rock, and the hydrocarbons being extracted through the inside of the casing. A n y hydrocarbons that leak f rom a produc ing well can cause problems for the environment. These inc lude blow outs, damage to subsurface ecosystems, po l lu t ion of aquifers, and leakage of poisonous gases. In the event of a leak, the source of the fluid migrat ion can be very diff icult to locate in the well and any ensuing remedial work, such as squeeze cementing, can be very 2 Chapter 1. Introduction Drill new Remove Insert Pump Pump Displace End of stage of drillpipe steel spacer lead & tail mud in operation well casing fluid slurry annuius Figure 1.2: Schematic of the primary cementing process, showing the various stages (left to right) in cementing a new casing. expensive. Leakage problems can persist not only during the extraction phase of the well operation, but also after wells are shut-down for abandonment. Leaking wells are often very difficult to seal and require continual monitoring long after the well has been shut-in. Sustained casing pressures at the surface are a common indicator of cementing failure. In 1996, of the 13,573 active completions in the Gulf of Mexico, some 8,122 have sustained casing pressures and flow through the casing to the surface [1]. There are several known causes of casing cementing failures. Firstly, small pockets of mud can become entrained in the cement and cause contamination. The small pockets increase the permeability of the cement and can allow fluid migration. Similarly, gelled drilling mud may remain stuck to the walls of the formation rock or casing pipe and again cause contamination of the cement. Often this is occurs as long fingers of drilling mud trapped on the narrow side of the gap. In the event of either of these two cases, the mud can become a fluid conduit. In cases of severe contamination, the final strength of the cement may be compromised. The Minerals Management Service believes the majority of excess casing pressure and surface flows are caused by the latter case, namely incomplete cement seals allowing fluid migration along long fingers of undisplaced drilling mud. The goal for all cementing jobs is that the drilling mud must be fully displaced from the annular 3 Chapter 1. Introduction gap, and that the displacement front, or interface, advances steadily up the well at the pumping velocity. Ideally, the wide and narrow sides of the interface progress at the same speed during the entire cementing job. However, maintaining a steady interface can be technically very diffi-cult and when the displacement front advances more rapidly on the wide side, then the interface elongates producing an unsteady displacement [2]. This can result in long fingers of drilling mud remaining undisplaced. As the cement cures, water is removed from the mud, producing a porous channel along which reservoir fluids can migrate upwards. The fingering is a phenom-ena common with viscous displacements, and is exacerbated by the viscoplastic rheology of the drilling mud. Specifically, the drilling mud readily forms a gel whenever the shear stress in the flowing fluid falls below the yield stress of the fluid. This can occur in the narrow side of the annulus where the drilling mud becomes static and enters a gel-like state. Once the drilling mud has become a gel, it can be technically challenging to return the gel to a fluid state [3]. In fact, if the gel remains static for long periods of time, it may become a dehydrated and form a deep porous filter-cake. Whilst permanent filter cakes are always present as relatively thin layers, if the gel remains static for long periods of time then a secondary filter cakes develops and these are very difficult to erode and remove. Finding strategies to break up and remove the gels and filter-cakes have been the focus of primary cementing research for the past 40 years. Understanding the causes of unsteady displacements can be understood by considering the geometry of the annulus. In an oil well, the annular gap is typically eccentric, with the ec-centricity increasing as the well becomes more deviated from a vertical orientation. Highly deviated wells have centralizers attached to the outer wall of the steel casing to prevent the casing slumping down towards the lower side of the formation hole. Centralizers are expensive, and difficult to place correctly which means that the ccntralizer spacing interval is often large [4]. Consequently, slumping is unavoidable between centralizers. In addition to the varying eccentricity, the formation hole is rarely uniformly smooth. Enlarged regions called "washouts" can be encountered, and the frequency and size of these sections depends on the fragility of local rock structures. Drilling mud in the washed out sections can be difficult to displace and hence difficult to seal effectively with cement. 4 Chapter 1. Introduction 1.3 Thesis Motivation The current cementing methods employed by industry predominantly based on. hands-on expe-rience with tried and tested field guide lines [2, 4]. For vertical wells, these rules state that the flow rate must be sufficiently high and preferably turbulent to prevent the formation of mud channels on the walls of the formation rock [5, 6]. If the flow is not turbulent, then these rules further state that there should be a increasing gradient of viscosity and density so that each successive fluid generates higher frictional pressure at the wall and is heavier than its prede-cessor [7]. Over the past decade, instrumentation has helped field engineers with measuring critical parameters such flow rates and pressure [8]. While this has helped engineers to keep pump rates so that the cementing job is conducted within the pore-fracture envelope, relatively few numerical tools are available to predict whether a static finger of drilling mud may be left behind. Providing industry with better prediction methods is critical to the success of the Canadian oil industry. Prediction tools have already been developed by Frigaard et al. and it is the purpose of this project to design and build apparatus that is capable of providing validation data. Although the motivation of this work is largely industrially based, the results also extend the scientific understanding of annular displacement flows. To date, extensive experimental work has been completed for single-fluid non-Newtonian axial flows in annular geometries. However, relevant literature for non-Newtonian displacement flows is more limited. This project aims to build an apparatus and produce an experimental methodology that is capable of replicating scaled displacement flows that will be able to extend the scientific understanding of displacement flows. 1.4 Review of previous experimental work Research into well bore displacement flows began in earnest during the 1970's with large scale laboratory based experiments to simulate primary cementing. Clark and Garter built a 10ft permeable sandstone tube to simulate formation rock, and varied the inner diameter to simulate 5 Chapter 1. Introduction a washed out section [9]. An inner pipe was inserted to simulate steel casing and drilling mud was injected into the annular gap. The temperatures and pressures were raised to simulate field conditions, and displacements with cement were run at various eccentricities and various inner pipe rotation and reciprocation rates. After the displacement, the cement was allowed to set, and the core was removed from the annulus. The core was then cut into seven sections and measurements were made of the area of cement and remaining mud. They concluded that to produce an efficient displacement, several strategies should be employed to "condition" the drilling mud immediately before cementing. These included reciprocation and rotation of the steel casing to help to dislodge static mud. For washed out sections of the annulus reciprocation was more effective than rotation. The next major experimental study was undertaken by Zuiderwijk et al [10]. They used an experimental set-up similar to Clark and Garter, but this time using impermeable walls, and a fixed inner pipe. They used a radioactive tracer, An 198, mixed with the drilling mud to determine the amount of mud left behind after a displacement. By measuring the residual radioactivity, they could determine the fraction of drilling mud displaced. By varying the eccentricity, they concluded that keeping the casing centralized was vital to optimizing dis-placements. They also experimented at high flow rates. To produce a turbulent flow regime, they increased the flow rate and lowered the viscosity of the displacement fluid. They concluded that efficiency increased when the displacement fluid was turbulent. For slow flows less than 0.5ft/s they concluded that the cement density and viscosity should be higher than that of the mud. They also recommended that a pre-flush be used to prevent mixing of the mud and cement. At the end of the decade, Haut et al. conducted extensive work exploring the relative im-portance of five of the major findings of previous work [11]. They compared the condition of the drilling fluid, pipe wall permeability, pipe centralization, flow rate, and density difference. Their apparatus and data acquisition was similar to Clark and Garter, except for they used a fixed inner pipe. Most of their findings confirmed the earlier work of Clark and Garter, but they did uncover some new phenomena. They found that with test sections with realistic per-6 Chapter 1. Introduction meability, mud that had lost its fluidity was never displaced. Also, they tested cement slurries with high yield points and showed that increasing the yield point did not promote efficient dis-placement. Interestingly, they found that increasing the density of the cement did not promote displacement for the case when the drilling mud had been static for long periods of time. They hypothesize that this is due to an increase in density of the gel due to water migration through the permeable outer wall. Finally, Haut et al. conducted further experiments to investigate the influence of higher axial velocities [12]. Using the same apparatus as before, they found that high flow rates provide better displacement than plug flow rates. This was attributed partly to turbulent conditions. During the late 1980's, Lockyear et al. continued the experimental research work on a large scale 15m annulus constructed for British Petroleum [13, 5]. The device was inclinable from vertical to horizontal, with a movable inner pipe to control the eccentricity. The test sections were 0.244m outer diameter, 0.178m inner diameter casing and designed to have interchangeable permeable and impermeable outer walls. Data was acquired using three sets of five conductivity probes which enabled them to measure interfacial velocity profiles. The findings are in good agreement with the previous findings, but they reinterpreted the dynamics of the flow regime. They believed that for an eccentric annulus, the concept of strictly laminar or turbulent flow was ambiguous , and that it is possible to achieve both flow regimes in the same pipe. However, they are not clear whether they are referring to transition flows, or spatial variability across the annulus (the wide section of an annulus will produce larger Reynolds number than the narrow side because of both higher flow rates and wider gaps). Later experiments showed that the use of spacer fluids was vital for optimizing displacement flows. They found that the viscosity of the spacers should be less that the cement, but could not determine whether turbulence in the spacer fluid was important. Experimental work was continued in the 1991 by Beirute, Sabins and Ravi [14]. Using the same technique as Lockyear, they investigated the role of drilling mud conditioning. Whilst they did not investigate displacement flows insofar as using cement to displace the mud, they 7 Chapter 1. Introduction instead looked at methods to reduce the amount of mud in a gelled state. The experimental • technique consisted of filling the annulus with drilling mud and then circulating tagged drilling mud through the flow loop. Using conductivity measurements at the annulus wall, they could determine the fraction of untagged fluid at, sections along the annulus. They confirmed earlier findings with regards to rotation of the casing to breaking the mud gel, and agreed that in-creasing the flow circulation rate helps to dislodge gelled mud on the narrow side of the annular gap. They also showed that the use of cable loop wall cleaners (scratchers) across zones of high permeability helped to remove the filter cake. They issued a series of recommendations for industry where they stressed the importance of maintaining drilling fluid movement at all times because any extended period of fluid quiescence would promote gel build up. The final industrial study to be examined here is the work completed by Jakobsen et al. They investigated the consequence of buoyancy effects in eccentric deviated wells. For a 60 degree deviated well, with an offset of 55 % towards the lower side of the axial center, they showed that a high density displacing fluid can increase efficiency. Even with a density difference of of only 5 %, the dense displacing fluid was observed to slump under the lighter drilling mud and displace it from the narrow gap [15]. This concludes the review of industrial research. In fact, very little experimental industrial research work had been completed since the early 1990's. Instead, the academic community continued on with the experimental research, and placed the focus of work on exploring the fundamental fluid mechanics driving the displacement flows. Much of the academic work com: pleted examines the non-Newtonian flow dynamics without displacement. However, this work is pertinent to our discussion of interface dynamics since the system is predominantly driven by axial flow. Escudier et al: investigated two cases of fluid flow [16]. Using Laser Doppler Aneomometry, they examined the velocity flow field for an eccentric annular flow with 0.1 % CMC and 0.1 % Xanthan gum in a water based solution. At Reynolds number 263, they found 8 Chapter 1. Introduction that the axial velocities have flattened profiles relative to Newtonian flows. They also compare their experimental results to the work of Nouri et al. at higher Reynolds number (Re = 599), which show velocity peaks far in excess of Newtonian flows on the narrow side [17]. However, this could be explained by the discrepancies in Reynolds number, with a possible change over to turbulent flow affecting the velocity profiles. For displacement flows, Tehrani, Bittleston and Long published some interesting experimental results in 1993 [18, 19]. Using the conductivity technique pioneered by Lockyear, they investigated the role of density in displacement flows. They found that increasing the density of the cement can promote azimuthal flow near the interface. However, they observed that gravity driven azimuthal flow is prone to severe insta-bilities which accelerate the displacement process but may leave behind an immobile strip in the narrow gap. In order to assist the cementing industry to attain zonal isolation for these more challenging boreholes, predictive numerical tools have been developed by Frigaard et al. Two models have been constructed to simulate displacement fluid flow during cementing. The first model is a 2D Hele-Shaw model which unwraps the annular gap into a slot of varying thickness. Figure 1.3 shows the eccentric annular gap as a non-uniform thickness slab where axial length >> circumference >> annular gap. Figure 1.3: Set-up for numerical models 9 Chapter 1. Introduction The boundaries conditions on the edges of the slot are matched to ensure that symmetry is imposed in the azimuthal direction. To simplify the equations, they average the fluid velocity across the gap to reduce the problem from 3D to 2D flow in the axial and azimuthal directions. The 2D Hele-Shaw model consists of solving a nonlinear elliptic inequality equation for the stream function, coupled to an equation for interface advection [20]. They investigate the effects of varying eccentricity and show that as the annular gap offset increases then interface becomes unsteady. They also show that for low eccentricities, the viscoplastic fluids arc often fully yielded, but as the eccentricity increases further the narrow side fluid remains unyielded. For large eccentricities the interface becomes static and a narrow side mud channel forms. Although the 2D model is capable of producing a physical shape for the interface, the cost is in computational time. However, Frigaard et al. also constructed a fast solver based on a ID lubrication model. The ID model is computationally less expensive, but does not provide any physical information other than wide side and narrow side velocity. However, this is sufficient for industry, where the main design requirement is an indication for whether the flow is "steady" or "unsteady". The model starts with an assumption that a long finger has already formed where fluid one (the displacing fluid) occupies the wide side of the annulus, and fluid two occupies the narrow side. The model then determines the velocity in both fluids based on fluid rheologies, eccentricity and buoyancy ratio. The velocities are then used to generate a flux function that appears in the lubrication/thin film style equations for the interface motion. Analysis of the flux function shows the elongation rate of the interface which then allows us to determine whether the displacement would result in an unstable or stable interface. If the model shows that the interface speed on the wide side is faster than the narrow side, then any interface would be unsteady. If the interface speed is calculated to be faster on the narrow side, then the flow regime must be steady (faster flow on the narrow side is actually not possible. Physically, for steady displacements there is a restoring azimuthal flow which keeps the narrow side and wide speeds the same). It should be noted that limitations outlined are i) assumption of homogeneity of fluids across the gap; ii) assumption of laminar flow in the entire annular domain; iii) neglecting dynamics close 10 Chapter 1. Introduction to the interface and concentrating on the bulk flow. 1.5 Thesis Overview This thesis summerizes the experiments performed to generate steady and unsteady displace-ments by visualizing the interface between pairs of fluid as the flow along an annulus. We experimentally investigate the interface between displacing fluids in a vertical annulus with fixed length and diameter. We vary the flow rate and eccentricity and investigate the effect of fluid rheology on the efficiency of displacements. The outline of the thesis is as follows. A brief background for the physics behind displacement flow is presented in chapter 2, including the scaling arguments used to match the field conditions to the laboratory apparatus. We discuss the experimental apparatus built and the capabilities and limitations of the device. In chapter 3, we discuss the experimental setup and give an overview of analysis techniques. In chapter 4, both qualitative and quantitative results are presented and analysed. Thereafter, a comparison of the experimental results with the numerical simulations is undertaken. 11 Chapter 2 Experimental design consideration A number of different factors influence our experimental apparatus and design. These include geometrical, rheological and kinematic scalings. Each is discussed in detail, along with simpli-fying assumptions. 2.1 Wellbore geometry The dynamics of displacement flows in eccentric annuli are driven by the geometrical configura-tion of the annulus and the fluid rheologies for the fluids. The relevant geometrical parameters to consider are annular eccentricity, angle of well inclination, annular wall porosity, and wall roughness. For a concentric displacement the fluid flow divides equally around the gap and the fluid velocity is radially symmetric. For an eccentric displacement, the fluid will instead preferentially flow in the widest gap region where the flow is less impeded by frictional drag from neighbouring walls. The ensuing fluid velocities will be highest in the wide gap region and lowest in the narrow region. In contrast to the concentric case, the fluid flow is radially asymmetrical with respect to the narrow and wide gap regions. Nevertheless, for stable flows the symmetry is retained in reflection across the bisection of these regions. The next geometrical consideration is the the angle of inclination. This is critical for non-vertical displacements where density differences in the fluids becomes important. Here, fluids with high densities are driven by buoyancy forces which cause them to slump and flow under 12 Chapter 2. Experimental design consideration lighter fluids. This becomes important for inclined and horizontal cementing jobs, and criti-cal for horizontal extended reach wells where slumping can dominate displacement dynamics. Even though this study focuses on vertical well displacements, subsequent projects will look at inclined wells. Therefore it was nessesary to incorporate both inclination and eccentricity variations into the apparatus design. The next geometrical parameter to be considered is wall roughness and washouts. This is important for cementing in the field, where abraded walls in the formation rock can encourage thick layers of gelled drilling mud to persist even under high flow rates. However, for the purposes of this experimental study, we investigate the case of a 'gun barrel' hole where wall roughness length scale is very small compared to the gap size. Finally, although porosity is not purely a geometrical parameter, it is worth briefly considering here. Porous walls are commonly found in the field which, under certain conditions, can allow a net flux of fluid through the borehole wall. Indeed, much attention is given to the drilling fluid rheology and density to ensure that pore pressure is matched as closely as possible at the drilling depth of the rig. A net inflow of formation fluids will cause contamination of the drilling fluid, and a net outflow can induce cake formation on the borehole wall. However, for this study it is assumed that the axial fluid velocity is much greater than the pore fluid velocity which justifies the use of non-porous pipe material. 2.2 Rheological Considerations Fluid rheologies control the dynamics of the flow regime in the gap and are especially important for the local fluid dynamics at the interface. In all cases, the fluids used in drilling and cementing operations are non-Newtonian. 2.2.1 Fluid characteristics In order to characterize the fluids, the following constitutive equations are used to model the behaviour of the fluids during deformation. For spacer fluids and liquid cement, the Power law model is frequently used, where f is the stress, 7 is the shear rate, and k is the consistency and 13 Chapter 2. Experimental design consideration n is the power law number. Note we adopt the convention that a hat above a variable donates a physical quantity (with units), and a star denotes a characteristic scale. f = «^yn (2.1) For drilling mud, which is viscoplastic, the model choice is less straight forward. Typically, a Hershel-Bulkley (HB) fluid model is preferred, which can be considered as a Power law fluid with a yield stress. When the drilling mud is deformed with a shear stress above the yield stress then the fluid will flow. When the shear stress falls below the yield stress, the fluid will cease to flow, and enter a gel state with strength ry f = fy + AV (2.2) As previously discussed, the rheology of drilling muds does exhibit some HB behaviour, but in reality the rheology is considerably more complex than this simple model. Drilling fluids have strong thixotropic properties which mean that the characteristics of the fluid vary with time. Of particular importance is the build-up in gel strength that occurs if the drilling mud is left uncirculated in the well. When combined with fluid loss to the formation rock and mud dehydration, this can mean the gel is difficult to remove if the well is shut down for any extended period time. The rheology of drilling mud also varies with temperature, which although is not a significant issue at the well head, can be important at increasing depth into the formation rock. Since viscosity decreases with increasing temperature, the ability to entrain particulate matter will decrease with depth. For the purposes of this experiment, we will assume that temperature varies slowly with depth and thermal effects are ignored. 2.2.2 Selection of laboratory fluids The selection of experimental fluids was driven by the need to simulate the behaviour of field fluids. To simulate cement (or spacer fluids), which is typically characterized by a power-law model, an aqueous xanthan gum solution was mixed. Xanthan gum is a transparent, high 14 Chapter 2. Experimental design consideration molecular weight polysaccharide commonly used in the food industry to increase the viscosity of organic fluids. Xanthan has been widely used for research purposes, partly because it is relatively inexpensive, but also because the rheology varies little with respect to pH and water purity. Finding a fluid suitable to simulate the visco-plastic properties of drilling fluid is more chal-lenging. In general, aqueous Carbopol© polymer solutions are frequently chosen over other viscoplastic fluid such as bentonite suspensions. Carbopol© is unique in that it is highly trans-parent, a property vital to visualizing interface dynamics in the annulus. Nevertheless the cost is in the difficulty of finding a good rheological model, and obtaining reproducible results. It is a poorly understood material, indeed there are several research efforts underway searching for a description of the micro-structure. However, it is outside of the scope of this project to con-duct a detailed analysis. Instead, only an outline of the macroscopic properties of Carbopol© is discussed. Carbopol® polymers are long cross-linked polyacrylic acid polymers which are highly sensitive to local pH values in aqueous solutions. The polymer chains become entangled when subjected to a pH environment between 4-8 causing a dramatic increase in viscosity and yield stress. Carbopol® has an additional advantage of being non-toxic and inexpensive, two properties that lend itself to its frequent use as a thickening agent in cosmetic products. The constitutive model used to characterize Carbopol® is given as a Casson fluid by the manufac-turer, Noveon chemicals. However, the academic community generally uses a Hershel-Bulkley model which as discussed can simply be regarded as a power law fluid with a yield stress. 2.3 Modelling flow in eccentric annuli One of the fundamental difficulties of investigating the interface dynamics in the field is the difficulty encountered when attempting to observe or measure underground flows. The chal-lenging environmental conditions make direct observation almost impossible, whereas creating a small scale experiment in order to visualize the interface is relatively straightforward. A brief outline of typical interface dynamics is outlined below, along with geometrical considerations and scaling arguments. 15 Chapter 2. Experimental design consideration From the perspective of laminar displacement flow models [21] [20], there are three broad classfications of displacements. From an initial condition of a flat interface, Figure 2.1(a), the interface will develop into either an unsteady or steady displacement. Secondly, for (a) (b) (c) (d) Figure 2.1: Displacement classification. (a)Initial condition, (b)Static, (c)Unsteady, (d)Steady viscoplastic fluids, a subset of unsteady displacements called static displacements can occur where the fluid on the narrow side remains in an immobile gel state, as shown in Figure 2.1(b). Unsteady displacement occurs when the wide side advances more rapidly than the narrow side, see Figure 2.1(c). Steady displacements occur when the fluid flow on the narrow side and wide side advance at the same speed. Figure 2.1(d). The overall scope of our study is to design and construct a scaled experimental annulus in which we can perform displacement experiments that distinguish the above flow regimes. Therefore we are largely influenced by the modelling work in our design. We intend to measure the shape and velocity of the interface using optical techniques, and then compare the results to those produced by numerical model predictions. 2.3.1 Scaling field conditions In order to scale the field conditions to a practical laboratory experiment, each of the physical parameters are scaled kinematically and/or geometrically. The geometrical scalings determine 16 Chapter 2. Experimental design consideration the design of the experimental apparatus and the rheological scalings determine the design of the fluids used in the laboratory settings. The scaling arguments closely follow those outlined in [22], where they are derived from the non-dimensionalization of the Navier Stokes equations. In the field, the cross-section of the well is assumed to be that of an eccentric annulus, with inner radius fi(z), corresponding to the outer radius of the casing and outer radius f0(z) corresponding to the inner radius of the drilled hole. At each depth z, the mean radius fa(z) and the mean half-gap width d(z) are defined by: ra=l(Ro + Ri) (2.3) d=l(Ro-Ri) (2.4) The most important geometrical radial scaling is as the dimensionless gap scale S = -^ which is defined as ( R 0 - Ri) <5=-V^ ^ (2-5) ir^Ro + Ri) The field range for this value is typically in the range 0.01 to 0.1. This scaling physically means that the gap must be much smaller than radius of the borehole. The last geometrical consideration is that the axial dimension must be much larger than the borehole radius, such that the ratio of radius to axial length is small. The physical dimensions of the experimental annulus is designed within these constraints. The kinematic scalings include the velocity of the fluids and the fluid rheologies. First of all, the bulk velocity, density and rheologies are scaled. = p* = max[pfe]. kk = (2.6) 17 Chapter 2. Experimental design consideration f* = m a x [ f f c , y + K f c ( 7 * ) , * ] ) k (2.7) In regions where the fluid is yielded, the effective viscosity 77 is: •nk , Tk,Y Vk = Kkl + ^ T -T (2.8) The Reynolds number is defined as the ratio of inertial forces to viscous forces. Due to the extensive use of non-Newtonian fluids, this means that the Reynolds number must be expressed with non-constant viscosities p. Note, that the relevant length dimension is the average gap width. In the field, the range of Re is typically 0 - 100, but our focus is on recreating slow flow conditions that exist for cementing deep vertical and ERD wells where Re typically falls in a smaller range of 0 - 10. This means that the viscous forces dominate and the flow is laminar. In order to compare our experimental, results to the output of the numerical models then a weaker condition can be placed by the Hele-Shaw model derived by Frigaard et al. [20]. Here, 3-D flow dynamics can only be ignored in the model if JJ?e<<l. Since f5<<l then this condition is satified by all our experiments. Now we consider the Bingham scaling which is specific to viscoplastic fluids. For drilling mud with a yeild stress, a Bingham number must be defined to scale the yield point of the drilling mud to the experimental fluid. ^ (2.9) (2.10) 2Vp 18 Chapter 2. Experimental design consideration Physically, the Bingham number gives the ratio of force required to yield the fluid relative to the viscous forces. Consequently, a large Bingham number means that the fluid has a high gel strength compared to the viscous forces of the same fluid in motion. The field range is typically The Stokes number is considered which compares the force due to visous stress relative to the gravitational forces on the fluid. The field range is 1-100. Finally, buoyancy forces are considered by examining the buoyancy number b. Here the yield strength of the fluid in the gap is compared to the density difference of the fluids, where p\ and p2 are the densities for fluid 1 and fluid 2 respectively. For values of b less than -1.5, buoyancy is the dominant parameter in most displacements, and dominates over fluid viscosities. <"» Now that the scalings have been defined, the parameters defining the fluid rheologies can be correctly designed. However, the scaling equations are coupled via the viscosity and yield stress terms. This means that the correct ranges can only, be achieved by iterating through various values for each of the parameters. Once the values were determined, and were checked to be physically possible to achieve in a laboratory setting, the design process for the fluids was com-plete. The choice of scalings to be examined during the data analysis was limited to the eight most relevant parameters. Of the scalings discussed, the most pertinent to the Hele-Shaw model and to the experimental data analysis are the following parameters: The parameters e and b contain all geometrical, buoyancy and kinematic conditions. The groups 771, r/2 contains all six 0 - 10. St = T* (2.11) 19 Chapter 2. Experimental design consideration rheological parameters, namely T\, T2, KI, K2, ni and n2. For the remainder of the thesis, the groups e, b, rji and r]2 will be discussed. 20 Chapter 3 Experimental Set-up In the previous chapter the design parameters were determined'and discussed. The next step in the design process is to build the annulus, support structure and instrumentation for data acquisition. Details are now given for the annulus construction, flow loop, optical set-up, and a description of the post processing techniques required to extract velocity data from the captured images. The final section of the chapter includes details of the fluid design. The rheological measurements for xanthan and Carbopol are discussed and we examine the factors that can cause the rheologies to vary in our laboratory environment. 3.1 Overview The two pipes of the annulus are mounted in a fully adjustable support frame. The frame itself is mounted on a bearing which connects the frame to a static leg mounted to the laboratory floor. The support frame pivots on the bearing and allows inclination adjustments from 0 -90 degrees. The support frame also houses the mirror used to assist in observing the fluids flowing inside the annulus. During experimental trials, the fluid is pumped by a progressive cavity pump (PCP) and flow speed is regulated with PID control to ensure repeatable flow rates. The apparatus is equipped with a magnetic flow meter, two pressure transducers, and a thermocouple. Data logging from these peripheral devices and pump control is achieved with a National Instruments Data Acquisition system. 21 Chapter 3. Experimental Set-up 3.1.1 Annulus Now that the geometr ical constraints are calculated v ia the scal ing arguments, the annulus could then be sized w i th f c at 1.91cm, r i at 1.11cm and a length of 188.3cm. The outer p ipe is constructed f rom acryl ic tub ing w i th a wal l thickness of 12.7mm. T h e inner p ipe is an a lu-m i n i u m pipe w i th wal l thickness of 1.58mm. It is mounted on two adjustable stainless steel bol ts that al low the inner p ipe eccentricity to be adjusted relative to the fixed outer pipe. The eccentr ici ty is easily adjusted by rotat ing the mount ing screws unt i l the desired eccentrici ty is achieved. Eccent r ic i ty is measured w i th micrometer depth gauges mounted on either end of the annulus. Sma l l adjustments are usual ly required to ensure that the inner p ipe is perfectly al igned w i th the outer p ipe. See F igure 3.1 for a schematic and image of the set-up. T h e annulus is immersed in a ta l l tank w i th a square crossection and fi l led w i th glycer in. T h i s tank reduces opt ica l d is tor t ion by match ing the index of refract ion on the curved acryl ic p ipe to the glycer in. Cross-sectional View Front View Adjustment Depth port Gauge Figure 3.1: Schematic of the eccentrici ty control for the inner p ipe. T h e crossection shows the inner p ipe mounted on the adjustment screws and a front view image shows the adjustment por t and micrometer depth gauge. 22 Chapter 3. Experimental Set-up 3.1.2 Flow Loop The main flow loop can be broadly broken into two groups of components, a main loop that is used during experimental trials and a secondary loop that is used for preloading and draining the apparatus between experiments. The main flow loop consists of a progressive cavity pump (PCP) supplying the annulus with displacing fluid and an outlet pump that drains the fluid for recirculation or waste disposal. All fluid flow occurs in the annular gap that is bounded at the outer diameter of the aluminium pipe at 22.2mm and the inner diameter of the acrylic pipe at 44.5mm. The flow rate is controlled by the use of a series of valves, and by the PCP Seepex MD Dosing with a maximum flow rate of 2 L/minute and driven by with a 1 1/2 hp single-phase AC motor. Variable speed operation is achieved through variable frequency drive speed commands sent from the computer via RS-232 connection. See Figure 3.2 for a schematic of the main flow loop. A detailed schematic of the secondary loop is included in Appendix C. Pressure Transducer Camera Drain Pump Flow Meter Figure 3.2: Schematic of main flow loop 23 Chapter 3. Experimental Set-up 3.1.3 Control and measurement devices All devices are calibrated before experimentation. The two pressure transducers are cali-brated with static water columns of increasing height and their outputs are A-D converted and monitered by a computer acquisition system. A thermocouple is mounted inside the inlet pipe of the annulus but found to be very sensitive to noise generated by the motor controller. However, because the temperature of the fluids does not very rapidly with respect to time, the readings were time averaged and calibrated against a standard thermometer. The flow rate is measured with a Cole Parmer pilot-scale magnetic flowmeter EM101-038 which the manufac-turer specifies as accurate to 2 percent. This was checked with a simple calibration experiment by measuring the mass of fluid pumped over a fixed time interval. Finally, the output of the flow meter was directed to the control computer via the A-D conversion of a 4mA variable current. A scale measures the weight of the displacement fluid container and can be either be used for calibration of the flow rate or for weighing fluids and solids required for mixing the Carbopol and xanthan solutions. The output of the scale is acquired by a direct R.S232 connection to the computer. See Appendix C for a schematic of the control and measurement system design. 3.1.4 Optical Set-up In order to observe and record the interface dynamics, a digital camera is mounted on a linear actuator and which images the flow dynamics. The camera lens is a 35mm SLR compatible lens, mounted to the camera body with anF-C adaptor. The linear actuator is controlled via serial commands to a micro controller connected to the computer. Commands that control actuator location, velocity and acceleration are controlled via LabView. The camera images 270 degrees of the annulus by the use of a mirror. See Figure 3.3. Before presenting images acquired during experimental trials, a brief overview of the image set-up will given. Each acquired image contains a reflected and an unreflected image. The reflected image is simply the view of the annulus taken via a mirror mounted at 45 degrees. See Figure 3.4 24 Chapter 3. Experimental Set-up Figure 3.3: Schematic of the opt ica l set, showing a cross-section of the annulus and opt ical rays ar r iv ing di rect ly f rom the annulus and indirect ly v ia the mirror . F igure 3.4: Schematic of image set up. (a) Shows the two f luids stacked one above the other at t = 0, and (b) shows the displacement progressing after pump ing starts 25 Chapter 3. Experimental Set-up The images are captured at a frame rate that depends on the flow rate. The fastest frame rate (10Hz) is required for high flow rates and 4Hz for slow flows. To ensure that the ROI in each succesive experiment is consistent, a ruler is permanantly attached to the annulus and the ROI located relative to this fixed position. The images are captured in uncompressed 8 bit monochrome format, with a signal to noise ratio of 50dB. Figure 3.5 shows an example image. Figure 3.5: Sample image from displacement experiment. (a)Direct view (b)Reflected image, (c) Line profile located at 210mm above gate valve Before any analysis or image registration can be completed, the images must be first transformed by relating the projected image on the CCD to the circumferential location in the annulus. Assuming the gap is small, then the following simple geometrical transformation shown in Figure 3.6 can be completed. For a small gap scale, the transformation is described by Although the transformation effectively unwraps the acquired images, the noise is increased at the image edges, and information close to the annulus wall is disregarded. Next, the information from both direct image and the reflected image can be combined to (a) (3.1) 26 Chapter 3. Experimental Set-up R - x JR 1—• y ——• CCD Xi Figure 3.6: Schematic for image transformation generate data for 270 degree of the annulus. First the edges are located on the transformed image, and then the two images are colocated and registered. See Figure 3.7. Horizontal [cm] 2 4 Horizontal [cm] 4 6 8 Horizontal [cm] Figure 3.7: Transformed images with edge detection. (a)Renected image (b) Direct image, (c) Registered image showing 270 degree of annulus However, for symmetrical displacements, only 180 degrees of imformation is required and the remainder is inferred by data reflection. Once the images are acquired, and transformed, further post-processing techniques are used to extract the velocity of the interface. 27 Chapter 3. Experimental Set-up 3.2 Analysing interface dynamics We have already discussed the post processing steps required to process raw image data into filtered and transformed .images. The next step is to calculate the velocity from these trans-formed images. The method is based on watching a finger of displacement fluid flowing past a fixed camera and finding the width of the finger as it passes the camera. 3.2.1 Methodology for finding velocity of Interface The saturation curves are a result of a simple analysis technique that involves examining a line profile at a fixed height and measuring w(t) which is the width of the advancing finger of displacement. As the interface progresses in each frame, the integrated line profile is plotted against time. Figure 3.5(c) shows the location-of the line profile. As the experiment progresses, the interface moves upwards and the front and rear of the interface can be identified where the slope of the saturation curve changes rapidly. Figure 3.8 shows a typical saturation curve. The front of the interface is defined as occurring when the finger width w(t), normalized by the annulus width d, first exceeds 0.05 and the back of the interface is defined as the point where w(t)/d exceeds 0.95 The analysis is completed for images acquired at z = 210mm and also a location downstream at z — 1140mm. The saturation curves can be directly compared, and with the time and location known for the interface at both positions, then the velocities can be compared. In fact, by examining the saturation curves in Figure 3.8 we can immediately see that the interface is more elongated by the time it reaches z = 1140mm. This indicates that the displacement is unsteady. By comparing the velocity magnitudes for the front and the rear of the interface, we can identify steady, unsteady and static displacements. If the velocity for the front, Vw, and rear of the interface, Vn, are identical then the interface is steady, if > then the interface is unsteady. If Vn is undefined because the rear of the interface never passes the ROI then the 28 Chapter 3. Experimental Set-up 1 0.9 0.8 0.7 0.6 S 0.5 5 0.4 0.3 0.2 0.1 xo o Z = 1140mm Z = 210mm oieeP-0 1000 2000 3000 4000 5000 6000 7000 Time [msec] Figure 3.8: Typical saturation curve showing cut off thresholds displacement is static. The final step is to find the velocity difference by calculating Vdiff = v,„ (3.2) where Vdiff is a measure of "unsteadyness" and varies between zero and unity. Vdiff can now be analysed with respect to the eight parameters already outlined. As discussed, the eight parameters are succinctly contained in the scaling groups e, b, rji, rj2- We are interested in the relationship between these scalings and the measure of unsteadyness indicated by the velocity difference 3.3 Experimental fluid rheologies Both fluids were mixed according to the manufacturers methodology, however extra precautions were exercised with the Carbopol© to minimize the variability between batches. The fluids were dyed with ordinary Parker pen ink, at a concentration of 2.3 ppm. Rheology measurements showed that the effect of dye on the rheology of fluids was negligible. See Appendix A for mixing 29 Chapter 3. Experimental Set-up methodology. After each batch of fluid was mixed, they were then tested for shear viscosity and yield stress using-a Bohlin C-VOR rheometer, with a concentric vane and cup geometry. The temperature was fixed to be isothermal at a temperature of approximately 25C. The xanthan measurements were highly repeatable, with an error of less than 2% between succesive measurements. However, the Carbopol® was much more challenging to measure; it was found to exhibit thixotropic properties due aging and polymer restructuring. A rigorous technique was developed by A. Putz, where all tests were subjected to a preshear of 30 seconds in the rheometer before data acquisition. This ensured that all samples had an identically sheared structural configuration which greatly improved repeatability, especially at low shear rates. After pre-shearing, the stress values were acquired by the rheometer for increasing shear rates. The range of shear rates chosen was between O-TOs-1, this was chosen to cover the expected range of shear rates in the experiment 0-5s_1. To generate a density difference between the two fluids, sucrose (Rogers© white sugar) was added alternately to one fluid of interest whilst the other was left as a pure aqueous solution. Although adding sucrose had the desired effect of increasing the density, but unfortunately it also increased the viscosity. The resulting coupling of the density to viscosity caused some difficulty in keeping the fluid rheological parameters within the envelope of allowable values (determined by the scaling conditions). The final xanthan, Carbopol© and sugar concentrations were determined by trial and error. Suitable concentration were determined to be 0.3 and 0.5 wt % percent for Xanthan, and 0.07, 0.10 and 0.12 wt% percent for Carbopol®. An appropriate sugar concentration was found to be 33 wt %, which increased the density of either fluid to 1089± 2kg/m3 The rheological parameters of the fluids were determined by analysing the rheology data. For xanthan, the consistency and power law number were found by fitting the curves for shear rate versus shear stress to a power law model. Figure 3.9 shows the measured experimental rheological data with theoretical curves superimposed. In all cases, the fitting error is below 3.5% for the consistency, and 4% for the Power law number. A similar technique was used for the Carbopol®, but with the additional step of finding the yield stress. This was determined finding shear stress at the global maxima of viscosity. When 30 Chapter 3. Experimental Set-up w 0-1 I • 0.7% Xanthan • 0.5% Xanthan 0.3% Xanthan + 33% Sucrose j : i i i — . — i i i i i 1 10 Shear Rate [1/s] Figure 3.9: Rheological properties of xanthan fitted with power law model the data shows a sharp decline in the viscosity, then Carbopol is deemed.to be flowing as a fluid. Figure 3.10 shows a example of this technique. Here, a yield point is determined for a 0.07% Carbopol. As shown, the error is typically quiet large. Once the yield stress was found, then this value was subtracted from the remaining data, which was then fitted to a power law curve. Figure 3.11 shows the data for fluids used in the experiments. The error on the fitting is larger on Carbopol, particularly with regards to the yield point which has an error in the range of 20-50%. This is partly due to complications caused by the microstructure of Carbopol. The behaviour of Carbopol is less known than xanthan, and two tests were conducted to check for variability to temperature and shear damage. The rheology was measured for temperature ranges that would typically be found in a research laboratory (15-30C0). Figure 3.12 shows the variation is small with only a 7% decrease in viscosity over the 15C° range. The actual lab temperature range is small, and only varies by approximately 5C°, which further reduces the error. The other problem reported for Carbopol is shear damage that can occur during mixing and 31 Chapter 3. Experimental Set-up I > r i | I i i i i i i i . : + + + r + + + 1 r ++ +*S/IH* t (a) -0.01 0.1 1 10 Shear Stress [Pa] Figure 3.10: Yield determination of Carbopol. (a) shows the range in error. 1 * i 1 i • 0.07% Carbopol + 33% Sucrose , r . 1 10 Shear Rate [1/s] Figure 3.11: Rheological properties of Carbopol. 32 Chapter 3. Experimental Set-up Yield V Point • A T * A T •* • • A * 4 03 0.01 0.1 1 Shear Rate [1/s] • 15C • 18.8C • A 22.5C • 26.2C < 30C Figure 3.12: Rheology of Carbopol with varying temperature. Data below the yield value is noise pumping. For Carbopol, and other long chain polymers, shear damage occurs when the chains get chopped and shortened by the mechanical action pump and impellor blades. To check for damage that would occur during an experimental trial, a test was run with the Seepex PCP at the highest rate expected during a trial. Carbopol 940 was tested before and after pumping. Figure 3.13 shows that almost no rheological change has occured Finally, the remaining factor to consider for the fluid rheologies is the problem of aging. The rheology of both Carbopol and xanthan change with time, but for different reasons. Xanthan exhibits aging due to bacterial growth in the fluid. This can be prevented by repeated steril-ization with Chlorine at a concentration of 60ppm, and by keeping the fluid as cool as possible. On the other hand, Carbopol spontaneously ages due to a slow and irreversible change in the micro structure [23]. Since there is no method to prevent Carbopol aging, the fluid was mixed, measured and tested as quickly as possible, typically within a three day period. 33 Chapter 3. Experimental Set-up Yield Point 1 w 0.1 • Before Pumping A After Pumping 10 Shear Rate[1/s] Figure 3.13: Rheology of Carbopol before and after pumping 34 Chapter 4 Results and Discussion Experimental data were aquired for a diverse range of fluid types. The fluid pairs examined include Herschel-Bulkley (HB), Power-Law (PL) and Newtonian (NT). All combinations were examined, with varying rheology and density differences. The various fluid types, rheological and geometrical parameters are listed in detail in Appendix A. A major experimental challenge is that, even after the simplications gained from scaling for the Hele-Shaw model, there are still eight dimensionless parameters to be considered. The geometrical parameters are relatively easy to control, but the rheological parameters are coupled and much more difficult to control independently. Specifically, the consistency and power law number are linked through polymer concentration and in the case of Carbopol the yield stress is also coupled. In view of this, a wide variety of experiments were conducted to observe all possible displacement types under differing flow speeds and eccentricities. This chapter is composed of four sections. First, a typical experiment is briefly described and then we present phenomenological descriptions of some of the interesting and unexpected observations. Next we examine and give an overview of all completed experiments and discuss general trends. These results are then compared to the results of the ID lubrication code. The final section discusses selected experiments in more detail, with a focus on transitions from steady to unsteady displacements. 35 Chapter 4. Results and Discussion 4.1 Procedure: A typical experiment For brevity, only the basic steps are described but more details is included in Appendix C. Before every experiment the annulus is emptied and flushed. Next, the fluids are agitated with an impellor to remove any stratification layers that have developed over time. The annulus is then filled with two fluids which are kept separated by a sluice gate valve located at the entry point of the annulus. Care is taken to prevent any air bubbles getting trapped in the annulus during pumping. At the beginning of each trial, the pump is controlled to produce the desired flow rate and the gate valve opened. Image acquisition occurs just before the displacing fluid enters the camera's region of interest (ROI), and is continued until the rear of the interface exits the ROI. For image acquisition at other locations on the annulus, the camera is moved to a new position for imaging while the pumping continues uninterrupted. After the experiment is finished, the pump is switched off and entire annulus is imaged before draining. 4.2 P henomenology Following the basic procedures above, experiments were run with a range of flow rates, rheolo-gies, density differences, eccentricities and yield strengths. In all cases, the annulus was imaged after the flow was fully developed. There are two.development lengths to consider, namely, a gap width scale development length and an annulus diameter scale. The gap width scale can be calculated by examining the diffusive time scale t = ^ '(4.1) P* and the length scale is determined by L = ^ l = d R e (4.2) Here dRe « Trfa so that the gap development length is very short compared to the annular 36 Chapter 4. Results and Discussion radial distance, which in turn is smaller than the first observation point with the camera. This means the gap flow is fully developed before being imaged. The annulus development length, in the azimuthal direction, is evaluated with a different rational. The flow can be thought of as occurring in the Hele-Shaw assumption of gap flow. However, since the boundaries at the edge of the slots are symmetrical and fully mobile, there are no azimuthal fixed boundary conditions to generate a boundary layer flow. Therefore, for steady displacements the development length is merely the length required for the front and rear of the interface to pass the entry point. This length was observed to be never more than r r r a . All images where taken approximately five times this distance downstream to ensure that fully developed conditions exist for steady displacements. As expected, all three types of interface where observed. See Figure 4.1 which shows two example displacements. For high flow rates and eccentricities, unsteady displace-ments were observed as seen in Figure 4.1(a) and (b). The image shown in (b) was taken early in the displacement process after travelling 210mm from the gate valve which is located at the z = 0, t = 0 position. The image shown in (a) was taken after the interface had travelled 1150mm from the gatevalve, and the interface has clearly become more elongated. The images taken in (c) and (d) are taken at the same heights as (a) and (b). In this case, the interface shape has not changed which indicates a steady displacement. However, the interface was not always well-defined. In the case where the viscosities were low then extensive mixing occured which prevented measurement of the interface velocity. 4.2.1 Gap Phenomena During most displacement experiments, three dimensional in-gap flow variations were observed. Of course, these effects are not included in the numerical models where gap averaging means that gap effects are ignored. However, displacement experiments revealed that there were two types gap flow phenomena. The first effect was observed when xanthan displaces Carbopol. A long thin finger of displacement fluid was observed to propagate at approximately 5 times the wide side velocity. The finger was approximately 0.5mm x 7mm for slow rates, thickening to approximately 1mm x 7mm for high flow rates. The effect was observed for unsteady displace-37 Chapter 4. Results and Discussion Figure 4.1: Example Displacements, (a) and (b) Images of unsteady displacement after travel-ling 940mm at flowrate 3.0xl0 _ 5m 3/s, (c) and (d) Steady displacement after travelling 940mm at 2.40xl0~6m3/s. The rheologies are pi = 1048kg/m3 p2 = 997kg/m3; ryi = 0.57Pa; TY2 = OPa; ki = 0.32Pa.sn; k2 = 0.91Pa.s"). Power index m = 0.51, n2 = 0.36. 38 Chapter 4. Results and Discussion merits for high flow rates and eccentricit ies, however, the volumetr ic flow rate in the finger was very smal l relat ive to the bu lk flow. Th i s means that fluid transfer f rom the displacement fluid to the th in finger d id not affect the bulk flow speed of the interface. See F igure 4.2 which shows an example of the finger. T h e finger is hard to see i n (a) which shows that it is very th in . T h e side v iew shows the finger clearly, here the finger is seen at (c) as a vert ical wisp. The finger was often observed to be unstable w i th a wavelength of approximately 1/4 of a circumference w id th . A quest ion to be considered is the effects of local dynamics on far field flow. For steady displacements, the flow dynamics d id not alter the flow at large distances upstream or down-stream f rom the interface. However, for unsteady displacements the fingering effects extended far downstream and the influence of this on far field flow has yet to be determined. The f inal considerat ion is the influence of surface tension. For al l of our experiments, the fluid pairs were miscible which means that forces due to surface tension were negligible. F igure 4.2: T h i n finger propagat ing at approximately 5 t imes the wide side interface velocity, (a) Direct v iew, (b) Side view in mirror , (c) T h i n finger. T h e rheologies are once again p\ = 1048kg/m3 p2 = 9 9 7 % / m 3 ; m = 0 .57Pa ; r y 2 = O P a ; fc] = 0 . 3 2 P a . s " ; k2 = 0.91Pa.sn. Power index n\ = 0.51, n2 = 0.36. T h e second major gap phenomena was dispersion. T h i s effect was observed to be most pro-nounced dur ing Newton ian-Newton ian displacements, but was evident to a lesser degree for other fluid combinat ions. In general, the leading edge of the gap flow was observed to propa-39 Chapter 4. Results and Discussion gate rapidly in the wide side of the gap. This has significance for velocity measurement because this causes the velocity difference between the wide and narrow gaps to be elevated. For New-tonian displacements the effect was large, but for the bulk of the experiments this effect was small. Figure 4.3 shows the progression of the dispersion for two identicle Newtonian fluids. The camera was held constant, and each progressive image shows the displacement experiment proceeding. The dispersion can be clearly seen in Figure 4.3(a) as a thin dark finger propagating up the wide gap (located to the left of the inner pipe). For this Newtonian displacement, we would expect a parabolic shape to the gap interface. Instead we see a sharp point, which is due to the fact that the gap dispersion is superimposed on the annular dispersion which eventually wraps the interface around the inner pipe in (c). In (d), all that remains of the clear fluid two (annulus fluid) is a thin wall layer which becomes thinner, but never fully displaced, with increasing time. 4.2.2 Wall Effects In addition to the expected phenomena, several other important dynamical effects were ob-served. This includes a thin wall layer as predicted by Allouche et al. [24]. They show that for plane-channel displacement flow of two visco-plastic fluids, it is possible for there to be a static residual layer of the displaced fluid left stuck to the walls of the channel. The static wall layer was clearly visible for steady displacements when Xanthan fluids displaced Carbopol. Surprisingly, the static wall layer was not clearly visible between two Carbopol fluids, but this could have been due to a larger stress ratio. Figure 4.4 shows a static layer of 0.57mm in thickness. The fluids were p \ = 1048fcg/m3 p2 = 997%/m3; Tyi = 0.50Pa; ry 2 = 0.17Pa; kx = l.72Pa.s^n; k2 = QAlPa.s1/71. Power index ni = 0.36, ni = 0.56 4.3 Part I: Overview of experimental results By using the saturation curve technique, the velocity differences for all the experiments were calculated. Note that for image sequences where the interface was ill-defined, the saturation 40 Chapter 4. Results and Discussion Figure 4.3: Images taken w i th the camera in fixed posi t ion. The dispersion is visible on the wide side of the annular gap. (a) t = 0, (b) t = 4.1s, (c) t = 16s, (d) t = 32s. T h e fluids are ident ical w i th rheologies fci = k2 = 0 . 3 1 P a . s 1 / / r l ; and for th is Newton ian fluid the Power index is s imply n\ = n2 = 1 41 Chapter 4. Results and Discussion Figure 4.4: Stat ic wal l layer observed dur ing an xanthan and Ca rbopo l displacement. a lgor i thm could not find the edge w i th sufficient precision. In these cases the image sequences were viewed image by image and the interface located manual ly. A plot of e w i th respect to b is shown in F igure 4.5. Here we can see that as the buoyancy number becomes increasingly negative ( imply ing that the displacing fluid is becoming denser) then the velocity difference becomes smaller and the displacements become more steady. T h i s t rend is most clear w i t h e = 0.00, 0.13, and 0.50. Note that to a t ta in stabal izat ion of the interface in a highly eccentered annulus requires a larger buoyancy number. Th i s makes physical sense when consider ing that the az imutha l backflow must be stronger to displace fluid f rom the a very narrow gap. W h e n b becomes smal l then this signifies that the stress due to the density difference is insignif-icant when compared to the stress due to the viscosity difference. For b = 0, e = 0.25, a l l the displacements are unsteady. Fur ther experiments need to be completed to find fluid pairs that w i l l generate steady displacements wi thout buoyancy stabi l izat ion. T h e dimensionless parameter settings for the entire da ta set were input into the I D lubr ica t ion 42 Chapter 4. Results and Discussion 1 0.9 0.8 0.7 g 0.6 £ 0.5 ^ 0.4 0.3 0.2 0.1 0 + e = 0.00 X e = 0.13 • ' o O e = 0.25 A e = 0.50 o o o o A O O o o A O -e-'. o o , o + O -8 -6 Figure 4.5: Distribution of experimental data showing buoyancy parameter b with respect to degree of unsteadyness Vdiff for all rheologies 43 Chapter 4. Results and Discussion numerical model and the calculated interface velocity, dQ/dy, was plotted with repect to the normalised azimuthal coordinate y where 0 > y > 1. Two example plots are shown in Figure 4.6. Note that the curve does not represent the actual velocity profile. The location y = 0 represents the wide side and y = 1 represents the narrow side location. The results can be interpreted by first inspecting for monotonacity. If the curve is monotonic, then the global minima and maxima are located at the endpoints of the curve and the wide side and narrow side velocities can be read off the curve directly. As previously mentioned, iff Vw > Vn then the flow is unsteady. Two example outputs are shown in Figure 4.6. The fluids are (a) pi = 1048fcg/m3, p2 = 997fcfir/m3; -ryi = 0.50Pa; -ry2 = 0.17Pa; fci = 1.72Pa.s"; k2 = 0A7Pa.sn. Power index ni = 0.36, n 2 = 0.56, e = 0.13, b = -0.7 (b) P l = p2 = 1261fcg/m3 ; ryi = ry 2 = OPa; k\ = k2 = 0.31Pa.s". Power index n\ — n2 = 1, e = 0.25, 6 = 0. The plot shown in (a) shows a steady displacement because Vw < Vn whereas the plot in (b) is unsteady because Vw > Vn 0 0.2 0.4 0.6 0.8 I ' O 0.2 0.4 0.6 0.8 1 y y Figure 4.6: Output of lubrication model However, in our experiments steady displacements rarely exactly Vw = Vn along the interface. In fact, it was observed that for the bulk of the experiments there was always a small discrep-ancy between the wide and narrow gap velocities. These were typically less than approximately 6%. In order to attempt to account for the discrepancy, gap dispersive effects were examined in the wide and narrow sides of the annulus. Gap dispersion has already been described phe-nomenalogically in the previous section. Experimentally, gap dispersion was clearly visible in the wide side of the annulus, but only partially visible in the narrow gap. 44 Chapter 4. Results and Discussion The concern is that the wide side dispersion causes the edge finding algorithm to trigger slightly early. In order to quantify the extent of this in-gap dispersion a numerical model was created specifically for slot flow in the gap. The dispersive effects were calculated for a sample group of cases, of which two are shown in Figure 4.7. The variable r is the distance across the gap (equivalent to following a radial line from the inner pipe wall to the center of the gap), r = 0 is the center line of the gap flow, and r = 1 is the inner wall of the slot (represents the inner pipe wall of the gap). The average difference for the tip region, defined as the zone one quarter gap width either side of the center flow, is calculated and the difference between is found to be 3% for case (a) and 1% for case (b). This gives us an error bound due to dispersive effects, but this is not enough to account for the original 6% observed difference. However, a second examina-tion of the experimental data reveals a 8% difference between the bulk flow rate (as measured with the flowmeter) and the average interface speed of an experimentally steady displacement. This could be an indication of dispersive effects not predicted by the gap numerical model and this motivates further investigation. 2.5 (a) -Wide Gap Velocity Narrow Gap Velocity Wide Gap Velocity Narrow Gap Velocity Figure 4.7: Gap dispersion plots. Experiment (a)pi = 1048fcp/m3 p2 = 997%/m3; Tyi = 0.50Pa; ry 2 = 0.17Pa; h = 1.72Pa.sn; k2 = 0.47Pa.s". Power index nx = 0.36, n 2 = 0.56, e = 0.13, b = -0.7 (b)pi = 997fc5/m3; p2 = 1048fcp/m3 ; ryi = 0.17Pa; ry 2 = 0.5Pa; fci = 0.47Pa.sn; fc2 = 1.73Pa.sn.' Power index ni = 0.56, n 2 = 0.36, e = 0.13, , b = 0.56 45 Chapter 4. Results and Discussion A plot of e with respect to the velocity difference is shown in Figure 4.8. The results of the nu-merical model are indicated with markers. The numerical model can be seen to perform reliably when predicting experimentally steady displacements. For experimentally steady displacements with a velocity difference of less than 0.06, or 6%, the numerical model correctly predicts the outcome 92% of the time. This corresponds to the following experimental conditions: slow flows 0.01 < Re < 0.86, small eccentricities 0.0 < e < 0.5 and large magnitude and negative buoyancy displacement with —14.0 < b < 0.7. Unfortunately, for experimentally unsteady displacements the numerical model is less accurate, with only 75% of the displacements correctly predicted to be unsteady. 1 0.9 0.8 0.7 0.6 > U.3 i 0.4 0.3 0.2 0.1 4* i o1 © o x O o O Numerically Steady x Numerically Unsteady 0.2 0.4 0.6 0.8 Figure 4.8: Distribution of experimental data showing eccentricity with respect to degree of unsteadyness 46 Chapter 4. Results and Discussion 4.4 Part II: Details of individual experiments We now discuss a selection of individual experiments which were found to show interesting results. This section is composed of four experiments. We first discuss a control experiment where the interface between Newtonian fluids (glycerin) is examined. We then investigate a displacement involving two Carbopol solutions for various flow rates. This data set is interesting because it shows a clear transition from unsteady to steady flow. Next, we then describe a similar experiment where xanthan gum displaces Carbopol for varying eccentricity and flow rate and discuss the effect of these parameters on the interface steadyness, V~di/f For the last two experiments, we depart from the usual Vdiff based analysis. We discuss an experiment that investigates two consecutive interfaces sandwiched between three fluids. The analysis examines the saturation curves directly to discuss the displacement in terms of efficiency. The final experiment looks at a long static finger of Carbopol, and we analyse the efficiency of displacement after a fixed volume of displacement fluid has been pumped. We also analyse an instability that was observed at the contact line between the finger and the displacing fluid. 4.4.1 Newtonian control experiments The first experiments to be discussed are control trials with Newtonian fluids. These experi-ments are important to check for symmetry in the annulus, and to check for gap scale dispersion. Glycerin solutions of various concentrations were used to simulate the displacement fluid 1 and the annular fluid 2. The first experiment was conducted with 100% glycerin displacing 100% glycerin dyed with a small amount of Parker ink. The viscosities were measured and found to be equal (both pip = 0.54Pa.s, pip = 1261fc#/m3) which means that the displacement would be purely dispersive. Two other trials were conducted with 90% (fa = 0.31Po.s, pi = 1235/cg/m3) and 50% (fa = 0.039Pa.s, p2 = 1126fc#/m3) concentration and the interface velocities mea-sured. The additions of water for the latter two trials means that both the viscosity and density difference is varied. Results are shown in Figure 4.9. The eccentricity was held constant at e = 0.25 and only the flow rate was varied. 47 Chapter 4. Results and Discussion 0.8 1 C * 0.4 | 5. • 100% Glycerin v 90% Glycerin x 50% Glycerin HJH '"J"1 T T T T HH HH HIH 0.0 0.2 0.4 0.6 0.8 1.0 Flow Rate [Us] 1.6 Figure 4.9: Newtonian control experiments. The displacing fluid for all three trials is 100% glycerin, p,\ = 0.54Pa.s, p\ = 1261fc<?/m3) The results contain large errors due to the difficulty in locating the front and rear of the interface. This is due to gap dispersion producing diffuse clamping points near the walls. However, there are general trends that are clearly observable. For pure dispersion, the displacements are always unsteady which makes physical sense because the interface is continuously elongating; there is no density differences to stabilize the interface, and the viscosity is constant everywhere. For the cases where there is a negative density gradient, the density difference helps to stabilize the interface with a restoring azimuthal backflow from the wide side to the narrow side. The negative viscosity gradient also helps to displace fluid 2 due to the increased shear force at the interface. When the density and viscosity difference is large enough then the displacements becomes steady. In fact when fluid 2 is only 50% glycerin then the displacements are steady for all flowrates. 4.4.2 Herschel-Bulkley displacements The next set of experiments investigated a displacement, flow for two yield stress fluids. This can occur in industry when a cement with a weak yield point displaces a drilling mud with 48 Chapter 4. Results and Discussion a much stronger yield strength. The fluids used were two Carbopol solutions with identical measured, with fluid 1; h = 0A7Pa.sn; m = 0.56 Tyi = 0.17Pa, pi = 998kg/m3;, and fluid 2; p2 = 1058kg/m3; T y 2 = 0.49Pa; fc2 = lJBPa.s1^; and Power index n 2 = 0.36. The first experiments investigated the displacement of a positive gradient of densities (high density displacing low density). The eccentricity was held constant at e = 0.13 and the flow rate was varied. Experiments were completed with fluid 1 displacing fluid 2, and then a second set of trials with fluid 2 displacing fluid 1. In either case, the interface was clearly observable, and the front and back of the interface were well defined. The velocities for the front and back of the interface were computed from the acquired images, and the velocity ratios calculated. The results are shown in Figure 4.10. The case where fluid 1 displaces denser fluid 2 shows unsteady behaviour for all flow rates. However, the effect of buoyancy alone cannot be attributed to this behaviour since the yield point and consistency are also increased by the addition of the weighting agent (sucrose). The series of displacements of the same fluids in reverse order show quite different dynamics. For slow flow rates, the displacement is steady but as the flow rate increases the displacement becomes strongly unsteady. concentrations at 0.07% but one fluid was weighted with 33% sugar. The rheologies were 40 [• 30 \-I > 20 i 10 0 0.0 0.5 1.0 1.5 2.0 Flow Rate [L/s] Figure 4.10: Velocity differences for pairs of Herschel-Bulkley displacements 49 Chapter 4. Results and Discussion 4.4.3 Power Law fluids displacing Herschel-Bulkley fluids The next data set to be examined looks at the interface between a densified xanthan solution displacing a Carbopol solution. In previously discussed experiments, only the flowrate was varied, but now we vary both the flowrate Q and eccentricity e. The fluid rheologies and displacement order was held constant. A densified 0.3% Xanthan solution always displaced 0.07% Carbopol. The rheologies are pi = 1048kg/L p2 = 997kg/ra3; -ryi = 0.57Pa; 7 y 2 = OPa; fci = 0.32Pa.s"; fc2 = 0.91Pa.s". Power index nx = 0.51, n 2 = 0.36. The interface was well defined and the Vdi// errors are less than 2% for steady displacements and approximately 8% for the fastest displacements. The results are plotted in Figure 4.11 Figure 4.11: Summary of data showing regions of steady displacement (near origin) 'and un-steady displacement (high flow rates and high eccentricity) An examination of the Q-e plot shows that for small eccentricities there are almost always steady displacements. Similarly, for small flow rates less than O.lL/s there are numerous steady displacements. Physically, the high density of the displacing fluid is helping to stabilize the interface by creating a strong azimuthal flow in the displacing fluid from the wide side of the annulus to the narrow side. At slow flow rates, tins effect dominates the local flows at the interface. However, at higher pumping rate the the Carbopol viscosity drops more quickly than the xanthan consistency. Therefore, in regions of high shear stress, such as the widest region of 50 Chapter 4. Results and Discussion the gap, the xanthan more readily fingers into the Carbopol. See Figure 4.12 which shows the change in consistency with respect to shear stress. • Fluid 2 + Fluid 1 0.1 1 Shear Stress [Pa] Figure 4.12: Consistency curves for Fluid 1 (0.067% Carbopol) and Fluid 2 (0.3% Xanthan) 4.4.4 Double Interface Experiment In the previous set of experiments, only one interface between two fluids was investigated. The next data set investigates two interfaces between three fluids. The viscosity of the fluids were increased, along with the yield point of the Carbopol. Fluid 2 was 0.12% Carbopol and fluids 1 and 3 were 0.5% Xanthan. The rheologies are p\ —p2 =P3 = 997kg/m3; TVI = r y 3 — OPa; 7Y2 = 4Pa; fci = fc3 = l.OPa.s"; k2 = 14Pa.s". Power index ni = n 3 = 0.47, n2 = 0.30. The displacement order was Carbopol displacing xanthan, followed by an identicle xanthan displacing Carbopol. The first interface was generated by using the slice sluice gate valve. The second interface was generated by injection of xanthan into the Carbopol loading chamber. See Figure 4.13 which shows a schematic of the setup. The volume of the middle 'pill' of fluid (Carbopol) is fixed at 1.05 L in volume and the eccentricity fixed at e = 0.50. Note that the densities are identical which means the b — 0. In the previous experiments, the camera was relocated from the lower position to an upper 51 Chapter 4. Results and Discussion (a) (b) Front View (x-y plane) V Fluid 2 Fluid 3 Fluid 1 z X Active Displacement Figure 4.13: Schematic showing double interface set-up posi t ion to gain a l l the images required to calculate the interface velocity. However, in this exper iment the dynamics are occur ing at mult ip le posit ions on the annulus so the camera was held f ixed at the lower posi t ion for the entire durat ion of the experiments. Unfor tunately, this precludes gather ing velocity in format ion but the results are st i l l informative. T h e results are shown in the form of saturat ion curves, where the t ime axis has been scaled w i t h F igure 4.14 shows that the curves do not quite collapse. T h i s shows that the interface stretches axia l ly for higher flow rates. T h e dynamics of the second interface clearly show a long stat ic finger of C a r b o p o l is left beh ind by the d isplac ing xanthan. T h i s is in agreement w i th the predict ion that low viscosity d isplacing fluids cannot displace high viscosity, high y ie ld point fluids. The result for the first interface is unexpected. Here, the C a r b o p o l was observed to be slow at d isplacing the xan than even though the C a r b o p o l solut ion is an order of magni tude higher i n viscosity. A reason for the inefficient displacement dyanmics could be due to the relat ively close prox imi ty 52 Chapter 4. Results and Discussion 0 50 100 150 Advection Time Figure 4.14: Three interface experiment. The first displacement front passing at (a) is not distinct, as indicated by the gradual slope (b) up to the maximal displacement by Carbopol (c). The second interface is shown by the sudden change in slope terminating at (d). of the two interfaces. If the interfaces are coupled hydrodynamically then this could have industrial implications where sequences of fluids are sometimes pumped in rapid succession. This motivates further investigation in order to find out the minimum pill length required to decouple the interfaces. 4.4.5 Efficiency of displacement for a static finger The next experiment was designed to find the final efficiency of displacement after a fixed volume of fluid is pumped. The displacement proceeded until the given amount of fluid was pumped and then the pump was halted and the annulus was imaged at 90mm increments along the vertical axis. The efficiency for each height was found by normalizing the image intensity for successive line profiles across the vertical axis. Trials were conducted for two offsets, a small eccentricity of 0.13, and moderate eccentricity of 0.50. For this experiment, the difference between the consistencies and the difference between the power law numbers for the fluids was minimized and only the density and yield stress differed. The displacing fluid, was 0.5% Xanthan with 53 Chapter 4.- Results and Discussion • o o o o o o o o . o D D n D D D o n n X • 0.36 L/s I o 0.72 L/s . A 1.08 L/s v 1.44 L/s -0 5 10 15 20 25 0 5 10 15 20 25 Z/Annulus Diameter Z/Annulus Diameter (a) Eccentricity 0.13 (b) Eccentricity 0.50 Figure 4.15: Displacement efficiency at various flow rates. the same concentration and rheology as before k\ = l.OPa.s"; hi = 0.47, p\ — 998kg/m3;, and fluid 2 was Carbopol 0.07%, p2 = 1058/cg/m3; T y 2 = 0.5Pa; k2 = l.lPa.s"; and Power index n.2 = 0.45. Figure 4.15 shows the results for both trials. Interestingly, Figure 4.15(a) show that efficiency for small eccentricities is higher than those in Figure 4.15(b) which shows the results for eccentricity 0.50. The analysis shows that the final long finger static finger of Carbopol contains periodic waves which indicate an instability occurred during the displacement process. This can be seen in Fig-ure 4.15 as a variation in displacement efficiency with respect to vertical height. The amplitude increases as the displacement progresses axially along the annulus, but most importantly the magnitude of displacement efficiency variation is larger for the smaller eccentricity, as show by Figure 4.16. The first wave node always starts at the gate valve which acts as a perturbation. Interestingly, the wavelength is constant at 9.5 pipe widths for both eccentricities. A further experiment was conducted by rerunning the same experimental procedure as before, but this time reversing the order of fluids. The densified 0.07% Carbopol was found to fully displace the 0.5% Xanthan with 100% efficiency for an eccentricity of 0.13 at all flow speeds. These clearly demonstates that even when there are no consistency and power law number vari-ations across the interface, the effect of density and and yield stress are significant during the -T —\— -- ' y A o A O X A O w A O V 0 X •c o X X o V V o ° --0 A V X 0.36 Us 0.72 Us 1.08 L/s 1.44 Us 1 J 54 Chapter 4. Results and Discussion - i — i — i — i — i — i — i — i — i — 0.0 0.2 0.4 0.6 0.8 1.0 112 1.4 1.6 1.8 2.0 Flow Rate [L/s] Figure 4.16: Amplitude of instability displacement process. Figure 4.17 shows a comparison of the two displacements. The displace-ment in Figure 4.17(a) shows that all the Xanthan is easily displaced, whereas Figure 4.17(b) shows a finger propagating up the wide side of the annulus. 4.5 Summary of experiments From the observation of this set of experiments, the interface dynamics were discovered to be more rich in physical phenomena than originally thought. The gap dispersion was not expected and should be investigated further. A challenge to the experiment was the extensive parameter space. However, by looking at the buoyancy and eccentricity with respect to the velocity difference, we can see that in general buoyancy dominates over the effective viscosity. This can be also seen when we fit e, b, T]eff = r]effi/'neff2 in a non-linear regression model in Matlab. Using an unconstrained nonlinear optimization which minimizes a multivariable function the final model gives a fit in terms of Vrfj//. Vdiff = 5(2.5 - b)-2V°eff(1.5 - e)-3 (4.3) " ; . ' \ The model fits with an r 2 of approximately 0.8 which indicates a trend, but not a strong fit. However, it does indicate that the.effective viscosity ratio only has a minor influence compared 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 v 1 mm offset o 4mm offset T _1 i I i_±J • L 55 Chapter 4. Results and Discussion Figure 4.17: V i s u a l compar ison of C a r b o p o l and X a n t h a n displacements, (a) shows densified C a r b o p o l d isp lac ing X a n t h a n , and (b) shows the X a n t h a n fingering into densified C a r b o p o l to the buoyancy and eccentricity. W h e n the buoyancy effects where smal l , many stat ic fingers and unsteady displacements were observed. In these cases, the viscosity rat io was impor tant and for a l l exper iments these tr ials r/e// > 1. For the next set of experiments, invest igat ing Veff < 1 wou ld be invaluble for understanding the effect of a posit ive viscosity rat io. 56 Chapter 5 Conclusion In this thesis we have described and analyzed new experiments in non-Newtonian displacement flows in eccentric vertical annuli. The project involved designing and building apparatus that is capable of reproducing scaled field conditions for a variety of flow rates and annulus eccen-tricities and inclinations. We have extended on previously completed work in the experimental displacement flows by visualizing and analysing interface shape, velocity and location. We have attained the following main achievements: 1. We have designed and built apparatus that can replicate displacement flows. The fluid temperature and pressure drop can be monitored during any given displacement and the interface imaged with a camera mounted on a Lab View controlled linear actuated mount. The flow rate can be controlled by a PID controlled PCP pump that can produced replicatable flow rates for any single phase non-Newtonian fluid. 2. We calibrated and tested the apparatus for a variety of fluid displacements and built a Matlab analysis tool to extract velocity information from the front and rear of the interface. 3. We conducted a range of experiments with a variety of non-Newtonian fluids and ran con-trol experiments with Newtonian fluids. We showed that for the fluid rheologies examined, the buoyancy and density differences dominated the effective viscosity ratios. 4. We compared the experimentally observed interface types with the results of the numerical model and showed that the numerical model correctly predicts steady displacement 92% of the time and unsteady results 75%. The correct predictions occurred for the experimental 57 Chapter 5. Conclusion conditions: slow flows 0.01 < Re < 0.86, small eccentricities 0 < e < 0.5 and large magnitude and negative buoyancy displacement with —14 < b < —0.7. There was a single steady displacement at b — 0 that was correctly predicted, mainly due to the large negative in viscosity gradients which helped to stabilize the flow. Reasons for incorrect predictions were difficult to pinpoint, but they were observed for nejj < 0.3 with —1.56 < b < —0.86 and 0.13 < e < 0.38. Specifically, for moderate b and e, and small rjeff the model appears to be more vulnerable to incorrect predictions. The reason for this is not known, and motivates further work. 58 Chapter 6 Recommendations The recommendations include some minor modifications to the apparatus, image acquisition system. We also include improvements to the experimental methodology and possible augmen-tations to the existing data set. 6.1 Image acquisition System It is possible to continue experimentation with the single CCD camera, but the speed of the linear actuator places an upper bound to the interface speed. In the current set-up, the camera has to wait for the lower part of the interface to cross the lower ROI, while the upper part of a fast moving unsteady interface has already passed the upper ROI. A second camera would enable simultaneous image acquisition at the upper and lower locations of the annulus. An extension of this idea would be place a series of lower resolution cameras along the entire length of the annulus (approximately 6 cameras required) and then stitch the acquired images together. The latter solution would be invaluable for unsteady and static displacements at high flow rates. Note that high frame rate cameras are not required; most interface dynamics can be captured at less than 10 fps. 6.2 Experimental Methodology One of the major challenges for the experiment was handling the large number of variables. In hindsight, experiments should have been conducted with more parameters switched off. Of particular interest would be to eliminate the overwhelming influence of density differences. A 59 Chapter 6. Recommendations new set of experiments could easily be run with the buoyancy effects suppressed. For vertical displacements, this would reduce the effects of azimuthal backflow hence allowing the effects of rheology to dominate. 6.3 Apparatus The apparatus is fully operational, but there are several improvements possible for the flow loop. After experimental runs, emptying the annulus is slow for high viscosity fluids. The outer diameter of the main drainage line could be increased to lower the fluid velocity in the line, hence lower the frictional losses. During operation, the magnetic flowmeter occasionally cuts out. Possible causes include external noise interfering with the output signal and the remedy would be to improve the line shielding. 6.4 Gap dispersion In order to reduce the gap dispersion, the average width of the gap could be reduced by obtaining an inner pipe of larger diameter. Reducing the gap would supress other 3D flow effects hence bringing the experiment closer to the ideals of the numerical models. 60 Bibliography J.R. Smith. Lsu well control. The Brief, pages 28-29, December 1997. D. Guillot, D. Hendriks, F. Callet, and B. Vidick. Well Cementing, chapter Mud Removal (Chapter 5). Schlumberger Educational Services, Houston, 1990. K.V. Ravi, K.V. Beirute, and R.L. Covington. Erodability of partially dehydrated gelled drilling fluid and filter cake. Society of Petroleum Engineers, 1992. Paper number SPE 24571. CW. Sauer. Mud displacement during cementing: A state of the art. Journal of Petroleum Technology, pages 1091-^ 1101, September 1987. C F . Lockyear and A.P. Hibbert. Integrated primary cementing study defines key factors for field success. Journal of Petroleum Technology, December 1989. C F . Lockyear, D.F. Ryan, and D.F. Gunningham. Cement channeling: How to predict and prevent. Society of Petroleum Engineers, 1989. Paper number SPE 19865. M. Couturier, D. Guillot, H. Hendriks, and F. Callet. Design rules and associated spacer properties for optimal mud removal in eccentric annuli. Society of Petroleum Engineers, 1990. Paper number SPE 21594. K.V. Ravi, R.M. Beirute, and R.L. Covington. Improve primary cementing by continious' monitoring of circulatable hole. Society of Petroleum Engineers, 1993. Paper number SPE 26574. CR. Clark and G.L. Carter. Mud displacement with cement slurries. Society of Petroleum Engineers, 1973. Paper number SPE 4090. J.J.M. Zuiderwijk. Mud displacement in primary cementation. Society of Petroleum En-gineers, 1975. Paper number SPE 4830. R.C Haut and R.C Crook. Primary cementing: The noncirculatable mud displacement process. Society of Petroleum Engineers, 1979. Paper number SPE 8253. R.C. Haut and R.J. Crook. Primary cementing: Optimizing for maximum displacement. World Oil, pages 105-116, November 1980. C F . Lockyear and A.P. Hibbert. A novel approach to primary cementation using a field scale flow loop. Society of Petroleum Engineers, 1988. Paper number SPE 18376. R.M. Beirute, F.L. Sabins, and K.V. Ravi. Large-scale experiments show proper hole con-ditioning: A critical requirement for successful cementing operations. Society of Petroleum Engineers, 1991. Paper number SPE 22774. J. Jakobsen, N. Sterri, A. Saasen, and B. Aas. Displacements in eccentric annuli during primary cementing in deviated wells. Society of Petroleum Engineers, 1991. Paper number SPE 21686. 61 Bibliography [16] M.P. Escudier, P.J. Oliveira, F.T. Pinho, and S. Smith. Fully developed laminar flow of non-newtonian liquids through annuli: Comparison of numerical calculations with experiments. Experiments in Fluids, 33, pages 101-111, 2002. [17] J. M Nouri, H. Umur, and J. H. Whitelaw. Flow of newtonian and non-newtonian fluids in concentric and eccentric annuli. Journal of Fluid Mechanics, 253, pages 617-641, 1993. [18] A. Tehrani, S.H. Bittleston, and P.G.J. Long. Flow instabilities during annular displace-ment of one non-newtonian fluid by another. Experiments in Fluids 14, pages 246-256, 1993. [19] A. Tehrani, J. Ferguson, and S.H. Bittleston. Laminar displacement in annuli: A combined theoretical and experimental study. Society of Petroleum Engineers, 1992. Paper number SPE 24569. [20] I.A. Frigaard and S. Pelipenko. Two-dimensional computational simulation of eccentric annular cementing displacements. J. Appl. Math, 2003. [21] LA. Frigaard, S.H. Bittleston, and J. Ferguson. Mud removal and cement placement during primary cementing of an oil well, 2004. [22] LA. Frigaard and S. Pelipenko. Effective and ineffective strategies for mud removal and cement slurry design. Society of Petroleum Engineers, 2003. Paper number SPE 80999. [23] M. Cloitre, R. Borrega, and L. Leibler. Rheological aging and rejuvenation in microgel pastes. Physical Review Letters, v 85, n 22, pages 4819-4822, 2000. [24] M. Allouche and LA. Mewis, Frigaard. Static wall layers in the displacement of two visco-plastic fluids in a plane channel. Journal of Fluid Mechanics, pages 243-277, 2000. 62 Appendix A A . l Mixing methodology for polymer solutions Both Carbopol® and xanthan solutions were prepared by slowly dissolving the appropriate amount of polymer into a 60L cylindrical container of warm tap water agitated by a 105watt Heidolph mixer mounted with an 8cm double bladed impellor. The impellor speed is controlled carefully to ensure that no air was entrained during the mixing period. The polymers are added very slowly to minimize the clumping and mixed at least 24hours at 22C°. Additional steps are required for Carbopol® solutions. Sodium hydroxide is added to the solution until the pH is between 6.8 and 7.2. This step activates the polymers and is essential for building a yield stress in the fluid. Note that if the Carbopol® solution is above 1.0 wt.% then the final mixing must be manually completed by stirring the solution with a paddle. Xanthan and Carbopol solutions are then pumped through a Moyno 3320467 PCP pump to ensure,the solution is thoroughly mixed. Caution must be taken with the pump for this last step; the fluids sitting in the pump must be first flushed out. If either fluid density is to be increased by the addition sucrose (Rogers white sugar) then the powdered sugar is added before neutralization (or as a last ingredient for Xanthan solutions) and left 12 hours to fully dissolve. Shear viscosity measurements were taken of both liquids using a Bohlin C-VOR rheometer, with a concentric vane and cup geometry, at a temperature of approximately 20C°. The rheologies and settings for the experiments are summarized in Table A . l and Table A.2 63 Appendix A. n T 2 K l K.2 n i n 2 Pi P2 e Q K / s ] Fluid Prod. [Pa] [Pa] [Pa.s'1] [Pa.s"] [kg/mJ] [kg/ma] [ma/s] [Type] 0 0 0.54 0.54 1 1 1261 1261 0.25 0.562 3.00 E-06 NT.NT UNST 0 0 0.54 0.54 1 1 1261 1261 0.25 0.574 6.01E-06 UNST 0 0 0.54 0.54 1 1 1261 1261 0.25 0.640 1.20E-05 UNST 0 0 0.54 0.54 1 1 1261 1261 0.25 0.659 1.80E-05 UNST 0 0 0.54 0.54 1 1 1261 1261 0.25 0.677 2.40E-05 UNST 0 0 0.31 0.10 1 1 1261 1235 0.25 0.297 3.00E-06 NT.NT ST 0 0 0.31 0.10 1 1 1261 1235 0.25 0.507 1.20E-05 ST 0 0 0.31 0.10 1 1 1261 1235 0.25 0.582 2.40E-05 ST 0 0 0.31 0.04 1 1 1235 1126 0.25 0.000 3.00E-06 NT.NT ST 0 0 0.31 0.04 1 1 1235 1126 0.25 0.000 1.20E-05 ST 0 0 0.31 0.04 1 1 1235 1126 0.25 0.000 2.40E-05 ST 0 2.0 0.31 2.21 1 0.3 1235 998 0.25 0.107 3.00E-06 NT.HB ST 0 2.0 0.31 2.21 1 0.3 1235 998 0.25 0.261 6.01E-06 ST 0 2.0 0.31 2.21 1 0.3 1235 998 0.25 0.715 1.20E-05 ST 0 2.0 0.31 2.21 1 0.3 1235 998 0.25 0.617 1.80E-05 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0 0.006 6.01E-06 PL.HB ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0 0.017 1.20E-05 ST 0 ' 0.6 0.32 0.91 0.51 0.36 1058 998 0 0.009 1.80E-05 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0 0.048 2.40E-05 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0 0.105 3.00E-05 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.13 0.017 3.00E-06 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.13 0.047 6.01E-06 ST 0 0.6 0.32 0.91 . 0.51 0.36 1058 998 0.13 0.007 1.20E-05 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.13 -0.011 1.80B-05 ST 0 0.6 0.32 0.91 0:51 0.36 1058 998 0.13 0.059 2.40E-05 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.13 0.059 3.00E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.13 0.065 3.60E-05 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.25 -0.013 6.01E-06 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.25 0.157 9.01E-06 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.25 0.106 1.20E-05 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.25 0.223 1.80E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.25 0.297 2.40E-05 . UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.25 0.336 3.00E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.38 0.060 3.00E-06 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.38 0.233 6.01E-06 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.38 0.471 1.20E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.38 0.318 1.80E-05 M UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.38 0.125 3.00E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.50 0.061. 3.00E-06 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.50 0.226 6.01E-06 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.50 0.485 1.20E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.50 0.581 1.80E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.63 0.189 3.00E-06 ST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.63 0.355 6.01E-06 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.63 0.437 1.20E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.63 0.575 1.80E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.63 0.690 2.40E-05 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.76 0.475 1.80E-06 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.76 0.600 6.01E-06 UNST 0 0.6 0.32 0.91 0.51 0.36 1058 998 0.76 0.702 1.20E-05 UNST 0.2 0.5 0.47 1.73 0.56 0.36 998 1058 0.13 0.601 6.01E-06 HB.HB UNST 0.2 0.5 0.47 1.73 0.56 0.36 998 1058 0.13 0.170 1.20E-05 UNST 0.2 0.5 0.47 1.73 0.56 0.36 998 1058 0.13 0.350 1.80E-05 " UNST 0.2 0.5 0.47 1.73 0.56 0.36 998 1058 0.13 0.474 2.40E-05 UNST 0.5 0.2 1.73 0.47 0.36 0.56 1058 998 0.13 0.001 6.01E-06 ST 0.5 0.2 1.73 0.47 0.36 0.56 1058 998 0.13 0.000 1.20E-05 ST 0.5 0.2 1.73 0.47 0.36 0.56 1058 998 0.13 0.090 2.40E-05 ST 0.5 0.2 1.73 0.47 0.36 0.56 1058 . 998 0.13 0.176 3.00E-05 ST 0 0 0.54 1.02 1 0.47 1261 998 0.25 0.018 6.01E-06 NW-PL ST 0 0 0.54 1.02 1 0.47 1261 998 0.25 -0.005 1.20E-05 ST 0 0 0.54 1.02 1 0.47 1261 998 0.25 0.007 2.40E-05 ST 64 Appendix A. n T2 K.2 ni n 2 Pi P2 c Q [ma/s] Fluid Pred. [Pa) [Pa] [Pa.s»| [Pa.s"] [kg/m*] [kg/W] K/>1 [Type] 0 0 0.54 1.02 1 0.47 1261 998 0.25 0.014 3.60E-05 ST 0 0 1.02 0.54 0.47 1 998 1261 0.25 0.900 1.20E-05 UNST 0 0 1.02 0.54 0.47 1 998 1261 0.25 0.900 2.40E-05 " UNST 0 0 3.75 1.02 0.25 0.47 998 998 0.25 0.480 3.00E-06 PL.PL ST 0 0 . 3.75 1.02 0.25 0.47 998 998 0.25 0.541 6.01E-06 ST 0 0 3.75 1.02 0.25 0.47 998 998 0.25 0.527 1.20E-05 ST ' 0 0 3.75 1.02 0.25 0.47 998 998 0.25 0.553 2.40E-05 " ST 0 0 1.02 3.75 0.47 0.25 998 998 0.25 0.900 1.20E-05 " UNST 0 0 1.02 3.75 0.47 0.25 998 998 0.25 0.900 2.40E-05 " UNST 0 6.2 3.75 9.10 0.25 0.32 998 998 0.25 1.000 6.01E-06 UNST 0 6.2 3.75 9.10 0.25 0.32 998 998 0.25 1.000 1.20E-05 UNST 0 6.2 3.75 9.10 0.25 0.32 998 998 0.25 1.000 1.80E-05 UNST 0 6.2 3.75 9.10 0.25 0.32 998 998 0.25 1.000 2.40E-05 UNST 0 6.2 3.75 9.10 0.25 0.32 998 998 0.25 1.000 3.00E-05 UNST 6.2 0 9.1 3.75 0.32 0.25 998 998 0.25 0.059 1.20E-05 ST 0 6.2 1.1 9.10 0.47 0.32 1058 998 0.25 1.000 3.00E-06 UNST 0 6.2 1.1 9.10 0.47 0.32 1058 998 0.25 1.000 6.01E-06 " UNST 0 6.2 1.1 9.10 0.47 0.32 1058 998 0.25 1.000 2.40E-05 " UNST Table A . l : Table of parameter settings for all experiments conducted up to January 2007. The final column shows the result of the ID lubrication model. UNST indicates an unsteady displacement was predicted and ST indicates a steady 65 Appendix A. n T2 K l . K.2 m m left Re St e b Vdiff 0.00 0.00 1.00 1.00 1.00 1.00 1.00 0.003 0.04 0.01 0.25 0.00 0.56 0.00 0.00 1.00 1.00 1.00 1.00 1.00 0.006 0.07 0.02 0.25 0.00 0.57. 0.00 0.00 1.00 1.00 1.00 1.00 1.00 0.012 0.15 0.04 0.25 0.00 0.64 0.00 0.00 1.00 1.00 1.00 1.00 1.00 0.019 0.22 0.07 0.25 0.00 0.66 0.00 0.00 1.00 1.00 1.00 1.00 1.00 0.025 0.30 0.09 0.25 0.00 0.68 0.00 0.00 1.00 0.32 1.00 0.32 3.10 0.005 0.06 . 0.01 0.25 -3.25 0.30 0.00 0.00 1.00 0.32 1.00 0.32 3.10 0.022 0.26 0.03 0.25 -0.81 0.51 0.00 0.00. 1.00 0.32 1.00 0.32 3.10 0.043 0.52 0.05 0.25 -0.41 0.58 0.00 0.00 1.00 0.13 1.00 0.13 7.95 0.005 0.06 0.01 0.25 -13.61 0.00 0.00 0.00 1.00 0.13 1.00 0.13 7.95 0.021 0.25 0.03 0.25 -3.40 0.00 0.00 0.00 1.00 0.13 1.00 0.13 7.95 0.042 0.51 0.05 0.25 -1.70 0.00 0.00 0.47 0.07 0.53 0.07 1.00 0.07 0.000 0.00 0.09 0.25 -2.19 0.11 0.00 0.42 0.13 0.58 0.13 1.00 0.13 0.001 0.02 0.10 0.25 -1.95 0.26 0.00 0.37 0.23 0.63 0.23 1.00 0.23 0.005 0.06 0.11 0.25 -1.72 0.72 0.00 0.35 0.32 0.65 0.32 1.00 0.32 0.010 0.12 0.12 0.25 -1.59 0.62 0.00 0.33 0.26 0.67 0.26 1.00 0.26 0.003 0.04 0.04 0.00 -1.33 0.01 0.00 0.28 0.31 0.72 0.31 1.00 0.31 0.011 0.13 0.05 0.00 -1.12 0.02 0.00 0.25 0.35 0.75 0.35 1.00 0.35 0.022 0.26 0.06 0.00 -1.01 0.01 0.00 0.23 0.37 0.77 0.37 1.00 0.37 0.036 0.43 0.06 0.00 -0.93 0.05 0.00 0.22 0.39 0.78 0.39 1.00 0.39 0.053 0.63 0.06 0.00 -0.88 0.10 0.00 0.39 0.22 0.61 0.22 1.00 0.22 0.001 0.01 0.04 0.13 -1.57 0.02 0.00 0.33 0.26 0.67 0.26 1.00 0.26 0.003 0.04 0.04 0.13 -1.33 0.05 0.00 0.28 0.31 0.72 0.31 1.00 0.31 0.011 0.13 0.05 0.13 -1.12 0.01 0.00 0.25 0.35 0.75 0.35 1.00 0.35 0.022 0.26 0.06 0.13 -1.01 -0.01 0.00 0.23 0.37 0.77 0.37 1.00 0.37 0.036 0.43 0.06 0.13 -0.93 0.06 0.00 0.22 0.39 0.78 0.39 1.00 0.39 0.053 0.63 0.06 0.13 -0.88 0.06 0.00 0.21 0.41 0.79 0.41 1.00 0.41 0.072 0.86 0.07 0.13 -0.83 0.06 0.00 0.33 0.26 0.67 0.26 1.00 0.26 0.003 0.04 0.04 0.25 -1.33 -0.01 0.00 0.30 0.29 0.70 0.29 1.00 0.29 0.007 0.08 0.05 0.25 -1.21 0.16 0.00 0.28 0.31 0.72 0.31 1.00 0.31 0.011 0.13 0.05 0.25 -1.12 0.11 0.00 0.25 0.35 0.75 0.35 1.00 0.35 0.022 0.26 0.06 0.25 -1.01 0.22 0.00 0.23 0.37 0.77 0.37 1.00 0.37 0.036 0.43 0.06 0.25 -0.93 0.30 0.00 0.22 0.39 0.78 0.39 1.00 0.39 0.053 0.63 0.06 0.25 -0.88 0.34 0.00 0.39 0.22 0.61 0.22 1.00 0.22 0.001 0.01 0.04 0.38 -1.57 0.06 0.00 0.33 0.26 0.67 0.26 1.00 0.26 0.003 0.04 0.04 0.38 -1.33 0.23 0.00 0.28 0.31 0.72 0.31 1.00 0.31 0.011 0.13 0.05 0.38 -1.12 0.47 0.00 0.25 0.35 0.75 0.35 1.00 0.35 0.022 0.26 0.06 0.38 -1.01 0.32 0.00 0.22 0.39 0.78 0.39 1.00 0.39 0.053 0.63 0.06 0.38 -0.88 0.13 0.00 0.39 0.22 0.61 0.22 1.00 0.22 0.001 0.01 0.04 0.50 -1.57 0.06 0.00 0.33 0.26 0.67 0.26 1.00 0.26 0.003 0.04 0.04 0.50 -1.33 - 0.23 0.00 0.28 0.31 0.72 0.31 1.00 0.31 0.011 0.13 0.05 0.50 -1.12 0.48 0.00 0.25 0.35 0.75 0.35 1.00 0.35 0.022 0.26 0.06 0.50 -1.01 0.58 0.00 0.39 0.22 0.61 0.22 1.00 0.22 0.001 0.01 0.04 0.63 -1.57 0.19 0.00 0.33 0.26 0.67 0.26 1.00 •0.26 , 0.003 0.04 0.04 0.63 -1.33 0.35 0.00 0.28 0.31 0.72 0.31 1.00 0.31 0.011 0.13 0.05 0.63 -1.12 0.44 0.00 0.25 0.35 0.75 0.35 1.00 0.35 0.022 0.26 0.06 0.63 -1.01 0.57 0.00 0.23 0.37 0.77 0.37 1.00 0.37 0.036 0.43 0.06 0.63 -0.93 0.69 0.00 0.43 0.18 0.57 0.18 1.00 0.18 0.000 0.00 0.03 0.76 -1.74 0.47 0.00 0.33 0.26 0.67 0.26 1.00 0.26 0.003 0.04 0.04 0.76 -1.33 0.60 0.00 0.28 0.31 0.72 0.31 1.00 0.31 0.011 0.13 0.05 0.76 -1.12 0.70 0.06 0.18 0.26 0.82 0.32 1.00 0.32 0.002 0.02 0.07 0.13 0.86 0.60 0.05 0.15 0.31 0.85 0.36 1.00 0.36 0.007 0.08 0.08 0.13 0.70 0.17 0.04 0.13 0.34 0.87 0.38 1.00 0.38 0.013 0.16 0.09 0.13 0.61 0.35 0.04 0.12 0.36 0.88 0.40 1.00 0.40 0.022 0.26 0.10 0.13 0.56 0.47 0.18 0.06 0.82 0.26 1.00 0.32 3.15 0.002 0.02 0.07 0.13 -0.86 0.00 0.15 0.05 0.85 0.31 1.00 0.36 2.81 0.007 0.08 0.08 0.13 -0.70 0.00 0.12 0.04 0.88 0.36 1.00 0.40 2.48 0.022 0.26 0.10 0.13 -0.56 0.09 0.11 0.04 0.89 0.38 1.00 0.42 2.38 0.032 0.38 0.11 0.13 -0.52 0.18 0.00 0.00 0.77 1.00 0.77. 1.00 0.77 0.005 0.06 0.03 0.25 -7.23 0.02 0.00 0.00 1.00 0.90 1.00 0.90 1.11 0.012 0.15 0.04 0.25 -4.71 -0.01 0.00 0.00 1.00 0.63 1.00 0.63 1.60 . 0.025 0.30 0.09 0.25 . -2.36 0.01 66 Appendix A. 0.00 0.00 1.00 0.50 1.00 0.50 1.98 0.037 0.44 0.13 0.25 -1.57 0.01 0.00 0.00 0.90 1.00 0.90 1.00 0.90 0.012 0.15 0.04 0.25 4.71 0.90 0.00. 0.00 0.63 1.00 0.63 1.00 0.63 0.025 0.30 0.09 0.25 2.36 0.90 0.00 0.00 •1.00 0.27 1.00 0.27 3.67 0.000 0.00 0.10 0.25 0.00 0.48 0.00 0.00 1.00 0.32 1.00 0.32 3.15 0.001 0.01 0.11 0.25 0.00 0.54 0.00 0.00 1.00 0.37 1.00 0.37 2.71 0.004 0.05 0.14 0.25 0.00 0.53 0.00 0.00 1.00 0.43, 1.00 0.43 2.32 0.014 0.16 0.16 0.25 0.00 0.55 0.00 0.00 0.37 1.00 0.37 1.00 0.37 0.004 0.05 0.14 0.25 0.00 0.90 0.00 0.00 0.43 1.00 0.43 1.00 0.43 0.014 0.16 0.16 0.25 0.00 0.90 0.00 0.35 0.25 0.65 0.25 1.00 0.25 0.000 0.00 0.45. 0.25 0.00 1.00 0.00 0.30 0.26 0.70 0.26 1.00 0.26 0.001 0.01 0.53 0.25 0.00 1.00 0.00 0.28 0.26 0.72 0.26 1.00 0.26 0.002 0.03 0.58 0.25 0.00 1.00 0.00 0.26 0.26 0.74 0.26 1.00 0.26 0.004 0.04 0.62 0.25 0.00 1.00 0.00 0.25 0.26 0.75 0.26 1.00 0.26 0.005 0.06 0.65 0.25 0.00 1.00 0.30 0.00 0.70 0.26 1.00 0.26 3.84 0.001 0.01 0.53 0.25 0.00 0.06 0.00 0.40 0.07 0.60 0.07 1.00 0.07 0.000 0.00 0.37 0.25 -0.15 1.00 0.00 0.35 0.09 0.65 0.09 1.00 0.09 0.000 0.00 0.43 0.25 -0.13 1.00 0.00 0.26 0.12 0.74 0.12 1.00 0.12 0.004 0.05 0.58 0.25 -0.10 1.00 Table A.2: Table of dimensionless parameter settings for all experiments conducted up to January 2007. 67 Appendix B B . l Images of apparatus Figure B. l shows an image of the entire apparatus. The annulus is positioned at a tilted inclination. The apparatus stands 2.8m high when in the vertical position, and 2.4m across when inclined to a horizontal position. It is currently located in Leonard S.Klink building at the Complex Fluids Laboratory. Figure B.2 shows the annulus mounted in the supporting frame. 68 Appendix B. Appendix B. Imaging Camera Lighting System Mirror Data Acquisition "Fish "Tank Annulus inside "Fish Tank " Gate Valve Figure B.2: Close-up image of apparatus 70 Appendix C C . l Secondary Flow Loop Figure C.l shows a schematic of the secondary flow loop. The secondary group of components consists of pipes and valves used to fill the preload chamber, to generate secondary interfaces and to clear airpockets. (9) 0 > * (h) (i) (7) (b) (11) L X H (12) (c) (4) (f) (2) (d) - O -(8) (9) X V (a) - C X — ' (6) (5) V (e) S1QL Figure C.l: Schematic of secondary flow loop 71 Appendix C. The design of the secondary flow loop allows the user to fill either the annulus from either pump, but the most practical methodology is the following simple procedure. After the annulus at (c) is drained and flushed, filling can proceed from tank (a). The slice gate valve at (f) must be closed and valves at (2),(3) must be off. Pumping from (1) can proceed until the annulus is full. Next, the preload chamber below (f) should be filled from pump at (5). Valve (6) should be off and the air release valve at (4) MUST BE OPEN. Failure to open this valve could cause a rapid pressure build-up in the pre-fill chamber at (f) and damage the apparatus. Once all the the pre-fill chamber is full, and the air is expelled, pumping from (5) can be stopped and valve (4) closed. Valves (12),(2) and (7) must remain closed. To begin a displacement, the slice sluice gate valve is opened and the pump at (5) operated. The temperature at (9) and the flow rate at (8) can be recorded via the computer interface, whilst the pump speed of (5) and the scale at (10) can be monitored. As the displacement proceeds, fluid is expelled through line (g) into bucket (h) for recycling or disposal. Valve (3) must be open during an experiment to prevent pressure build up. When an experiment is finished, the sluice gate valve is opened, along with valves (6) and (7). The air line (d) is opened at valve (2) to prevent suction. The system may be flushed by either connecting the line (d) to the mains water supply, or by pumping from (a) at a high flow rate. In the event of Carbopol being jammed in the annulus, then pumping with a high (or low) pH fluid wash will cause the polymer chains in the Carbopol to rapidly degrade, and the polymer can be easily removed. C.2 Instrumentation and Control Schematic Figure C.2 shows a schematic of the control system and other analytical measuring devices. The pressure transducers at (b) require A-D conversion, as does the thermocouple at (e) and the flow meter at (d). These are inputted into the SCB-68, and sent to the NI DAQ card for processing. The microcontroller (m) for the linear stepper motor (i) sends and receives serial commands from Lab View at (n). Finally, the scale at (g) is connected by to second serial port and can be monitored by Lab View after the scale resetting in output mode. This step is only required for calibration exercises, and in general the onboard display is sufficient for weighing 72 Appendix C. (c) (i) r © ©-c (b) 6f (i) (a) (d) 0 (f) (h) SCD-68 Microcontroller (m) j (g) (n) LabView Figure C.2: Schematic of measurement and control system 73 Appendix C. incredients for mixing polymer solutions. The only device not connected to Lab View is pump (a). Since this pump is only required to fill the annulus before experimentation, the pump is simply switched on/off and the pump rate set by a variable speed controller. 74 

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