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An experimental study of two multi-fluid flows of interest to the oilfield cementing industry Malekmohammadi, Sardar 2009

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An Experimental Study of Two Multi-fluid Flows of Interest to the Oilfield Cementing Industry by Sardar Malekmohammadi B.Sc., Amirkabir University of Technology, 2007 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 (Vancouver) July 2009 c Sardar Malekmohammadi 2009  Abstract In this work two multi-fluid flows are studied. In the first project, displacement flows in an eccentric annulus were studied experimentally. Displacement flows occur in the oil and gas industry during well construction when drilling mud is displaced by a cement slurry. To remove mud effectively, it is important to design the fluid rheology so that a steady displacement front can be achieved when displacing along an eccentric annulus. In our research, we have investigated the effects of viscosity, density and eccentricity of inner pipe on the interface dynamics. An experimental matrix is devised in a manner to capture the boundary between steady and unsteady displacements for specific pairs of fluids, and to compare against previously published models. Reasonable qualitative agreement was achieved however a systematic discrepancy was observed due to the presence of secondary flows and dispersion in the experiments. These effects have not been studied carefully so far and more sophisticated models are needed to predict this type of flow. In the second project, slumping flows of two non-Newtonian fluids in horizontal closed pipes were studied. This type of flow occurs during the abandonment of horizontal oil and gas wells, when sealing the well through the setting of cement plugs. We have studied the effects of changes in density difference and of small deviations from a perfectly horizontal inclination. The effects of these parameters on the slump length versus time were analyzed. Comparison of numerical and experimental results shows broadly similar trends, but with some qualitative differences also observed, possibly due to interfacial effects.  ii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Tables  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Co-authorship statement  . . . . . . . . . . . . . . . . . . . . . . .  x  1 Introduction . . . . . . 1.1 Cementing services . 1.2 Background . . . . . 1.3 Experimental set-up 1.4 Literature review . . 1.5 Thesis overview . . . 1.6 Bibliography . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  1 1 3 6 8 8 9  2 Displacement flows in eccentric annuli . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental methodology . . . . . . . . . . . . . . 2.2.1 Interface shape analysis . . . . . . . . . . . . 2.2.2 Experimental design and process related issues 2.2.3 Selection of fluids . . . . . . . . . . . . . . .  . . . .  . . . . . . .  . . . . . .  . . . . . .  . . . . . .  10 10 17 19 21 26 iii  Table of Contents  2.3  2.4  2.5 2.6  2.2.4 Experimental plan . . . . . . . . . . . . . . . . . Experimental results . . . . . . . . . . . . . . . . . . . . 2.3.1 Illustrations of typical displacements . . . . . . . 2.3.2 Parametric results: newtonian fluids . . . . . . . 2.3.3 Parametric results: non-newtonian fluids . . . . . Secondary flows and dispersion . . . . . . . . . . . . . . 2.4.1 Dispersive effects on the scale of the annular gap 2.4.2 Large-scale dispersion . . . . . . . . . . . . . . . 2.4.3 Combined effects: spikes and tails . . . . . . . . 2.4.4 Quantifying dispersion . . . . . . . . . . . . . . . 2.4.5 Other interesting phenomena . . . . . . . . . . . Discussion and conclusions . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  . . . . . . . . . . . . .  29 31 31 33 37 39 39 41 46 49 52 53 59  3 Buoyancy driven slump flows . 3.1 Introduction . . . . . . . . . 3.2 Analytical predictions . . . . 3.3 Methodology . . . . . . . . . 3.3.1 Experimental method 3.3.2 Numerical modeling . 3.4 Results . . . . . . . . . . . . 3.4.1 Transient results . . . 3.4.2 Quasi-Static results . 3.5 Discussion . . . . . . . . . . 3.6 Bibliography . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  65 65 67 70 70 74 76 76 79 79 86  4 Conclusions . . . . . . 4.1 Summary . . . . . 4.2 Future work . . . 4.3 Bibliography . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  89 89 91 93  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  iv  Table of Contents  Appendices A Horizontal displacement . . . . . . . . . . . . . . . . . . . . . . 94 A.1 Apparatus and method . . . . . . . . . . . . . . . . . . . . . 94 A.2 Typical result . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 B Slump test images  . . . . . . . . . . . . . . . . . . . . . . . . . 99  v  List of Tables 2.1 2.2 2.3  Properties of displacing fluids used in the experimental series. 30 Properties of displaced fluids used in the experimental series. . 30 Experimental condition for the different series. . . . . . . . . . 30  3.1 3.2  Series of experiments in slump tests . . . . . . . . . . . . . . . 72 Properties of the fluids used. . . . . . . . . . . . . . . . . . . . 73  vi  List of Figures 1.1 1.2  Displacement flow in a vertical eccentric annulus . . . . . . . . Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .  2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21  Schematic of the primary cementing process . . . . . . . . . Schematic of the experimental setup . . . . . . . . . . . . . Schematic of the optical set-up [42]. . . . . . . . . . . . . . . Schematic of interface shape and residence time variations . Examples of steady and unsteady displacements . . . . . . . Spatio-temporal plots of a steady displacement . . . . . . . . Spatio-temporal plots of an unsteady displacement . . . . . Residence time distribution for displacements . . . . . . . . Variation of σ∆t /µ∆t for Newtonian fluid series 1-3 . . . . . . Variation of σ∆t /µ∆t for non-Newtonian fluids . . . . . . . . Dispersive finger in the displacement of two Newtonian fluids Moving frame streamlines computed from the model of [37] . Edge-detected interface for a Newtonian fluid displacement . Recirculation observed in a Newtonian displacement . . . . . Spikes on the wide side in non-Newtonian displacements . . Spikes on the narrow side in Newtonian displacements . . . . Interface shape . . . . . . . . . . . . . . . . . . . . . . . . . Interface detection from the saturation curve . . . . . . . . . Effect of eccentricity on the saturation time growth rate . . . Effect of flow rate on the saturation time growth rate . . . . A typical drainage wall layer . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  2 7 11 18 20 22 32 33 34 35 36 38 40 44 45 46 47 48 49 51 51 52 53 vii  List of Figures 2.22 Static channel on the narrow side for an experiment in series 5 54 2.23 Comparison of experimental results with the lubrication model 57 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12  Prediction of slump length . . . . . . . . . . . . . . . . . . . . Schematic of the experimental set-up . . . . . . . . . . . . . . Flow curves of fluids used in slump tests . . . . . . . . . . . . Range of yield stresses of Carbopol . . . . . . . . . . . . . . . Inertial effects in a typical simulation. . . . . . . . . . . . . . . Comparison of interface shape . . . . . . . . . . . . . . . . . . Experimental and numerical results for the slump progression The experimental errors in Series 1 . . . . . . . . . . . . . . . The predicted slump length by the analytical method . . . . . he effect of low shear rate viscosity on slump length . . . . . . The effect of yield stress on slump length . . . . . . . . . . . . Transient behavior of slumping . . . . . . . . . . . . . . . . .  69 70 72 73 76 77 78 78 80 82 83 84  A.1 A.2 A.3 A.4  Visualization system for horizontal experiments Schematic of geometrical transformation . . . . A typical horizontal experiment . . . . . . . . . unwrapped image of the annulus . . . . . . . . .  95 96 97 98  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  B.1 Experimental setup used for slump tests . . . . . . . . . . . . 99 B.2 A typical slump test . . . . . . . . . . . . . . . . . . . . . . . 100  viii  Acknowledgements This work could not have been done without the support and guidance of my supervisors. I would like to thank Dr. Ian Frigaard and Dr. Mark Martinez for all their support, help, inspiration and encouragement over the last two years. A special thanks to Stefan Storey for provding figures 1.2, 2.3, 2.11, 2.13, 2.17, 2.21 and 2.22. I would also like to thank Dr. Monica Naccache for numerous discussions during her visit in 2008. The financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Schlumberger is also gratefully acknowledged.  ix  Co-authorship statement Chapter 2 is a manuscript co-authored with M. Carrasco-Teja., S. Storey, I.A. Frigaard and D.M Martinez. I.A. Frigaard, S. Storey and myself are the primary authors in all regards. The theoretical analysis was carried out by I.A. Frigaard and M. Carrasco-Teja, some part of experimental results was prepared using the experiments performed by S. Storey. Chapter 3 is a manuscript co-authored with M.F. Naccache, I.A. Frigaard and D.M Martinez. The numerical analysis was carried out by M.F. Naccache and the analytical work was performed by I.A. Frigaard.  x  Chapter 1 Introduction In this work we study the flow of multiple Newtonian and non-Newtonian fluids in ducts, and more specifically in annuli and pipes. Although the work has application in a number of different industrial processes, this work is motivated from one unit operation in the oil and gas industry, namely cementing. In this chapter we shall briefly introduce this operation, below in §1.1. Thereafter, in §1.2 we give an overview of the types of fluid used in cementing and those used in our experiments to mimic the rheological properties. Section 1.3 presents an overview of the experimental apparatus used. The chapter closes with a guide to the thesis (§1.5).  1.1  Cementing services  Cementing services mainly deal with constructing, protecting, maintaining zonal isolation and abandonment of any oil and gas well. In general, cementing services involves three main operations: (i) “Primary Cementing” is the first and most important cementing operation. The purpose of primary cementing is to construct a physical barrier around an oil or gas well and seal the well thoroughly. The common practice in the oil industry is to place a steel casing inside the wellbore after drilling and then fill the annulus gap between the bed rock and the casing with cement, see Figure 1.1. To prevent mixing and maintain the hydraulic seal along the well during the productivity life of the well, the well should be lined with cement. However for long 1  Fluid 2 Fluid 1  Flow Direction  1.1. Cementing services  g Figure 1.1: Displacement flow in a vertical eccentric annulus wells it is very difficult to properly place the cement all around the casing. Primary cementing is usually not free of defects and many voids and micro channels may be left in the cemented annulus. These defects may allow reservoir fluids to percolate to surface, reducing the ability of the well to produce oil. Defects in the cement also allow water to come into contact with the steel casing, leading to corrosion and eventually well leakage. The first project in this thesis, (discussed in Chapter 2), is motivated by the primary cementing displacement flows. (ii) Local defects in the cemented annulus can be repaired by “Squeeze Cementing”. This operation usually occurs a few years after the well has been put into production, when defects or loss of productivity are first noticed. In this process, cement slurry is forced into the cemented annulus and the surrounding by applying high pressure. Designing the cement slurry with an appropriate particle size and understanding the particle flow under pressure are the most challenging tasks in squeeze cementing, see [2]. we do not consider squeeze cementing flow further in this thesis.  2  1.2. Background (iii) The final cementing operation is “Plug Cementing”. This process is used to seal an oil well when the well is abandoned. In this process a plug of cement is placed inside the well. This plug then sets, to create a solid plug inside the well. The second project in this thesis (in Chapter 3) is motivated by the plug cementing process. Further details of primary cementing and plug cementing processes are given in §2.1 and §3.1, respectively.  1.2  Background  A cement slurry is a thick suspension of cement, water and additives used in well construction. It consists of solid particles suspended in an aqueous phase. The aqueous phase is a water based solution of additives and ionic materials. In general, the viscosity of cements depends on the solid volume fraction, which varies from 0.2 to 0.7, and polymeric additives. By using polymer based additives known as plasticizers and superplasticizers, the rheological properties of cement slurries can be adjusted for different process conditions. The optimized rheological properties of cement depend mostly on drilling fluids properties, geometry of the well and the interaction between cement and the formation. Lignosulfonates and hydroxycarboxylic acids like tartaric acid and citric acids, which are common Plasticizers in cement industry, also act as effective cement retarders. Since lignosulfonates have strong retarding properties, they are rarely used in cold areas. Superplasticizers include sulfonated polymers such as polycarboxylate and polystyrene sulfonate compounds. Polynaphtalene is the most commonly used superplasticizer in designing rheological properties of cement slurries. However, due to its toxicity to algae and nonbiodegradability, its application is limited to non-marine areas. In these areas where the use of Polynaphtalene is not allowed, polymelamine sulfonate and polycarboxylate-based polymers are used instead.  3  1.2. Background In order to drill a well as efficiently as possible, drilling mud is used during the drilling process. It suspends the cuttings and cools the drill bit. In general, there are two types of drilling mud: polymer muds and bentonite muds. Polymer muds are thinned by adding water and used on mineral formations. Bentonite muds are easier to displace and their rheology can be adjusted widely by dispersants. The cement slurry and the drilling mud are usually incompatible and can form a highly viscous material during displacement process. This incompatibility arises due to the chemical composition of both cement and drilling mud. In order to prevent mixing of these two fluids, buffers known as preflushes or spacers are used to separate them. Chemical washes are the main solutions used as buffers in well cementing. The density of a spacer fluid can be adjusted by adding weighting agents. Silica flour, calcium carbonate and barite are the common weighting agents used in spacer fluids. Densifying the spacer fluid can improve displacement of the mud. In contrast, washes have density and rheological properties very similar to those of water. Washes erode and disperse mud layers. They also wet the surfaces to provide a better bonding for cement. Depending on well inclination and fluid properties, different sequences of the above fluids are circulated in the well to achieve a better displacement. However, the usual sequence for a cementing job is to displace mud with a highly viscous spacer first. Then washes are pumped to clean both the casing and formation walls. At the final stage the mixture of mud and washes is displaced by another spacer fluid. The purpose of the last step is to prevent cement slurry contamination with other fluids [3]. Although some washes are water based fluids with Newtonian behavior, in general, the fluids used in cementing are mostly non-Newtonian. Spacer fluids are shear-thinning. The rheology of cement and drilling mud is fairly complex. Drilling mud in general has yield stress and a gel is formed whenever the applied shear stress, is lower than its yield stress. Characterizing  4  1.2. Background drilling mud is quite difficult. Drilling muds show shear-thinning behavior at high shear rates above the yield stress. The Hershel-Bulkley model is commonly used to characterize both drilling mud and cement slurry. This model is composed of two parts: a yield stress part and a power law part. Whenever the shear stress is below the yield stress of the fluid, the fluid remains unyielded and the shear rate is zero. However at shear stress above the yield stress, the model shows shear-thinning behaviour following a power law model. Drilling muds exhibit some thixotropic properties as well. In other words, their rheological properties vary with time which makes them difficult to be modeled very accurately. Running experiments with real cement and drilling fluids in a laboratory is not an easy task. These fluids are hard to work with and washing and cleaning them in an apparatus is time consuming. Furthermore, visualizing the interface of the two fluids is almost impossible. Drilling fluids and cement are opaque. These facts and other problems associated with working with them imply that studying the fluid dynamics of cement and drilling fluids during displacement is almost impossible experimentally. However, Xanthan gum and Carbopol solutions are widely used to mimic shear-thinning and yield stress behaviour in experimental studies of flows. These solutions exhibit non-Newtonian behavior quite similar to those of cement slurries and drilling fluids. Xanthan gum and Carbopol are water soluble polymers which create transparent fluids by dissolving small amount of their powder in water. The rheological properties of these fluids mostly depends on concentration of the polymer. This helps experimenters to run experiments with a wide range of rheological properties. In addition to that, these polymeric solutions are transparent, so that analyzing the interface between the fluids is much easier than with the real fluids. Also, since these fluids are washable with water, experiments can be carried out at a faster rate than with real fluids. Beside these advantages, Xanthan gum and Carbopol solutions are non-toxic and safe for disposal.  5  1.3. Experimental set-up In this experimental study we have used Xanthan gum, Carbopol and Glycerin solutions to simulate the real fluids used in cementing oil and gas wells. The preparation and characteristics of these fluids are described, in §2.2.3.  1.3  Experimental set-up  The experimental setup we have used was built by previous MASc. student (S. Storey), in 2006. Although he conducted some experiments with nonNewtonian fluids, the main focus of his work was on building the apparatus, calibration and checking the apparatus performance in displacement flows. He designed the apparatus to replicate displacement flows for a wide range of flow rates and annulus inclination. The largely calibration experiments he performed were all conducted in a vertical annulus and in a non-systematic manner. This made meaningful physical conclusion hard to make, based on the experiments performed. In addition, the fluid combinations used were not enough to check the effect of all parameters. Finally, the analysis method did not provide a clear understanding of the interface dynamics around the annulus. However, the previous analysis method was capable of distinguishing steady from unsteady displacements, at least approximately. Figure 1.2 shows the experimental set up built in 2006. The research in this thesis, on displacement flows, started with the objective of completing the previously started experimental work with a narrower annular gap size. The main goal was to perform a consistent set of experiments to study displacement flows and explore the interface dynamics of steady and unsteady displacements in detail. Initially, the new set of experiments was intended to be conducted in a horizontal annulus, to study horizontal well cementing. In order to compare the experimental results with the predictions of Hele-Shaw model, some modification were made in the first step. The inner pipe of the annulus was replaced by one with larger 6  1.3. Experimental set-up  Figure 1.2: Experimental setup built by Stefan Storey [4]. diamater to better satisfy the Hele-Shaw assumption of narrow gap size [1]. The visualization system was also modified to unwrap the whole annulus. This was made by placing tall acrylic mirrors in the apparatus. However, initial control experiments showed some difficulties in maintaining the constant eccentricity along the annulus in horizontal position. The inner pipe deflects in most experiments due to its weight and buoyancy effects. In order to keep eccentricity constant, a few supports were required along the inner pipe which were tedious to design and mount. There was also the possibility of flow disturbance by mounting those supporters. Details of our horizontal displacement experiments are given in Appendix 1. Instead of horizontal tests, it was decided to run experiments in vertical position with a narrower gap and to study the interface dynamics with a robust analysis method. In addition to that, besides our main project on primary cementing, the plug cementing process was also explored as a second  7  1.4. Literature review project. In this project, we analyzed the slump flows in nearly horizontal wells. The slump tests were conducted with the same apparatus, with minor modifications. The inner pipe was removed and a high resolution SLR camera was used. The slump tail of fluids are usually long and a wider lense was required to capture the slump length of fluids. The images of experimntal set up used for slump experiments and a typical test are given in Appendix 2.  1.4  Literature review  The literature review of displacement flows and slump flows are given in §2.1 and §3.1 of this thesis. The information provided in those sections is not repeated here and the reader is referred to them.  1.5  Thesis overview  In this thesis two separate, yet complementary studies, are performed to gain insight into the cementing process. In Chapter 2, the primary research effort of this thesis is presented. Here both experimental and numerical results are presented on the displacement of slowly-moving fluids in narrow eccentric vertical annuli. Both Newtonian and non-Newtonian fluids are considered. In the second study, i.e. Chapter 3, we attempt to study buoyancy driven slump flows in near-horizontal pipes and initiate this work by examining the behaviour of the interface of two fluids in an exchange flow. In Chapter 4 an overall summary of the work is presented.  8  1.6. Bibliography  1.6  Bibliography  [1] S.H. Bittleston, J. Ferguson, and I.A. Frigaard. Mud removal and cement placement during primary cementing of an oil well; laminar nonNewtonian displacements in an eccentric Hele-Shaw cell. Journal of Engineering Mathematics, pages 229–253, 2002. [2] Roscoe Moss Company. Handbook of Ground Water Development. Wiley, 1990. [3] E.B. Nelson and D. Guillot. Well Cementing, 2nd Edition. Schlumberger Educational Services, 2006. [4] S. Storey. Experimental study of non-Newtonian dispacement flows in vertical eccentric annului, 2007. MASc Thesis, University of British Columbia, Canada.  9  Chapter 2 An experimental study of displacement flow phenomena in narrow vertical eccentric annuli 1 2.1  Introduction  In this chapter, we present an experimental study of slow laminar miscible displacement flows in vertical narrow eccentric annuli. The underlying motivation for the study comes from the oilfield process of primary cementing, which we explain briefly below. The objectives of our study are partly to provide a controlled set of experiments, suitable for exploring the validity of mathematical models of the displacement flow, and partly to consider displacements in parameter ranges having some overlap with field conditions. There is a large scientific literature on flows in an annular geometry in which the walls are stationary or moving. Some of the main problems studied include: instability and transition in Taylor-Couette type flows; rotational flows and interfacial instabilities in horizontally stratified fluids, driven around a vertical annulus; thermal convection and generation of secondary flows. In each of these areas the literature is extensive. The annular geometry is also widely used in industrial applications: in heat exchangers, food processing, 1 A version of this chapter has been submitted for publication. Malekmohammadi, S. Carrasco-Teja, M. Storey, S. Frigaard, I.A. and Martinez, D.M.  10  2.1. Introduction  D  r  i  l  l  n  e  w  R  e  m  o  v  e  I  n  e  r  s  e  s  w  t  a  e  g  o  f  d  r  i  l  l  p  i  p  e  l  P  e  p  p  e  s  l  i  a  m  P  u  m  p  D  i  p  s  l  s  l  t  c  u  t  e  s  n  f  g  a  l  u  r  l  e  d  c  i  &  a  d  l  s  i  t  u  r  r  l  m  e  a  u  d  E  n  d  o  f  c  i  n  l  u  o  p  e  r  a  t  i  o  n  a  y  n  a  n  u  s  Figure 2.1: Schematic of the primary cementing process, showing the various stages (left to right) in cementing a new casing. fluidized beds, pulp screening, extrusion and oil well construction, which motivates our work. Primary cementing is described at length in the recent text by [29]. In this process a steel casing is cemented into a wellbore, ensuring a tight hydraulic seal with the outer rock formation. The annular space to be filled with cement is initially full of drilling fluid (or other fluids), which must be removed during the cement placement. The primary cementing process proceeds as follows, see Figure . 2.1. A new section of the well is drilled. The drillpipe is removed from the wellbore, leaving drilling mud inside the wellbore. A steel tube (casing or liner) is inserted into the wellbore, typically leaving a gap of ≈ 2cm between the outside of the tube and the inside of the wellbore, i.e. the annulus. The tubing is inserted in sections of length ≈ 10m each. At certain points, centralizers are fitted to the outside of the tube, to prevent the heavy steel tubing from slumping or sagging to the lower side of the wellbore. However, it is still very common that the annulus is eccentric, especially in inclined wellbores. Once the tube is in place, with drilling mud on the inside and outside, a sequence of fluids are circulated down the  11  2.1. Introduction inside of the tubing reaching bottom-hole and returning up the outside of the annulus. Typically, a wash or spacer fluid is pumped first, followed by one or more cement slurries. The rheologies and densities of the spacer and cement slurries can be designed so as to aid in displacement of the annulus drilling mud, within the constraints of maintaining well security. The fluid volumes are designed so that the cement slurries fill the annular space to be cemented. Drilling mud follows the final cement slurry to be pumped and the circulation is stopped with a few metres of cement at the bottom of the inside of the casing, see final figure in Figure 2.1, and the cement is allowed to set. The final part of cement inside the tubing is drilled out as the well proceeds. From the fluid mechanics perspective, since the volumes of fluids pumped are relatively large, so that successive interfaces are separated, it is reasonable to consider alone the displacement flow between a single pair of fluids. Equally, the geometry changes slowly in the axial direction, relative to the scale of the annular gap or circumference, so that consideration of a uniform annulus is also reasonable. Thus, we consider displacement flows through a uniform eccentric annulus. The fluids used in cementing and those that we study are both Newtonian and non-Newtonian. In the latter case we focus mainly on fluids where the behaviour is dominated by a nonlinear shear viscosity, i.e. shear-thinning and yield stress effects. Rudimentary hydraulics-style studies of annular flows for this type of fluid may be found in the technical literature of various industries, dating back to the 1960’s or earlier. However, detailed experimental studies of this type of fluid flow in annular geometries are more recent. Probably the best known of these studies are those by Nouar and Lebouch´e, e.g. [28, 31, 32], those by Nouri and Whitelaw, e.g. [33, 34, 35], and the extensive studies of Escudier and co-workers, e.g. [9, 10, 11, 12]. Globally these studies consider fluids similar to those we use here: CMC, Xanthan, Carbopol, etc., which are the most commonly used fluids for this type of experiment, and cover  12  2.1. Introduction a wide range of eccentricities, aspect ratios, inner-body rotation rates and Reynolds numbers. Detailed LDA/LDV measurements of velocity profiles have been made and in many cases these have been compared favourably with computational results. There are also numerous computational studies for these flows and some analytical solutions. The reader is referred to [13], (which also contains an excellent and comprehensive bibliography), for an overview of this area. Thus, the experimental study of a single generalised non-Newtonian fluids, flowing in laminar regime through an annulus, is a mature and well studied area. In terms of eccentric annular displacement flows, the experimental literature is much smaller than for single phase flows. The first detailed study that we know of was carried out by Tehrani and co-workers: [23, 44, 45]. These experiments were carried out in a narrow annulus, (aspect ratio: δ = 0.035), of 3m in length, fully inclinable. Various flow rate and eccentricity combinations were tested, using Xanthan as the base non-Newtonian fluid. The main measurement method consisted of adding a conductive tracer to one fluid and measuring the fluid conductivity at 8 azimuthal positions around the annulus, close to the exit. The conductivity data was used to give the displacing fluid concentration at the exit. This data was compared with model-based output, i.e. in the form of a final displacement efficiency at the end of the experiment. The use of conductivity has some advantages over visualisation, in terms of objectivity and the ability to use opaque fluids. On the other hand, the use of a single displacement efficiency to characterise the flow has drawbacks in terms of generalisations to longer annuli. Reasonable qualitative agreement was however found between model predictions and experimental results; see [44]. A number of interesting flow phenomena were also reported in these studies. Other than the studies by Tehrani and co-workers, there have been only occasional experimental results reported, e.g. [8, 18, 30, 46]. Our initial interest in annular displacement flows came from revisiting  13  2.1. Introduction the studies of Tehrani and co-workers. A simple dimensionless analysis of this type of flow, between two non-Newtonian fluids, revealed that as well as Reynolds number, buoyancy number, P´eclet number, density ratio, viscosity ratio, eccentricity, aspect ratio and annular inclination, up to 4 other dimensionless rheological parameters need considering, i.e. 8-12 dimensionless parameters. Without some simplifying focus, it was clear that it would be infeasible to study such flows effectively, either experimentally or computationally. The focus chosen was to look at the narrow gap (Hele-Shaw) limit in which inertial effects are negligible and at the high P´eclet number limit, that is anyway commonly found. This simplified the parametric dependence to 5-9 parameters, which is still large for an experimental study. In place of the classical “displacement efficiency” approach it was decided to first try to understand the dynamics of the displacement flows in this simpler regime, via mathematical modeling, and then use more limited experimental studies to validate the dynamical understanding and illuminate any major shortcomings. The modeling approach that we have used for these flows is outlined in [2], although the underlying idea of using a Hele-Shaw/porous media approach dates back to [24] and [44]. This model has been analysed in depth by Pelipenko & Frigaard in the sequence of papers [36, 37, 38], which focus principally at near vertical wells. The dynamics are dominated by the existence (or not) of steady traveling wave solutions, i.e. for certain parameter values the displacement front advects along the annulus at the mean pumping speed. When this does not occur, the front tends to advance faster on the wider side of the annulus and elongates into a finger. Where yield stress fluids are concerned, (as is the typical case industrially), it is possible for the fluids to become stuck in the narrow part of the annulus, bridging between inner and outer walls. For limited parameter ranges (near concentric annuli) it is possible to construct analytic solutions to the displacement problem, exhibiting the steady traveling wave behaviour; see [36]. These steady states  14  2.1. Introduction are in fact found computationally for a much wider range of parameters than those for which it is possible to find analytical solutions, [37], and it is possible to approximately predict steady and unsteady displacements using a lubrication-style displacement model; see [38]. Another body of work is directed at understanding the stability of these displacement flows. The basic transition from having a steady traveling wave displacement to an elongating displacement front is of course one type of instability. However, in such flows the interface does not necessarily become locally unstable. A typical industrial situation involves displacements of less viscous fluids by more viscous fluids. Thus, classical viscous fingering which is commonplace in Hele-Shaw geometries is not a key concern. Some prediction of viscous fingering regimes is however made in [38]. Instead, the approach taken by Moyers-Gonz´alez & Frigaard is to consider the stability of parallel flows in the Hele-Shaw setting; see [26, 27]. This analysis is directed at the flows that evolve from unsteady displacements, where at longer times the interface becomes pseudo-parallel to the annulus axis. More recently we have started to consider horizontal well cementing. Although the underlying Hele-Shaw model is similar to that used for vertical cementing, the physical phenomena observed are different. Horizontal well annuli are nearly always eccentric. When there are strong density differences between the fluids displaced there is a competition between the effects of buoyancy and eccentricity. These regimes have been studied by the authors in [4]. Somewhat surprisingly, it is found that buoyancy has an essentially passive role in these displacements: the interface tends to “slump” under the effects of buoyancy, as it advances. At long times buoyancy simply determines the axial length-scale of the interface: whether or not the displacement gives a steady solution depends on the fluid rheologies, annular eccentricity and inclination from horizontal. In the absence of steady state displacement fronts the interface elongates, (with the possibility of static fluid on the narrow side in the case of yield stress fluids). The very recent work in this  15  2.1. Introduction domain, considers the effects of rotating or reciprocating the inner pipe of the annulus; see [3]. In all the above we have worked under the Hele-Shaw model assumptions, (which we outline more precisely below in §2.2.2). These assumptions and this style of modeling are however strictly valid for single phase flows. For multi-phase systems a variety of phenomena can impact the validity of the model assumptions at the interface. In the first place, under suitable conditions on the mobility ratio, it is known that local instabilities arise, i.e. viscous fingering; see e.g. [17]. In the second place, dispersive effects are always present in a miscible displacement. Thirdly, the local velocity is nearly always 3D at the interface. The combination of the above 3 phenomena can be complex and their impact on the validity of the Hele-Shaw approach is subtle. Regarding viscous fingering, for the most part this is not a concern for the flows considered as we typically have positive viscosity ratios. Dispersive effects are present in our flows, in particular due to significant azimuthal current close to the interface, driven by annular eccentricity. These effects will form a significant part of our study. We note however that the experimental timescales considered are relatively short, meaning that we are very far from diffusive dispersion regimes. Regarding 3D effects at the interface, these are unavoidable. As with dispersion, these effects occur and will not be eliminated by e.g. working with smaller aspect ratio annuli. Instead the key question is whether or not these local phenomena have an impact on the global dynamics of the system or whether the effects remain local, as is often the case, e.g. [49] found that the local details of the concentration front close to the interface were relatively unimportant in comparison to the bulk pressure gradients, when considering anisotropic porous media displacements. There is a growing literature on miscible displacements in pipes and channels. Whilst relevant, here we have strong geometric effects on the base flow, so that direct comparison with this literature is hard. For brevity, in place of  16  2.2. Experimental methodology a review, we refer the reader to [38] or [4] for an overview of relevant studies. Instead, we highlight within our results when we have observed similar effects to those already published. An outline of this chapter is as follows. In §2.2 following, we describe the experimental setup. The first sequence of results is given in §2.3, where we classify the displacements as either steady or unsteady, in each of our 6 series of experiments. Section 2.4 examines the significant role of dispersion and secondary flows in our experiments, illustrating the various observed phenomena. We close with a discussion, comparing with model results and assessing the overall validity of the Hele-Shaw modeling approach.  2.2  Experimental methodology  A schematic of the experimental setup is given in Figure 2.2. The annulus dimensions are rˆo =1.91 cm, rˆi =1.27 cm, and a length of 188.3 cm. The outer pipe is constructed from acrylic tubing with a wall thickness of 12.7 mm [42]. The inner pipe is an aluminium pipe with wall thickness of 1.58 mm. It is mounted on two adjustable stainless steel bolts that allow the inner pipe eccentricity to be adjusted relative to the fixed outer pipe. Eccentricity is measured with micrometer depth gauges mounted on either end of the annulus. The annulus is immersed in a tall tank with a square cross-section filled with glycerin, to reduce optical distortion by matching the index of refraction of the curved acrylic pipe with that of the glycerin. The main flow loop consists of a progressive cavity pump (PCP) supplying the annulus with displacing fluid from the bottom and another pump that fills the annulus with displaced fluid from the top. The flow rate is controlled by the use of a series of valves and by the PCP (Seepex MD Dosing). The pump has a maximum flow rate of 2 L/min and is driven by a 1.5 hp singlephase AC motor. Variable speed operation is achieved through a variable frequency drive. 17  2.2. Experimental methodology  Figure 2.2: Schematic of the experimental setup All devices are calibrated before the experiment. A thermocouple is mounted inside the inlet pipe of the annulus. The flow rate is measured with a Cole-Parmer pilot-scale magnetic flowmeter (EM101-038) which the manufacturer specifies as accurate to 2%, the output of which was acquired by the computer. The flowmeter accuracy was cross-checked in a simple calibration experiment by measuring the mass of fluid pumped over a fixed time interval. Finally, a weigh-scale measures the mass of the displacement fluid container, which is used to check the accuracy of the flow control, (as well as during fluid preparation). To run a typical displacement experiment the upper section of the annulus is filled initially with displaced fluid while the gate valve is closed. Then, the bottom section of the annulus, below the gate valve, is filled with displacing fluid, while bleeding out any trapped air. A displacement experiment starts by slowly opening the gate valve and pumping the displacing fluid, dyed black using pen ink, from the bottom of the annulus to the top, see Figure 2.2. The 18  2.2. Experimental methodology interface is tracked by a digital camera which is mounted on a linear actuator that moves parallel to the axis of the annulus. The camera continuously captures images until the interface reaches the top of annulus, at which time the pumping stops. At the end of each experiment both fluids are drained to the waste container.  2.2.1  Interface shape analysis  All images are taken with a digital camera mounted to a linear actuator, the location, velocity and acceleration of which are controlled via LabView. The camera has a 35 mm SLR compatible lens, mounted to the body with an F-C adaptor. The camera images 270 degrees of the annulus by the use of a mirror; see Figure 2.3. Each image thus consists of a reflected image from the side of the annulus and a non-reflected image from the front. Images are captured at a frame rate that depends on the flow rate, ranging from 10 Hz at high flow rates to 4 Hz at slow flows. The images are captured in uncompressed 8 bit monochrome format, with a signal to noise ratio of 50 dB. The two images are unwrapped via a simple geometric transformation. The edges are located on the transformed images, and then the two images are collocated and registered. This geometric transform increases the level of noise at the image edges, one of which is at the narrow side of the annulus. There is also some loss of information close to the annulus walls in each image. In order to measure the steadiness of the interface shape, two fixed vertical locations were selected at 300 mm and 1000 mm above the gate valve. Pixel values of the images were recorded along the circumference at these two heights. Pixel values varied in grayscale from 0 (white) to 255 (black), and a total of 130 pixels were located on each circumferential line. Before starting any analysis, an initial background image is subtracted from each image, to correct for local lighting variations. As the interface passes each fixed height, the value of each pixel on the circumferential line increases from zero 19  2.2. Experimental methodology Displacing Fluid  Inner pipe  Outer pipe  Fish Tank  Optical Ray Path CCD Camera Optical Ray Path Mirror  Figure 2.3: Schematic of the optical set-up [42]. until it reaches a maximum. This process at each pixel can be described by a saturation curve in which darkness intensity is plotted against time. By normalizing with the local maximum pixel value, the normalised darkness intensity varies from 0 to 1 as the interface passes. The saturation time, when an interface is regarded to have passed a given position is the time at which the normalised darkness intensity is equal to 0.95, i.e. because the interface is never completely sharp in a miscible displacement. In an ideal case where the interface is a horizontal line all pixels are saturated simultaneously. However in an eccentric annulus, a flat interface is rarely formed and pixels are saturated at different times. By subtracting the saturation time measured for pixel j at the upper (downstream) location from that at the lower (upstream) location, we arrive at a residence time ∆tˆj , which is the time taken for the interface to traverse between lower and upper positions at the azimuthal position corresponding to pixel j. On assuming a “piston-like” displacement at the mean speed of the flow,  20  2.2. Experimental methodology ˆ say wˆ ∗ = Q/π(ˆ ro2 − rˆi2 ), the idealised mean residence time is denoted ∆tˆp : ∆tˆp =  ∆ˆ y , wˆ ∗  (2.1)  where ∆ˆ y =700 mm, is the vertical distance between upper and lower measurement locations. Throughout this chapter we shall use the “hat” notation, i.e. ˆ·, to denote variables that are dimensional. The piston-like residence time is used to scale the individual residence times, resulting in: ∆tj =  ∆tˆj , ∆tˆp  j = 1, 2, ..., 130.  (2.2)  In a steady displacement, where the interface travels in a steady manner, the normalized ∆tj should give the same constant value for each pixel j, (see Figure 2.4a). It should be noted that the interface itself does not need to be horizontal in order for ∆tj to be constant. The interface can be any shape so long as it maintains that shape during displacement. In an unsteady displacement, see Figure 2.4b, the fluid on the wide side flows faster than the fluid on the narrow side. Thus, the interface stretches as the flow progresses. Different pixels along the circumference have different residence times and the distribution of ∆tj suggests the unsteadiness of the displacement. For an objective measure of the unsteadiness we may consider the standard deviation of the ∆tj distribution, say σ∆t . A large standard deviation suggests an unsteady displacement while a small standard deviation suggests a steady displacement.  2.2.2  Experimental design and process related issues  As explained in §2.1 the objectives of our study were partly to provide a controlled set of annular displacement experiments, suitable for exploring the validity of the Hele-Shaw modeling approach adopted previously, and partly to consider displacements in parameter ranges having some overlap with 21  2.2. Experimental methodology  3  2  2  × 3  ×  ×  1  ×  1  ×  ×  '  '  3  1  3  2  1  1 ×  2  ×  ×  1 ×  ×  ×  ×  ×  ×  ×  ×  ×  y  y  x  x  Figure 2.4: Schematic of interface shape and residence time variations for: a) steady displacement b) unsteady displacement. If all points lie on ∆t = 1, they move at exactly the bulk mean velocity.  22  2.2. Experimental methodology field conditions. We therefore briefly review the dimensionless parameters of relevance to the modeling approach and field conditions. For further detail on the modeling approach the reader is referred to [2] or [38]. The main simplifications that we adopt, with respect to industrial conditions, are to consider a single uniform section of the annulus, fix the orientation at vertical and to consider only 2 fluids in our displacement. The fluid types that we consider are shear-thinning and yield stress fluids, although half of the experiments are conducted with Newtonian fluids. These fluids are characterised in the oilfield cementing industry by rheological models such as the Herschel-Bulkley model, (including the Bingham, power law and Newtonian models as sub-cases). This model contains 3 parameters: the yield stress, τˆk,Y , the consistency, κ ˆ k , and the (shear-thinning or) power law index, nk , (where k = 1, 2 denotes the fluid). The fluid densities are denoted ρˆk . In order to follow the Hele-Shaw approach of [2], we define the aspect ratio of circumferential and radial length-scales, δ, in the following way: δ=  rˆo − rˆi . π(ˆ ro + rˆi )  (2.3)  Azimuthal and axial distances are scaled with 0.5π(ˆ ro + rˆi ), whereas radial distance from the annular centreline is scaled with 0.5(ˆ ro − rˆi ). For a velocity ∗ scale we take the mean flow velocity, wˆ : w ˆ∗ =  ˆ∗ Q . π(ˆ ro2 − rˆi2 )  (2.4)  Axial and azimuthal velocity components are scaled with wˆ ∗ , and radial velocity with δ wˆ ∗ . A representative shear rate is γˆ˙ ∗ = 2wˆ ∗ /(ˆ ro − rˆi ), which ∗ is used for the viscous stress scale τˆ : τˆ∗ = max[ˆ τk,Y + κ ˆ k (γˆ˙ ∗ )nk ]. k=1,2  (2.5)  23  2.2. Experimental methodology The viscosity scale is, µ ˆ∗ = τˆ∗ /γˆ˙ ∗ , and finally, densities are scaled with the maximum density: ρˆ∗ = maxk=1,2 {ˆ ρk }. Primarily the Hele-Shaw approach relies on the neglect of terms in the Navier Stokes equations that are of O(δ) and O(δRe), where Re =  0.5(ˆ ro − rˆi )wˆ ∗ ρˆ∗ , µ ˆ∗  (2.6)  is the Reynolds number. The field range for δ is typically in the range 0.01 to 0.1. Our annular radii give δ = 0.064, which is in this range. We have also conducted a limited number of experiments in a slightly larger aspect ratio annulus, δ = 0.084. Moving to a much narrower annular gap presents problems in both cleaning of the apparatus and in terms of controlling the uniformity when eccentric, i.e. small deflections of the inner or outer wall become very significant. Regarding the Reynolds number, this has a very wide range in field applications, ranging from near-creeping flows to strongly turbulent flows. However, for very large Re flows the Hele-Shaw approach is anyway not applicable, so here we focus mainly on experimental laminar flows in the Reynolds number range of 0 − 10. This is restrictive from the industrial perspective but does represent a limiting parameter regime that is easier to understand. There are two development lengths to consider in our apparatus. Firstly, we have a development length-scale associated with the width of the annular gap. The timescale for viscous diffusion across the annular gap is: ρˆ∗ (ˆ ro − rˆi )2 tˆv = 4ˆ µ∗  (2.7)  and the length-scale is thus: wˆ ∗ tˆv ≪ 0.5π(ˆ ro + rˆi ) ≈5 cm, since we have δRe ≪ 1 for the flows considered. Secondly, we may consider development of the Hele-Shaw type flow in the azimuthal-axial directions. The boundaries at the ends of the Hele-Shaw cell (annulus) are fully mobile and there are no  24  2.2. Experimental methodology fixed boundaries in the azimuthal direction to generate boundary layer flows. Therefore, we expect flow developments to take place on the shortest lengthscale of the Hele-Shaw cell, i.e. the azimuthal length-scale: 0.5π(ˆ ro + rˆi ). For steady displacements another development length relates the length required for the front and rear of the interface to pass the entry point. Finally, at the inflow, there is also a development from the initial shape of the interface at the gate valve towards a steady state shape. Typical steady displacements we have observed experimentally in vertical annuli do not have axial extensions greater than the azimuthal length-scale 0.5π(ˆ ro + rˆi ). Thus, it appears that the choice of lower observation point at 30cm is reasonable. Other than the Reynolds number which does not appear in reduced models, the main flow-controlling parameter is the dimensionless buoyancy number, b = (ρ2 − ρ1 )/St∗ . Here St∗ is the Stokes number of the flow, defined in terms of the dimensional parameters by: St∗ =  τˆ∗ . 0.5ˆ ρ∗ gˆ(ˆ ro − rˆi )  (2.8)  The Stokes number can vary in the range 0.1-100 in the field setting, but the dimensionless density difference (ρ2 − ρ1 ) is also small. In terms of dimensional quantities the buoyancy parameter b is defined by: b=  0.5[ˆ ρ2 − ρˆ1 ]ˆ g (ˆ ro − rˆi ) , ∗ τˆ  (2.9)  which clearly reflects the balance between buoyant and viscous stresses. Typical sizes of b may range from 0-10, and typically the displacing fluid is denser, meaning b < 0. Intuitively, we expect that |b| ≫ 1 indicates the dominance of buoyancy over viscous effects. With respect to the dimensionless geometric parameters in the flow, having fixed the inclination at vertical, the only other geometric parameter is the eccentricity, e. Although in strongly inclined wells values of e close to 1 do occur, in vertical wells this rarer. A more common range would be 25  2.2. Experimental methodology e ∈ [0, 0.6], which we can adequately cover. A practical difficulty with larger eccentricities (experimentally) is mentioned earlier in the context of smaller δ, i.e. small imprecisions in the apparatus geometry become significant with respect to the annular gap size, on the narrow side of the annulus. In the modeling approach that we seek to validate, apart from b and e all other dimensionless parameters are rheological. For fluid k, the dimensionless rheological parameters are defined in terms of their dimensional analogues by: κ ˆ k (γˆ˙ ∗ )nk τˆk,Y κk = , τ = . (2.10) k,Y τˆ∗ τˆ∗ In the case that the fluids are Newtonian, note that the consistency is simply the viscosity. In a field setting, power law indices in the range 0.3-1 are fairly commonplace, effective viscosities when sheared are 1-2 orders of magnitude larger than water. Yield stresses can range from 0-20 Pa. The Bingham number Bk = τk,Y /κk , gives an indication of how plug-like the local velocity profile is, when viewed across the annular gap. A typical range is 0-10. It is worth commenting that at large values of Bingham number the flow is likely to become locally stationary on the narrow side of the annulus.  2.2.3  Selection of fluids  Although ideally one would like to be able to select fluids to match given dimensionless parameter ranges, in reality this is very difficult, and particularly as we wish to work with transparent and relatively inelastic fluids. The easiest parameter to vary experimentally between experiments, is the flow rate. For Newtonian fluids, change of the flow rate affects only b. Therefore, we may select a given fluid pair with desired viscosity ratio and fixed densities, then explore the space (e, b) at a fixed viscosity ratio: κ1 /κ2 . This was the approach adopted for the first 3 series of experiments, using glycerol solutions as a Newtonian fluid. Glycerol solutions were prepared by 26  2.2. Experimental methodology diluting pure glycerol with water. Density and viscosity of glycerol solutions are very sensitive to water content and temperature. Fluid densities were measured with a hydrometer, accurate to ±1 kg/m3 . Viscosities of fluids were tested before each experiment to check the self-consistency of experiments in each series. For our non-Newtonian experiments Xanthan gum and Carbopol 940 were used. These fluids are complex long chained polymers and were mixed according to the manufacturers’ methodology. The fluids were dyed with ordinary (Higgins Eternal) pen ink, at a concentration of 300 ppm. Rheology measurements showed that the effect of dye on the rheology of either fluid was insignificant. The rheological properties of each solution were determined using a Bohlin C-VOR digital controlled shear stress-shear rate rheometer. The temperature was fixed to be isothermal at a temperature of approximately 23o C. The Xanthan measurements were highly repeatable, with an error of less than 2% between successive measurements. The data were fitted to a power law model: τˆ = κ ˆ γˆ˙ n , (2.11) which is known to give a reasonable representation of the flowcurve data, over a restricted shear rates ranges. Carbopol 940 was more challenging to characterise. It was found to exhibit thixotropic properties due to aging and polymer restructuring. In order to “reset” the structure of the polymer between samples, all tests were subjected to a pre-shear 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 an increasing ramp of shear rates. The rheological parameters of the fluids were determined by analysing the flowcurve data and fitting to a Hershel-Bulkley model. τˆ = τˆY + κ ˆ γˆ˙ n . (2.12) 27  2.2. Experimental methodology The yield stress was determined finding the shear stress value at the global maximum of the viscosity. Once the yield stress was found, this value was subtracted from the remaining data, which was then fitted to a power law curve. The error on the parameter fitting is larger with Carbopol, particularly with regards to the yield stress, which can have an error, in the worst case, in the range of 20-50% since the global maximum is often not well defined. This apparently large error in τˆY has a relatively small impact on model usage for flowing fluids as the uncertainty occurs at very small strain rates. Once the yield stress is fixed, the fitting error is below 3.5% for the consistency, κ ˆ , and 4% for the power law index, n. The behaviour of Carbopol is less documented than that of Xanthan. Thus two additional tests were conducted to check for variability to temperature and shear damage. The rheology was measured for the temperature range of 20-30o C, with only a small (7%) decrease in viscosity observed over this range. The actual lab temperature variation range is less than 5o C, so that temperature effects are not significant. The other problem reported for Carbopol is shear damage that can occur during mixing and pumping. For Carbopol, and other long chain polymers, shear damage occurs when the chains get chopped and shortened by the mechanical action of pump and impellor blades. To check for damage that would occur during an experimental trial, a test was run with the PCP at the highest rate expected during a trial. Carbopol 940 was tested before and after passing through the pump. We found that almost no rheological change had occurred, i.e. degradation does result from repeated pumping/circulation, but in displacement experiments it appears this effect may be ignored. A final complication to consider was 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 sterilization with Calcium Hypochlorite at a concentration of 100 ppm, and by keeping the fluid as cool as possible. On the other  28  2.2. Experimental methodology hand, Carbopol spontaneously ages due to a slow and irreversible change in the micro structure; see [6]. Since there is no method to prevent Carbopol aging, the fluid was mixed, measured and run through the test series as quickly as possible, typically within a three day period.  2.2.4  Experimental plan  We aimed to study the effects on the displacement flows of variations in the dimensionless governing parameters of the Hele-Shaw model. Six series of experiments with Newtonian and non-Newtonian fluids were performed. Each experimental series corresponded to a fixed pair of fluids with the displacement performed at a range of different eccentricities and flow rates. The flow rate ranges were selected to maintain comparable ranges of |b| (between 0 and 6.3) for each series. The fluids used and their physical properties are listed in Tables 2.1 and 2.2. For the Newtonian-Newtonian displacements (series 1-3), the viscous stress scale is unaffected by the flow rate, which acts only on the buoyancy number. Glycerol solutions were used, at different concentrations in each series, to control the viscosity ratio. For the non-Newtonian fluids, we attempted to match the power-law indices of the two fluids, (series 4 and 5). In this case the stress scale also changes with the n-th power of the flow rate. This maintains a constant ratio of κ1 /κ2 , in the case of two power law fluids. Series 4 consisted of 2 different Xanthan solutions of the same power law index, but with a density and consistency difference. In series 5 we have looked at displacing Carbopol with Xanthan in the absence of a density difference, (but with identical power law indices). Finally, series 6 considered displacing Carbopol with Xanthan in the presence of a density difference. Table 2.3 summarizes the experimental conditions for each series of experiments.  29  2.2. Experimental methodology  Series 1 2 3 4 5 6  Fluid Additive ρˆ1 (kg/m3 ) κ ˆ 1 (P ) 78% Gly 0.4% Salt 1200 0.049 88% Gly 1234 0.15 98% Gly 1254 0.64 0.3% Xan 30% Sug 1118 1.45 0.8% Xan 1000 3.6 0.3% Xan 13% Sug 1048 0.32  n1 1 1 1 0.38 0.3 0.51  Table 2.1: Properties of displacing fluids used in the experimental series.  Series 1 2 3 4 5 6  Fluid ρˆ2 (kg/m3 ) τˆY (P a) κ ˆ 2 (P ) 78% Gly 1192 0.049 78% Gly 1192 0.049 78% Gly 1192 0.049 0.38% Xan 1000 1.00 0.1% Car 1000 2 2.2 0.07% Car 1000 0.6 0.91  n2 1 1 1 0.38 0.3 0.36  Table 2.2: Properties of displaced fluids used in the experimental series.  Series Flow Rate (l/min) 1 0.17 - 0.72 2 0.17 - 0.72 3 0.17 - 0.72 4 0.17 - 0.72 5 0.17 - 0.72 6 0.10 - 2.16  0 0 0 0 0 0  e - 0.5 - 0.5 - 0.5 - 0.5 - 0.5 - 0.5  |b| 0.9 - 3.7 1.5 - 6.3 0.5 - 2.2 1.2 - 2.2 0 0.7 - 1.4  κ1 /κ2 1 3.1 13.1 1.45 1.6 0.37 - 0.46  Table 2.3: Experimental condition for the different series.  30  2.3. Experimental results  2.3  Experimental results  Before presenting parametric results from each series of experiments, we illustrate typical experimental results in the cases when a displacement is steady or unsteady.  2.3.1  Illustrations of typical displacements  We commence by showing in Figure 2.5 typical images from both a steady and unsteady displacement. Front and side images are presented at a sequence of different times, when the interface passes the lower and upper vertical positions. The steady displacement is taken from experimental series 3 and the unsteady displacement from series 1. The first observation we make is that, although miscible, the bulk of the two fluids remains separate and unmixed. The interface itself is somewhat diffuse and we can observe dispersive currents within the flow, but there is no large-scale mixing in evidence. Qualitatively, the shape of the interface in Figure 2.5a translates axially while remaining constant, whereas that in Figure 2.5b elongates progressively. Due to background lighting variations these images are obscured to the left of the side views, but once the background light is subtracted, the darkness intensity data is much clearer. To quantify more precisely this notion of steady/unsteady displacement and to measure the amount of dispersion, we construct spatio-temporal plots at the upstream and downstream locations, using the normalised darkness intensity obtained via the method explained in §2.2.1. These are shown in Figs. 2.6 & 2.7, for the experiments of Figure 2.5a & b respectively. These plots are created by combining the images from the side and the front of the annulus. The azimuthal distance shown thus corresponds to roughly 3/4 of the annulus. The left hand side of the images corresponds to the narrow side of the annulus. The x-axis in these plots is measured in pixels, with the first 65 pixels coming from the side view and the last 65 from the frontal view. 31  2.3. Experimental results  S  i  d  e  y  F  =  3  r  o  n  t  S  i  d  e  F  r  o  n  t  S  i  d  e  F  6  s  1  1  8  s  1  2  4  s  1  1  6  s  8  s  2  s  1  1  4  s  4  s  0  s  1  1  2  s  0  s  0  0  m  m  y  =  1  0  0  0  m  m  y  =  3  r  o  n  t  S  i  d  e  F  s  5  5  5  4  0  0  m  m  y  =  1  0  8  s  4  s  0  s  6  s  0  0  m  r  o  n  t  m  Figure 2.5: Examples of steady and unsteady displacements: a) e = 0.5, ˆ =0.17 l/min, fluid properties from series 3; b) e = 0.2, Q ˆ =0.34 l/min, Q fluid properties from series 1. In each of a & b the left 2 columns of the figures shows successive time frames at the lower (upstream) position, y = 300mm, (front and side views), whereas the last two columns show images at the upper (downstream) vertical position, y = 1000mm, (front and side views). The zero reference time is at the start of the first image.  32  2.3. Experimental results Fluid 1 Concentration 1 200  60  0.9  Fluid 1  0.8  50  180  0.7 0.6  Time (sec)  Time (sec)  40 30  160 0.5 0.4 140  20  0.3  Fluid 2  0.2  120  10  0.1 0 0  a)  20  40  60 80 100 Azimuthal Distance (pixel)  0  120  20  40  60 80 100 Azimuthal Distance (pixel)  120  b)  Figure 2.6: Spatio-temporal for the displacement of Figure 2.5a: a) Upstream location; b) Downstream location. In Figure 2.6 we observe at initial times that the cross-section is full of fluid 2. We see a large diffuse cloud of intermediate colour scale emerging at around tˆ =20 s. This effect comes from opening of the gate valve, which entrains fluid 1 into fluid 2 as can be observed in the images of Figure 2.5a at the upstream (lower) position. Other than this effect, we observe that the temporal frontier between fluids 1 and 2 remains horizontal with a very small azimuthal variation in these figures. In fact the interface at the upper position appears flatter than that at the lower position, since the entrainment effects of valve opening have dissipated further downstream. By comparison with Figure 2.6, the spatiotemporal plot for the unsteady displacement, Figure 2.7, shows strong azimuthal variations at both lower and upper positions. The arrival times at the upper location are much shorter at the wide side than at the narrow side. We again have some localised entrainment and mixing at the lower position, just after the gate valve is opened.  2.3.2  Parametric results: newtonian fluids  We now present the results of our different series of experiments, in particular focusing on the question of whether the displacement may be considered 33  2.3. Experimental results 50  Fluid 1 Concentration 1  95  0.9 40  Fluid 1  0.7  30  Time (sec)  Time (sec)  0.8  85  20  0.6  75  0.5 0.4 65  0.3  10  0.2 55  Fluid 2  0.1  0 0  a)  20  40  60 80 100 Azimuthal Distance (pixel)  0  120  20  40  60 80 100 Azimuthal Distance (pixel)  120  b)  Figure 2.7: Spatio-temporal for the displacement of Figure 2.5b: a) Upstream location; b) Downstream location. steady or unsteady. In order to make this type of classification, we need to have a meaningful measure of the unsteadiness. Using the method explained in §2.2.1 we compute the scaled residence times ∆tj at each pixel value j, i.e. the time taken for the interface to pass from upstream to downstream measurement locations, divided by the mean theoretical time of travel as computed from the flow rate. Typical frequency distributions of residence times are shown below in Figure 2.8 for the displacements of Figure 2.5a & b. Evidently, the standard deviation of the residence time distribution, σ∆t , is much smaller for the displacement of Figure 2.5a than for that of Figure 2.5b. Perhaps a less obvious effect is that the mean residence time, µ∆t , of the displacement of Figure 2.5a is significantly larger than that of Figure 2.5b, but is still less than unity. This effect is due to dispersive fluid currents, which we shall discuss at length later. For all steady displacements we have systematically found µ∆t < 1, which discounts the possibility of a random experimental error. We have carried out some concentric test displacements in which glycerol displaces air, where dispersion is minimal and still |µ∆t − 1| ≈ 0.02, which is some measure of  34  2.3. Experimental results 140  140  Experiment 120  Theory  100  100  80  80  Frequency  Frequency  120  60  40  20  20  0.0  a)  Theory  60  40  0  Experiment  0 0.2  0.4  0.6  t  j  0.8  1.0  1.2  0.0  b)  0.2  0.4  0.6  0.8  1.0  1.2  t  j  Figure 2.8: Residence time distribution for the displacements of Figs. 2.5a & b, respectively. In a) µ∆t =0.95, σ∆t /µ∆t =0.0026. In b) µ∆t =0.77, σ∆t /µ∆t =0.06. the imperfection of our apparatus. There are also more random errors due to pump flow rate fluctuations, geometrical imperfections, image processing, etc., which ensure that σ∆t > 0. The size of these errors is indicated by the value of σ∆t /µ∆t , when measured in steady displacements, and is typically <1%. For each experimental series we use σ∆t /µ∆t to compare between experiments, as e and |b| = −b are varied. In Figs. 2.9a-c we present the values of σ∆t /µ∆t for each of series 1-3, respectively. The experimental values are also used to construct the shaded contour plots using 2D linear interpolation and extrapolation. Given that the matrix of experimental points in each series remains fairly sparse, we interpret these contours mostly as a qualitative indication of the variation of σ∆t /µ∆t , except close to each data point. In particular, the extrapolation to |b| = 0, outside of the range of experimental |b|, appears non-physical in Figure 2.9c, but otherwise is at least consistent with our physical intuition. By comparison between Figs. 2.9a-c, we observe that increasing the viscosity ratio has the effect of reducing σ∆t /µ∆t , i.e. stabilising the flow and promoting steadiness. Increasingly steady displacements are also found for less eccentric annuli and larger buoyancy |b|. 35  2.3. Experimental results  3.5  6  0.16  5  0.14  0.50 3 2.5  0.40  |b| 2  0.30  0.12 4 0.10  |b| 3  0.08  0.20  2  0.06  0.10  1  1.5 1  0.04 0.5 0 a) 0  0.1  0.2  e  0.3  0.4  0.02 0  0.5  b)  0  0.1  0.2  e  0.3  0.4  0.5  0.10  0.14  2  e = 0.2  0.08  0.12 1.5  0.1  0.06 t  |b|  0.08 t  /  1 0.06 0.04  0.5  0.04  0.02  0.02 0  c) 0  0.00  0.1  0.2  e  0.3  0.4  0.5  0  d)  1  2  |b|  Figure 2.9: Variation of σ∆t /µ∆t for Newtonian fluid series 1-3: a) Series 1 (κ1 /κ2 = 1); b) Series 2 (κ1 /κ2 = 3.1; c) Series 3 (κ1 /κ2 = 13.1). The contour plots are constructed from interpolation and extrapolation and each experimental point is classified as either steady (squares) or unsteady (triangles). In d) we show a typical variation in σ∆t /µ∆t with |b| for e = 0.2, from series 3.  36  2.3. Experimental results We have also classified each experimental displacement as either steady (squares) or unsteady (triangles). Figure 2.9d shows a typical variation of σ∆t /µ∆t as b is varied. In a regime which is categorized as steady there is very little variation in σ∆t /µ∆t , about some small constant value. However, we consistently observe a rapid change in σ∆t /µ∆t as |b| is decreased and the interface becomes unsteady. We classify by identifying this rapid change in σ∆t /µ∆t with b at each fixed e. It is not possible to specify an exact and universal transition threshold. This notion is anyway problematic since some dispersion is always present and the amount is fixed. However, typically we have recorded values σ∆t /µ∆t 1% before the transition to unsteadiness, and displacements become obviously unsteady for σ∆t /µ∆t in the range 3-5%. For series 1 (Figure 2.9a) both fluids have the same viscosity and the displaced fluid is less than 1% lighter than the displacing fluid. Steady displacement was only achievable in the concentric annulus or at small eccentricity with very low flow rate. In series 2 (Figure 2.9b), the displacing fluid is three times more viscous than the displaced fluid and is 3.5% denser than the displaced fluid. Although the density difference was very small, steady displacement were achieved even at high eccentricities with sufficiently low flow rates, |b| > 3. In series 3, with larger viscosity ratio, steady displacements were found at smaller values of |b|. Over the ranges of fluid properties and flow rates tested it appears that the buoyancy stress vs viscous stress balance, captured in b, has a more significant stabilising effect than viscosity ratio alone.  2.3.3  Parametric results: non-newtonian fluids  Three series of experiments were conducted with non-Newtonian fluids using Xanthan gum and Carbopol 940 solutions. In the first series of nonNewtonian experiments, Figure 2.10a, a power law fluid displaced another power law fluid with lower consistency. It should be noted that both fluids have the same power law index so that the effects of shear thinning, (as the 37  2.3. Experimental results  2.2  0.06  1.4  2  0.05  1.2  0.04  |b| 1  0.90 0.80 0.70  1.8  0.60  |b|  0.50  1.6 0.03 1.4  a)  0.30 0.20  0.02  1.2 1 0  0.40 0.8  0.6 0.1  0.2  e  0.3  0.4  0.5  b)  0.10 0.2  0.3  e  0.4  0.5  Figure 2.10: Variation of σ∆t /µ∆t for non-Newtonian fluid series: a) series 4, (power law fluid displacing power law fluid, Xanthan-Xanthan); b) series 6, (power law fluid displacing visco-plastic fluid, Xanthan-Carbopol). flow rate is increased), are similar in both fluids and the dimensionless ratio, κ1 /κ2 also remains invariant as the flow rate changes. In Figure 2.10a the contours show the same qualitative trends as for the Newtonian displacements. A little buoyancy is required to make displacements steady when there is some eccentricity. In the second series of non-Newtonian displacement experiments, a viscoplastic fluid (Carbopol) was displaced by a power law fluid (Xanthan). Again by matching the power law indices, the effects of shear thinning in each fluid are broadly similar. In this series of experiments, the fluids were of the same density, (|b| = 0), the displacing fluid had a larger consistency value, but no yield stress. Steady displacement of the Carbopol was never achieved even at e = 0. Long static channels of Carbopol were observed on the narrow side of the annulus for e > 0.1. Due to the extensive channeling it was not possible to quantify σ∆t /µ∆t . In the last series of non-Newtonian experiments, by adding 5% density difference to the displacing fluid, steady displacements resulted in both eccentric and concentric annuli; see Figure 2.10b. Although the displacing fluid  38  2.4. Secondary flows and dispersion has a smaller consistency than the displaced fluid, the density difference is sufficient to displace the yield stress fluid from the narrow side of the annulus.  2.4  Secondary flows and dispersion  It is immediately apparent when observing these displacement flows that a number of secondary phenomena influence the flow. These fluids are miscible, but the timescale of the experiments (∼100 s) is very much shorter than that for molecular diffusion, acting alone. Thus, many of the observed effects are essentially dispersive and we use this terminology to loosely describe these phenomena. On the other hand, note that we are also parametrically far from the laminar Taylor dispersion regime, so that the effects observed are local and not averaged by cross-gap dispersion in any way.  2.4.1  Dispersive effects on the scale of the annular gap  To start with, if we consider the idealisation of the narrow annular gap as a plane channel, along which we are displacing vertically upwards, we expect to see a displacement finger advance in the centre of the channel faster than the mean flow. This type of symmetric duct displacement flow has been studied in some depth, in both tubes and plane channels, experimentally by [15, 16, 19, 20, 21, 22, 39, 41], and computationally by [1, 5, 14, 40, 47, 48]. Both Newtonian and non-Newtonian fluids have been considered. Since we deal primarily with “stable” viscosity ratios, κ1 > κ2 , as is common in the industrial setting, the advancing displacement finger is expected to be locally stable, i.e. this is not a viscous finger. This may be thought of as a form of dispersion in which the dispersive effects are modulated by the positive viscosity ratio, κ1 > κ2 and possibly by buoyancy b < 0, both of which act via modifying the velocity field of the underlying Poiseuille flow,  39  2.4. Secondary flows and dispersion  (a)  (b)  (c)  (d)  Figure 2.11: Dispersive finger in the displacement flow of two identical Newtonian fluids: κ ˆ1 = κ ˆ 2 =0.31Pa.s: a) tˆ =0s; b) tˆ =4.1s; c) tˆ =16s; d) tˆ =32s; The annular eccentricity is e =0.25 and the aspect ratio δ =0.084. close to the interface. Amongst locally stable displacements, dispersive effects are most prominent when the two fluids are identical. An example of such a displacement is shown in Figure 2.11, taken from a Newtonian experiment in the larger aspect ratio annulus, δ = 0.084, which aids visualisation. The fluids are identical, except for the colouring of the displacing fluid. The annulus is eccentric, e =0.25, and therefore we observe the finger advancing first on the wide side of the annulus Figure 2.11a. As the displacement advances, in Figure 2.11c we first see the front on the narrow side. Optically we do not see this as a finger since the displacing fluid finger at other azimuthal positions partly masks the interface. At later times we observe near-complete displacement with thinning residual wall layers. In the case just examined, taking the analogy of the plane channel flow, we would expect the tip of the displacement front to advance at 1.5 times the mean flow, as the interface is simply advected by the plane Poiseuilee velocity profile. For two identical power law or Herschel-Bulkley fluids this same ratio may be easily calculated, and is reduced due to both shear-thinning and yield stress effects. When the fluids are not identical, some idea of the front speed at the channel centre can be gained from a lubrication/thin40  2.4. Secondary flows and dispersion film style of model. Such models have been developed by [22] for the case of 2 Newtonian fluids and by [1] for 2 Bingham fluids. These models give qualitative information concerning the effects of rheological and buoyancy parameters, but do also over-predict the front speed as they ignore the multidimensional nature of the flow at the front. We have developed such a model for 2 Herschel-Bulkley fluids, i.e. including Newtonian and power-law models, with a density difference, and have run various exploratory computations. In general and as might be expected, for stable viscosity ratios and moderate buoyancy number, the front velocity vs mean velocity ratios lie between 1 and that of the identical fluid case. It is only for dimensionless parameters that are more extreme than those considered in our experiments that the centreline front velocity approaches close to 1, with ratios in the range 1.15-1.5 being more common. If this type of gap-scale dispersion was acting alone in an unmodified fashion, we would be observing residence times smaller than those typically observed for steady displacements. For example, in the context of the steady displacements of series 1-3, (Figs. 2.9a-c), the mean scaled residence times are of the order of 0.95 ≈ 1/1.05.  2.4.2  Large-scale dispersion  As discussed above, it is likely that gap-scale dispersion effects are modified by other secondary flows present in the annulus. That such secondary flows exist has been known for many years, see e.g. [2, 36, 45]. Indeed, in the case of a steady displacement in an eccentric annulus, it is clear that there can be no stable steady front without azimuthal flows. To see this, consider the flow sufficiently far either upstream or downstream of a steadily propagating front. These flows are single phase driven by an axial pressure gradient, uniform in the radial and azimuthal directions. In the narrow gap limit, δ ≪ 1, the ratio of mean axial velocity through the wide side of the annulus, to the mean velocity through the entire annulus, is proportional to (1 + e)2 , (i.e. the 41  2.4. Secondary flows and dispersion gap-averaged velocity scales with the square of the gap width). The narrow side ratio is proportional to (1 − e)2 . Since the steady interface propagates dimensionlessly at speed 1, all around the annulus, it is clear that fluid on the wide side must decelerate and that on the narrow side must accelerate. Fluid ahead of the interface moves from the narrow side to the wide side, whereas that behind the interface moves from wide to narrow side. The size of the azimuthal velocities generated in this way are clearly of O(e) as e → 0. To illustrate the effects of these azimuthal currents we present a short series of result from the large aspect ratio annulus, δ =0.084, in which dispersive effects are amplified. For this series the displaced fluid 2 was a white corn syrup (Crown, ACH food Companies) and the displacing fluid 1 was an undiluted gold corn syrup. The density difference was varied by adding 8% water (saturated with 20% NaCl table salt) to fluid 2, resulting in densities: ρˆ1 =1398 kg/m3 and ρˆ2 =1363 kg/m3 . The viscosities were κ ˆ 1 =6.24 Pa.s ˆ was inand κ ˆ 2 =0.76 Pa.s. In the 3 experiments shown, the flow rate Q ˆ = 3 × 10−6 m3 /s, 6 × 10−6 m3 /s, 12 × 10−6 m3 /s. The annulus had creased: Q eccentricity, e =0.25. This resulted in the following set of dimensionless numbers: κ1 = 1, κ2 = 0.122, and −b = 0.222, 0.111, 0.056, for the 3 increasing flow rates. First we present results from the Hele-Shaw model of [37], shown in Figure 2.12a-c. Each figure shows the interface and the moving frame stream lines at a dimensionless time, t = 10, by which time the displacement is in steady state. Each figure shows only half of the annulus, with the wide side at φ = 0 and the narrow side at φ = 1. Moving left to right in this figure the flow rate increases, hence |b| decreases. These are the streamlines relative to the mean velocity. The counter-current secondary flow is evident: from wide to narrow side under the interface and from narrow to wide side above the interface. To explore the effect of eccentricity, we also present in Figure 2.12d-i, equivalent results at an eccentricity of 0.38 and 0.5, with the same fixed values of b. The strength of the secondary flow increases in line  42  2.4. Secondary flows and dispersion with the eccentricity, as suggested by the argument outlined above. Figure 2.13 shows the experimental results corresponding to Figure 2.12ac. The figure shows snapshots of the displacement front and below it are the results of an edge detection algorithm, with five interfaces imaged at increasing distance along the annulus. By comparison with Figure 2.12a-c, it is immediately apparent that there are some significant differences. At this large viscosity ratio, the interfaces are stable and the displacements all appear steady. As |b| decreases however, the effects on the model results are minimal, but very noticeable in the experiment. Smaller |b| corresponds to larger flow rate and the size of the secondary azimuthal flows scale in proportion to the flow rate. What appears to be happening is that dispersion on the gap-scale promotes an axially advancing local finger towards the centre of the annular gap, at each azimuthal position, i.e. simply because the velocity is larger in the centre of the gap than at the walls. This finger must extend ahead of the mean position of the interface, and is thus influenced by the secondary azimuthal flows that exist. The displacing fluid is therefore swept azimuthally towards the wide side of the annulus, advancing the interface on that side. At larger flow rates the secondary flows are larger and the interface becomes increasingly elongated, although remaining steady. It is hard to make direct comparisons with the model results in terms of interface shape, due to optical distortion in processing the images. But we may approximately compare the axial extension of the interface along the annulus. In the model this is approximately 14 mm, at eccentricity of e = 0.25, which compares reasonably well with the extension at the largest value of |b|, but is much smaller than the axial extensions at smaller |b|. Clearly dispersion can have a significant effect. Further experimental evidence of large scale dispersion is found at the beginning of the displacement. The initial fluids are static and separated by a gate valve. As the experiment starts the gate valve is opened, which inevitably causes some entrainment and mixing locally. As the flow rate  43  6.5  6.5  6  6  6  5.5  5  4.5 0  ξ−t  6.5  ξ−t  ξ−t  2.4. Secondary flows and dispersion  5.5  5  0.2  0.4  0.6  0.8  4.5 0  1  5  0.2  0.4  φ  4.5 0  1  6  6  6  5.5  ξ−t  6.5  5.5  5  0.2  0.4  0.6  0.8  4.5 0  1  0.2  0.4  0.6  0.8  4.5 0  1  6  6  ξ−t  6  ξ−t  6.5  5.5  5  0.4  0.2  0.4  0.6  0.8  4.5 0  1  φ  0.8  1  0.6  0.8  1  5.5  5  0.2  0.4  0.6  0.8  4.5 0  1  φ h)  0.6 φ  6.5  0.2  1  f)  6.5  5  0.8  5.5  φ e)  5.5  0.6  5  φ d)  4.5 0  0.4 φ  6.5  4.5 0  0.2  c)  6.5  ξ−t  ξ−t  0.8  b)  5  ξ−t  0.6 φ  a)  g)  5.5  0.2  0.4 φ  i)  Figure 2.12: Moving frame streamlines and interface (heavy line) computed from the model of [37], at time t = 10, for 2 Newtonian fluids with κ1 = 1, κ2 = 0.122: (a)-(c) e = 0.25 with b = −0.222, −0.111, −0.056, respectively; (d)-(f) e = 0.38 with b = −0.222, − 0.111, − 0.056, respectively; (g)-(i) e = 0.5 with b = −0.222, − 0.111, − 0.056, respectively. Contour spacing for the moving frame streamlines is at intervals 0.02 for (a)-(f), and at 0.04 for (g)-(i).  44  2.4. Secondary flows and dispersion  (a)  (b)  W  (c)  N  Figure 2.13: Displacement of two Newtonian fluids with κ1 = 1, κ2 = 0.122, e = 0.25, (δ =0.084): a) b = −0.222; b) b = −0.111; c) b = −0.056. Below each image are the results of edge-detected interface at successive times along the annulus. The dimensionless distance between successive interfaces is: a) 0.26, b) 0.25, c) 0.14; (dimensional lengthscale is 47.4mm). Physical parameters given in the text.  45  2.4. Secondary flows and dispersion  Figure 2.14: Front view of the wide side of the annulus in Newtonian displacement; images were taken at 2 second intervals, beginning 36 seconds after opening the gate valve; fluids properties are ρˆ1 = 1254kg/m3 , ρˆ2 = 1192kg/m3 ; κ ˆ 1 = 0.64P a.s, κ ˆ 2 = 0.049P a.s; n1 = n2 = 1. starts, this entrained fluid is visible and is advected downstream. We show an example of this, (an expanded version of Figure 2.5a), below in Figure 2.14, 36 seconds after opening the gate valve. The entrained fluid is not distributed around the annulus, but has been advected largely over to the wide side and ahead of the interface. Recirculating streamlines are visible in the traces of displacing fluid. In the displacement series we have run there was no evidence that the small amount of initially entrained fluid affected the stability of the interface.  2.4.3  Combined effects: spikes and tails  Although we have introduced dispersive effects above, by considering gapscale and large-scale effects separately, in practice the effects occur simultaneously. This leads to a number of interesting flow observations. In the first place, the azimuthal secondary flow causes a focusing of fluid on the wide side, where it is swept ahead of the advancing front. As the flow across the gap is locally Poiseuille-like, the fluid that is focused on the wide side advances fastest in the centre of the gap in the form of a protruding spike. 46  2.4. Secondary flows and dispersion  Figure 2.15: Spikes on the wide side in non-Newtonian displacements from ˆ =0.17 l/min; b) Q ˆ =0.34 l/min; c) Q ˆ = 0.72 l/min. series 4, at e = 0.5; a) Q An example of this is shown in Figure 2.15, (see also Figure 2.13). These images are from series 4, at the highest eccentricity, e = 0.5, and show the phenomenon at increasing flow rates, for Figs. 2.15a-c respectively. At the 2 lower flow rates the displacement is steady but for the highest flow rate the displacement is unsteady, due to a diminished |b|. The spike on the wide side occurs for most displacements, regardless of rheology and whether steady or unsteady, with a varying degree of visibility. When the flow is unsteady it is hard to distinguish the advancing spike from the underlying unsteady interface. More interesting and varied is the behaviour on the narrow side of the annulus. If we consider first a steady displacement, on the narrow side of the annulus we have a direct competition between gap-scale dispersion and the annular secondary flows. The former is advancing displacing (black) fluid ahead of the mean front position. The secondary flows on the other hand move backwards relative to the mean flow, hence stripping off displacing fluid from the sides of the advancing central finger and advecting this fluid towards the wide side. The net effect of this competition is a small spike, that sticks out ahead of the front, see Figure 2.16. The spike is slightly longer and more visible at higher flow rates. It was observed in almost all Newtonian exper47  2.4. Secondary flows and dispersion  Figure 2.16: Spikes on the narrow side in Newtonian displacements from ˆ =0.17 l/min; b) Q ˆ =0.34 l/min; c) Q ˆ = 0.72 l/min. series 3, at e = 0.5; a) Q iments, but not in non-Newtonian experiments, see e.g. Figure 2.15. This may be because the gap-scale dispersion is reduced by both shear-thinning and yield stress effects, and hence the azimuthal currents dominate. The occurrence of spike-like interfaces has been observed before, e.g. by [39, 41] in capillary tubes and by [19, 20, 22] in plane channel/Hele-Shaw geometries. Petitjeans & Maxworthy [39] relate the occurrence to a transition in the pattern of streamlines, as suggested by [43]. Lajeunesse and co-workers [20] have developed a predictive methodology based on a lubrication displacement model. The regime in which they delineate spike formation is that in which buoyancy is dominant. Our case is different to both of these, in that the spikes appear to be governed by the secondary azimuthal flow, and the base tendency for the fluid to move faster in the centre of the channel, rather than by buoyancy or by the recirculation dynamics on the gap scale. In the case of unsteady displacements the narrow side behaviour is quite different as the mean interface position moves slower than the mean displacement speed. The interface elongates and the rear of the interface, at the narrow gap, typically showed either a ‘V’ or ‘U’ shape. The ‘V’ shape indicates that the interface is continually elongating in the narrow gap, whereas 48  2.4. Secondary flows and dispersion  (a)  (b)  Figure 2.17: Examples of interface shape at rear of an unsteady displacement on the narrow side: a) an example ‘V’ shape; b) an example ‘U’ shape. the ‘U’ shape indicates that the narrowest part is moving at a steady speed, over some range of azimuthal angles. These two features are illustrated in Figure 2.17. In the lubrication model developed by [38], there are parameter ranges for which the narrow side interface elongates progressively and others for which a shock forms and steady propagation is found. These 2 possibilities may correspond to the ‘V’ and ‘U’ shapes, but we have not carried out any systematic study. [45] report observing a variety of behaviours on the narrow side of the annulus, including a form of ribbing instability. As the interface elongates they identify a hydrostatic pressure imbalance as the driving force for azimuthal flows that may destabilise the narrow side. We have not observed this type of phenomenom in our experiments.  2.4.4  Quantifying dispersion  In order to quantify the amount of dispersion a number of measures are possible. Following the procedure described in §2.2.1, suppose we consider a 49  2.4. Secondary flows and dispersion fixed azimuthal position, or pixel value j, and observe the normalised darkness intensity as the interface passes at each observation height. We may expect a saturation curve qualitatively like Figure 2.18. We define the front of interface as the location at which the normalised darkness intensity of displacing fluid is 0.05 and the back of interface is defined as the location at which the normalised darkness intensity of displacing fluid is 0.95. If tˆ1 is the time at which the front of interface just passed the upstream detection line and tˆ2 is the time at which the back of interface passed this line, tˆ2 − tˆ1 is the amount of time that corresponds to the dispersion at the lower (upstream) measurement locations, i.e. a saturation time. Similarly, tˆ4 − tˆ3 indicates the saturation time at the upper (downstream) measurement location. Although we can extract these measurement from our data straightforwardly, it is less clear how to use them. In particular, due to the short timescale of the experiment, we are far from any classical diffusive regime of dispersion, (i.e. Taylor-dispersion), so it is not sensible to attempt to fit a diffusivity/dispersivity. On the timescale of the experiment, dispersion is dominated by advection, but the advective currents are clearly complex. We present therefore a simple indicator of the rate of growth of the saturation time. Subtracting these two saturation times and normalising with the mean travel time of the piston like displacement, ∆tˆp , leads to the a normalized growth rate of the saturation time, at pixel j, say ∆tD,j : ∆tD,j =  [(tˆ4 − tˆ3 ) − (tˆ2 − tˆ1 )]j . ∆tˆp  (2.13)  Examining the distribution of ∆tD,j allows us to compare the effects of dispersion at different points around the annulus. Figure 2.19 shows the effect of eccentricity on the saturation time growth rate, in the displacements of series 3. In this figure, the x-axis shows the azimuthal distance around the outer pipe, starting from the narrow side, measured in pixels. The total domain over which we have measurements corresponds to approximately 270 50  2.4. Secondary flows and dispersion  0  .  9  5  0  .  0  5  1  D  i  s  p  e  r  s  i  o  n  2  a  t  l  o  w  e  r  3  p  o  s  i  t  i  o  n  D  4  i  s  p  e  r  s  i  o  n  a  t  u  p  p  e  r  p  o  s  i  t  i  o  n  Figure 2.18: Interface detection from the saturation curve. 1.0  e = 0 e = 0.2  0.8  e = 0.5  t  D, j  0.6  0.4  0.2  0.0  0  10  20  30  40  50  60  70  80  90  100 110 120 130  Azimuthal Distance (pixel)  Figure 2.19: Effect of eccentricity on the saturation time growth rate, ∆tD,j , ˆ = 0.72 l/min. in series 3, at Q degrees of the annulus. The saturation time growth rate on the wide side is significantly larger than that on the narrow side. In a concentric annulus, the growth rate is almost the same for every point around the annulus. To investigate the effects of flow rate, we have computed the saturation time growth rate, ∆tD,j , for each experimental series and in Figure 2.20 we show these effects for series 2 and 3 at fixed eccentricity, e = 0.5. From the limited data that we have, it appears that at higher flow rates the saturation time growth rate distribution is relatively unaffected by the flow rate. Other than the saturation time growth rates, which indicate local varia51  2.4. Secondary flows and dispersion 1.0  1.0  Q = 0.17 L/min Q = 0.17 L/min 0.8  Q = 0.34 L/min  0.8  Q = 0.34 L/min  Q = 0.52 L/min  Q = 0.52 L/min  Q = 0.72 L/min  Q = 0.72 L/min  0.6  t  t  D, j  D, j  0.6  0.4  0.2  0.2  0.0  0.0  a)  0.4  0  10  20  30  40  50  60  70  80  90  100  110  120  130  b)  0  10  20  Azimuthal Distance (pixel)  30  40  50  60  70  80  90  100  110  120  130  Azimuthal Distance (pixel)  Figure 2.20: Effect of flow rate on the saturation time growth rate, ∆tD,j , at e = 0.5 in: a) series 2; b) series 3. tions in dispersion, we may consider a more global measure of the dispersion. Previously, we have used the distribution of the scaled residence times, ∆tj , to characterise whether or not the flow is steady, via the ratio σ∆t /µ∆t . If the displacement is steady, then a global indicator of the amount of dispersion is given simply by 1 − µ∆t , but when unsteady this includes non-dispersive effects.  2.4.5  Other interesting phenomena  In addition to the dispersion phenomena reported, several other effects were observed. Firstly, in many situations we have observed a thin drainage layer adjacent to the outer pipe wall, i.e. in the plane of the annular gap. Figure 2.21 shows a drainage layer of 0.6-1mm in thickness. It is likely that a similar layer exists at the inner wall, but this is not visible. The drainage wall layer was most clearly visible for steady displacements when Xanthan displaced Carbopol. In experiments in our larger aspect ratio annulus, with lower viscosity pairs of Newtonian fluids, wall layers were observed to destabilise, with aperiodic wave undulations of wavelength 8-20mm. In displacements of yield stress fluids (series 5) at larger eccentricities it  52  2.5. Discussion and conclusions  0.6mm  Figure 2.21: Drainage wall layer observed during a Newtonian displacement. was common to find a channel of Carbopol left behind on the narrow side of the annulus. This phenomenon was first highlighted by [25] in the cementing context as a potential process problem. An example is shown in Figure 2.22, taken form series 5. It is worth noting that in this flow, as the displacement is unsteady, azimuthal secondary flows are minimal and also the displaced fluid is static. Thus, dispersive currents are greatly reduced and we observe a very clean and sharp interface.  2.5  Discussion and conclusions  This chapter has presented the results of 6 series of displacement flow experiments in narrow eccentric annuli. Each series consisted of displacements with the same fluid pair, repeated at different eccentricities and different flow rates. The underlying results largely confirm the qualitative picture that underpins field-based “rules of thumb” for the primary cementing of vertical wells, e.g. [7, 29]. This same scenario has also been extensively studied in the context of a Hele-Shaw style displacement model, by [2, 36, 37, 38]. The overall trends observed are that: (i) it is possible to have steady traveling wave displacement fronts in eccentric annular geometries, as well as unsteady displacements for which the interface elongates along the annulus; (ii) steadi-  53  2.5. Discussion and conclusions  Figure 2.22: Static channel on the narrow side for an experiment in series 5: ˆ =0.34 l/min. Images are shown at: 10s, 45s and 75s after opening e = 0.5, Q of the gate valve ness is promoted by a positive ratio of viscosity and density, (i.e. displacing fluid more viscous and heavier), and by a smaller eccentricity. The above falls into the realm of existing knowledge. More interesting and novel has been the observation of various secondary flow structures, that modify the displacement via dispersion. Two underlying processes combine to drive these flows. First, on the scale of the annular gap a Poiseuille-like velocity profile across the gap causes displacing fluid to advance in the centre faster than the mean position of the “gap-averaged” interface. Secondly, secondary azimuthal flows produce a counter-current shear across the interface, in the case of a steady displacement. The size of the azimuthal secondary flow increases with the eccentricity. The secondary flow transports the advancing fingers of displacing fluid around to the wide side, where they frequently form a long finger/spike advancing locally ahead of the mean interface speed. The advance of this spike is due to the two dispersive tendencies acting together. A smaller spike may also sometimes form on the narrow side of the annulus, but does not grow in time. Here the azimuthal secondary flow opposes the gap-scale dispersive effects. Both effects are amplified by the flow rate.  54  2.5. Discussion and conclusions We have characterised our experiments as either steady or unsteady by using the ratio of standard deviation to mean of the residence time distribution. We have also used other measurements from the local saturation curves to indicate how dispersive effects vary azimuthally. In the parameter regime where we operate we are far from a diffusive representation of dispersion, (i.e. Taylor dispersion), but due to the geometric complexity of the flows it is hard to provide a simple characterisation of advective effects. These measurements therefore give mostly qualitative information, principally confirming that dispersion manifests predominantly on the wide of the annulus, in the presence of eccentricity. In our Newtonian experiments, we note that increasing the flow rate has the effect of reducing |b|, which promotes unsteadiness, and also increases the amount of azimuthal dispersion. Both destabilising effects thus act together. We suspect therefore, that an unsteady displacement will elongate faster experimentally than predicted by the Hele-Shaw type of model, (which has no gap-scale dispersion). On the other hand, the driving force for the azimuthal secondary currents is the mismatch between the far-field gap-averaged annular Poiseuille flow, (which moves faster on the wide side), and a steadily propagating interface. Once the interface is not propagating steadily, we may expect that the azimuthal current decays as the interface elongates. This leads naturally to the observation that in an eccentric annulus, even every steady interface is unsteady. The azimuthal secondary flow continues to pump displacing fluid towards the wide side, which then accelerates ahead of the front in the form of a spike. Whilst this can not be disputed we note that the spike is a local phenomenon and would not be present if there was no dispersion on the gap-scale. The absence of gap-scale dispersion leads to exactly the Hele-Shaw type of displacement flow. Therefore, we see that gapscale dispersion is strongly modified by the azimuthal secondary flow. The reverse coupling is however not evident. Over wide ranges of flow parameters we have computed residence time distributions that have relatively constant  55  2.5. Discussion and conclusions σ∆t /µ∆t ≪ 1, over a broad range of parameters, only increasing sharply at some threshold value: see e.g. Figure 2.9d. This suggests that the net result of the dispersion does not change the dynamics of the underlying largescale Hele-Shaw flow. The failure of local “interfacial” effects to modify the global flow is relatively commonplace in Hele-Shaw (and Darcy) flows, see e.g. [49]. On the other hand there is some uncertainty about this conclusion since our experimental timescale is much shorter than that in the industrial application. Whilst we suspect that the experimental displacement flows will be more unsteady than the computed Hele-Shaw displacements, with the same parameters, this is not straightforward to test. Experimentally there are numerous restrictions on the set of feasible experimental parameters. From the modeling perspective, it is time consuming and imprecise to determine a stability frontier from repetitive time-dependent simulations. Not least, this is because as the frontier is approached, growth rates approach zero, requiring very long times to infer stability or instability from simulation results. In [38] a semi-analytical approach is followed, in which a lubrication-style model is developed from the Hele-Shaw displacement model. This approach assumes an elongated interface has already developed and questions whether it would continue to grow. This gives semi-analytical bounds on sufficient conditions to be satisfied for the interface to grow indefinitely. In Figure 2.23 we compare the predictions from this model with the experimental results. The regions that are predicted to be unsteady by the model in [38] are found to be unsteady for our experiments, but the prediction is clearly conservative with respect to the experiments. In conclusion, the underlying dynamics of the Hele-Shaw style of model from [2] do appear relevant to the experimental displacements. However, the experiments also expose a number of interesting dispersive effects that are simply not accounted for by the Hele-Shaw style of model. Our future plans in this domain include the attempt to include gap-scale dispersion within our  56  2.5. Discussion and conclusions  4 6 5  3  Steady 4  |b |  |b |  2  Steady  3 Unsteady  1  2 1  0  0  0.2  0.4  a)  e  0.6  0.8  1  0 0 b)  0.2  2.5  2.5  2  2  1.5 |b |  1.5 1  0.5  0.5  c)  0  0.6  e  0.2  0.4  e  0.6  0.8  1  0  d)  0  0.8  1  Steady  |b |  Steady  1  0  0.4  Unsteady 0.2  0.4  e  0.6  0.8  1  Figure 2.23: Comparison of the classified steady and unsteady experiments with the lubrication model predictions from [38]: a) Series 1; b) Series 2; c) Series 3; d) Series 4.  57  2.5. Discussion and conclusions present Hele-Shaw model, and then study its effects on the displacement. 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SPE Annual Fall Technical Conference and Exhibition, 1-3 October 1978, Houston, Texas, SPE paper 7590, 1978. [25] R.H. McLean, C.W. Manry, and W.W. Whitaker. Displacement mechanics in primary cementing. Society of Petroleum Engineering, (1488), 1966. 61  2.6. Bibliography [26] M.A. Moyers-Gonz´alez and I.A. Frigaard. Kinematic instabilities in twolawyer eccentric annular flows, part 2: shear thinning and yield stress effects. Journal of Engineering Mathematics, DOI 10.1007/s10665-0089260-0, 2008. [27] M.A. Moyers-Gonz´alez and I.A. Frigaard. Kinematic instabilities in two-layer eccentric annular flows, part 1: Newtonian fluids. Journal of Engineering Mathematics, 62:130–131, 2008. ´ [28] M. Naimi, R. Devienne, and M. Lebouch´e. Etude dynamique et thermique de lecoulement de couette-taylor-poiseuille; cas dun fluide pr´esentant un seuil d´ecoulement. Int. J. Heat Mass Trans., 33:381–391, 1990. [29] E.B. Nelson and D. Guillot. Well Cementing, 2nd Edition. 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Physics of Fluids A, 3:1859–1872, 1991.  64  Chapter 3 Buoyancy driven slump flows of non-Newtonian fluids in pipes 2 3.1  Introduction  The motivation for this study comes from the industrial process of plug cementing, which is a common process in the oil industry; see e.g. [23]. In this process dense cement slurry is placed into an oil or gas well that is already partially filled with another fluid. The aim of placing the cement plug is to form an impermeable hydraulic seal of the well and a hard mechanical barrier. A typical cement slurry density is around 1900 kg/m3 , and the in-situ fluids, (drilling mud, viscous pill, spacer fluid, or simply residual reservoir fluids: oil/water), could have significantly lower density, say 1000 − 1700 kg/m3 . These density differences result in buoyancy induced fluid motion, which needs to be understood if the process is to be effective. Since the early 1990’s there has been a massive increase in the numbers of oil wells constructed horizontally, primarily to increase productivity by aligning the well with the reservoir. The increase in horizontal wells was made possible by advances in directional drilling technology. The early 1990’s saw a continual pushing of the horizontal reach of wells up to around 10 km, see e.g. the detailed description in [25]. The 10 km barrier was broken in a number of wells drilled at Wytch Farm, UK. The limits of “extreme” extended reach wells are now much higher, but these are not common. With 2  A version of this chapter has been submitted for publication. Malekmohammadi, S. Naccache M.F. Frigaard, I.A. and Martinez, D.M.  65  3.1. Introduction present day technology it is feasible to reliably construct wells with horizontal extensions in the 7 − 10 km range. Whereas one of the initial drivers for horizontal wells was offshore multi-lateral drilling, reducing rig time and production footprint, these wells are now also common onshore. They occur both for conventional oil and gas wells and also in oil-sands development with the large-scale adoption of production processes such as SAGD (steamassisted gravity drive) and others based on dissolving the heavy oil, typically with horizontal reaches of 1 − 3 km. Combined with the anticipated shift into increased gas production as world energy consumption climbs and oil reserves peak and decline over the next 20 years, issues of horizontal well abandonment will become increasingly topical. This is the focus of this chapter. In abandoning a horizontal production well, we are typically placing a cement plug into low density Newtonian fluids. The buoyancy induced slumping motion can only be arrested by the yield stress of the cement slurry, which is typically insufficient to prevent initial motion during and after placement. Thus, slumping is inevitable and the yield stress acts to stop the flow only as gravitational forces are diminished by the slump extending down the pipe, reducing the slope of the interface. The final static state has been analysed by [13] who predicted both the maximum slump length and the shape of the slump. However, the slumping process is itself transient. In the initial transient phase of the flow inertial effects are potentially important and later there is the possibility of mixing, both of which may compromise the accuracy of the static slump estimates in [13]. Thus, in this chapter we investigate transient buoyancy-driven slumping flow inside a closed near-horizontal pipe. We use both numerical and experimental methods, as well as making comparison with the static predictions from [13]. We present results showing the effects of density difference and pipe inclination on the slump length. We remark that there is a large literature on plug cementing in general and on this type of flow, e.g. [4, 5, 6, 8, 9, 10, 11, 12, 14, 17, 28]. However,  66  3.2. Analytical predictions nearly all of these studies focus on plug-cementing in inclined-vertical wells, where the role of buoyancy is quite different. There have also been a large number of works in which thin-film/lubrication theory has been used to model the flow of a single visco-plastic fluid along a sloping surface, often inclined close to horizontal. In general the underlying momentum balance in these single fluid situations is much simpler to solve than in the 2-fluid case here. Consequently much progress has been made. For example, geophysical flows (lava, mudslides, etc) have been studied extensively; see [7] and the reviews of [3, 16]. Other interesting studies of thin film flows of yield stress fluids are by [2, 21, 26, 27, 29]. An outline of this chapter is as follows. Below in §3.2 we give an overview of the predictions of [13], on maximal static slump lengths and shapes, as it relates to our experiments and numerical modelling. Section 3.3 describes the methodologies that we have used in our study. In §3.4 we present our results, comparing first transient phenomena and then against the static predictions. This chapter concludes in §3.5 with a discussion of the principal results.  3.2  Analytical predictions  In [13] the exchange flow of two Bingham fluids in a near-horizontal pipe, has been analyzed in the thin-film asymptotic limit. Although the transient problem is also formulated in [13], the principal results derived relate to the prediction of the maximal static slump length and shape. This maximal length is achieved when the shear stresses due to the density differences, slope of the interface and inclination of the pipe balance the yield stresses in each fluid. Since the fluid is static when the maximal length is achieved, the length of the slump depends only on the yield stresses and not on the effective viscosity of the fluids when yielded. Thus, the results in [13] are valid for any generalised Newtonian fluids with a yield stress. For the experimental situation we consider later, only one fluid (Carbopol) 67  3.2. Analytical predictions has a yield stress, τy , which simplifies the results. For our situation, the thinfilm analysis in [13] requires that δ ≪ 1, where δ=  τy , [ρX − ρC ]gD  (3.1)  D is the diameter of the pipe and ρX , ρC , denote respectively the densities of the Xanthan and Carbopol solutions. In our experiments we have δ typically between 0.03 and 0.1, so this requirement is approximately satisfied. The shape of the slump is described by the curve z(y), where y ∈ [0, D] denotes the height of the interface above the bottom of the pipe: dz D β0 (h) = , dh δ βm (h) + χβ0 (h)  (3.2)  where h = y/D, where β0 (h) & βm (h) are geometric functions defined in [13], (see Figure 3.1a & b), and where χ = cos β/δ. The length of the slump, say Lstatic , is found from integrating the above: Lstatic  D = δ  1 0  β0 (h) dh. βm (h) + χβ0 (h)  (3.3)  We note that dimensionless slump shapes and lengths are calculated by dividing through by D/δ = [ρX − ρC ]gD2 /τy . Figure 3.1c plots the dimensionless slump shape for various χ ∈ [−1, 1]. For β < 90◦ we have positive χ. The slump is “uphill” and reduced by the inclination of the pipe. As we move to β > 90◦ we have χ > 0 and the effect of pipe inclination is to increase slump length. For the experiments conducted, values of χ lie within this interval. Figure 3.1d plots the dimensionless slump length as a function of χ.  68  3.2. Analytical predictions  0.2  1.2 1  0.15  βm (h)  β0 (h)  0.8 0.1  0.6 0.4  0.05 0.2 0 0  0.2  0.4  0.6  0.8  0 0  1  h  a)  0.2  0.4  0.6  0.8  1  h  b)  1  χ=0 Carbopol  0.6  0.25  χ=1  δLstatic /D  h = y/D  0.8  χ = −1  0.4  0.2  0 −0.35 c)  0.15  Xanthan  −0.3  −0.25  0.2  −0.2 −0.15 −δz(y)/D  −0.1  −0.05  0  0.1 −1 d)  −0.5  0 χ  0.5  1  Figure 3.1: Static analysis from [13]: a) β0 (h); b) βm (h); c) dimensionless slump shape for different χ ∈ [−1, 1]; d) variation in dimensionless slump length with χ.  69  3.3. Methodology  i  L  u P  m u  g  l  F  P  h  u  t  i  d  p m  p  A  A  n  g  l  e  m  e  t  e  c  r  y  l  i  c  P  i  p  e  r  G  a  t  e  V  G  P  r  e  s  s  u  r  e  T  r  a  n  s  d  u  c  e  i  D  a  g  i  m  t  a  e  c  q  u  i  s  i  t  i  o  l  e  v  e  V  a  r  e  s  s  u  r  e  T  r  a  n  s  d  u  c  e  r  e  a  l  u  v  i  y  d  u  m  p  F  s  t  e  e  n  P  y  v  a  F  s  l  l  r  H  A  a  t  r  P  C  a  l  o  w  m  e  t  e  r  m  Figure 3.2: Schematic of the experimental set-up.  3.3 3.3.1  Methodology Experimental method  The experimental set up consists of a long acrylic pipe with length of 1.88 m and inner diameter of D = 38 mm, an SLR digital camera, lighting system, two pumps, and a frame to support the pipe. A bearing connects the frame to the base steel structure and allows the pipe to be inclined from 0 to 93 degrees (from vertical). Figure 3.2 shows the schematic of the experimental set up. To perform the experiments the pipe is filled in the vertical position by a progressive cavity pump. The light fluid (Carbopol solution) is filled from the top and the heavy fluid (Xanthan solution), dyed black, is then filled 70  3.3. Methodology from the bottom, leaving the initial interface flat in the middle of the pipe. To minimize breakage of polymeric chains, pumps are driven with the lowest speed (5 Hz). The frame which holds the pipe is rotated to the horizontal position while the camera is capturing images continuously at a rate of 1 frame per second. Since most of slumping happens in the first 2 minutes, the camera speed is reduced gradually from 1 frame per second to 4 frames per hour. A mirror system allows views from the front and side of the pipe. The images are later processed to find the slump length in each time-frame. The length of slump is measured with a measuring tape that is mounted on the mirror system. The accuracy of measurements is 1 mm. To study the effect of pipe inclination 3 different angles were investigated: 89◦ , 90◦ and 91◦ . Four sets of fluids were used in this work. Carbopol 940 was used in each set of experiment with constant aqueous concentration of 0.12% and a pH equal to 6.0. In order to see the effects of rheology and density difference 4 different Xanthan solutions were prepared, with different concentrations of sugar. Sugar was used to increase both the density and viscosity of the Xanthan solutions. These fluids are described in in Table 3.1. ∆ρ is the density difference between Heavy Fluid and Light Fluid and ρc is the density of Carbopol solution (Light Fluid). Note that of the rheological parameters, the yield stress is the most problematic to measure reliably and repeatably. By using the same Carbopol solution for each experiment and varying instead the Xanthan properties, we ensure consistency of any error, i.e. according to the static analysis the principal dimensionless parameter is the ratio of yield stress to buoyancy stress. The rheological properties of each solution were determined using a Bohlin C-VOR digital controlled shear stress-shear rate rheometer. Flow curves of a typical pair of experimental fluids, for 0.12% Carbopol 940 and 0.1% Xanthan+50% Sugar solutions, are shown in Figure 3.3. Table 3.2 summarizes the measured rheological properties and density of each solution. The Levenberg-Marquardt method was employed for curve fit-  71  3.3. Methodology  Light Fluid Heavy Fluid Series Carbopol (wt.%) Xanthan (wt.%) Sugar (wt.%) 1 0.12 0.1 50 2 0.12 0.15 40 3 0.12 0.15 30 4 0.12 0.15 20  ∆ρ/ρc 0.224 0.160 0.120 0.072  Table 3.1: Series of fluids used in experiments.  2  10  10  Shear Stress (Pa)  Shear Stress (Pa)  1  1  (a)  0  10  -1  0.1 0.1  10  10 1  10  Shear Rate (1/s)  100  -4  10  (b)  -3  10  -2  10  -1  10  0  10  1  10  Shear Rate (1/s)  Figure 3.3: Typical flow curves: (a) 0.1% Xanthan solution at 23o C. (b) 0.12% Carbopol 940 solution at pH 6.0 and 23o C.  72  3.3. Methodology Fluid τy (Pa) κ ( ) 0.12% Carbopol 940 3-6 21 0.10% Xan.+50% Sugar 0.6 0.15% Xan.+40% Sugar 0.3 0.15% Xan.+30% Sugar 0.2 0.15% Xant+20% Sugar 0.2  n ρ (kg/m3 ) 0.6 1000 0.7 1224 0.6 1160 0.7 1120 0.5 1072  Table 3.2: Properties of the fluids used. 100000  y  ~ 3 - 6 Pa  Viscosity (Pa.s)  10000  1000  100  10 1  10  100  Shear Stress (Pa)  Figure 3.4: Range of yield stresses of Carbopol ting of rheometric data. Xanthan solutions showed shear-thinning behaviour and were modeled by the power-law model. The Herschel-Bulkley model is commonly used to describe the rheological behavior of Carbopol solutions. However, determining the yield stress of a visco-plastic fluid is not an easy task. Figure 3.4, suggests that there is a range of values for the yield stress rather than a single value. Although, there is not a single value for the yield stress, a constant value was needed to perform numerical simulation. As explained later, a value of 3.2Pa was chosen for the yield stress of Carbopol in our numerical simulations. The rheological parameters of the fluids used are given in Table 3.2. 73  3.3. Methodology  3.3.2  Numerical modeling  The numerical solution was obtained using a finite volume method, [24], with the 2 fluids modelled via the volume of fluid method (VOF) method. The VOF method solves a single set of momentum conservation equations in the domain and tracks the volume fraction of each fluid (αi , i = 1, 2). The solution variables and fluid properties, say φ, are shared by the fluids and represent volume-averaged variables, defined by: φ = α1 φ1 + α2 φ2  (3.4)  where φi is the solution variable or property of fluid i. Tracking of the fluids (hence the interface) is obtained by the solution of the continuity equation for phase 2: ∂α2 ρ2 + ∇ · (α2 ρ2 v) = 0 (3.5) ∂t where t is time, ρ2 is the density of fluid 2 and v is the velocity vector. Note that (3.5) also represents a concentration equation for the case in which molecular diffusion between fluids is vanishingly small. The volume fraction of fluid 1 is obtained by the condition: α1 + α2 = 1. The momentum equation is given by: ∂(ρv) + ∇ · (ρvv) = −∇p + ∇ · η ∇v + ∇vT ∂t  + ρg  (3.6)  where ρ is the density, p is the pressure, g is the gravitational acceleration vector and η is the viscosity function. Again note that ρ and η need be interpolated from the properties of the individual phases. For our numerical simulations we always denote the heavier fluid (Xanthan) by fluid 1. The rheological model for fluid 1 is the power-law model: η1 = κ1 γ˙ n1 −1  (3.7)  74  3.3. Methodology For fluid 2, (Carbopol), a modified bi-viscosity function was used to model the visco-plastic behavior:  τ   y + κ2 γ˙ n2 −1 if γ˙ > γ˙ small  γ˙ η2 =  η  otherwise 0  (3.8)  where τy is the yield stress, η0 is a low shear rate viscosity constant and γ˙ small = τy /η0 . Apart from an initial stage of the flow, inertial effects are minimal and although the flow is 3D the assumption of symmetry about a vertical plane cutting the pipe along the centerline of the pipe is reasonable. Boundary conditions are the usual no-slip and impermeability conditions at the solid boundaries, and symmetry conditions on the central plane of the pipe. In the experiments the tube was filled with both fluids in a vertical position and then rotated to the horizontal position. To mimic this, at t = 0 the (numerical) tube was in vertical position with interface a horizontal interface which was perpendicular to the direction of gravity. For t > 0 the numerical domain was rotated to horizontal at constant angular speed during the initial 3 seconds of the simulation. Although inertial effects were included in all simulations, after the initial few seconds the inertial terms in the equations are negligible. Numerical results (Figure 3.5) showed that inertia effects were only important in the beginning of the simulations and did not affect the eventual evolution of the slump length. The tube diameter in the computations was identical to that in the experiments, D = 38 mm, but the length was about 1/3 that in the experiments, (i.e. 0.6 m), in order to reduce computational times. Computational tests were run to ensure that the end walls of the tube were not affecting flow at the interface. Mesh tests were performed on 3 unstructured meshes: Meshes 1-3 with 124080, 60900 and 26474 control volumes, respectively. The difference in slump length over time between Mesh 2 and 3 were up to 10%; those between Mesh 1 and 2 were below 5%. Three-dimensional computations with 75  3.4. Results  N  u  X  m  a  e  n  t  r  h  i  a  0 ý  þ  /  þ  c  c  a  n  l  0  .  2  r  .  2  e  1  s  0  u  l  t  s  %  S  u  g  a  r  5  0  %  4  =  1  0 =  0  9  ☎  L  *  0  .  1  i  n  e  n  0  .  0  o  r  t  i  n  i  a  e  r  t  i  a  1  1  0  1  0  0  1  0  0  0  * t  Figure 3.5: Inertial effects in a typical simulation. complex fluids are costly and those on Mesh 1 took approximately 10 times longer than those on Mesh 2. Thus as a compromise between speed and accuracy, Mesh 2 was used for our comparative results. The software Fluent [1, 18] is used for all computations.  3.4 3.4.1  Results Transient results  The effects of pipe inclination and density difference between fluids on the slump length were investigated. The rheological parameters used in the numerical simulations for the lighter fluid (Carbopol solution) were: η0 = 10000 Pa.s, τy = 3.2 Pa, n = 0.7, and κ = 21 . To choose a reasonable value for yield stress within the range indicated in Figure 3.4, different values for the yield stress were tried in the numerical simulations. The numerical results of different simulations suggested the value of τy = 3.2 Pa gives the closest results to the experimental results for all 4 series. With reference  76  3.4. Results  Figure 3.6: Comparison of interface shape after 1.5 hours: (a) Experimental Result: Series 1, ∆ρ/ρc = 0.224; (b) Numerical Result: Series 1, ∆ρ/ρc = 0.224; (c) Experimental Result: Series 2, ∆ρ/ρc = 0.160; (d) Numerical Result: Series 2, ∆ρ/ρc = 0.160; to Figure 3.4, we observe that this value is also close to the maximum. As mentioned before, four different heavier fluids (Xanthan solutions) were analyzed, each modeled by the Power-law equation. The rheological parameters used in the simulations were as those given in Table 3.2. Figure 3.6 shows the interface shapes after 1.5 hours, in series 1 and series 2 of experiments, at 90◦ inclination of pipe (Heavier fluid is the dark one). It can be observed that for higher density differences the slump length is longer and the interface shape is more stretched, as expected. The flow visualization shows an irregular interface, which could suggest some possible mixing close to the interface. A reasonable agreement between numerical and experimental results is observed. Figure 3.7 shows (a) experimental and (b) numerical results of the dimensionless slump length increase with time, for three pipe inclinations. The dimensionless slump length and time are defined as: L∗ =  L D  t∗ =  t [D/(∆ρgDn+1 /κ)1/n ]  (3.9)  The experimental errors for a typical case are shown in Figure 3.8. The highest and lowest values of slump length are shown with dots in this figure. The solid line is the curve fitting of average values of slump lengths. It is 77  3.4. Results  5  5  E  x  p  e  r  i  m  e  n  t  a  l  r  e  s  u  l  t  s  N  X  a  n  t  h  a  n  0  .  1  0  %  S  u  g  a  r  0  5  u  ✱  /  ✱  c  .  2  2  m  e  r  i  a  n  t  h  a  a  l  n  r  0  .  e  s  1  u  0  %  l  t  s  S  u  g  a  r  0  5  %  4  =  0 ❯  L  c  %  X  0 ✰  4  4  3  3  ❱  /  ❱  c  .  2  2  4  =  *  L  *  2  2  ✍  =  8  9  =  8  9  0  ✾  =  9  0  0  ✾  =  9  1  0  1  ✍  =  ✍  9  =  0  9  1  0  0  0  0  0  1  1  0  4  2  1  0  4  3  1  0  4  4  1  0  4  0  b  1  1  0  4  2  1  0  4  3  1  0  4  4  1  0  4  )  *  t  a  ✾  0  1  )  t  *  Figure 3.7: (a) Experimental results for the slump length versus time for Series 1, ∆ρ/ρc = 0.224. (b) Numerical results for the slump length versus time for Series 1, ∆ρ/ρc = 0.224.  5  4  3  L  *  2  X  a  n  t  h  a  q  r  /  r  c  .  2  .  2  1  0  %  S  u  g  a  r  5  0  %  4  =  0 =  0  n  0 1  0  9  ✈  0  0  1  1  0  4  2  1  t  0  4  3  1  0  4  4  1  0  4  *  Figure 3.8: Experimental results for the slump length versus time for Series 1, ∆ρ/ρc = 0.224, α = 90◦ .  78  3.5. Discussion observed that there is a steep increase in the slump length at the beginning of the process, but as flow goes on the slump length tends to an asymptote. At the onset, the buoyancy driven stresses are high enough to overcome the yield stress, but as the interface stretches buoyancy effects decrease and eventually the flow stops. We also observe a slight increase in slump length as the pipe inclination varies from 89◦ to 91◦ . Some differences between numerical and experimental results are observed. The numerical solution gives lower slump lengths, but a higher slump velocity at later times. This behavior could be explained by a mixing process that may be going on, changing the actual fluids properties close to the interface, or could be due to the modeling methodology: either the use of a viscosity regularization technique or the multiphase modeling using VOF. Moreover, it is noted that the experimental errors are larger for lower times, where the comparison is worse.  3.4.2  Quasi-Static results  Figure 3.9a & b shows the slump length as a function of the density difference for three pipe inclinations, after 1.5 hours. Figure 3.9c shows the slump length predicted by the analytical method. A linear relation is observed, for the analytical, numerical and experimental results. It is noted that the qualitative behavior between these results is similar, but some discrepancy in the quantitative values is observed. Since the slump flow is buoyancy driven, it is to be expected that pipe inclination effects become more important at higher density differences.  3.5  Discussion  The results we have presented show reasonable quantitative and qualitative agreement between numerical and experimental results. However, there are still obvious discrepancies that need explaining. 79  3.5. Discussion  5  5  E  x  p  e  r  i  m  e  n  t  a  l  N  L  4  4  3  3  u  m  e  r  i  c  a  l  *  L  *  2  2  ⑨  =  8  9  =  8  9  →  =  9  0  →  =  9  1  0  1  ⑨  =  ⑨  9  =  0  9  0  0  1  0  0  0  0  0  .  0  5  0  .  1  0  .  1  0  5  .  2  0  .  2  0  5  b  a  →  0  1  .  0  0  5  .  1  0  .  1  0  5  .  2  0  .  2  5  )  )  ❽  ❾  /  ❾  c  ➟  ➠  /  ➠  c  5  A  n  a  l  y  t  i  c  a  l  4  3  L  *  2  ➚  =  8  9  0  ➚  =  9  0  0  ➚  =  9  1  1  0  0  0  c  .  0  5  0  .  1  0  .  1  0  5  .  2  0  .  2  5  )  ➱  ✃  /  ✃  c  Figure 3.9: (a) Experimental results for the slump length after 1.5 hours versus ∆ρ/ρc for α = 89◦ , 90◦ and 91◦ ; (b) Numerical results for the slump length after 1.5 hours versus ∆ρ/ρc for α = 89◦ , 90◦ and 91◦ ; (c) Analytical results for the slump length versus ∆ρ/ρc for α = 89◦ , 90◦ and 91◦  80  3.5. Discussion Apart from purely numerical artifacts, we feel that two different physical explanations could lead to the discrepancies observed between the numerical and experimental results. First note that, after an initial phase in which inertia may be significant, we expect that the chief physical balance in the flow is between buoyancy and yield stress, (i.e. viscous stresses are decaying as the flow decelerates, as are inertial stresses). Thus, the first potential source of error comes from the low shear rheology. Measurement of low shear rate rheology is a difficult task and, in the case of yield stress fluids, there is a ongoing debate about the true nature of the material behaviour at sub-yield stress values. In essence the question is whether the sub-yield behaviour of a given fluid is better described by slow viscous flow or by elastic deformation. For the purpose of numerical models such as Fluent, the former of these behaviours is assumed. Low shear rheological effects are governed by both τy and η0 : as the shear stress drops below τy the fluid behaves as a very viscous fluid with viscosity η0 . The effect of low shear rate viscosity on slump length through time is shown in Figure 3.10. It is observed that by increasing η0 ten times, slump length does not change initially. However, for larger times, the velocities obtained for very low η0 fluid are much larger than the ones observed experimentally, where the flow became quasi static after almost 2 hours. It is also observed that for η0 > 5000 Pa.s, the results seem to be unaffected by η0 . Thus, taking η0 = 10000 Pa.s is a reasonable approximation for η0 . As mentioned in §3.3.1, in our numerical simulations we have used a value of 3.2 Pa close to the lower limit of the range of experimental yield stress values (3 Pa). This is partly because of the possibility of mixing. If little mixing happens very close to the interface, it would cause a decrease of the effective yield stress of the lighter fluid, and also a decrease of the effective density difference between the two fluids. Measuring the rheological properties and densities of the fluids at the interface during and after the experiments is not possible. However the effect of this assumption can be  81  3.5. Discussion  1  0  ì  ð  Y  =  ñ  3  /  .  ñ  2  P  a  ß  0  =  ß  0  =  ß  0  =  1  0  0  0  P  a  .  0  0  0  P  a  .  0  0  0  s  c  =  0  .  2  4  8  0  õ  =  5  s  0  9  1  0  P  a  .  s  6  L  *  4  2  0  0  1  1  0  4  2  1  0  t  4  3  1  0  4  4  1  0  4  *  Figure 3.10: Experimental and numerical results for the slump length versus time, for Series 1, ∆ρ/ρc = 0.224 and α = 90◦ . Effect of the lighter fluid’s low shear rate viscosity. investigated in the numerical simulations. Figure 3.11 shows how decreasing the yield stress of the lighter fluid affects the slump length. It is observed that with a very high value, τy = 6.0 Pa, the discrepancy between experimental and numerical results is large. However, as we decrease the yield stress, the numerical results approach the experimental ones. It is worth mentioning that at lower times the experimental errors are larger. The second possible explanation for the discrepancy is related to possible mixing of the two fluids close to the interface, during the experiments (see Figure 3.6). This mixing could both reduce the local yield stress, as discussed above, and lower the density difference between fluids. There is notably less mixing in the simulations than the experiment, essentially due to the numerical VOF model which attempts to keep the interface sharp. We may however simulate this mixing effect by artificially decreasing the density difference between the fluids after some fixed (arbitrary) time in numerical  82  3.5. Discussion  5  E  4  x  p  e  r  i  m  e  n  t  a  l  N  u  m  e  r  i  c  a  l  ,  ✎  Y  =  2  .  6  P  a  N  u  m  e  r  i  c  a  l  ,  ✎  Y  =  3  .  2  P  a  N  u  m  e  r  i  c  a  l  ,  ✎  Y  =  6  .  0  P  a  3  L  *  2  1  X  a  n  t  h  a  n  0  ✫  ✬  /  ✬  c  0  =  .  0  .  2  2  1  %  0  S  u  g  a  r  5  0  %  4  =  0  9  ✰  0  0  1  1  0  4  2  1  t  0  4  3  1  0  4  4  1  0  4  *  Figure 3.11: Experimental and numerical results for the slump length versus time, for Series 1, ∆ρ/ρc = 0.224 and α = 90◦ . Effect of the lighter fluid’s yield stress. simulation. In doing so, lower velocities were obtained at larger times and a similar final transient behavior between numerical and experimental results could be observed; see Figure 3.12. Being able to get a very close fit between experimental and numerical results this way, does support the idea of mixing at the interface being responsible, but the fitting method is evidently somewhat ad hoc. Possibly this could be improved by introducing a proper miscible fluids formulation. It should however be mentioned that mixing can not progress quickly. Carbopol solution is a gel and consists of large molecular networks which hinder the mixing of the fluids. In the experimental results shown earlier we have seen an apparently asymptotic increase in the slump length towards some finite limit. It is worth asking if there is some kind of simplified model in which the same characteristics are found. One option here is to attempt to derive and solve a thin-film style model, for which the results in section 3.2 provide the static  83  3.5. Discussion  5  X  a  N  n  u  N  t  m  u  m  h  a  e  r  e  n  i  r  0  c  a  i  c  .  l  a  1  ❘  l  %  Y  ❘  S  =  Y  3  =  u  .  3  g  2  .  a  ,  2  r  ❉  5  ❊  ,  ❉  /  0  ❊  ❊  %  /  (  =  c  ❊  0  =  c  .  0  ❉  2  .  ❊  /  ❊  c  =  0  .  2  4  )  4  2  4  (  t  *  <  2  4  4  *  >  2  4  4  1  )  4  ❉  ❊  /  ❊  c  =  0  .  1  2  (  t  1  )  3  L  *  2  1  0  ❭  =  0  9  0  0  1  1  0  4  2  1  0  4  3  1  0  4  4  1  0  4  * t  Figure 3.12: Experimental and numerical results for the slump length versus time, for Series 1 and α = 90◦ . Effect of the variation of ∆ρ/ρc in time limit. This is made difficult by the pipe geometry, but would be feasible for a plane channel geometry. A transient model of this type has been examined in [9], for planar channels that are inclined close to the vertical. In that model it is possible to show that there is a finite stopping time. There are in fact a large number of examples in the literature of such finite time decay results, e.g. [15, 19]. For strongly inclined channels, the effects of the interface slope are neglected at leading order. This leads to a hyperbolic system for the interface motion. We are thus able to bound the rate of spreading of the interface and derive estimates of this kind. For a strictly horizontal pipe (or other duct), there is no driving force for the fluid motion apart from the buoyancy gradient generated through the slope of the interface. This results in a parabolic problem for the interface height and the velocity is locally a function of the interface slope as well as its height. Recently, in [20, 22] it has been shown that for thin-film motions driven by the interface slope there is no finite stopping time. Instead, in 84  3.5. Discussion approaching the final static steady state the flow variables decay like t−n , with n denoting the power law index. The flow considered in [20, 22] is the classical thin film flow, for a Herschel-Bulkley fluid. Here we have a more complex geometry and a 2-fluid flow satisfying a zero flux condition. However, the driving force for the flow is still buoyancy rather than a forced flux and we believe that a similar result should hold. For both our experimental and numerical results, we do not appear to have a finite time stopping. We did not collect extensive data at long times, but approximately the slump length appears to asymptote to a constant value, with discrepancy decaying algebraically, like t−φ where φ ∈ (1, 2), but varies between experiments.  Acknowledgments This research was supported by NSERC Canada (SM, IAF & DMM) and by CNPq Brazil (MFN).  85  3.6. Bibliography  3.6  Bibliography  [1] Fluent user’s guide. 2006. v. 6.2. Fluent Inc., New Hampshire. [2] N.J. Balmforth and R.V. Craster. A consistent thin-layer theory for Bingham plastics. Journal of Fluid Mechanics, pages 65–81, 1999. [3] N.J. Balmforth, R.V. Craster, A.C. Rust, and Sassi R. Viscoplastic flow over an inclined surface. Journal of Non-Newtonian Fluid Mechanics, 142:219–243, 2007. [4] R. M. Beirute. Flow behaviour of an unset cement plug in place. Society of Petrolium Engineers, 1978. Paper number SPE 7589. [5] D.L. Bour, D.L. Sutton, and P.G. Creel. Development of effective methods for placing competent cement plugs. Society of Petrolium Engineers, 1986. Paper number SPE 15008. [6] D.G. Calvert, J.F. Heathman, and J.E. Griffith. Plug cementing: Horizontal to vertical conditions. Society of Petrolium Engineers, 1995. Paper number SPE 30514. [7] P. Coussot. Mud flow rheology and dynamics. 1997. IAHR/AIRH Monograph. [8] J.P. Crawshaw and I. Frigaard. Cement plugs: Stability and failure by buoyancy-driven mechanism. Society of Petrolium Engineers, 1999. Paper number SPE 56959. [9] H. Fenie and I.A. Frigaard. Cpreventing buoyancy driven flows of two Bingham fluids in a closed pipe:fluid rheology design for oilfield plug cementing. Mathematical and Computer Modelling, 30:71–91, 1999. [10] I.A. Frigaard. Stratified exchange flows of two Bingham fluids in an inclined slot. Journal of Non-Newtonian Fluid Mechanics, 78:61–87, 1998. 86  3.6. Bibliography [11] I.A. Frigaard and J.P. Crawshaw. Uniaxial exchange flows of two Bingham fluids in a cylindrical duct. IMA Journal of Applied Mathematics, 61:237–266, 1998. [12] I.A. Frigaard and J.P. Crawshaw. Preventing buoyancy driven flows of two Bingham fluids in a closed pipe: Fluid rheology design for oilfield plug. Journal of Engineering Mathematic, 36:327–348, 1999. [13] I.A. Frigaard and N.G. Ngwa. Upper bounds on the slump length in plug cementing of near-horizontal wells. Journal of Non-Newtonian Fluid Mechanics, pages 147–162, 2004. [14] I.A. Frigaard and O. Scherzer. The effects of yield stress variation on uniaxial exchange flows of two Bingham fluids in a pipe. Journal on Applied Mathematics, 6:1950–1976, 2000. [15] R. Glowinski. Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, 1983. [16] R.W. Griffiths. The dynamics of lava flows. Annual Review of Fluid Mechanics, 32:477–518, 2000. [17] K. Harestad, T.P. Herigstad, A. Torsvoll, N.E. Nodland, and A. Saasen. Optimization of balanced-plug cementing. Society of Petrolium Engineers, 1997. Paper number SPE 33084. [18] C.W. Hirt and B.D. Nichols. Volume of fluid (vof) method for the dynamics of free boundaries. Journal of Computational Physics, 39:204– 225, 1997. [19] R.R. Huilgol, B. Mena, and J.M. Piau. Finite stopping time problems and rheometry of Bingham fluids. Journal of Non-Newtonian Fluid Mechanics, 102:97–107, 2002.  87  3.6. Bibliography [20] R.R. Huilgol, B. Mena, and J.M. Piau. Two-dimensional dam break flows of Herschel-Bulkley fluids: The approach to the arrested state. Journal of Non-Newtonian Fluid Mechanics, 142:79–94, 2007. [21] K.F. Liu and C.C. Mei. Slow spreading of a sheet of Bingham fluid on an inclined plane. Journal of Fluid Mechanics, 207:505–529, 1989. [22] G.P. Matson and A.J. Hogg. Slumps of viscoplastic fluids on slopes. submitted to Journal of Non-Newtonian Fluid Mechanics, 2008. [23] E.B. Nelson and D. Guillot. Well Cementing, 2nd Edition. Schlumberger Educational Services, 2006. [24] S.V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, 1980. [25] M. L. Payne. Recent advances and emerging technologies for extended reach drilling. S.P.E., 1995. [26] J.M. Piau. Flow of a yield stress fluid in a long domain. application to flow on an inclined plane. Journal of Rheology, 40:711–723, 1996. [27] A.B. Ross, S.K. Wilson, and B.R. Duffy. Thin-film flow of a viscoplastic material round a large horizontal stationary or rotating cylinder. Journal of Fluid Mechanics, 430:309–333, 2001. [28] R.C. Smith, R.M. Beirute, and G.B. Holman. Improved method of setting successful whipstock cement plugs. Society of Petrolium Engineers, 1983. Paper number SPE 11415. [29] S.K. Wilson, B.R. Duffy, and A.B. Ross. On the gravity-driven draining of a rivulet of viscoplastic material down a slowly varying substrate. Physics of Fluids, 14:555–571, 2002.  88  Chapter 4 Conclusions 4.1  Summary  In this thesis we have studied two projects related to the cementing of oil and gas wells. In the first project, miscible displacement flows in an eccentric annulus were studied. Both Newtonian and non-Newtonian fluids were tested in displacement experiments. Previous experimental works showed that there are many different parameters which contribute to displacement flows. However there are two main difficulties in examining those parameters and understanding how they act together. Firstly, considering all the parameters needs a very large experimental matrix and analyzing the displacement tests would be overly complicated. Secondly, running experiments on the same scale as real displacement flows in the oil industry is not possible. In order to investigate the effects of the main parameters in displacement flows by laboratory scale experiments, dimensional analysis was used. By simplifying the Navier Stokes equations according to [1], three dimensionless parameters were chosen that govern displacement flows. Many process conditions used in the field are included in these three parameters and experiments were designed based on these dimensionless parameters: eccentricity, buoyancy and viscosity ratio. The main focus of this research was to understand the fluid mechanics behind the displacement flows and investigate the interface dynamics between the two fluids. In order to study the effects of these parameters, 6 series of experiments were conducted and the interface of the two fluids in each experiment was analyzed. The interface 89  4.1. Summary was tracked by a CCD camera both at upstream and downstream locations in the flow loop. A method was developed to analyze the captured images. This method is based on the estimate of the residence time distribution and the deviation of the shape of interface. In addition to that, spatiotemporal plots of the interface were constructed to characterize dispersion. The results showed that a steady displacement could be achieved by a proper combination of eccentricity, viscosity ratio and buoyancy. Buoyancy and eccentricity were found to have the most significant effects on the flow. Contour plots of buoyancy versus eccentricity were constructed at different viscosity ratios using 2D interpolation. However, more experiments are required for more accurate contour plots. Contour plots with more experimental points could be used later to define a boundary between steady an unsteady displacements. This boundary could be compared later with the boundary predicted by numerical tools. Furthermore, a number of interesting phenomena were observed in this project. Dispersion at the interface and secondary flows predicted by previous theoretical models were visualized during experiments. The amount of dispersion was quantified using the spatiotemporal plots and the results can aid development of more sophisticated numerical tools for the industry in the future. In addition to our main study of displacement in eccentric annului, slump flows of viscoplastic fluids were also studied as a second project in this thesis. The slump flow we studied here was a type of exchange flow in which a heavy fluid displaces a lighter one in a closed pipe. Flow occurs due to the density difference between the two fluids. The driving force for the fluid movement is the buoyancy force which is resisted by the yield stress of one or both fluids, as well as by viscous stresses. The slump experiments were conducted in a near-horizontal pipe to simulate the plug cementing process in near-horizontal wells. Experiments were carried out to investigate the transient slump length for different pairs of fluids and pipe inclinations. Results  90  4.2. Future work were obtained for different values of density, rheology and pipe inclinations. The experimental results were compared with numerical results and some discrepancies were observed. However in general, the experimental results were in good agreement with numerical results.  4.2  Future work  Besides the development of dispersion models, more experiments with improved experimental tools, such as a camera with higher resolution and a wider angle lens to cover all the length of the annulus, are recommended. Exploring dispersion with different fluid combinations and different annuli gap sizes could result in better understanding of dispersion in displacement flows. Although we have analyzed the flow in two different locations, the height of each image is limited and the time evolution of dispersion could not be fully tracked for each fluid combination. The main problem in selecting these tools is the fact that by widening the region of interest, the quality of the image is reduced. In addition to using better visualization tools, replacing the current gate valve with a solenoid valve could provide a cleaner interface of the two fluids. The gate valve is operated manually and there is evidence that the interface disturbed during opening, see Figure 2.6a. A solenoid valve which is actuated electrically might reduce the amount of interface disturbance. Although we have studied displacement flows in this thesis, our experimental results are limited to vertical displacements. Real oil and gas wells are not fully vertical, especially in areas where high extraction rate is required. Larger volumes of oil can be extracted by aligning the well with the reservoir. This requires construction of near-horizontal wells. The apparatus used in this work was first designed for running experiments at different pipe inclinations. However, it was realized that small variations of the annulus gap along the annulus significantly distorts the interface during horizontal 91  4.2. Future work experiments. In other words, the experiment is too sensitive to errors in the inner pipe offset. This small variation of inner pipe eccentricity is caused by the slight bend observed along all “straight” pipes tested and also by the small deflection of inner and outer pipe caused by the buoyancy effect of the fluids. The inner and outer pipes of the annulus are fixed at two different locations at a distance of 160 cm from each other. The bending of inner and outer pipe could be prevented by mounting some supports beneath the inner and outer pipe. Designing these supports is quite challenging especially for the inner pipe. These supports should be designed in a way not to disturb the flow inside the annulus. Another option could be filling the inner pipe with some metal plugs or plastic beads to balance the buoyancy force exerted on the inner pipe. With regard to current work, displacement flows could be explored further in inclined wells. Using the above recommendations for modifying the apparatus, new sets of experiments could be conducted with the goal of understanding the effect of well inclination on displacement flows. Also, using different fluid combinations, future work could focus on investigating dispersion effects and the secondary flows inside annuli. In conclusion, in this thesis we were able to design and run a controlled set of experiments to explore the displacement flows of two fluids. In addition to that, we developed a method to analyze the results of these experiments and compare them with each other. The experimental results were compared with the current displacement models and qualitative similarities were found. However it seems a more developed model is required to capture the dispersion for quantitative comparison. By exploring the results of this study, some interesting effects were discovered which can guide researchers in developing a more realistic model for displacement flows in the future.  92  4.3. Bibliography  4.3  Bibliography  [1] S.H. Bittleston, J. Ferguson, and I.A. Frigaard. Mud removal and cement placement during primary cementing of an oil well; laminar nonNewtonian displacements in an eccentric Hele-Shaw cell. Journal of Engineering Mathematics, pages 229–253, 2002.  93  Appendix A Horizontal displacement Here we present a brief overview of preliminary displacement experiments conducted as a part of this thesis work.  A.1  Apparatus and method  Horizontal displacement tests were conducted using the apparatus built by S. Storey. However, the apparatus was modified for better visualization and higher accuracy. The new visualization system consists of 4 mirrors instead of 1 with the configuration is shown in Figure A.1(a). In this configuration mirrors where placed at 45◦ angles so that the sides and back of annulus in addition to front view can be viewed. The resulting four images can be combined together to unwrap the whole annulus, as shown in Figure A.1(b). A geometrical transformation was required to combine the images. Figure A.2 describes the geometrical transformation used to unwrap each view of the annulus.  A.2  Typical result  In a typical horizontal experiment, first the annulus is filled with both fluids in vertical position. These fluids are separated by the gate valve. In the second step, the annulus is tilted to horizontal position from vertical, while the gate valve is still closed. To start the experiment the gate valve is opened and the displacing fluid is pumped with the desired pump speed. An SLR  94  A.2. Typical result  Mirror  SLR Camera  mera  a)  b)  Figure A.1: Visualization system in horizontal displacements. a) Configuration of mirrors; b) Combining the images from mirrors and unwrapping the annulus.  95  A.2. Typical result  x  θ = cos−1 ( ❵  φ = R .θ  R− x ) R  Unwrapped Image  Figure A.2: Geometrical transformation used to unwrap the annulus camera mounted on top of the annulus is used to capture the images from all views of the annulus. Figure A.3 shows a typical horizontal experiment in which a Newtonian fluid is displacing another Newtonian fluid. A MATLAB code was developed to combine the 4 views included in each image into a single image, and to unwrap the annulus using the geometrical transformation described in Figure A.2. Figure A.4 shows the picture of the unwraped annulus shown in Figure A.3(c). In this picture, the wide side of the annulus is located in the middle, and the narrow side is located on the left and right side. Displacing fluid is dyed black. As mentioned in Chapter 1, the main problem in horizontal experiments is the deflection of inner pipe in horizontal position which creates inaccuracy in the results.  96  A.2. Typical result  Right View Back View Front View Left View  a)  b)  c)  Figure A.3: Horizontal displacement of two identical Newtonian fluids at ˆ =0.72 l/min;. Fluids properties are ρˆ1 = ρˆ2 = 1116kg/m3 ; e = 0.2 and Q κ ˆ1 = κ ˆ 2 = 0.007P a.s; n1 = n2 = 1. Images are taken at: a) 3 sec, b) 4 sec 97 and c) 10 sec after openning the gate valve.  Flow Direction  A.2. Typical result  Figure A.4: The unwrapped image of the annulus showed in Figure A.3(c) using a MATLAB code.  98  Appendix B Slump test images Here we include, as examples, some sample images from the slump flow experimens in Chapter 3. These help illustrate better the apparatus and experimental procedure.  a)  b)  Figure B.1: Experimental setup used for slump tests: a) General view; b) Top view. 99  Appendix B. Slump test images  Side View Top View  a)  b)  c)  d)  Figure B.2: A typical slump test. Images are taken after: a) 1 sec, b) 2 sec, c) 5 min, d) 15 min. Fluids properties are ρˆ1 = 1224kg/m3 ; ρˆ2 = 1116kg/m3 ; κ ˆ 1 = 0.6P ;ˆ κ2 = 21P ; τy1 = 0; τy2 = 3 − 6; n1 = 0.7; n2 = 0.6.  100  


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