Primary Cementing of a Highly Deviated Oil Well by Mariana Carrasco-Teja B.Sc., Instituto Tecnológico Autónomo de México, 2000 M.Sc., University of Washington, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) January 2010 c© Mariana Carrasco-Teja 2009 Abstract In this thesis we study laminar displacement flows of one fluid by another in a horizontal annulus. The study comes from the primary cementing of highly deviated oil and gas wells. Highly deviated wells are those in which part of the wellbore is nearly horizontal. Primary cementing is a critical process in the construction of a well. The objective is to provide zonal isolation, i.e., a hydraulic seal between the well and the surrounding rock. This is essential to protect the environment and increase the productivity of the well. Therefore, an understanding of the process is indispensable. We model primary cementing displacement flows using a Hele-Shaw approach, and provide simple scientific tools to improve the design of cementing jobs. The contribution of the thesis comes in three parts. Firstly, we analyse the dis- placement of one viscoplastic fluid by another in a near-horizontal eccentric annulus with a fixed inner pipe. We present examples that illustrate the differences between vertical and horizontal displacements. We then derive a 1D lubrication model which gives analytical conditions that predict when the flow will stratify, according to the fluid properties and the annulus geometry. Secondly, we derive a 2D displacement model for Newtonian fluids which includes rotation and reciprocation of the inner cylinder. This is a common practice in the industry and not well understood. Using an asymptotic approach, we find steady-state traveling wave solutions for nearly-flat interfaces. Then we use numerical simulations to understand the flow dynamics for more elongated interfaces. In particular, we show that casing rotation can lead to local instabilities and mixing, which can shorten the length of the interface. Finally, we generalise this moving casing model to viscoplastic fluids. Using a lubrication- type model we explore the effects of casing motion, again deriving conditions for there to be steady solutions. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Co-authorship Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Primary cementing of highly deviated oil wells . . . . . . . . . . . . 2 1.2 Thesis objectives and introduction to the process modeling . . . . . 5 1.2.1 Objectives and methodology . . . . . . . . . . . . . . . . . . 5 1.2.2 Generalised Newtonian fluids . . . . . . . . . . . . . . . . . . 7 1.2.3 Flow in a Hele-Shaw cell . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Planar displacement flows and the Muskat problem . . . . . 16 1.3 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.1 Laminar single-phase annular flows . . . . . . . . . . . . . . 21 1.3.2 Viscoplastic displacement flows in Hele-Shaw cells or porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.3 Modeling and process design of cementing displacements . . 24 1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 iii Table of Contents 2 Stratification and traveling waves in narrow eccentric annuli . . 38 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Hele-Shaw modelling of cementing displacements . . . . . . . . . . . 43 2.2.1 Interface tracking . . . . . . . . . . . . . . . . . . . . . . . . 47 2.3 Horizontal displacement flows . . . . . . . . . . . . . . . . . . . . . 48 2.3.1 Heavy displacing fluids, b̃ < 0 . . . . . . . . . . . . . . . . . 50 2.3.2 Light displacing fluids, b̃ > 0 . . . . . . . . . . . . . . . . . . 56 2.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3.4 Computational issues . . . . . . . . . . . . . . . . . . . . . . 59 2.4 A lubrication model for gravitational spreading with |b̃| ≫ 1 . . . . 60 2.4.1 Lubrication model derivation . . . . . . . . . . . . . . . . . . 60 2.4.2 Diffusive nature of (2.29) . . . . . . . . . . . . . . . . . . . . 64 2.4.3 Steady traveling wave solutions . . . . . . . . . . . . . . . . 67 2.4.4 Light displacing fluids, b̃ > 0 . . . . . . . . . . . . . . . . . . 71 2.4.5 Parametric results . . . . . . . . . . . . . . . . . . . . . . . . 73 2.5 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . 77 2.5.1 Steady state stability . . . . . . . . . . . . . . . . . . . . . . 79 2.5.2 Industrial implications . . . . . . . . . . . . . . . . . . . . . 81 2.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3 Displacement flows with a moving inner cylinder . . . . . . . . . . 87 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.2 Model outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.2.1 Dimensionless parameters . . . . . . . . . . . . . . . . . . . . 94 3.2.2 Existing results for the stationary casing . . . . . . . . . . . 96 3.3 Steady traveling wave displacements: |b̃| ≪ 1 . . . . . . . . . . . . . 97 3.3.1 Concentric annuli . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3.2 Small eccentricities . . . . . . . . . . . . . . . . . . . . . . . 104 3.4 Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.4.1 Effects of casing rotation . . . . . . . . . . . . . . . . . . . . 109 3.4.2 Effects of constant axial casing motion . . . . . . . . . . . . 114 3.4.3 Combined reciprocation and rotation . . . . . . . . . . . . . 116 3.5 Stability and instability . . . . . . . . . . . . . . . . . . . . . . . . . 118 iv Table of Contents 3.5.1 Viscous fingering and casing motion . . . . . . . . . . . . . . 119 3.5.2 A criterion for buoyancy driven fingering . . . . . . . . . . . 121 3.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . 124 3.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4 Displacements in horizontal annuli with moving inner cylinder . 132 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.2 Model outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.2.1 Fluid concentration formulation . . . . . . . . . . . . . . . . 138 4.2.2 Interface tracking formulation . . . . . . . . . . . . . . . . . 143 4.2.3 Summary of scaling and dimensionless numbers . . . . . . . 144 4.3 Displacements at high buoyancy numbers . . . . . . . . . . . . . . . 146 4.3.1 Example Newtonian simulations for |b̃| ≫ 1 . . . . . . . . . . 147 4.3.2 Derivation of the lubrication displacement model . . . . . . . 148 4.3.3 Computational verification and comments . . . . . . . . . . 156 4.4 Steady traveling wave solutions . . . . . . . . . . . . . . . . . . . . . 158 4.4.1 Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.4.2 Power law fluids . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.4.3 Bingham fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . 170 4.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.1 Scientific contributions . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.1.1 Steady travelling waves . . . . . . . . . . . . . . . . . . . . . 179 5.1.2 Stability of the flows . . . . . . . . . . . . . . . . . . . . . . 181 5.1.3 Computational advances . . . . . . . . . . . . . . . . . . . . 182 5.2 Industrial perspective . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.3 Other contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.4 Critique of the methodology and future work . . . . . . . . . . . . . 186 5.4.1 Hele-Shaw model . . . . . . . . . . . . . . . . . . . . . . . . 186 5.4.2 Lubrication modelling approach . . . . . . . . . . . . . . . . 186 5.4.3 Numerical diffusion and dispersion . . . . . . . . . . . . . . . 187 v Table of Contents 5.4.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.4.5 Computational methods . . . . . . . . . . . . . . . . . . . . 188 5.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Appendices A Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . 191 A.1 Augmented Lagrangian method . . . . . . . . . . . . . . . . . . . . 191 A.2 Flux Corrected Transport scheme . . . . . . . . . . . . . . . . . . . 196 A.3 Discretisation and solution . . . . . . . . . . . . . . . . . . . . . . . 197 A.4 Moving casing algorithms . . . . . . . . . . . . . . . . . . . . . . . . 199 A.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 B Closure of the moving casing model . . . . . . . . . . . . . . . . . . 201 B.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 C Experimental study of displacement flows . . . . . . . . . . . . . . 210 C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 C.2 Experimental methodology . . . . . . . . . . . . . . . . . . . . . . . 216 C.2.1 Interface shape analysis . . . . . . . . . . . . . . . . . . . . . 217 C.2.2 Experimental design and process related issues . . . . . . . . 220 C.2.3 Selection of fluids . . . . . . . . . . . . . . . . . . . . . . . . 224 C.2.4 Experimental plan . . . . . . . . . . . . . . . . . . . . . . . . 226 C.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 C.3.1 Illustrations of typical displacements . . . . . . . . . . . . . 227 C.3.2 Parametric results: Newtonian fluids . . . . . . . . . . . . . 229 C.3.3 Parametric results: Newtonian fluids . . . . . . . . . . . . . 229 C.3.4 Parametric results: non-Newtonian fluids . . . . . . . . . . . 233 C.4 Secondary flows and dispersion . . . . . . . . . . . . . . . . . . . . . 234 C.4.1 Dispersive effects on the scale of the annular gap . . . . . . . 234 C.4.2 Large-scale dispersion . . . . . . . . . . . . . . . . . . . . . . 237 C.4.3 Combined effects: spikes and tails . . . . . . . . . . . . . . . 240 C.4.4 Quantifying dispersion . . . . . . . . . . . . . . . . . . . . . 243 vi Table of Contents C.4.5 Other interesting phenomena . . . . . . . . . . . . . . . . . . 245 C.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . 247 C.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 D List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 vii List of Figures 1.1 Primary cementing process . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Relationship between strain rate and shear stress 1 . . . . . . . . . . 9 1.3 Relationship between strain rate and shear stress 2 . . . . . . . . . . 10 1.4 Hele-Shaw Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Closure relationship between the pressure gradient and the velocity field for different power law numbers . . . . . . . . . . . . . . . . . . 15 1.6 Vertical Displacement in a Hele-Shaw cell with the interface denoted by ŷ = f̂(x̂, t̂). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7 Finger in the interface. . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 Schematic of the Primary cementing displacement in a horizontal well: process schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Schematic of mechanically stable separated flow configurations in a horizontal displacement . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 An example of slumping along the bottom of a horizontal concentric annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Geometry of the narrow eccentric annulus, mapped to the Hele-Shaw cell geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Test problem, comparing 2D computations with the analytical solu- tion from [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.6 Newtonian displacement at St∗ = 0.05 with positive buoyancy gradi- ent in mildly eccentric annulus . . . . . . . . . . . . . . . . . . . . . 51 2.7 Newtonian displacement at St∗ = 0.01 with positive buoyancy gradi- ent in mildly eccentric annulus . . . . . . . . . . . . . . . . . . . . . 52 2.8 Newtonian displacement at St∗ = 0.002 with positive buoyancy gra- dient in mildly eccentric annulus . . . . . . . . . . . . . . . . . . . . 53 viii List of Figures 2.9 Non-Newtonian displacement with positive buoyancy gradient in an eccentric annulus. Parameters . . . . . . . . . . . . . . . . . . . . . . 54 2.10 As Fig. 2.9 but with eccentricity, e = 0.4. . . . . . . . . . . . . . . . 55 2.11 Newtonian displacement with negative buoyancy gradient in eccentric annulus. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.12 Newtonian displacement with negative buoyancy gradient in eccentric annulus. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.13 Newtonian displacement with negative buoyancy gradient in eccentric annulus. Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.14 Conditions required for 2 Newtonian fluids to have a steady state displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.15 The effects on the steady state shape of increasing the eccentricity at fixed consistency ratio for 2 Newtonian fluids . . . . . . . . . . . . . 75 2.16 The effects on the steady state shape of increasing the consistency ratio at fixed eccentricity for 2 Newtonian fluids . . . . . . . . . . . . 76 2.17 Conditions required to have a steady state displacement at fixed n1, for 2 power law fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.18 Conditions required to have a steady state displacement at fixed n2, for 2 power law fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.19 The effects on the steady state shape of increasing the consistency ratio at fixed eccentricity and power law indices . . . . . . . . . . . . 79 2.20 The effects on the steady state shape of increasing the eccentricity at fixed consistency ratio and power law indices . . . . . . . . . . . . . 80 2.21 Examples of AL(Φi) and AH(Φi) . . . . . . . . . . . . . . . . . . . . 81 2.22 Conditions required to have a steady state displacement at fixed: n1, n2, and e for yield stress fluids . . . . . . . . . . . . . . . . . . . . . 82 2.23 The effects on the steady state shape of increasing the yield stress of the displacing fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.24 Examples of convergence to the steady state profile for the lubrication displacement model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.25 a) Schematic of the streamlines for a steady state . . . . . . . . . . . 84 3.1 Schematic of the process geometry . . . . . . . . . . . . . . . . . . . 91 ix List of Figures 3.2 Effects of casing motion on the steady state amplitude and on the phase shift, e = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3 Effects of casing motion on the steady state amplitude and on the phase shift, e = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.4 Comparison of analytical and numerical solutions . . . . . . . . . . . 108 3.5 Comparison of perturbation solution with computational solution . . 109 3.6 Example displacement with increasing casing rotation . . . . . . . . 110 3.7 Detailed evolution of the instability for the parameters of Fig. 3.6c. . 112 3.8 Spatiotemporal plot of c̄(ξ − t, t) with rotation. . . . . . . . . . . . . 113 3.9 Spatiotemporal plot of c̄(ξ − t, t) with axial casing motion I. . . . . . 115 3.10 Spatiotemporal plot of c̄(ξ − t, t) with axial casing motion II. . . . . 116 3.11 Effects of casing reciprocation wC(t) = w0 sin ( 2pit tr ) . . . . . . . . . . 117 3.12 The stream function Φ and interface, plotted over 1 period of recip- rocation for the parameters of Fig. 3.11c. . . . . . . . . . . . . . . . 118 3.13 Contours of the stream function Φ with combined rotation and recip- rocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.14 Isodensity displacements in a concentric annulus with an adverse vis- cosity ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.15 Isodensity displacements in a mildly eccentric annulus with an adverse viscosity ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.16 Schematic of interfacial fingering in an unwrapped annulus, in regions where heavy fluid lies on top of lighter fluid. . . . . . . . . . . . . . . 122 4.1 Schematic of the process geometry . . . . . . . . . . . . . . . . . . . 138 4.2 Streamlines and interface, (c̄(φ, ξ) = 0.5), at a fixed time from the 2D numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.3 Examples of an unsteady displacement. . . . . . . . . . . . . . . . . 149 4.4 Schematic of the asymmetric interface . . . . . . . . . . . . . . . . . 152 4.5 Comparison between the 2D numerical solution and the lubrication displacement model interface . . . . . . . . . . . . . . . . . . . . . . 157 4.6 Critical consistency ratio above which we have a steady state dis- placement solution for two Newtonian fluids for different values of α, and steady state examples . . . . . . . . . . . . . . . . . . . . . . . . 161 x List of Figures 4.7 Steady state shapes for two Newtonian fluids with varying casing rotating speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.8 Critical consistency ratio above which we have a steady state dis- placement solution for two power law fluids for different values of vC , and steady state examples . . . . . . . . . . . . . . . . . . . . . . . . 164 4.9 Critical consistency ratio above which we have a steady state dis- placement solution for two power law fluids for different values of wC , and steady state examples . . . . . . . . . . . . . . . . . . . . . 166 4.10 Critical consistency ratio above which we have a steady state dis- placement solution for two Bingham fluids for different values of vC , and steady state examples . . . . . . . . . . . . . . . . . . . . . . . . 168 4.11 Critical consistency ratio above which we have a steady state dis- placement solution for two Bingham fluids for different values of wC , and steady state examples. . . . . . . . . . . . . . . . . . . . . . . . 169 4.12 The effects of increasing rotation on interfacial stability for 2 New- tonian fluids; comparison of predicted steady states with numerical computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.13 Close-up of the interfacial region in Fig. 4.12 . . . . . . . . . . . . . 172 A.1 Schematic picture of the control volume CVi,j. . . . . . . . . . . . . 198 C.1 Schematic of the primary cementing process . . . . . . . . . . . . . . 211 C.2 Schematic of the experimental setup . . . . . . . . . . . . . . . . . . 217 C.3 Schematic of the optical set-up. . . . . . . . . . . . . . . . . . . . . . 218 C.4 Schematic of interface shape and residence time variations . . . . . . 220 C.5 Examples of steady and unsteady displacements . . . . . . . . . . . . 228 C.6 Spatio-temporal for the displacement of Figure C.5a . . . . . . . . . 230 C.7 Spatio-temporal for the displacement of Figure C.5b . . . . . . . . . 230 C.8 Residence time distribution for the displacements of Figs. C.5a & b . 231 C.9 Variation of σ∆t/µ∆t for Newtonian fluid series 1-3 . . . . . . . . . . 232 C.10 Variation of σ∆t/µ∆t for non-Newtonian fluid series 4 and 6 . . . . . 235 C.11 Dispersive finger in the displacement flow of two identical Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 xi List of Figures C.12 Moving frame streamlines and interface (heavy line) computed from the model of [38] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 C.13 Displacement of two Newtonian fluids . . . . . . . . . . . . . . . . . 240 C.14 Front view of the wide side of the annulus in Newtonian displacement 241 C.15 Spikes on the wide side in non-Newtonian displacements from series 4 242 C.16 Spikes on the narrow side in Newtonian displacements from series 3 243 C.17 Interface shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 C.18 Interface detection from the saturation curve. . . . . . . . . . . . . . 245 C.19 Effect of eccentricity on the saturation time growth rate . . . . . . . 246 C.20 Effect of flow rate on the saturation time growth rate . . . . . . . . . 246 C.21 Drainage wall layer observed during a Newtonian displacement. . . . 247 C.22 Static channel on the narrow side for an experiment in series 5 . . . 248 C.23 Comparison of the classified steady and unsteady experiments with the lubrication model predictions from [39] series 1-4 . . . . . . . . . 251 xii Acknowledgements The realisation of this work wouldn’t have been possible without the guidance, sup- port, help and patience of Prof. Ian Frigaard. To him I am really grateful. I am very thankful to Prof. Brian Seymour, who sat with me through numerous discus- sions and who made time for it. Parts of my research were supported financially by Schlumberger and NSERC. This support is also gratefully acknowledged. Finally I want to thank Andreas Putz that always lent me a hand. I deeply appreciated it. xiii Dedication To Rafael, who far away or close by has always been there for me. I thank him that he kept me going all those times that I had given up on myself. To my sweet Alfredo, who always has and always will inspire me. To my precious Fernanda, who I was so anxious to meet and is finally here. I enjoyed having her in and out the belly while working on the last bits of it. To my mother, who always supported me, no matter what I decided to do. To my father’s memory that will always be here. Finally, to all my dear friends in Vancouver and my dear Jihyoun, thank you for being my family, when family is so far away. xiv Co-authorship Statement The technical contribution of the thesis is given mostly in chapters 2, 3 and 4, which comprise 3 co-authored manuscripts. Additional contributions are listed in the appendices, and in particular Appendix C consists of a further co-authored manuscript. The identification and design of the overall research program has been carried out jointly with my supervisors, I.A. Frigaard and B.R. Seymour. Chapter 2 is a manuscript, co-authored with I.A. Frigaard, B.R. Seymour and S. Storey. I.A. Frigaard and myself are the primary authors in all regards. The analysis was carried out by M. Carrasco-Teja, I.A. Frigaard and B.R. Seymour, the numerical simulation by M. Carrasco-Teja and I.A. Frigaard. S. Storey was an MASc student (under the direction of D.M. Martinez and I.A. Frigaard) who had conducted experimental studies on these flows. The paper contains example images from these experiments. Chapter 3 is a manuscript, co-authored with I.A. Frigaard. The analysis has been carried out jointly. Code development and running of the computational results in this paper I have performed. We have jointly prepared the manuscript. Chapter 4 is a manuscript, co-authored with I.A. Frigaard. The analysis has been carried out jointly. Running of the computational results in this paper I have performed. We have jointly prepared the manuscript. Appendix C is a manuscript, co-authored with S. Malekmohammadi, S. Storey, D.M. Martinez and I.A. Frigaard. This is primarily a study in experimental fluid mechanics, performed by S. Malekmohammadi and S. Storey, two MASc students under the co-supervision of D.M. Martinez and I.A. Frigaard. My contribution has been limited to performing comparative studies. xv Chapter 1 Introduction In this thesis we consider the process of laminar displacement flows of one fluid by another in a narrow, eccentric, nearly-horizontal annulus. The motivation comes from the industrial process of primary cementing of highly deviated oil and gas wells. The main results of the thesis are contained in chapters 2-4, which are largely self-contained in terms of literature review and introduction. Therefore, here we focus on giving a more general introduction. In the next sections we introduce the reader to the process and to the different elements we have used in our model of the process. In the first section we give an overview of highly-deviated oil wells. We describe the primary cementing process, focusing on the differences between vertical and horizontal wells, and the extra challenges that are encountered in the later ones. In the next section we state the thesis objectives and explain the methodologies used to achieve them. The cementing process is idealised as the displacement flow of one fluid by another in a narrow, eccentric annulus. In general, the fluids involved in the cementing process are very viscous, shear thinning fluids, usually with a yield stress. We give an overview of some of the different mathematical models that describe the behavior of these fluids. Later we use a Hele-Shaw approach to model the displacement, so here we derive the equations for the flow of viscoplastic fluids in a Hele-Shaw cell. We conclude the section with a description of the basic theory of viscous fingering in a planar displacement flow in such a geometry. The following section is a literature review of background work related to the thesis. As many topics are covered in the introduction to each chapter, we just fill in some of the gaps. We concentrate on laminar annular flows for single-phase fluids, cementing displacements and on Hele-Shaw/porous media displacement flows of viscoplastic fluids. The final section gives an outline of the thesis. 1 1.1. Primary cementing of highly deviated oil wells 1.1 Primary cementing of highly deviated oil wells Highly deviated wells are those in which part of the wellbore is inclined approxi- mately 90o with respect to the vertical. Drilling of horizontal wells started in Texas in the 1920’s, followed by the U.S.S.R. in the 1930’s and China in the 1960’s. There was a slow start until the late 1970’s when the commercial viability was understood: although the cost of drilling and completing an horizontal well may be 2 or 3 times higher, the equivalent production from vertical wells would require up to ten times as many wells [49, 50]. Through 2001, more than 34,777 horizontal wells had been drilled in 72 countries spanning the globe and thousands more have been built ev- ery year. Today, horizontal and extended-reach wells are used to overcome various challenges imposed by reservoir geometry, fluid characteristics, economic conditions, or environmental constraints. These includes: coning of gas or water, water flood conformance, improved recovery from thin beds, economic and technical limitations, environmental restrictions, heterogeneous reservoirs and recovery from tar sands. Technology used to be limited by depth, as long wells required a large angle build rate [49]. However, with newer technology there is little limitation on depth, length of a well or angle build rates. The longest wells drilled up to 2008 were around 37,000 ft of horizontal length with a total vertical depth of 3,100 to 3,500 ft [65]. A few wells may be left as open holes, but most require some kind of lining to prevent collapse or sloughing. If a well is not cemented, it must be lined with a slotted liner, a pre-perforated liner, or with wire-wrap sand control liner. For these methods, a good cementing job in the previous segment is required anyway to protect the intermediate string from produced fluids, and to provide isolation between the upper cased-off zones and the lower producing intervals, (see [49, 50]). Whenever zonal isolation is needed, it is done using cemented liners or casings, although other methods have been tried, (e.g. pre-drilled liners). All of these methods must seal the casing completely to provide long-term zonal isolation. Well construction, for a cased well, starts with drilling the borehole and insertion of a casing or liner. Both the operations of drilling and running casing are out of the scope of this thesis. The following step is cementing of the casing of liner. This process is called primary cementing as it is the first layer of cement, (and hopefully the last), to cover the walls of the well. If the cement job is poor, secondary 2 1.1. Primary cementing of highly deviated oil wells cementing jobs, (called remedial cementing), are required. The primary cementing process proceeds in the following way. After the borehole is drilled and the drillpipe removed, (see Fig. 1.1(a)), a steel tube (casing or liner) is inserted into the hole, typically leaving a gap of approximately 2 cm between the outside of the tube and the inside of the wellbore, i.e., forming the annulus, (see Fig. 1.1(a)). The tubing is inserted in sections of about 10 meters each. At certain points, centralizers are fitted to the outside of the tube to prevent the tubing from falling to the lower side of the wellbore. Even with the centralizers, it is very common to have an eccentric annulus, being even more common in horizontal wells. There are centralizers that have been designed specifically for the cementing of horizontal wells, (see e.g. [29]). Once the tube is in place, surrounded by mud in the annular region and inside the casing, a sequence of fluids are pumped down the inside of the casing reaching the end of the hole and returning up the outside of the casing, (i.e. the annulus). Usually, a wash or spacer is pumped in first, followed by one or more cement slurries, (see Fig. 1.1(b,c)). The rheologies and densities of the spacer and cement slurry can be designed to aid the displacement of the drilling mud in the annulus, within certain constraints [6]. The fluid volumes are designed so that the cement slurries fill the annular space to be cemented. The circulation is stopped with a few meters of cement at the bottom of the inside of the the casing, and the cement is allowed to set, (see Fig. 1.1(d)). The final part of cement inside the tubing is drilled out as the well proceeds. The role of the cement job is zonal isolation, but this may be impaired by several means. In both vertical and horizontal wells, shrinkage/thermal stress effects and formation gases may produce mechanical defects. These processes are beyond the scope of this review. Another cause of impaired zonal isolation is that the mud is not fully removed from the annulus during the displacement process, remaining on the annulus wall in thin layers, or in a channel filling the narrow side of the annulus. Design methodologies for primary cementing that consider the rheology of the fluids have a long history. An overview can be found in [6]. The process is the same for vertical and horizontal wells, though the design is different. Extra attention has been given to horizontal wells as there are additional difficulties due to the gravitational acceleration acting perpendicular to the flow. The main areas where problems are encountered when cementing horizontal wells 3 1.1. Primary cementing of highly deviated oil wells Figure 1.1: Primary cementing process: (a) Insertion of the casing into the borehole; (b,c) A wash or spacer is pumped in the casing followed by one or more cement slurries;(d) Cement is allowed to set. are as follows, (see also [62]): • Hole-cleaning and displacement mechanics. These are more complicated in horizontal well as fluids may stratify, and cuttings of drilled solids settle in the narrow side of the casing. The settling is associated with a balance between buoyant settling and viscous/plastic drag from the fluid viscosity and yield stress. Thus, for a more deviated well, a higher yield stress is suggested to remove the cuttings during the displacement. Rotation and reciprocation have been suggested to deal with both these problems. • It is usually accepted that displacing the well in turbulent flow regime is op- timal. However, in long horizontal wells there are greater limitations due to either the equipment or to the risk of fracturing the surrounding rock. Con- sequently, many horizontal wells are displaced in laminar regimes [61]. 4 1.2. Thesis objectives and introduction to the process modeling • Cement slurry design. Besides having a high yield stress to drag the cuttings, other properties must be considered. For example, there should be no free water on the top of the annulus as a water channel can form through the cemented interval allowing communication of fracturing or reservoir fluids, impeding zonal isolation, (see [49, 50]). • Pipe centralization. It is evident that in a highly-deviated oil well the casing tends to collapse toward the bottom, creating a very eccentric annulus. Cen- tralizers for horizontal wells must provide standoff while also supporting pipe movement such as reciprocation and rotation. Common problems with cen- tralizers are that an insufficient number are used, they can cause an excessive amount of drag, and the spacing is not optimal. 1.2 Thesis objectives and introduction to the process modeling In the first part of this section we explain our objectives and describe the method- ologies. This is followed by a background of the methodologies used. This includes: (i) a brief overview of the different models used to describe the behavior of vis- coplastic fluids; (ii) the derivation of the field equations of the flow of Newtonian and Herschel-Bulkley fluids in a uniform vertical Hele-Shaw cell; (iii) a description of the planar displacement flow of in a vertical Hele-Shaw cell, and a simple analysis of the Muskat-type problem to determine fingering instability in such flows. 1.2.1 Objectives and methodology The focus of this thesis is to study laminar displacement flows of one viscoplastic fluid by another in a narrow, nearly-horizontal, eccentric annulus. Our objective is to understand the dynamics of the flow close to the interface between the fluids. In particular, we want to understand the implications of the inclination of the well, as well as the effects that rotation and reciprocation of the inner cylinder have on the displacement. We focus our attention on the deformation of the interface between the fluids. It is not our intent to establish its stability, but to be able to predict whether the fluids will stratify or travel steadily along the annulus. The latter case 5 1.2. Thesis objectives and introduction to the process modeling implies that the displaced fluid will eventually be removed from the whole annulus. Horizontal annular displacements are not well studied in general. Specifically, the effects of the casing movement are not well documented. We also want to provide simple scientific tools to aid the design of a cementing job, as well as insight into similar industrial processes. Due to the narrow aspect ratio of the gap to the length and the circumference of the annulus, and the assumption of a laminar regime, we adopt a Hele-Shaw modeling approach. The eccentric annulus is unwrapped into a Hele-Shaw cell with a nonuniform but periodic gap width. First we assume that both walls are fixed, but later in the thesis we include wall motion in the model. The fluids are modeled as shear thinning fluids with a yield stress, using the Herschel-Bulkley model. We assume that the fluids are miscible, but in the large Péclet number where molecular diffusion is negligible. We have two strategies to model the interface. The first one is to define a gap- averaged fluid concentration and advect it. In this case, the interface is interpreted as a level curve of the concentration, and the fluid properties are also expressed as functions of the averaged concentration. This approach is used mainly for 2D computations. We use the computations to explore different regimes intrinsic to the process. These simulations also motivate our analytical work. The second strategy is to track the interface between the fluids using a kinematic condition. We use this approach in the limit where the interface is either very flat or very elongated. Both cases allow the use of asymptotic methods. In chapters 2 and 4 we derive 1D lubrication-type models and perform an extensive analysis of the effects of the different variables in the model such as buoyancy, eccentricity, fluids rheologies and casing movement. In chapter 3 we consider fully 2D flows and steady states via a domain perturbation method. Even though stability is not our main concern, we use the 2D numerical simu- lations to understand when instabilities may be an important factor in the process. For the case of the displacement of Newtonian fluids with a moving wall, we derive a simple Muskat-type analysis to determine the stability of steady traveling waves in the asymptotic limit of small buoyancy and small eccentricity. 6 1.2. Thesis objectives and introduction to the process modeling 1.2.2 Generalised Newtonian fluids Fluids may be categorised as Newtonian or non-Newtonian. Let τ̂ij be the devia- toric stress tensor, and ˆ̇γij = ∂ûi ∂x̂j + ∂ûj ∂x̂i the rate of strain tensor, where dimensional quantities are denoted with a ,̂ (this notation is followed throughout the thesis). Newtonian fluids are those that, when sheared, the deviatoric stress tensor is pro- portional to the rate of strain tensor, i.e., τ̂ij = µ̂ˆ̇γij (1.1) where µ̂ the viscosity, (see Fig. 1.2(a)). These fluids will flow even with the minimum amount of applied deviatoric stress, e.g., like air or water. The simplest type of non-Newtonian fluid are those for which the viscosity, µ̂, is generalised so as to depend on the invariants of the rate of strain tensor. Fluids that have a so-called effective viscosity η̂ which depends only on the strain rate are referred to as generalised Newtonian fluids, i.e., τ̂ = η̂ ( ˆ̇γ ) ˆ̇γ (1.2) where ˆ̇γ denotes the second invariant of ˆ̇γij , defined as ˆ̇γ = 1 2 3∑ ij ˆ̇γ2ij 1/2 . (1.3) Similarly, we define τ̂ by τ̂ = 1 2 3∑ ij τ̂2ij 1/2 . (1.4) Other phenomena, as viscoelasticity and thixotropy are also often present in non- Newtonian fluids, but are not considered here. Usually the fluids involved in the cementing process are classified as generalised Newtonian fluids. They are very viscous, shear thinning fluids, often with a yield stress (see [25, 55, 60, 73]). There are many models that are used to approximate the behavior of such fluids. Here we review some of them, including the Herschel- 7 1.2. Thesis objectives and introduction to the process modeling Bulkley model, which is adopted in [6] and later in the thesis. The rheological behavior of a generalised Newtonian fluid is characterised by the relation between the strain rate and the effective viscosity, η̂ ( ˆ̇γ ) . Common exam- ples of generalised Newtonian fluids are power law fluids, Bingham fluids, Herschel- Bulkley fluids, Casson fluids and Carreau fluids. We describe each of these models below. • A power law fluid is one that exhibits a power law dependency of the effective viscosity, i.e., η̂ ( ˆ̇γ ) = κ̂ˆ̇γn−1, (1.5) where n is called the power law index, and κ̂ the consistency. They are called shear thinning fluids when n < 1, and shear thickening fluids when n > 1, (see Fig. 1.2(b)). • A Bingham fluid is a viscoplastic fluid that requires a certain minimum stress to initiate flow, called the yield stress, but the relation between the strain rate and the shear stress is still linear [5]. This can be written as η̂ ( ˆ̇γ ) = µ̂+ τ̂Yˆ̇γ , if τ̂ > τ̂Y , ˆ̇γ = 0, if τ̂ ≤ τ̂Y , (1.6) with τ̂Y being the yield stress (see Fig. 1.2(c)). • A Herschel-Bulkley fluid is a power law fluid with a yield stress [23], that is, η̂ ( ˆ̇γ ) = κ̂ˆ̇γn−1 + τ̂Yˆ̇γ , if τ̂ > τ̂Y , ˆ̇γ = 0, if τ̂ ≤ τ̂Y , (1.7) (see Fig. 1.2(d)). • The Carreau model describes a fluid which behaves like a Newtonian fluid at very low strain rates and at very high strain rates, with shear-thinning behavior at the intermediate strain rates. The constitutive law is η̂ ( ˆ̇γ ) = µ̂∞ + (µ̂0 − µ̂∞) ( 1 + (λ̂ˆ̇γ)2 )n−1 2 , (1.8) 8 1.2. Thesis objectives and introduction to the process modeling 0 ˆ̇γ τ̂ 0 shear thinning shear thickenning ˆ̇γ τ̂ 0 ˆ̇γ τ̂ τ̂Y 0 shear thinning shear thickenning ˆ̇γ τ̂ τ̂Y (a) (b) (c) (d) Figure 1.2: Relationship between strain rate ˆ̇γ and shear stress τ̂ for: (a) Newtonian Fluids; (b) Power law fluids; (c) Bingham fluids; (d) Herschel-Bulkley fluids. where µ̂0 and µ̂∞ are the viscosities at zero and infinity strain rates, n is a shear-thinning index and λ̂ is the time constant that controls the transition between low and high shear. Typically µ̂0 ≫ µ̂∞ so that globally speaking, the fluid thins as ˆ̇γ increases. For a small λ̂ the transition to power law behavior is slower and vice versa for large λ̂, (see Fig. 1.3(a)). It is widely used to describe fluid and semi-fluid foods. • The Casson model is based on a structural model of the interactive behavior of solid and liquid phases of a two-phase suspension, (see [9]). The model describes the flow of viscoplastic fluids, with the effective viscosity function 9 1.2. Thesis objectives and introduction to the process modeling given by √ η̂ ( ˆ̇γ ) = √ τ̂Y ˆ̇γ + √ µ̂, if τ̂ > τ̂Y , ˆ̇γ = 0, if τ̂ ≤ τ̂Y , (see Fig. 1.3(b)) This model is popular in blood flow models, as blood is a two-phase suspension. 0 ˆ̇γ τ̂ τ̂Y 0 λ̂ = 4 λ̂ = .25 λ̂ = 1 ˆ̇γ τ̂ (a) (b) Figure 1.3: Relation between strain rate ˆ̇γ and shear stress τ̂ for: (a) Carreau Fluids; (b) Casson fluids. There are many other models, but these considered above are the most common. 1.2.3 Flow in a Hele-Shaw cell A Hele-Shaw cell is a device for investigating the two-dimensional flow of a viscous fluid in a narrow gap. It consists of two parallel plates of length L̂, width M̂ ∼ O(L̂), and separated by a distance 2Ĥ ≪ L̂, (see Figure 1.4). The plates are stationary relative to one-another and the flow between the plates is driven either by pressure being applied or by a flow rate. Let Re be the Reynolds number based on Ĥ and the imposed flow velocity, p̂ the pressure, and (û, v̂, ŵ) the velocities in the (x̂, ŷ, ẑ)- direction respectively. 10 1.2. Thesis objectives and introduction to the process modeling Figure 1.4: Hele-Shaw Cell Using thin-film theory, these flows are dominated by the viscous forces as long as ( Ĥ L̂ )2 Re≪ 1, and H L ≪ 1, (1.9) (see e.g. [53]). Then the reduced mass conservation and momentum equations are ∂û ∂x̂ + ∂v̂ ∂ŷ + ∂ŵ ∂ẑ = 0, (1.10) −∂p̂ ∂ẑ = 0, (1.11) −∂p̂ ∂x̂ + ∂τ̂x̂ẑ ∂ẑ = 0, (1.12) −∂p̂ ∂ŷ − ρ̂ĝ + ∂τ̂ŷẑ ∂ẑ = 0, . (1.13) We will use these equations to find the relationship between the gap-averaged velocity field and the pressure. We start with the simple Newtonian case, and finish up with the equations for a flow of a Herschel-Bulkley fluid. 11 1.2. Thesis objectives and introduction to the process modeling Laminar flow of a Newtonian fluid in a Hele-Shaw cell Using equations (1.10)-(1.13) to find the relationship between the pressure gradient and the velocity field is rather simple when the fluid is Newtonian. Equations (1.12) and (1.13) are simplified to −∂p̂ ∂x̂ + µ̂ ∂2û ∂ẑ2 = 0, −∂p̂ ∂ŷ − ρ̂ĝ + µ̂∂ 2v̂ ∂ẑ2 = 0. Using non-slip boundary conditions at ẑ = ±Ĥ yields, û = (ẑ + Ĥ)(ẑ − Ĥ) 2µ̂ ∂p̂ ∂x̂ , v̂ = (ẑ + Ĥ)(ẑ − Ĥ) 2µ̂ ( ∂p̂ ∂ŷ + ρ̂ĝ ) , which show that the fluid flows in the direction of the modified pressure gradient( ∂p̂ ∂x̂ , ∂p̂ ∂ŷ + ρ̂ĝ ) . Averaging across the gap yields ( ˆ̄u, ˆ̄v ) = −Ĥ 2 3µ̂ ( ∂p̂ ∂x̂ , ∂p̂ ∂ŷ + ρ̂ĝ ) , where ˆ̄u = 1 2Ĥ ∫ Ĥ −Ĥ ûdẑ and ˆ̄v = 1 2Ĥ ∫ Ĥ −Ĥ v̂dẑ. Therefore, integrating equation (1.10) we have ∇ · (ˆ̄u, ˆ̄v) = 0, with ∇ = ( ∂ ∂x̂ , ∂ ∂ŷ ) . Finally, denoting ˆ̄u = ( ˆ̄u, ˆ̄v ) we have ∇ · ˆ̄u = 0, (1.14) ˆ̄u = −Ĥ 2 3µ̂ ( ∂p̂ ∂x̂ , ∂p̂ ∂ŷ + ρ̂ĝ ) . (1.15) 12 1.2. Thesis objectives and introduction to the process modeling Using the continuity equation (1.14) we can define a stream function Ψ̂ such that Ĥ ˆ̄u = −∂Ψ̂ ∂ŷ , and Ĥ ˆ̄v = ∂Ψ̂ ∂x̂ . (1.16) Now we have the option to work with either the pressure or the stream function. Either way it results in a Laplace equation, i.e., ∇2p̂ = 0, (1.17) for the pressure, or ∇2Ψ̂ = 0 (1.18) for the stream function. For Newtonian fluids both equations are well defined, so the choice usually de- pends on the boundary conditions imposed in a given problem, or on the type of questions one wants to answer. We will now find the equivalent equations for flows of Herschel-Bulkley fluids where we will find that the pressure field is not well defined in regions where the flow is unyielded. Laminar flow of a Herschel-Bulkley fluid in a Hele-Shaw cell To find the Hele-Shaw model for a Herschel-Bulkley fluid, we assume that the flow is symmetric with respect to ẑ = 0 and integrate equations (1.12) and (1.13) three times over the half-gap width to get ˆ̄u = −∂p̂ ∂x̂ 1 Ĥ ∫ Ĥ 0 ∫ Ĥ ẑ ξ̂ η̂(ξ̂) dξ̂dẑ, (1.19) ˆ̄v = − ( ∂p̂ ∂ŷ + ρ̂ĝ ) 1 Ĥ ∫ Ĥ 0 ∫ Ĥ ẑ ξ̂ η̂(ξ̂) dξ̂dẑ, (1.20) where the integrand on the right hand side is zero where the fluid is unyielded. Thus, again, the gap-averaged velocities are parallel to the modified pressure gradient defined as Ĝ ≡ ( ∂p̂ ∂x̂ , ∂p̂ ∂ŷ + ρ̂ĝ ) , (1.21) 13 1.2. Thesis objectives and introduction to the process modeling with Ĝ = ∣∣∣Ĝ∣∣∣, |(a, b)| = √a2 + b2. Let û = (û, v̂) be the velocity vector in the (x̂, ŷ)−direction. Then, in a local coordinate direction that is aligned to the direction of the gap-averaged flow, equations (1.12) and (1.13) become ∂ ∂ẑ τ̂f = −Ĝ (1.22) where τ̂f is the single non-zero component to leading order of the deviatoric stress in the direction of the flow. In the same way, let |û| is the single positive component of the velocity in the direction of the flow. As in the Hele-Shaw cell geometry the leading order strain rate is given by ˆ̇γ = ∣∣∣∂û∂ẑ ∣∣∣, then τ̂f and |û| satisfy the constitutive law τ̂f = ( κ̂ ∣∣∣∂û∂ẑ ∣∣∣n−1 + τ̂Y/∣∣∣∂û∂ẑ ∣∣∣ ) ∂|û| ∂ẑ , if |τ̂f | > τ̂Y ,∣∣∣∂û∂ẑ ∣∣∣ = 0, if τ̂f ≤ τ̂Y . (1.23) Using the symmetry condition τ̂f = 0 at ẑ = 0, we can integrate equation (1.22) from 0 to ẑ to get τ̂f = −Ĝẑ. Integrating one more time from ẑ to Ĥ using (1.23) and the no-slip boundary con- dition, yields |û| = (ĜĤ − τ̂Y )m+1 κ̂mĜ(m+ 1) − (Ĝẑ − τ̂Y ) m+1 κ̂mĜ(m+ 1) , Ĝẑ > τ̂Y , (ĜĤ − τ̂Y )m+1 κ̂mĜ(m+ 1) , Ĝẑ ≤ τ̂Y . (1.24) where m = 1/n is the inverse of the power law index. Thus, again we have û = −|û| Ĝ Ĝ, (1.25) If the coordinate direction is chosen correctly, the flow is simply a Poiseuille flow between two parallel plates. Averaging equation (1.24) across the half-gap width, the closure relationship between Ĝ and the gap-averaged speed ∣∣ ˆ̄u∣∣ is given by Ĥ ∣∣ ˆ̄u∣∣ = F (G), (1.26) 14 1.2. Thesis objectives and introduction to the process modeling where F (Ĝ) = 0 ĤĜ ≤ τ̂Y ,( (m+ 1)ĤĜ+ τ̂Y ) Ĝ2(m+ 1)(m + 2) ( ĤĜ− τ̂Y )m+1 κ̂m ĤĜ > τ̂Y . (1.27) Some examples of the function F (Ĝ)are shown in Fig. 1.5(a). Note that for a large Ĝ, increasing the shear-thinning of the fluid will speed up the flow compared to a Bingham (or Newtonian) fluid (m = 1). Rewriting equations (1.19) and (1.20) using F (Ĝ) yields Ĥ ˆ̄u = −F (Ĝ) Ĝ ∇ (p̂+ ρ̂ĝŷ) , (1.28) which coupled with the averaged continuity equation ∇ · ( Ĥ ˆ̄u ) = 0 (1.29) will govern the flow. Note that these equations are equivalent to (1.14) and (1.15) for a Newtonian fluid, i.e., m = 1, τ̂Y = 0. 0 F (Ĝ ) Ĝ 0 Ĝ = F -1 (|∇ Ψ̂ |) |∇Ψ̂|(a) (b) Figure 1.5: Closure relationship between the pressure gradient and the velocity field. Parameters: κ̂ = .8, τ̂Y = .5, m = 1 (–), 2 (−·), 4 (· · · ). 15 1.2. Thesis objectives and introduction to the process modeling Taking the divergence of both sides of equation (1.28) we get an equation for the pressure, i.e., ∇ · ( F (Ĝ) Ĝ ∇(p̂+ ρ̂ĝŷ) ) = 0. (1.30) On the other hand, similarly to the Newtonian case, we can use the continuity equation (1.29) to define a stream function Ψ̂. Rewriting equation (1.28) in terms of ∇Ψ̂ and cross-differentiating the terms in the modified pressure gradient (1.21), we get ∇ · F−1 (∣∣∣∇Ψ̂∣∣∣)∣∣∣∇Ψ̂∣∣∣ ∇Ψ̂ = 0. (1.31) Note that for fluids with a yield stress, p̂ is not determined in regions where the flow is unyielded (see Fig. 1.5(b)). Therefore we consider that in this case is a better choice to work with the stream function equation (1.31). 1.2.4 Planar displacement flows and the Muskat problem Suppose instead of a single fluid, we consider now the vertical displacement flow of two fluids along a vertical Hele-Shaw cell, (see Fig. 1.6). At ŷ = 0 fluids are pumped in with uniform mean speed, V̂ > 0. In each fluid we have the Hele-Shaw equations Ĥ ˆ̄u = −Fk(Ĝ) Ĝ ∇(p̂+ ρ̂kĝŷ), or (1.32) ∇(p̂+ ρ̂kĝŷ) = − F−1k ( Ĥ ∣∣ ˆ̄u∣∣)∣∣ ˆ̄u∣∣ ˆ̄u, (1.33) and the mass conservation equation ∇ · ˆ̄u = 0, (1.34) with ρ̂k the density of fluid k = 1, 2, and with Fk the closure relation (1.27) in each fluid, with properties mk, κ̂k and τ̂Y,k in fluid k. At the interface, defined as 16 1.2. Thesis objectives and introduction to the process modeling ŷ = f̂(x̂, t̂), the pressure and the normal velocity are continuous, i.e., [p̂]21 = 0, and [ ˆ̄u · n̂]21 = 0, where [q]21 denotes the jump at the interface from fluid 1 to fluid 2 of property q, and n̂ is the normal vector to the interface. The interface also satisfies the kinematic condition ∂f̂ ∂t̂ + ˆ̄u ∂f̂ ∂x̂ = ˆ̄v. (1.35) It is clear that there is a planar displacement (i.e. a flat interface perpendicular to the flow), such that: ˆ̄u = 0, (1.36) ˆ̄v = V̂ . (1.37) Arbitrarily initializing the interface position we have f̂(x̂, t̂) = V̂ t̂. Thus, the pressure in each fluid is given by p̂k(ŷ, t̂) = p̂i(t̂)− ( F−1k (ĤV̂ ) + ρ̂kĝ )( ŷ − f̂(x̂, t̂) ) , (1.38) with p̂i(t̂) an interfatial pressure. The Muskat problem A simple method to give insight into the stability of the interface (often called the Muskat problem, see [48]), is the following analysis. This method gives the same stability criteria as the Saffman-Taylor analysis. Assume that a long, narrow finger extends ahead of the interface (see Fig.1.7). At the interface, ˆ̄v = V̂ and the pressure is constant. Since the finger is long and thin, ∂p̂∂x̂ = 0, across the finger. Thus the pressure inside the finger is given by the pressure in fluid 2, i.e., p̂finger = p̂2 = p̂i(t̂)− ( F−12 (ĤV̂ ) + ρ̂2ĝ )( ŷ − f̂(x̂, t̂) ) . (1.39) 17 1.2. Thesis objectives and introduction to the process modeling Figure 1.6: Vertical Displacement in a Hele-Shaw cell with the interface denoted by ŷ = f̂(x̂, t̂). On the other hand, the finger is full of fluid 1. Thus, using equations (1.32), (1.33) and (1.39), the velocity in the finger is given by Ĥv̂finger = −sign (σ̂)F1 (|σ̂|) . (1.40) where σ̂ = (ρ̂1 − ρ̂2)ĝ − F−12 (ĤV̂ ). (1.41) The finger will elongate whenever v̂finger > V̂ . In order to gain more insight, we can write Ĥ ( v̂finger − V̂ ) = −sign (σ̂)F1 (|σ̂|)− F1 ( F−11 (ĤV̂ ) ) . (1.42) Note that when σ > 0 the finger won’t grow. This will happen, for example, when the displaced fluid is much lighter than the displaced fluid, and ĤV̂ is small. 18 1.3. Literature review Figure 1.7: Finger in the interface. When σ̂ < 0 we have Ĥ ( v̂finger − V̂ ) = F1 (|σ̂|)− F1 ( F−11 (ĤV̂ ) ) . As F1 is non-decreasing and positive, this means that the interface will be stable whenever |σ̂| > F−11 (ĤV̂ ) and unstable otherwise. For Newtonian fluids is easy to verify that the finger will grow when 0 ≤ µ̂2 − µ̂1 + (ρ̂2 − ρ̂1) 3V̂ ĝĤ2. For example, the interface will be unstable when µ̂1 < µ̂2 and ρ̂1 < ρ̂2. 1.3 Literature review In this thesis we study a model for laminar Hele-Shaw displacement flows in narrow- gap eccentric annuli, using shear-thinning yield stress fluids. It is evident that the subject area, although quite specialised, touches on many related areas. In the first place, disregarding displacements, the laminar flow of a single phase non-Newtonian fluid through an annular duct has a well developed literature, but much of the literature has been contributed in the past 20 years, made feasible by quick computations and motivated largely by industrial applications such as cementing. We therefore provide an overview of laminar annular flow studies first, in §1.3.1. 19 1.3. Literature review Secondly, in the miscible displacement context, we are naturally drawn in two directions: • There is an ever increasing interest in miscible liquid-liquid flows in ducts, whether Newtonian or non-Newtonian fluids are involved, in different regimes of Péclet number, Reynolds number, duct aspect ratio and buoyancy number. The literature is large and is developing. However, in the Hele-Shaw model that we study in this thesis, we suppress small scale effects of molecular dif- fusion and dispersion by assuming a uniform fluid concentration across the annular gap. By doing this, this large literature on Navier-Stokes displace- ment flows becomes not directly relevant to the results of our mathematical analysis. For this reason we do not attempt a review of this literature. We acknowledge that the physical relevance of these studies remains if one wants to explore the limits of validity of our Hele-Shaw approach. • Lastly, there is a vast literature on Hele-Shaw type displacements and their porous media analogues. The greater part of this literature is concerned with viscous fingering instabilities, which are reviewed in [24] and elsewhere. Al- though later in the thesis we will see that density-driven fingering does occur, viscous fingering is uncommon in laminar cementing displacements as one generally tries to preserve a positive viscosity ratio between fluids. Thus, the viscous fingering literature is not entirely relevant. Therefore, here we con- fine our review to the area of visco-plastic fluid displacements in Hele-Shaw geometries, and their porous media analogues. This literature is reviewed in §1.3.2. Thirdly, as with any industrial process practiced continuously for 80 years or more, there is an extensive industrial technical literature. This literature in a large part describes process specific details that are not relevant to displacement flows. That part of the literature that does deal with displacement (mud removal) is also not focused exclusively at the fluid mechanics of the process. Much of it consists of case studies and anecdotal evidence for process improvement, often hard to test or verify. Nevertheless, over the years there have been a number of attempts to understand primary cementing displacement flows from a mechanical viewpoint, and we review these studies in §1.3.3. 20 1.3. Literature review Finally, we comment that each of the chapters 2-4 contains a review of directly relevant literature. 1.3.1 Laminar single-phase annular flows Laminar flow in concentric and eccentric annuli has been studied since at least the 1950s. The early studies, [19, 22, 64] considered concentric annuli, where axisym- metry reduces the problem to 1D. Many of these early studies developed analytical expressions or semi-analytical algorithms for computing the velocity solutions and hence hydraulic quantities such as flow rates and pressure drops, Hanks, [22], ap- pears to have been the first to have used the Herschel-Bulkley model in this context. The oilfield drilling and cementing industries have been the motivation for many studies of annular flows. A number of the hydraulic-style closure expressions were developed within the industry (perhaps even independently to those cited above) and now appear routinely in standard books on drilling or cementing engineering, e.g. [8, 49]. They are also used extensively in proprietary hydraulics design software by many oilfield companies. Some of the first studies of the effects of annular eccentricity were also made within the technical literature, by approximating a narrow eccentric annulus as a plane channel of varying width: the so-called slot approximation. This idea was put on a firm footing by Walton & Bittleston, [71] who developed a fully asymptotic theory for narrow gap eccentric annular flows of Bingham fluids. They showed that in an eccentric annulus, the central plug region present in a concentric annulus could in fact break up into two plug regions, on the wide and narrow sides of the annulus. These two plugs would move at different axial speeds, separated by a pseudo-plug region in which the plug speed varied slowly in the azimuthal direction. They also predicted the shape of the yield surfaces and compared their solutions against finite element computations of the flow. These results were confirmed by Szabo & Hassager, [66], who also went further in developing a predictive theory for when the single plug of the concentric annular flow would break into two. The two papers [66, 71] mark the start of commonly available computational tools for axial annular flows of generalised Newtonian fluids, and many authors have since followed. Some of the better known examples are [4, 16, 26, 70, 72]. 21 1.3. Literature review Motion of the inner cylinder was first studied in the context of concentric annuli. In the first place, there are a number of analytical and semi-analytical approaches, exemplified by e.g. [7], which are focused at understanding base flows with some kind of symmetry. A second category of annular flows studies has been focused at the Taylor-Couette instability, (see e.g. [37, 38]), and the review in [32] for yield stress fluids in particular. For the rotation rates common in primary cementing of horizontal wells we are generally far from the Taylor-Couette regime, so these flows are less relevant. An excellent review of much of the numerical and analytical literature of annular flows is given in the review by Escudier et al., [15], which includes a very extensive bibliography. We have focused above on non-Newtonian fluids of generalised Newtonian type, as these are those that we consider later in our displacement model. We have only considered analytical and numerical studies. There is also an experimental literature, which we review in appendix C. It is worth mentioning that there is also a large literature for viscoelastic fluid flows in annular geometries. These are motivated by extrusion processes with poly- mer melts and solutions, and also by problems in tribology. Other industries do also consider annular flows of pastes and nonlinearly viscous fluids. For example, in [17] an annular heat exchanger from the food industry is modeled. This device consists of a concentric annulus through which food is slowly pumped. The cylinders co- rotate in order to shear-thin the food and the inner annulus is fitted with scrapers to maintain the fluids in motion. 1.3.2 Viscoplastic displacement flows in Hele-Shaw cells or porous media Viscoplastic displacement flows in Hele-Shaw cells have a relatively recent history, compared to their porous media analogue. In the porous media context, the yield stress divided by the gap width translates into a limiting pressure gradient. In the simplest case, (analogous to the Bingham fluid), we have a Darcy-type law in which the fluid speed is proportional to the difference between the pressure gradient and the limiting pressure gradient. This limiting pressure gradient can either be a property of the fluid or may be a property of the porous media. 22 1.3. Literature review These types of flows were studied mostly in the Russian literature, starting in the 1970s, being motivated largely by the physical properties of certain heavy oils that were difficult to extract. Since the problems were nonlinear and computational resources limited this literature consists mainly of analytical results. In particular there are a large number of elegant solutions to planar 2D flow problems, (often exploiting symmetry properties of the flow), that were solved by the use of complex variable methods. Other results include the characterisation of the stream function and pressure fields as minimisers of certain “dissipation potentials” (i.e. variational principles). These were used to prove comparison-type results between domains where one domain is an indented version of another domain. This Russian literature is summarised largely in the two texts [3, 20]. Many of the contributions were due to Entov. It is worth remarking that most of this early literature did not in fact consider liquid-liquid displacement, but simply the problem of removing heavy oil from a reservoir, minimising the amount of oil left behind. Classical porous media displacement instabilities for these types of fluid were first studied by Pascal, [54, 55, 56] who considered both planar and radially symmetric displacement fronts. His analysis is essentially that we have derived earlier for the vertical displacement flow, but for different geometric and rheological configurations. As we have seen, derivation of the base Hele-Shaw model for generalised Newto- nian fluids is relatively straightforward. For yield stress fluids in particular, approx- imate criteria for fingering in a planar Hele-Shaw displacement have been developed by Lindner, Coussot and Bonn; see [11, 34, 36]. There is still much ongoing work, e.g. [35, 40], which is largely experimental. Other analytical and computational work in this area includes the study of bubble propagation in a Hele-Shaw cell filled with a yield stress fluid, (see [1, 21]). Although we have focused exclusively on yield stress fluids here, we remark that there is also a large literature on porous media and Hele-Shaw displacements with shear thinning fluids. Some representative examples are: [2, 30, 31, 33]. 23 1.3. Literature review 1.3.3 Modeling and process design of cementing displacements Design methodologies for primary cementing that consider the rheology of the fluids have a long history. The possibility of a mud channel forming on the narrow side of the annulus was first identified in [42]. The reasoning used in [42] is essentially a hydraulic approach. Extensions have led to whole systems of design rules for laminar displacements, [59], also based on hydraulic reasoning. In general, these rule sets state that the flow rate must be sufficiently high to avoid a mud channel on the narrow side of the annulus, that there should be a hierarchy of the fluid rheologies pumped, (i.e. each fluid should generate a higher frictional pressure than its predecessor), and that there should be a hierarchy of the fluid densities, (each fluid heavier than its predecessor). These design rule systems were generally developed in the 1990s and before, and were focused initially at vertical wells. In the petroleum industry literature there are a number of published case studies that illustrate how such systems have been successfully applied. The two main criticisms of such systems are that: (i) the level of fundamental physical understanding is low and predictions made are generally conservative; (ii) these systems tend to be ineffective in highly deviated and horizontal wells. In particular, the practice of keeping a significant density difference between fluids clearly promotes stratification in horizontal wells, (see [13, 28, 62]) . Turning now to more modern studies of laminar cementing displacements, these can be put into two categories: • Those that exploit a Hele-Shaw approach, as we do later in the thesis. • Those that use CFD to simulate the displacement in 3D. The origin of the Hele-Shaw approach dates back to Martin et al., [41] who in fact uses a porous media analogue to provide a closure relationship for the frictional pressure in the annulus. This approach was used more explicitly in the models de- scribed in [39], used to make comparison with experimental studies of displacement. A complete derivation of this modeling approach was however first published in [6]. This model is very much focused at cementing. Varying annular geometries are in- cluded, as is the displacement of any number of fluids in sequence. Example results are presented, showing mud channel formation and also steady displacement flows. 24 1.3. Literature review They modeled the fluids as Herschel-Bulkley fluids, and assumed miscible fluids but in the large Péclet number limit, where diffusion is negligible. The numerical method used in [6] was a hybrid numerical-asymptotic method, exploiting the long length of the well. This underlying model from [6] was adopted by Pelipenko & Frigaard in a se- quence of papers, [57, 58, 59]. In [57] the theoretical background to the model was developed, i.e. existence and uniqueness of stream function solutions. They also found steady state displacement solutions in near-concentric annuli. A fully 2D nu- merical algorithm was developed in [58], using the augmented Lagrangian method and FCT scheme, and this was used to explore various features of displacements in vertical wells. Finally, in [59] a lubrication-style model was developed a 1D lubrica- tion type model for elongated interfaces in nearly-vertical wells, to find conditions on the fluids rheologies, densities and the eccentricity for which there were steady trav- eling wave solutions. The predictions from this lubrication model were compared with those from the rule-based system of [12], showing it to be very conservative in its requirements for a steady displacement. The work by [57, 58, 59] is directly relevant to the work in this thesis. The underlying 2D Hele-Shaw model is identical to that we consider in chapter 2 and the computational methods are also the same. However, the physical situation is very different in horizontal wells, compared to vertical wells, and this is the main point of the analysis in chapter 2 of this thesis. We also develop in chapter 2 a lubrication model, but this model results in a quasilinear advection-diffusion equation for the interface position, as opposed to the hyperbolic problem in [59]. Therefore, analysis of this problem is different. Later in the thesis, chapters 3 and 4, we again use the same underlying model as the starting point, but inclusion of casing motion is a non-trivial change in both the computation and analysis. The Hele-Shaw approach has also been adapted somewhat by Moyers-González [44]. He considers a variant of the Hele-Shaw approach in which linear accelerations are retained in the reduced momentum equations. This leads to a complex nonlinear evolution problem for the stream function, in place of the elliptic problem of [57], the theory of which is developed in [47]. The reason for the change is in order to study various transient problems in cementing. For example, in [47] the effect of flow pulsation on narrow side mud 25 1.3. Literature review channels is studied via numerical simulation. In [46, 45], the stability of multilayer annular flow is studied, i.e. those flows that emerge as the interface between fluids elongates along the annulus in an unsteady displacement. This analysis is performed for both vertical and horizontal wells. On the 3D simulation side of things, Szabo & Hassager, [67, 68], have computed a 3D immiscible displacement flow between 2 Newtonian fluids using the arbitrary Lagrange-Euler formulation. The model shows reasonable agreement with a film- draining model derived for concentric annular displacements and also shows that the displacement efficiency drops significantly with annular eccentricity. 3D approaches have also been taken by other authors, using general purpose CFD codes, e.g. [14, 27, 51, 52, 69]. As with [67, 68], such studies have value in understanding details of the flow near the interface but are of limited use in understanding flows on a larger scale, which is the advantage of the Hele-Shaw approach. A hybrid 2D/3D approach has recently been adopted by Savery et al., [63], which includes casing motion. As this is the only model we have found published that studies comparable flows, we review it below in some detail. Savery et al. hybrid 2D/3D model of the cementing process In [63] and followed up in [43], Savery et al. adopt a hybrid 2D/3D approach, in which the Navier-Stokes equations are simplified by ignoring radial velocities and azimuthal pressure gradients, but the model is still resolved in 3D. An outline of the model is the following. Using the general momentum and mass-conservation equations for unsteady single-phase fluids in cylindrical coordinates (r, θ, z) with corresponding radial, az- imuthal and axial velocities (u, v,w): ρ ( ∂u ∂t + u ∂u ∂r + u r ∂u ∂θ − v 2 r + w ∂u ∂z ) = Fr − ∂p ∂r + 1 r ∂rτ rr ∂r + 1 r ∂τ rθ ∂θ − 1 r τ θθ + ∂τ rz ∂z , (1.43) 26 1.3. Literature review ρ ( ∂v ∂t + u ∂v ∂r + v r ∂v ∂θ − uv r +w ∂v ∂z ) = Fθ − 1 r ∂p ∂θ + 1 r2 ∂r2τ θr ∂r + 1 r ∂τ θθ ∂θ + ∂τ θz ∂z , (1.44) ρ ( ∂w ∂t + u ∂w ∂r + v r ∂w ∂θ + w ∂w ∂z ) = Fz − ∂p ∂z + 1 r ∂rτ zr ∂r + 1 r ∂τ zθ ∂θ + ∂τ zz ∂z , (1.45) ∂u ∂r + u r + 1 r ∂w ∂θ + ∂w ∂z = 0, (1.46) where p is the pressure, Fr, Fθ and Fz are body forces, and τ is the deviatoric stress tensor. The authors assume the radial velocity u and the azimuthal pressure gradient ∂p∂θ in the tangential momentum equation are negligible. They claim the first is small compared with the azimuthal and axial velocities, and the second is negligible because the induced effect is small relative to the dragging motion associated with casing rotation. They also ignore the radial momentum. Then equations (1.43)-(1.46) become ρ ( ∂v ∂t + v r ∂v ∂θ + w ∂v ∂z ) = Fθ + 1 r2 ∂r2τ θr ∂r + 1 r ∂τ θθ ∂θ + ∂τ θz ∂z , (1.47) ρ ( ∂w ∂t + v r ∂w ∂θ + w ∂w ∂z ) = Fz − ∂p ∂z + 1 r ∂rτ zr ∂r + 1 r ∂τ zθ ∂θ + ∂τ zz ∂z , (1.48) 1 r ∂w ∂θ + ∂w ∂z = 0, (1.49) with the 3 unknowns v, w and ∂p∂z . No-slip conditions are applied to all solid surfaces and uniform velocities are given as initial conditions. In general terms, the equa- tions are solved using a finite-difference time-marching scheme though no further details are provided. The model accommodates geometric flexibility by transforming any annular cross-section into Cartesian coordinates (ξ, η, z), then re-expressing the surface in general boundary-conforming curvilinear coordinates using the mapping procedure described in [10]. As equations (1.47-1.49) only apply for a single-phase fluid, a concentration 27 1.3. Literature review function C that satisfies a transformed convection-diffusion equation, is used. The concentration satisfies C = 1 when only the displacing fluid, fluid 1, is present, and C = 0 when there is only displaced fluid, or fluid 2. The density used in the equation is a linear interpolation between the two fluids, i.e. ρ = Cρ1 + (1−C)ρ2, where ρk corresponds to fluid k. The rheological properties, consistency µ, power law number n and yield stress τY , are determined from laboratory data. The authors are able to include casing motion in this approach, but a justification of the assumptions and even details of the final simplified model are not given in [63]. Even though this model contains all the process variables in primary cementing, we note the following weaknesses: 1 There is a complete lack of details and open source references behind the work. 2 In particular, neglect of azimuthal pressure gradient is not justified in eccentric annuli, whenever there are steady traveling wave displacements, (which is common). 3 Solving in a 3D mesh increases the numerical work, requiring a very coarse mesh in the axial direction, while the velocity in the model is anyway only resolved in 2D. Therefore, there is little advantage in not adopting the Hele- Shaw approach. 4 With non-uniform geometries (like washouts), the flow is likely to be fully 3D. The reduced model, although having three dimensions cannot represent the actual 3D flow. Thus neither the radial velocity nor the azimuthal pressure gradient should be neglected. 5 The articles published with the model show very specific case examples, but no general analysis of the displacement. On the other hand, including the diffusion terms in the concentration equation, even though the diffusivity might be very small, gives a better understanding of the mixing between the different fluids, compared to the Hele-Shaw approach in [6]. The calculation of the fluid properties in the concentration equation using case-specific laboratory data may also increase the accuracy of the model. 28 1.4. Thesis outline 1.4 Thesis outline The content of this thesis is divided in three main chapters. In chapter 2 we adopt the model of an annular displacement flow derived in [6] described in the previous section. We use it to study the stratification and traveling wave solutions in nearly- horizontal, narrow, eccentric annuli. The aim is to study the difference between vertical and horizontal displacements, and to predict when stratification will occur in the latter ones. We use the augmented Lagrangian algorithm described in [18], and the Flux Corrected Transport scheme to solve a 2D model. A description of both algorithms is given in §A.1 and §A.2. We explore some of the numerical results to understand the effects of buoyancy and eccentricity, as well as the role of the rheology of the fluids. Then we derive a lubrication model that we use to predict the conditions under which a steady traveling wave solution can be found for the lubrication limit. Necessary and sufficient conditions for the existence of steady state traveling wave solutions are given and we survey the, (parametrically simpler), set of steady interface shapes and stability maps. The chapter closes with a summary of results and discussion of consequences. In chapter 3 we analyse the effects of rotation and axial motion of the inner cylinder of an eccentric annular duct during the displacement flow between two Newtonian fluids of differing density and viscosity. We restrict attention to Newto- nian fluids to gain some insight into the effects of casing movement by first looking at simpler fluids. We derive a Hele-Shaw type model, but with the necessary mod- ifications to include rotation and reciprocation of the casing. We investigate the possibility of finding steady traveling wave displacements in these geometries. We are able to show that if the interface is assumed to be close to flat (perpendicular to the axis), then we can find an analytical form for the steady state shape of the interface and the stream function, for both concentric and mildly eccentric annuli. Then we present computed results that explore the principal effects of casing rota- tion and reciprocation on displacements. Casing rotation induces a phase shift in the azimuthal position of the interface, which results in the positioning of heavy fluid above light fluid at certain azimuthal positions. This configuration is vulner- able to buoyancy driven fingering, which is observed in the computed results. The instability remains apparently local and over longer time scales we frequently attain 29 1.4. Thesis outline stable steady displacements with a diffuse interfatial region. Finally we consider various aspects of stability and instability of these flows. In chapter 4 we extend the model with a moving casing to viscoplastic fluids, with the main difference (and difficulty) being the computation of the closure between the velocity field and the modified pressure gradient. In the first part of chapter we address the model derivation and closure relations. The second part considers the limit of large buoyancy numbers, in which the interface elongates along the annulus. We derive a lubrication-style model for this situation, showing that the leading order interface is symmetric. We find that rotation of the inner cylinder only affects the length of the leading order interface, and this occurs only for non- Newtonian fluids via shear-thinning effects. At first order, casing rotation manifests in an asymmetrical “shift” of the interface in the direction of the rotation. We also derive conditions on the eccentricity, fluid rheology and inner cylinder velocity, under which we are able to find steady traveling wave displacement solutions. Chapter 5 of the thesis contains a summary of the results of the thesis, both from scientific and industrial perspectives. We also take a critical look at the methodology and make recommendations for future work. The thesis contains 4 appendices. Appendix A gives some of the details of the numerical method employed in chapters 2 and 3. Appendix B addresses the closure problem between applied pressure gradients and gap-averaged velocity field. In appendix C we include an experimental study of displacement flow phenomena in narrow vertical eccentric annuli. It considers both Newtonian and viscoplastic fluids. This experimental work is largely in line with predictions of a Hele-Shaw style of displacement model, which the author contributed the study. Finally, appendix D contains a list of publications of the author. 30 1.5. Bibliography 1.5 Bibliography [1] A. Alexandrou and V. Entov. On the steady-state advancement of fingers and bubbles in a Hele-Shaw cell filled by a non-Newtonian fluid. European Journal of Applied Mathematics, 8:73–87, 1997. [2] J. Azaiez and B. Singh. Stability of miscible displacements of shear thinning fluids in a Hele-Shaw cell. Physics of Fluids, 14:1557–1571, 2002. [3] G. Barenblatt, V. Entov, and V. Ryzhik. Theory of fluid flows through natural rocks. 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Journal of Fluid Mechanics, 380:363–376, 1999. [12] M. Couturier, D. Guillot, H. Hendriks, and F. Callet. Design rules and associ- ated spacer properties for optimal mud removal in eccentric annuli. SPE paper 21594, 1990. 31 1.5. Bibliography [13] R. Crook, S. Keller, and M. Wilson. Deviated wellbore cementing: Part 2 solutions. Journal of Petroleum Technology, pages 961–966, August 1987. [14] E. Dutra, M. Naccache, P. Souza-Mendes, C. Souto, A. Martins, and C. de Mi- randa. Analysis of interface between Newtonian and non-Newtonian fluids in- side annular eccentric tubes. SPE paper 59335, 2004. [15] M. Escudier, P. Oliveira, and F. Pihno. Fully developed laminar flow of purely viscous non-Newtonian liquids through annuli, including effects of eccentricity and inner-cylinder rotation. International Journal of Heat and Fluid Flow, 23:52–73, 2002. [16] P. Fang, R. Manglik, and M. Jog. Characteristics of laminar viscous shear- thinning fluid flows in eccentric annular channels. 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Physics of Fluids, 9:286–298, 1997. 37 Chapter 2 Visco-plastic fluid displacements in horizontal narrow eccentric annuli: stratification and traveling wave solutions1 2.1 Introduction The aim of this chapter is to derive conditions under which displacement flows of Herschel-Bulkley fluids along a narrow eccentric annulus may either advance as a steady traveling wave or may stratify gravitationally. The annulus is oriented hori- zontally, perpendicular to the direction of gravity, see Figure 1. The motivation for the study comes from the industrial process of primary cementing, used to complete oil wells prior to production. Since the early 1990’s there has been a massive increase in the numbers of oil wells constructed horizontally, primarily to increase produc- tivity 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 10km, see e.g. the detailed description in [6]. The 10km barrier was broken in a number of wells drilled at Wytch Farm, UK, around 2000. The limits of “extreme” extended reach wells are now being pushed into the 15− 20km range, but such wells are unusual and do not necessarily bring productivity benefits proportional to their technical challenges. In the present day it is routinely feasible to construct wells with horizontal extensions 1A version of this chapter has been published. Carrasco-Teja, M., Frigaard, I.A., Seymour, B.R. and Storey, S.(2008) Visco-plastic fluid displacements in horizontal narrow eccentric annuli: stratification and traveling wave solutions, J Fluid Mech 605:293-327. 38 2.1. Introduction in the 7− 10km range. Figure 2.1: Schematic of the Primary cementing displacement in a horizontal well: process schematic Many of the other well construction technologies have lagged behind in this surge in construction of long horizontal wells, cementing being one of these. Although many of the potential problems of cementing horizontal wells were identified some time ago, e.g. [10], the industrial response has been largely through technological advances, rather than by developing understanding of physical fundamentals that may affect the process. In this chapter we adopt the latter approach and consider the fluid dynamics of displacement during primary cementing of horizontal wells. The primary cementing process involves the displacement of one non-Newtonian fluid by another, pumped axially along the annulus at a constant imposed flow rate. The flows that we consider are laminar and the annuli considered have annular gaps that are narrow with respect to both circumferential and axial length-scales. Thus, a Hele-Shaw modelling approach is appropriate and we adopt a two-dimensional (2D) model derived by [1], outlined in §2.2. Those unfamiliar with this process may 39 2.1. Introduction consult [5] for background information. In cementing near-vertical wells it is advantageous to maintain a positive density and viscosity difference between successive fluid stages pumped. This helps to offset the destabilising effects of the annulus eccentricity and has long been advocated in the industrial literature, although without a firm scientific basis. In the sequence of papers [7, 8, 9] a more rational understanding of the displacement dynamics in near-vertical annuli was developed using a Hele-Shaw model. It was shown that under certain process conditions it is possible to obtain steady, traveling wave, dis- placement front, solutions that are stable. Conditions were derived that predict the occurrence of these steady solutions and, in certain simple situations, analytic ex- pressions are found for their shapes. Here this predictive understanding is extended to displacements in horizontal annuli. For brevity the reader is referred to the in- troductory sections of the above papers for a review of related work and industrial motivation. In primary cementing, see [5], successive casings or liners are fitted within one another, thus decreasing the size of the new hole that can be drilled. For this reason, longer extended reach wells tend to be of smaller diameter and also tend to be cemented increasingly in laminar flow regimes. This latter is due to smaller annular gap sizes and to the increased risk of high frictional pressures fracturing the surrounding formation at high flow rates. Thus, insofar as modeling the fluid mechanics of the displacement process, long horizontal wells are the process situation that offers the widest range of validity of the Hele-Shaw approach. In the absence of a density difference between fluids, displacements in horizontal annuli are the same as in vertical annuli. However, drilling fluids are typically 100−600kg/m3 lighter than cement slurries, and a chemically compatible spacer fluid designed to have intermediate density and rheological properties typically separates these two fluids. Therefore density differences are inevitable in the cementing of horizontal wells. A number of questions arise that are both of fluid mechanic interest and of practical industrial relevance. 1. Which dimensionless groups govern whether or not the flow will become strat- ified? In the worst case, stratification could allow the displacing fluid to com- pletely by-pass the displaced fluid rendering the displacement ineffective. 40 2.1. Introduction Figure 2.2: Schematic of mechanically stable separated flow configurations in a horizontal displacement: a) heavy fluid displacing; b) light fluid displacing. 2. With a density difference, is it possible to have steady, traveling wave, displace- ment fronts? What characteristics of these fronts can be predicted: shape, stability, etc? 3. What are the effects of an increased flow rate on a horizontal displacement? Finally, we might suspect that the typical shapes of displacement fronts are different in horizontal annuli than in vertical annuli. In vertical annuli, eccentricity tends to promote dispersive spreading on the wide side of the annulus, where the fluid velocities are larger. This effect is countered by positive viscosity and density differences, eventually resulting in steady traveling wave displacement fronts. In eccentric horizontal annuli the heavy steel casing is likely to lie to the lower side of the borehole. It appears that we may have two types of mechanically stable separated flow develop, see Fig. 2.2. First, suppose that there is a positive density difference strong enough to force the heavier displacing fluid azimuthally around to the narrow side of the annulus. In this case a heavy finger of displacing fluid advances along the bottom of the annulus ahead of the mean flow and we have the configuration of Fig. 2.2a. Second, suppose we have no density difference or that the displacing fluid is lighter than the displaced fluid. In this case we may expect a “classical ” form of separated flow to develop in which the displacing fluid advances ahead along the wide upper side of the annulus. These two basic types of flows can also be found experimentally. As an illustration, Fig. 2.3 shows an example result from recent laboratory experiments, with slumping along the bottom. The aim of our chapter is to answer some of the above questions and to develop a qualitative and quantitative understanding of these displacement flows. Our main 41 2.1. Introduction (b) (a) Figure 2.3: An example of slumping along the bottom of a horizontal concentric annulus: fluid 1 (ρ̂1 = 1398kg/m 3, κ̂1 = 6.24Pa.s, n1 = 1), coloured black with dye, displaces fluid 2 (ρ̂2 = 1363kg/m 3, κ̂2 = 0.76Pa.s, n2 = 1), moving from right to left at mean speed 1.6mm/s. Each subfigure shows a side view and an view from above the annulus, (centrebody is grey); image (b) is taken after image (a), after 116.5ml of fluid 1 has been pumped. tool for this is a simplified version of the Hele-Shaw displacement model from [1], restricted to a well section of constant inclination, eccentricity and radii, which we outline in §2.2. We then explore some typical results of numerical simulations conducted with the 2D model, (in §2.3). The 2D simulations are relatively time consuming as it is necessary to resolve potential regions of unyielded fluid in the annulus, which is achieved using an iterative approach; see [8]. Another difficulty with the 2D model is that these flows are governed by 11 dimensionless parameters, describing the rheology, density/buoyancy and well geometry, which makes broad parametric study of the flow regimes near-impossible. A last problem is that many of the displacement flows extend along the annulus axially a significant distance, which of course increases the size of numerical domain (and hence computational cost) needed for 2D simulation. The above factors motivate our development in §2.4 of a lubrication/thin-film model. Elongation of the interface is found when |b̃| ≫ 1, where b̃ denotes the ratio 42 2.2. Hele-Shaw modelling of cementing displacements of buoyancy and viscous forces. In this limit, an evolution equation for the propaga- tion of the volumetric interface location may be derived, which is a 1D quasi-linear advection-diffusion equation. This equation may be solved relatively quickly but may also be analysed. We show that it is possible to predict the conditions un- der which a steady traveling wave solution can be found for the lubrication limit. Necessary and sufficient conditions for the existence of steady state traveling wave solutions are given and we survey the (parametrically simpler) set of steady inter- face shapes and stability maps. The chapter closes with a summary of results and discussion of consequences in §2.5. 2.2 Hele-Shaw modelling of cementing displacements We consider the model developed by [1], simplified so that the annulus is locally uniform in the axial direction, i.e. over a length-scale that is long in comparison to the azimuthal scale, and only 2 fluids will be considered. The model and all parameters used in this chapter are dimensionless, with dimensional parameters only referred to to highlight the physical meaning of certain parameters. In this case, any dimensional quantity will be distinguished by the “hat” symbol, e.g. ρ̂1. A summary of the scaling used is given just before §2.2.1 below. Further details may be found in [1] or [7]. The axis is at a fixed angle of inclination, β ≈ π/2, to the vertical, see Fig. 2.4. As in the model used in [1, 7] averaging across the narrow annular gap eliminates radial variations, thus effectively unwrapping the annulus into a Hele-Shaw cell, see Fig. 2.4. The dimensionless spatial coordinates are (φ, ξ) ∈ (0, 1) × (0, Z). φ is the azimuthal coordinate with φ = 0 denoting the wide side of the annulus and φ = 1 the narrow (lower) side. The flow is assumed symmetric about φ = 0, and thus only half the annulus is considered. This assumption implies that the narrow side of the annulus is always lying on the low side of the well. The ξ coordinate measures axial depth along the annulus in the direction of the mean flow, see Fig. 2.4. Z denotes the length of the section of well to be cemented. Z ≫ 1, since the length-scale used for scaling the equations is one half the mean circumference, typically ≈ 0.5m, whereas lengths of cemented sections are in the 100 − 1000m range. The half-gap 43 2.2. Hele-Shaw modelling of cementing displacements Figure 2.4: Geometry of the narrow eccentric annulus, mapped to the Hele-Shaw cell geometry: a) near-horizontal annulus; b) annular gap geometry and eccentricity; c) unwrapped half-annular & half-gap Hele-Shaw geometry. width H varies only with φ and is defined by H(φ) = 1 + e cos πφ, (2.1) which is a narrow gap approximation. Here e ∈ [0, 1) is the annulus eccentricity, (see Fig. 2.2); e = 0 corresponds to a concentric annulus, e = 1 implies contact between casing and outer wall, on the narrow side of the annulus. The annulus is initially full of fluid 2, which is displaced by fluid 1. Although the fluids are miscible, the timescale for molecular diffusion to have significant effects is typically much longer than other process timescales. Thus the interface is essentially advected, and we consider two standard ways to model its motion. First, we consider a fluid concentration formulation in which c denotes the concentration of fluid 1, assumed not to vary across the annular gap width. The gap-averaged concentration equation (minus any diffusion) is simply: ∂ ∂t [Hc] + ∂ ∂φ [Hv c] + ∂ ∂ξ [Hw c] = 0, (2.2) 44 2.2. Hele-Shaw modelling of cementing displacements where the averaged velocity components in the (φ, ξ)-directions are (v,w). A second approach, that is more convenient later for our analysis, is to track the interface via a gap-averaged kinematic equation; this approach is described in §2.2.1. The fluids are incompressible so that (v,w) are written in terms of a stream function Ψ where ∂Ψ ∂φ = Hw, ∂Ψ ∂ξ = −Hv. (2.3) Boundary conditions for (2.2) and (2.3) are symmetry of concentration at φ = 0, 1, and specification of any inflowing fluid concentrations at the ends, ξ = 0, Z, i.e. c = 0 or c = 1, accordingly. The underlying fluids are modeled as Herschel-Bulkley fluids: m = 1/n is the inverse power law index, τY is the yield stress and κ is the consistency. For a Newtonian fluid m = 1 and τY = 0. We assume that the fluids are shear-thinning throughout, (m ≥ 1), since this is usual with oil field fluids. In the concentration approach, these fluid properties will depend on c, as will the fluid density, ρ. The stream function is found from the elliptic field equation ∇ · [S + f ] = 0, (2.4) that is derived by cross-differentiating the relevant Darcy-law to eliminate the pres- sure. This is the standard procedure in a Hele-Shaw flow law if one wishes to work with the stream function and not the pressure. For the stream function equation (2.4), boundary conditions are Ψ(0, ξ, t) = 0, Ψ(1, ξ, t) = 1, (2.5) ∂Ψ ∂ξ (φ,Z, t) = 0, ∂Ψ ∂ξ (φ, 0, t) = 0. (2.6) The relevant closure law is derived in [1], from where: S = [ χ(|∇Ψ|) + τY /H |∇Ψ| ] ∇Ψ ⇐⇒ |S| > τY H , (2.7) |∇Ψ| = 0 ⇐⇒ |S| ≤ τY H , (2.8) 45 2.2. Hele-Shaw modelling of cementing displacements where the function χ = χ(|∇Ψ|;H, τY , κ,m) is defined implicitly from the relation: |∇Ψ| = 0 χ ≤ 0, Hm+2 κm(m+ 2) χm+1 ( χ+ τY /H) 2 [ χ+ (m+ 2)τY (m+ 1)H ] χ > 0. (2.9) Underlying (2.7)-(2.9) is a model of the flow of a Herschel-Bulkley fluid along a plane channel in the direction of the modified pressure gradient. 2 If |S| ≤ τY /H then |∇Ψ| = 0, and there is no fluid flow. Physically the modified pressure gradient |S| is not strong enough to overcome the yield stress in that section of the annulus. At such points S is bounded but is indeterminate. At those points where |∇Ψ| > 0 the fluid is flowing. All buoyancy terms have been collected together in (2.4) in the term f : f = ( ρ(c) cos β St∗ , ρ(c) sin β sinπφ St∗ ) , (2.10) where St∗ is the global Stokes number for the flow, defined by: St∗ = µ̂∗ŵ∗ ρ̂∗ĝ(d̂∗)2 . To recover dimensional quantities, note that the axial and azimuthal veloc- ities have been scaled with the mean flow velocity, ŵ∗, lengths with the half- circumference, πr̂∗a, (here r̂ ∗ a is the mean radius). A rate of strain scale is obtained by dividing ŵ∗ by the half-gap width, d̂∗ = (r̂o − r̂i)/2, and this used with the constitutive laws to derive a viscosity scale, µ̂∗, defined as the maximal effective vis- 2For a complete model derivation the reader is referred to [1]. In outline, we use classical Hele- Shaw scaling arguments to simplify the momentum equations to the following reduced shear flow: ∂p ∂φ = ∂ ∂y τφy+gφ and ∂p ∂ξ = ∂ ∂y τξy+gξ, where y denotes a normalised annular gap coordinate, (v, w) are the azimuthal and axial velocity components, p is the pressure and (gφ, gξ) denotes the scaled gravitational acceleration vector. The fluid rheology is assumed to be that of a Herschel-Bulkley fluid: τij = (κγ̇ n−1 + τY /γ̇)γ̇ij ⇔ τ > τY ; γ̇ = 0 ⇔ τ ≤ τY . After scaling only the leading order shear stress and strain rate components, (φy and ξy), are considered in the rheological law. The rheological parameters are defined as functions of the base fluid parameters and fluid concentration. It follows that the gap-averaged velocity flows in the direction of the modified pressure gradient, (− ∂p ∂φ + gφ,− ∂p ∂ξ + gξ) and by orienting in the direction of this flow we may compute the relation (2.9) between mean flow rate and modified pressure gradient by solving a 1D flow problem across the annular gap. Cross-differentiation to eliminate the pressure results in (2.4). For example, for a Newtonian fluid we recover the familiar S = 3κ∇Ψ/H3, and (2.4) is simply a linear elliptic equation driven by the buoyancy gradient and imposed flow rate. 46 2.2. Hele-Shaw modelling of cementing displacements cosity at the mean shear rate. The pressure gradient balances with the leading order shear-stress scale, as always in a Hele-Shaw flow. Finally, ρ̂∗ is the density scale, (defined as the larger of the two unmixed fluid densities), and ĝ the gravitational acceleration. 2.2.1 Interface tracking The formulation with (2.2) requires that the interface is interpreted as a level line of the concentration field, e.g. c̄(φ, ξ, t) = 0.5, and also that closure laws be specified for the mixture fluid properties, i.e. as functions of c̄. The domain is divided into two fluid domains: Ω1 for the displacing (lower) fluid 1, and Ω2 for the displaced (upper) fluid 2, in each of which (2.4) is replaced by: ∇ · S1 = 0, (φ, ξ) ∈ Ω1, (2.11) ∇ · S2 = 0, (φ, ξ) ∈ Ω2, (2.12) with S1 and S2 defined as in (2.7-2.8), with properties ρ1, τ1,Y , κ1, m1 in fluid 1 and ρ2, τ2,Y , κ2, m2 in fluid 2. The interface is denoted by φ = φi(ξ, t), and satisfies the kinematic condition: ∂φi ∂t + w̄ ∂φi ∂ξ = v̄. (2.13) The leading order continuity conditions at the interface are that the stream func- tion Ψ and the pressure p are continuous across the interface. Assuming sufficient regularity of the interface, the former condition assures that the normal velocity, the derivative of Ψ along the interface, is well defined at the interface. Pressure continuity is expressed in terms of defining the jump in Sk across the interface: [( Sk,ξ ∂φi ∂ξ − Sk,φ ) + ( ρk sin β sinπφ St∗ ∂φi ∂ξ − ρk cos β St∗ )]2 1 = 0. (2.14) Equation (2.14) also defines the jump in normal derivative of Ψ across the interface. 47 2.3. Horizontal displacement flows 2.3 Horizontal displacement flows We now start to build our intuition of the types of displacement phenomena that we are likely to find in laminar horizontal well cementing flows. Treated in full generality, we see that displacements are governed by the following 11 dimensionless parameters: rheology τk,Y , κk, nk, k = 1, 2; density/buoyancy ρk, k = 1, 2, St ∗; well geometry, e and β. For the concentration-advection formulation, all 11 parameters are needed since intermediate values of c̄ are computed. In addition closure laws are needed for the rheology and density at intermediate concentrations. Although cumbersome in terms of the large parameter space, the concentration-advection formulation is the easiest to deal with computationally, and we present a number of results from this model below. For analytical results the interface tracking formulation is more convenient. In this formulation, ρk, k = 1, 2, and St ∗ may be replaced by a buoyancy number b̃: b̃ = ρ2 − ρ1 St∗ , (2.15) as this is the only combination in which these parameters appear in the jump condi- tions at the interface. For strictly horizontal wells β = π/2, and thus we are reduced to the 6 dimensionless rheological parameters, plus e and b̃. Here we present a number of results computed using the concentration-advection formulation. The numerical method used is that described in detail in [8]. At a given time step, assuming the concentrations and stream function are known, we differ- entiate the stream function to give the velocity field and advance the concentration one timestep. For this we use the FCT scheme, see [11], on a staggered rectangular mesh. The concentrations are approximated at the cell centres and the stream func- tion at the corners. Thus partial derivatives of the stream function along the edges give the required velocity components. Given concentrations at the new timestep, the stream function is computed from (2.4), which is solved using the augmented La- grangian approach with the Uzawa-like algorithm. Computations presented below have been solved with mesh spacings ∆φ = 1/40 and δξ = 1/20. As a test problem, we consider a concentric annulus displacement between two 48 2.3. Horizontal displacement flows Newtonian fluids, for which there is a steady traveling wave solution: ξ − t = − b̃ cosπφ 3π[κ2 − κ1] + constant, see [7]. Results of the computation for κ1 = 1, κ2 = 0.25, ρ1 = 1, ρ2 = 0.9, St∗ = 0.05, (=⇒ b̃ = −2), are shown in Fig. 2.5. In this and subsequent plots, the φ axis points vertically downwards, with the wide side at the top of each figure and the narrow side at the bottom, as would be the case in a horizontal well. The direction of flow is always from left to right, in the direction of increasing ξ. 0 5 10 15 20 1 0 φ 0 5 10 15 20 1 0 φ 16 16.5 17 17.5 18 18.5 19 19.5 20 1 0 ξ φ Figure 2.5: Test problem, comparing 2D computations with the analytical solution from [7], with parameters: e = 0, β = π/2, τY,k = 0, nk = 1, κ1 = 1, κ2 = 0.25, ρ1 = 1, ρ2 = 0.9, St ∗ = 0.05. Top figure: the contour c̄(φ, ξ, t) = 0.5 plotted at time intervals ∆t = 0.5. Middle figure: the analytical interface position, (steady state solution from [7]), plotted at time intervals ∆t = 0.5. Bottom figure: comparison of numerical and analytical solutions. The top figure shows the contour c̄(φ, ξ, t) = 0.5 plotted at successive time intervals ∆t = 0.5. We start the simulation with the annulus full of fluid 2 for ξ ≥ 1, and fluid 1 in ξ < 1. We see a quick evolution to a steadily advancing profile. The middle figure plots the above analytical solution at the same time intervals. 49 2.3. Horizontal displacement flows The lower figure overlays the computed and analytical profiles in the latter part of the annulus. We see that the agreement is good, with similar shapes and just a constant offset. If the same example is repeated with no density difference, a steady planar displacement front is found. The addition of the positive density difference evidently results in slumping of the heavy fluid towards the lower side of the well. It is interesting to note that with a significant buoyancy effect (i.e. |b̃| = O(1)) complete stratification does not occur in this case. In practice it is possible for the displacing fluid (fluid 1) to be either heavier or lighter than the displaced fluid. A heavier and more viscous fluid 1 might result when using a classical design for near-vertical wells, as positive density and viscosity gradients help a vertical displacement, see [9]. Regardless of the wellbore inclina- tion, at some point during a displacement there is always likely to be a positive density gradient between fluid stages, simply because the density of cement slurries is typically higher than that of drilling muds. For all positive density differences (heavy displacing light), we intuitively expect to see slumping of the displacing fluid to the bottom of the well, as in the test case above. Although less common, there are also situations in which the displacing fluid may be lighter, and either more or less viscous. For example, if a wash is pumped ahead of the cement slurry it will generally be lighter and less viscous than the drilling mud. Other than this, various lightweight spacer fluids have been used in displacing and these may be lighter but more viscous than the drilling mud. In either case, the expected effect of the density difference is to promote stratification, with the light fluid moving ahead along the top of the annulus. 2.3.1 Heavy displacing fluids, b̃ < 0 We first consider the more common case, where the displacing fluid is heavier than the displaced, (b̃ < 0). In Fig. 2.6 we present results of the same displacement as in Fig. 2.5, but with a mildly eccentric annulus, e = 0.1. The effect of eccentricity is to promote faster flow on the wide side of the annulus, φ = 0, and in the absence of any density difference the interface is usually found to advance further along the wide side than the narrow side. Thus, we can see that there is a competition between the effects of eccentricity and buoyancy, promoting interfacial advances on the wide 50 2.3. Horizontal displacement flows and narrow side respectively. Fig. 2.6a plots the evolution of the (interface) contour, c̄(φ, ξ, t) = 0.5, at time intervals ∆t = 0.5. The eccentricity-buoyancy competition is evident: the interface initially slumps then appears to recover slightly. The axial extent of the steady state interface is ≈ 0.4, whereas for e = 0 it was ≈ 0.5, i.e. eccentricity has reduced the extent of slumping along the wellbore axis. c) b) a) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ ξ φ 15 15.2 15.4 15.6 15.8 16 16.2 16.4 16.6 16.8 17 1 0 ξ − t φ 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 1 0 Φ = 0 Φ = 0 Figure 2.6: Newtonian displacement at St∗ = 0.05 with positive buoyancy gra- dient in mildly eccentric annulus. Parameters: e = 0.1, β = π/2, τY,k = 0, nk = 1, κ1 = 1, κ2 = 0.25, ρ1 = 1, ρ2 = 0.9. a) The contour c̄(φ, ξ, t) = 0.5 plotted at time intervals ∆t = 0.5. b) The moving frame stream function Φ(φ, ξ, t) at t = 15.5, with the contour c̄(φ, ξ, t) = 0.5 (heavy line); contours are Φ = 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06. c) Results from a moving frame compu- tation at t = 40: the moving frame stream function Φ(φ, ξ− t, t) at t = 40, with the contour c̄(φ, ξ−t, t) = 0.5 (heavy line); contours are Φ = 0.01, 0.02, 0.03, 0.04, 0.05. Fig. 2.6b plots contours of the moving frame stream function Φ(φ, ξ, t): Φ(φ, ξ, t) = Ψ(φ, ξ, t) − ∫ φ 0 H(φ̃) dφ̃, (2.16) 51 2.3. Horizontal displacement flows b) a) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ ξ − t φ 3 3.5 4 4.5 5 5.5 6 6.5 7 1 0 Figure 2.7: Newtonian displacement at St∗ = 0.01 with positive buoyancy gradient in mildly eccentric annulus. Parameters: e = 0.1, β = π/2, τY,k = 0, nk = 1, κ1 = 1, κ2 = 0.25, ρ1 = 1, ρ2 = 0.9. a) The contour c̄(φ, ξ, t) = 0.5 plotted at time intervals ∆t = 0.5. b) Results from a moving frame computation at t = 40: the moving frame stream function Φ(φ, ξ − t, t) at t = 30, with the contour c̄(φ, ξ − t, t) = 0.5 (heavy line); contours are Φ = 0.01, 0.02, 0.03, 0.04, 0.05, 0.06. at t = 15.5. The stream function Φ(φ, ξ, t) simply subtracts the effects of a uniform mean axial velocity field from Ψ(φ, ξ, t), and is appropriate for identifying steady traveling wave solutions. Fig. 2.6b shows an O(1) recirculation zone within the Φ(φ, ξ, t) = 0 contour. The interface contour, c̄(φ, ξ, t) = 0.5 (heavy line), is overlaid on the figure and lies within this zone. This large recirculatory zone is responsible for the slightly transitory behaviour in Fig. 2.6a. The two (near vertical) Φ = 0 contours divide the domain into three regions. In the two end regions Φ(φ, ξ, t) > 0 and the circulation is clockwise. The displacing fluid moves from wide to narrow side and the displaced fluid moves from narrow to wide side. In the central cell we have Φ(φ, ξ, t) < 0 and the circulation is counter-clockwise, but relatively slow. This O(1) central recirculation zone is not steady, but contracts slowly during the displacement, eventually becoming a single line between the wide and narrow sides. To avoid computations with a very large mesh, long-time computations are carried out in a steadily moving frame of reference. Fig. 2.6c shows the contours of Φ and the interface contour , c̄ = 0.5, in the moving frame at t = 40. In this frame of reference the streamlines and concentration are steady. We observe that buoyancy effects are dominant with the steady finger advanced along the bottom of the annulus. 52 2.3. Horizontal displacement flows b) a) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ ξ − t φ 0 2 4 6 8 10 12 14 1 0 Figure 2.8: As Fig. 2.7 with St∗ = 0.002. Two further Newtonian fluid computations are presented in Figs. 2.7 & 2.8, in which the Stokes number is decreased to St∗ = 0.01 and St∗ = 0.002 respectively. The parameters are otherwise the same as in Fig. 2.6. The decrease in St∗ physically corresponds to a reduction in dimensional flow rate, which reduces viscous stresses. In the Hele-Shaw model only viscous and buoyant forces are present and balance each other. Thus, as expected, reducing St∗ amplifies the relative effects of buoyancy and a slumping flow occurs. Figs. 2.7a & 2.8a show the evolution of the interface contour c̄ = 0.5. Figs. 2.7b & 2.8b show the streamlines Φ(φ, ξ − t, t), computed in the moving frame of reference, after a time t = 40. What is perhaps surprising is that even with this massively increased buoyancy effect, the flow settles down to a steady state, i.e. complete stratification does not occur. However, the steady state does stretch along the annulus axis a greater distance as St∗ is decreased. In the absence of an imposed flow rate, we would expect that the flow would stratify. Two natural questions arise. First, if we keep increasing the relative effects of buoyancy, in an example such as that in Figs. 2.6-2.8, will we always have steady state traveling wave interface solution albeit extending further along the annulus, or is there a point at which the fluids stratify? We investigate this limit and the associated question in §2.4. Secondly, we ask whether in all cases buoyancy will be dominant, or can the slumping motion be arrested via annular eccentricity and/or by rheological effects? The answer here is yes. In Fig. 2.9 we show a non-Newtonian displacement at 53 2.3. Horizontal displacement flows a) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ c) d) b) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ ξ φ 14 14.5 15 15.5 16 16.5 17 17.5 18 1 0 ξ − t φ 3 3.5 4 4.5 5 5.5 6 6.5 7 1 0 Φ = 0 Φ = 0 Figure 2.9: Non-Newtonian displacement with positive buoyancy gradient in an eccentric annulus. Parameters: e = 0.2, β = π/2, τY,1 = 0, τY,2 = 1, nk = 1, κ1 = 1, κ2 = 0.25, ρ1 = 1, ρ2 = 0.9. a) St ∗ = 0.05, contour c̄(φ, ξ, t) = 0.5 plotted at time intervals ∆t = 0.5. b) St∗ = 0.01, contour c̄(φ, ξ, t) = 0.5 plotted at time intervals ∆t = 0.5. c) St∗ = 0.01, the moving frame stream function Φ(φ, ξ, t) at t = 15.5; contours are Φ = 0, 0.03, 0.06, 0.09, 0.12, 0.15. d) St∗ = 0.01, results from a moving frame computation at t = 40: the moving frame stream function Φ(φ, ξ − t, t) at t = 30, with the contour c̄(φ, ξ − t, t) = 0.5 (heavy line); contours are Φ = 0.03, 0.06, 0.09, 0.12, 0.15. 54 2.3. Horizontal displacement flows a) 0 1 2 3 4 5 6 7 8 9 10 1 0 ξ φ c) d) b) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ ξ φ 14 14.5 15 15.5 16 16.5 17 17.5 18 1 0 ξ − t φ 3 3.5 4 4.5 5 5.5 6 6.5 7 1 0 Φ = 0 Φ = 0 Figure 2.10: As Fig. 2.9 but with eccentricity, e = 0.4. eccentricity e = 0.2, with a yield stress in the displaced fluid, τY,2 = 1, (all other parameters remaining as before). At St∗ = 0.05, Fig. 2.9a shows the evolution of the interfacial contour c̄ = 0.5. With the added eccentricity and yield stress, flow is retarded on the narrow side and we instead observe that the finger advances ahead on the upper wider side of the annulus. On reducing the Stokes number to St∗ = 0.01, buoyancy again dominates. Fig. 2.9b shows the evolution of the interfacial contour c̄ = 0.5. There is again slow transitory behaviour as eccentricity and buoyancy compete, characterised by a transitory recirculatory zone, (see Fig. 2.9c), before eventually the steady state profile is attained, see Fig. 2.9d. Fig. 2.10 shows results of the same simulation but at e = 0.4. The unsteady fingering at St∗ = 0.05 is now obvious, (Fig. 2.10a), but again decreasing St∗ results in a steady traveling wave, Figs. 2.10b-d. 55 2.3. Horizontal displacement flows 2.3.2 Light displacing fluids, b̃ > 0 Turning now to the situation where the displacing fluid is less dense, we first examine Newtonian displacements with a mildly eccentric annulus. The primary difference with the previous subsection is that now both buoyancy and eccentricity promote stratification along the top (wide) side of the annulus. Only rheological differences may be used to counter the tendency to finger along the wide side of the annulus. In the absence of any positive viscosity difference, we must expect that the interface will advance rapidly along the wide side, becoming quickly parallel with the annulus axis. An example of this is shown in Fig. 2.11, with parameters κ1 = 0.25, κ2 = 1, a 10% density difference, e = 0.2 and St∗ = 0.05, (Newtonian fluids). c) b) a) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ ξ φ 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ 0 2 4 6 8 10 12 14 16 18 20 1 0 Φ = 0.3 Φ = 0.3 Figure 2.11: Newtonian displacement with negative buoyancy gradient in eccentric annulus. Parameters: St∗ = 0.05, e = 0.2, β = π/2, τY,k = 0, nk = 1, κ1 = 0.25, κ2 = 1, ρ1 = 0.9, ρ2 = 1. a) Contour c̄(φ, ξ, t) = 0.5 plotted at time intervals ∆t = 0.5. b) The moving frame stream function Φ(φ, ξ, t) at t = 10; contours are Φ = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3. c) as figure b), but at t = 20. With the viscosity difference of Fig. 2.11 reversed, the effect is very different, see Fig. 2.12. The initial displacement advances along the wide side of the annulus, but does not succeed in fully stratifying. Over intermediate timescales there again 56 2.3. Horizontal displacement flows persists a zone of counter-clockwise circulation, Fig. 2.12b, which acts to partly steepen the interface and then contracts to a steady state profile, see Fig. 2.12c. It appears that this positive viscosity gradient (κ1 = 1, κ2 = 0.25) is sufficient to displace in a steady state at e = 0.2. b) c) a) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ ξ φ 14 14.5 15 15.5 16 16.5 17 17.5 18 1 0 ξ − t φ 3 3.5 4 4.5 5 5.5 6 6.5 7 1 0 Φ = 0 Φ = 0 Figure 2.12: Newtonian displacement with negative buoyancy gradient in eccentric annulus. Parameters: St∗ = 0.05, e = 0.2, β = π/2, τY,k = 0, nk = 1, κ1 = 1, κ2 = 0.25, ρ1 = 0.9, ρ2 = 1. a) Contours c̄(φ, ξ, t) = 0.5 plotted at time intervals ∆t = 0.5. b) The moving frame stream function Φ(φ, ξ, t) at t = 15.5, contours Φ == 0, 0.03, 0.06, 0.09, 0.12, 0.15. c) Results from a moving frame computation at t = 40: the moving frame stream function Φ(φ, ξ−t, t) at t = 30, with the contour c̄(φ, ξ − t, t) = 0.5 (heavy line); contours are Φ = 0.03, 0.06, 0.09, 0.12, 0.15. With non-Newtonian fluids little is changed fundamentally: eccentricity and buoyancy are destabilising but for certain rheological combinations it is possible to achieve a steady state displacement. One example of this is shown below in Fig. 2.13. The results are qualitatively similar to that of Fig. 2.12. An initial acceleration along the wide side is followed by a transitional motion of the interface via a slowly contracting weak counter-clockwise rotation and finally the profile converges to a steady state at longer times. 57 2.3. Horizontal displacement flows b) c) a) 0 2 4 6 8 10 12 14 16 18 20 1 0 ξ φ ξ φ 14 14.5 15 15.5 16 16.5 17 17.5 18 1 0 ξ − t φ 3 3.5 4 4.5 5 5.5 6 6.5 7 1 0 Φ = 0 Φ = 0 Figure 2.13: Non-Newtonian displacement with negative buoyancy gradient in ec- centric annulus. As Fig. 2.12 but with τY,1 = 1, κ2 = 0.05. 2.3.3 Summary We have run a number of computations for both positive and negative buoyancy differences, from which a number of conclusions may be drawn. First, the same solution types are found regardless of the sign of the buoyancy difference, (steady front or unsteady finger), and the form of approach to the steady state is also identical. Where the density difference is positive (heavy fluids displace light fluids) buoyant slumping and eccentricity compete, whereas for negative density differences they act together. However, in both cases significant rheological differences between the fluids can determine whether or not the displacement is steady. From the 2D simulations that we have run, we are in a position to at least postulate what sufficient conditions for a steady state should be. In a horizontal well displacement, under the Hele-Shaw assumptions of our model, buoyancy forces act only azimuthally. These contribute to the axial momentum balance only through the slope of the interface. As the interface elongates along the well the contribution of the azimuthal buoyancy forces is diminished. The only exception to this would be at the wide and narrow side of the annulus, where the interface may run perpendicular 58 2.3. Horizontal displacement flows to the annulus axis. However, the buoyancy force acts azimuthally with magnitude proportional to sinπφ, and thus vanishes on the wide and narrow sides. Therefore, we might postulate that a sufficient condition for the 2D simulation to have a steady state displacement solution is that the corresponding iso-density displacement has a steady state displacement. However, results such as those in Fig. 2.10 contradict this statement. Of key practical interest is to determine whether or not the displacement is steady. Our two-dimensional computations are not ideal for this purpose for a num- ber of reasons. Some of these reasons are computational, as discussed below in §2.3.4, and relate to computational times required to exhaustively explore a 10-dimensional parameter space. More generically, when stable steady structures occur transient solutions only converge at long times, (often also related to the initial conditions). In marginal cases separating stable and unstable states, convergence times become infinite and transient simulations are frequently quite simply inappropriate to de- termine marginal conditions with any accuracy. This motivates the development of a semi-analytical approach, below in §2.4. In this approach we first reduce the problem to 1D, so that computational simulations are indeed feasible. Second, we reduce the parametric dependency of the model from 10 dimensionless parameters to 8. Finally, this simplified approach allows analytical determination of conditions for steady state displacements. 2.3.4 Computational issues Finally, we mention some problems with the 2D simulation method. First, some spreading of the interface does occur. We start the computations with a jump in c̄ across the interface, but a small diffuse region emerges immediately. Although we have considered c̄ = 0.5 as the interface, this is arbitrary and any other level set could be used, giving rise to some uncertainty in computed interface position. Partly the spreading is due to numerical diffusion, partly due to dispersion and partly due to interpolation errors. The FCT scheme is reasonable in regard to numerical diffusion but cannot eliminate this effect. We see that in the moving frame of reference there are significant counter-current secondary flows close to the interface. These act to amplify any spreading via dispersion, but the size of the effect is hard to quantify. 59 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 Similarly, it is necessary to provide closure functions for the physical properties of the fluids. Here we have followed [1] and used linear interpolation. Effects of using different closure laws are unknown. An obvious solution to these numerical issues would be to use an interface tracking method directly. We do this below for a simplified model, but have not tried to use it with the 2D model. Secondly, although by comparison to other problems routinely solved in fluid mechanics, the computational load for a 2D problem may seem low, it is not so when yield stress fluids are involved (which is a situation of practical interest). The Uzawa algorithm used to solve the augmented Lagrangian problem converges relatively slowly and this must be repeated at each timestep. Run times are still several hours long on a 2007 desktop PC. This is sufficient to dissuade us from extensive explorations of parameter space. Linked in to the above point, we may observe that many of the flows we have computed do have interfaces that extend significantly along the annulus axis. Evidently, these elongated interfaces lead to larger computational domains and thus slower computational times. The approach that we explore below in §2.4 is an analytical effort to circumvent some of these difficulties. It might also be possible to look at improved computational techniques, such as adaptive meshing (note that much of the flow away from the front is 1D), or implementing a parallel algorithm. So far we have not investigated these approaches. 2.4 A lubrication model for gravitational spreading with |b̃| ≫ 1 We have seen that both positive and negative b̃ can lead to both steady trav- eling wave displacements and to unsteady displacements. Here we attempt to provide a quantitative understanding of these phenomena through analysis of the lubrication/thin-film limit, |b̃| ≫ 1, which we derive below. 2.4.1 Lubrication model derivation When |b̃| ≫ 1 the buoyancy tends to dominate and causes the interface to elongate in the axial direction. We adopt the interface tracking formulation of §2.2.1, and introduce a small parameter ε = 1/|b̃| ≪ 1. We will consider only mechanically 60 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 stable configurations, i.e. if b̃ < 0 it is assumed that the displacing fluid elongates along the bottom of the annulus (narrow side), and if b̃ > 0 the elongation is along the top of the annulus (wide side). Only the model with b̃ < 0 is derived; b̃ > 0 is stated afterwards. We assume that |b̃| represents the length-scale in the axial direction over which the streamlines, interface and modified pressure gradient field are all pseudo-parallel, and make the following assumptions. (i) Streamlines are pseudo-parallel to the annulus axis: so that the main velocity component is in the ξ-direction, and hence |∂Ψ ∂ξ | = O(ε), |∂Ψ ∂φ | = O(1). (2.17) (ii) Interface is pseudo-parallel to the annulus axis: i.e. is highly elongated in the ξ-direction. Denoting the interface by φ = φi(ξ, t) this translates into |∂φi ∂ξ | = O(ε). (2.18) (iii) Modified pressure gradient field: Sk has its main component in the φ-direction, driving the flow axially: |Sk,ξ| = O(ε), |Sk,φ| = O(1). (2.19) Note that this follows directly from (i) and the definition of S in the case that the fluids are yielded. Otherwise this is an additional assumption. With the above assumptions, we re-scale axial length and time variables by: z = εξ; t̃ = εt. (2.20) For the velocity and pressure we set W = w; V = εv; P = εp (2.21) and write Ψ(φ, z) for the stream function as before. Note that V = −Ψz = O(1) 61 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 following this re-scaling. After some manipulation, see [9], the kinematic equation for the interface φ = φi(z, t̃) becomes: ∂ ∂t̃ [∫ φi(z,t̃) 0 H(φ) dφ ] + ∂ ∂z [Ψ(φ, z, t̃)|φ=φi(z,t̃)] = 0. (2.22) The first term above is the time derivative of the volumetric interface position, Φi(z, t̃): Φi(z, t̃) = ∫ φi(z,t̃) 0 H(φ) dφ = φi(z, t̃) + e π sinπφi(z, t̃), (2.23) i.e. Φi(z, t̃) represents the volume fraction of fluid 1 at depth z. The relationship between φi and Φi is one-to-one, so we may write φi = φi(Φi). The components of Sk are given in terms of the pressure and gravitational acceleration by Sk,φ = −∂p ∂ξ − ρk cosβ St∗ , Sk,ξ = ∂p ∂φ − ρk sin β sinπφ St∗ . The assumption that |Sk,ξ| = O(ε) implies that 1 ε [ ∂P ∂φ − ρk sinβ sinπφ ρ1 − ρ2 ] = O(ε), and we shall assume that β ∼ π/2 +O(ε), so that to O(ε2): P (φ, z, t̃) = P (0, z, t̃) + ρ2[1− cosπφ] π(ρ1 − ρ2) , φ ∈ [0, φi], P (0, z, t̃) + ρ2[1− ρ1ρ2 cos πφ] π(ρ1 − ρ2) + cos πφi π , φ ∈ (φi, 1]. (2.24) We write cos β = αε with α = O(1) so that S1,φ = −∂P ∂z (0, z, t̃)− ρ1α ρ1 − ρ2 + ∂φi ∂z sinπφi, S2,φ = −∂P ∂z (0, z, t̃)− ρ2α ρ1 − ρ2 . 62 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 Note that S1,φ = S2,φ − (α − φi,z sinπφi) and that each Sk,φ is independent of φ. In order to find the stream function at each (z, t̃), we need only find A(z, t̃) = S2,φ, which we do by using the boundary condition at φ = 1 and the closure laws for the individual fluid phases: 1 = Ψ(1, z, t̃) = ∫ φi 0 ∂Ψ ∂φ ∣∣∣∣ k=2 dφ+ ∫ 1 φi ∂Ψ ∂φ ∣∣∣∣ k=1 dφ. (2.25) To evaluate the integrands, from (2.9) we have in fluid 1 ∂Ψ ∂φ ∣∣∣∣ k=1 = sgn(A− b) 0, |A− b| ≤ τ1,YH , Hm1+2(|A− b| − τ1,Y H )m1+1 κm11 (m1 + 2)|A − b|2 [ |A− b|+ τ1,Y H(m1 + 1) ] , |A− b| > τ1,YH , (2.26) where b = α− φi,z sinπφi, and in fluid 2 ∂Ψ ∂φ ∣∣∣∣ k=2 = sgn(A) 0, |A| ≤ τ2,YH , Hm2+2(|A| − τ2,Y H )m2+1 κm22 (m2 + 2)|A|2 [ |A|+ τ2,Y H(m2 + 1 ) ] |A| > τ2,YH . (2.27) Both (2.26) & (2.27) are monotone with respect to A, and from this we can de- duce that (2.25) has a unique solution. For the general case, this solution must be determined numerically. Having found A, we may integrate to find Ψ. Replacing φi with φi(Φi), we see that Ψ depends on (z, t̃) only via Φi(z, t̃) and b, which contains Φi(z, t̃) and Φi,z(z, t̃). The value of the stream function at the interface is denoted by q(Φi, b) = Ψ(φ, φi, b)|φ=φi(z,t̃) : b = α− φi,z sinπφi, φi = φi(Φi(z, t̃)). (2.28) Using this notation, (2.22) yields an evolution equation for the propagation of the 63 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 volumetric interface position, Φi(z, t̃): ∂Φi ∂t̃ + ∂ ∂z q(Φi, b) = 0. (2.29) 2.4.2 Diffusive nature of (2.29) Equation (2.29), with fixed constant b, is exactly the lubrication displacement flow conservation law studied in [9]. This is a 1st order, quasilinear, partial differential equation that admits a variety of different qualitative behaviours, including shocks and traveling wave solutions. Note that b = α − φi,z sinπφi and hence (2.29) is second order and can be written as ∂Φi ∂t̃ + ( ∂q ∂Φi + ∂q ∂b ∂b ∂Φi ) ∂Φi ∂z = −∂q ∂b ∂b ∂Φi,z ∂2Φi ∂z2 , (2.30) where ∂b ∂Φi,z = − sinπφi 1 + e cos πφi < 0, Φi ∈ (0, 1). Therefore (2.29) will be diffusive only if q is non-decreasing for all b. This is the case, as might be expected physically, i.e. gravity should act to spread the interface along the annulus. We expect that the effects of diffusion will be to smooth out sharp changes in interface shape, e.g. the shocks found for the 1st order model in [9], but it is interesting that on wide and narrow sides of the annulus the diffusion terms vanish. Lemma 1. q(Φi, b) is non-decreasing for all b. Proof. For brevity, let us write u(φ) = Ψ (φ, z, t) , u′(φ) = ∂Ψ∂φ , and observe that u(φi(Φi)) = q(Φi, b). We proceed formally by assuming that u ∈ V, where V is a suitable function space for what follows.3 Note that u(φ) satisfies boundary condi- tions u(0) = 0 and u(1) = 1. We denote by V0 the subspace of V containing all func- tions w(φ) ∈ V that satisfy homogeneous boundary conditions: w(0) = w(1) = 0. 3Following [7] we can expect that u ∈ W 1,p([0, 1]) with p = min{1 + n1, 1 + n2}. However, the results in [7] are for a general 2D flow, whereas here the base flows solved to give the lubrication model are 1D. It is thus likely that the solution is piecewise C1 (or smoother) and hence H1([0, 1]) could be used. 64 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 Then if u is the stream function solution, any other (test) solution v ∈ V can be written as v = u+w, for some w ∈ V0. For v ∈ V, we multiply the modified pressure gradients Sk,φ by (u − v)′ and integrate over φ ∈ [0, 1]. Firstly, by using the definition of A and b we arrive at: ∫ φi 0 (v − u)′S2,φ dφ + ∫ 1 φi (v − u)′S1,φ dφ = ∫ φi 0 (v − u)′A dφ+ ∫ 1 φi (v − u)′(A− b) dφ = b (v(φi)− u(φi)) (2.31) Considering first fluid 2, if fluid 2 is yielded we can use equation (2.7) directly to give ∫ φi 0 (v − u)′S2,φ dφ = ∫ φi 0 ( χ2 + τ2,y H |u′| ) (v − u)′u′ dφ ≤ ∫ φi 0 χ2 |u′| (v − u) ′u′ dφ + ∫ φi 0 τ2,y H (∣∣v′∣∣− ∣∣u′∣∣) dφ (2.32) with the last step following from the Cauchy-Schwarz inequality. Note that when unyielded we have that |S2,φ| ≤ τ2,y/H, u′ = 0 and χ2(|u′|) → 0 like |u′|1/(1+1/n2), (see [7]). Therefore, inequality (2.32) is valid also if fluid 2 is unyielded. We may write a similar inequality for fluid 1: ∫ 1 φi (v − u)′S1,φ dφ ≤ ∫ 1 φi χ1 |u′| (v − u) ′u′ dφ+ ∫ 1 φi τ1,y H (∣∣v′∣∣− ∣∣u′∣∣) dφ, 65 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 and combining with (2.31): b (v(φi)− u(φi)) ≤ ∫ φi 0 χ2 |u′|(v − u) ′u′ dφ+ ∫ φi 0 τ2,y H (∣∣v′∣∣− ∣∣u′∣∣) dφ + ∫ 1 φi χ1 |u′|(v − u) ′u′ dφ+ ∫ 1 φi τ1,y H (∣∣v′∣∣− ∣∣u′∣∣) dφ. (2.33) Now suppose that, for fixed Φi, the solution u1 corresponds to b = b1 and the solution u2 corresponds to b = b2. It is apparent that u2 is a test solution for u1 and vice versa. Inserting each into (2.33) and summing gives us: (b1 − b2) (u2(φi)− u1(φi)) ≤∫ φi 0 χ2 |u′1| (u2 − u1)′u′1 dφ + ∫ φi 0 χ2 |u′2| (u1 − u2)′u′2 dφ + ∫ 1 φi χ1 |u′1| (u2 − u1)′u′1 dφ + ∫ 1 φi χ1 |u′2| (u1 − u2)′u′2 dφ ≤ 0. (2.34) The last statement follows from the general properties of convex functionals, (see e.g. proposition 5.4 on page 25 of [2]). In [7] it is shown that if χk is defined implicitly by equation (2.9) then the functional Fk(x) = ∫ |x|2 0 χk(s 1/2) 2s1/2 ds (2.35) is Gateaux-differentiable and strictly convex, i.e. from this it follows that ∫ φi 0 χ2 |u′1| (u2 − u1)′u′1 dφ ≤ ∫ φi 0 F2(u2)− F2(u1) dφ, and the various expressions on the right-hand side of (2.34) cancel out. Therefore we have demonstrated that: (b2 − b1) (u2(φi)− u1(φi)) = (b2 − b1) (q(Φi, b2)− q(Φi, b1)) ≥ 0, (2.36) and q (Φi, b) is non-decreasing with respect to b. 66 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 2.4.3 Steady traveling wave solutions Probably the single most important feature of (2.29) to predict is whether or not steady traveling wave solutions can be found, since in their absence it is likely that the flow will stratify, implying the ineffectiveness of the displacement. Since hori- zontal well sections are constructed to lie along the reservoir, failure of a horizontal cemented section has a large impact on well productivity. We show that it is in fact possible to predict the conditions under which a steady traveling wave solution can be found. It is more convenient to work in a moving frame of reference and (due to the scaling adopted) a steady traveling wave solution that occupies the entire annulus will move with unit speed. Thus, we write x = z − t and seek a steady profile Φi = Φi(x). For heavy fluids displacing light fluids, (b̃ < 0), we expect an interface shape as illustrated schematically in Fig. 2.2a, so that Φi(x) is non-decreasing. Writing instead x = x(Φi) for the steady state, we seek x(Φi) such that the following physical conditions are satisfied: x′(0) = 0, x′′(0) > 0 x′(1) = 0, x′′(1) < 0 x′(Φi) > 0 Φi ∈ (0, 1). (2.37) Substituting for Φi = Φi(x) in the evolution equation (2.29) we find d dx [q ( Φi, b(Φi,Φ ′ i) )− Φi] = 0, and hence q ( Φi, b(Φi,Φ ′ i) )− Φi = 0, (2.38) must be satisfied by Φi(x), (or x(Φi)). Since we have that b = α− φi,z sinπφi = α− sinπφi(Φi) H(φi(Φi)) Φ′i(x) = α− sinπφi(Φi) H(φi(Φi))x′(Φi) , we see that equation (2.38) is in fact an algebraic equation for the derivative, x′(Φi), of x(Φi) with respect to Φi. 67 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 Evidently, we can integrate x′(Φi) to give the steady state shape provided that for all Φi ∈ [0, 1] there is a b for which (2.38) is satisfied and for which the conditions (2.37) on the physical shape of the interface are also satisfied. Since q(Φi, b) is continuous with respect to both arguments, this suffices to give a steady state shape. Theorem 1. The condition that q(Φi, α) ≥ Φi for all Φi ∈ [0, 1] (2.39) is a necessary and sufficient condition for the existence of a steady state traveling wave solution satisfying the conditions (2.37). Proof. First recall from lemma 1 that q(Φi, b) increases monotonically with respect to b. Therefore, if Φi ∈ (0, 1), from (2.37) we observe that α > b. Additionally, as Φi → 0 b→ α− π (1 + e)2x′′(0) +O(Φi) < α and as Φi → 1 b→ α+ π (1− e)2x′′(1) +O(Φi) < α. Therefore b < α for all Φi ∈ [0, 1] and lemma 1 implies that q(Φi, b) ≤ q(Φi, α) for all Φi ∈ [0, 1]. Thus, condition (2.39) is certainly a necessary condition to find a solution of (2.38). For sufficiency, let q(Φi, b) be given by (2.25)-(2.28) and suppose that (2.37) and (2.39) are satisfied. We shall now find b < α such that q(Φi, b) = Φi. For fixed Φi ∈ [0, 1] define gL(Ã) = ∫ φi 0 ∂Ψ ∂φ |2 dφ and gH(Ã) = ∫ 1 φi ∂Ψ ∂φ |1 dφ, i.e. we use (2.27) with A replaced by Ã for gL(Ã), and (2.26) with A − b replaced by Ã for gH(Ã). We note that ∂Ψ ∂φ in (2.26) & (2.27) increases continuously and monotonically with the modified pressure gradient Ã, approaching ±∞ as Ã→ ±∞. This monotonicity transfers to gH and gL. Using the monotonicity of gH and gL, 68 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 for any fixed α it follows that there is an Ã such that gL(Ã) + gH(Ã− α) = 1. This is the procedure for defining q(Φi, α) = gL(Ã) and, since (2.39) is satisfied, gL(Ã) ≥ Φi and gH(Ã− α) ≤ 1− Φi. Again from the monotonicity of gH and gL, it follows that we can find AL < Ã and AH > Ã− α such that gL(AL) = Φi, and gH(AH) = 1− Φi. (2.40) Now let b be defined by b = AL −AH < α. Therefore gL(AL) + gH(AL − b) = 1, and by setting gL(AL) = q(Φi, b) we have a solution to (2.38). To understand the physical meaning of theorem 1, note that q(Φi, b) is the volu- metric flux through the upper layer of light fluid and q(Φi, α) is the volumetric flux through the upper layer of light fluid, neglecting the effects of the interface slope. Condition (2.38) defines the distribution of volumetric flux for which the interface speed is everywhere identically equal to the mean speed of the flow. As the slope of the interface is positive and q increases with b, we can deduce that q decreases as the slope is increased. When condition (2.39) is satisfied, this implies that the flux is too large to satisfy (2.38). However, by increasing the slope of the interface from zero we can find a b for which (2.38) is satisfied. This constructive procedure gives the slope of the steady state shape, as well as the conditions for there to be a steady state. Conditions on the consistency ratio The procedure in the above theorem can be made constructive and yields conditions on the consistency ratio, κ1/κ2, that need to be satisfied for a steady state to exist. 69 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 First note that for a given modified pressure gradient Ã, (2.26) & (2.27) can be rewritten as ∂Ψ ∂φ (Ã) = sgn(Ã) |Ã|m κm(m+ 2) F+(B,m,H(φ)), (2.41) where F+(B,m,H) = 0, 1 ≤ B/H, Hm+2(1−B/H)m+1(1 +m+B/H) m+ 1 1 > B/H. (2.42) B = τY /|Ã| and in fluid k we use m = mk, τY = τY,k, κ = κk. The only dependency on φ enters through the function F+(B,m,H). For given Φi define AL and AH as Am2L κm22 (m2 + 2) IL = Φi and Am1H κm11 (m1 + 2) IH = 1− Φi, (2.43) where IL = ∫ φi 0 F+(B2,m2,H(φ)) dφ and IH = ∫ 1 φi F+(B1,m1,H(φ)) dφ, (2.44) with B1 = τY,1 AL and B2 = τY,2 AH . Thus if we set b(Φi) = ( (m2 + 2)Φi IL )n2 κ2 − ( (m1 + 2)(1 − Φi) IH )n1 κ1, (2.45) we have that q(Φi, b) = Φi. Using the definition of b from (2.28) x′(Φi) = sinπφi(Φi) H(Φi) (α− b(Φi)) , (2.46) and we note that to be physically plausible we require x′(Φi) > 0 for Φi ∈ (0, 1), see the conditions (2.37). This implies that κ1 κ2 > max Φi∈[0,1] FHL(Φi; e, n1, n2, B1, B2, α/κ2), (2.47) 70 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 where FHL = ( (m2 + 2)Φi IL )n2 ( (m1 + 2)(1 − Φi) IH )n1 − ακ2 1( (m1 + 2)(1 − Φi) IH )n1 , (2.48) and recall that mk = 1/nk. When condition (2.47) is satisfied, the shape of the steady traveling wave can be found by integrating (2.46). 2.4.4 Light displacing fluids, b̃ > 0 When the displacing fluid is lighter than the displaced fluid we expect the config- uration illustrated schematically in Fig. 2.2b to result, and again it is possible to derive a lubrication displacement model. The volumetric interface position, Φi(z, t̃), again evolves according to: ∂Φi ∂t̃ + ∂ ∂z q(Φi, b) = 0, (2.49) where now q(Φi, b) = ∫ φi 0 ∂Ψ ∂φ ∣∣∣∣ k=1 dφ, and this is computed in a similar way. The φ components of S in each fluid are constants satisfying S2,φ = S1,φ − b. Defining S1,φ = A, equations (2.26) and (2.27) are replaced by ∂Ψ ∂φ ∣∣∣∣ k=1 = sgn(A− b) 0, |A| ≤ τ1,YH , Hm1+2(|A| − τ1,Y H )m1+1 κm11 (m1 + 2)|A|2 [ |A|+ τ1,Y H(m1 + 1) ] , |A| > τ1,YH , (2.50) 71 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 in fluid 1, and in fluid 2 by ∂Ψ ∂φ ∣∣∣∣ k=2 = sgn(A− b) 0, |A− b| ≤ τ2,YH , Hm2+2(|A− b| − τ2,Y H )m2+1 κm22 (m2 + 2)|A − b|2 [ |A− b|+ τ2,Y H(m2 + 1) ] , |A− b| > τ2,YH , (2.51) where b is still given by b = α− φi,z sinπφi. The modified pressure gradient A is computed from 1 = Ψ(1, z, t̃) = ∫ φi 0 ∂Ψ ∂φ ∣∣∣∣ k=1 dφ+ ∫ 1 φi ∂Ψ ∂φ ∣∣∣∣ k=2 dφ. (2.52) Diffusive spreading and steady state profiles As before, we can show that q(Φi, b) is non-decreasing with b and that (2.49) is diffusive in nature. The methodology is similar to that used previously. The profile of the interface when the displacing fluid is lighter is expected to satisfy x′(0) = 0, x′′(0) < 0 x′(1) = 0, x′′(1) > 0 x′(Φi) < 0 Φi ∈ (0, 1), (2.53) which replaces (2.37). Under these assumptions, a necessary and sufficient condition for a steady state is that q(Φi, α) ≤ Φi. (2.54) This is satisfied if κ1 κ2 > max φi∈[0,1] FLH(φi; e, n1, n2, B1, B2, α/κ2), (2.55) 72 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 where FLH = ( (m2 + 2)(1 −Φi) IH )n2 ( (m1 + 2)Φi IL )n1 + ακ2 1( (m1 + 2)Φi IL )n1 . (2.56) In this case IL and IH are defined as IH = ∫ 1 φi F+(B2,m2,H(φ)) dφ (2.57) IL = ∫ φi 0 F+(B1,m1,H(φ)) dφ. (2.58) Note that in this case an increase in α requires an increase in the consistency ratio. 2.4.5 Parametric results Here we explore conditions (2.47) & (2.55) that dictate whether of not a steady travelling wave shape can be found. Note that the procedure leading to (2.47) & (2.55) is constructive and is fully explicit in the case that the fluids have no yield stress. When the fluids have a yield stress the same procedure can be followed, but since Bk depends on the modified pressure gradients this relationship is implicit. Newtonian fluids If the two fluids are Newtonian and the fluid 1 is heavier, the flux function q is qHL(Φi, b) = ( 3κ1 + b ∫ 1 φi H3(φ)dφ ) ∫ φi 0 H 3(φ)dφ 3κ2 ∫ 1 φi H3(φ)dφ+ 3κ1 ∫ φi 0 H 3(φ)dφ . (2.59) When the displacing fluid is lighter this expression becomes qLH(Φi, b) = ( 3κ2 + b ∫ 1 φi H3(φ)dφ ) ∫ φi 0 H 3(φ)dφ 3κ2 ∫ φi 0 H 3(φ)dφ+ 3κ1 ∫ 1 φi H3(φ)dφ . (2.60) The conditions required to have a steady state, (2.48) and (2.56), are plotted for different angles α in Fig. 2.14. Note that a positive angle helps the heavy-light 73 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 0 .4 1 2 3 4 e κ1/κ2 0.4 1 0 500 1000 e κ1/κ2 b)0 1 0 0.35 0.7 1.05 1.4 e κ1/κ2 a) Figure 2.14: Conditions required for 2 Newtonian fluids to have a steady state displacement: a) HL - heavy fluid displacing light fluid, (2.48); b) LH - light fluid displacing heavy fluid, (2.56). In each figure the curves are: α/κ2 = −0.6 △, − 0.3 ©, 0.0 , 0.3 ♦, 0.6 ▽. The consistency ratio must lie above the curve to have a steady traveling wave displacement. (HL) displacement, while it worsens the light-heavy (LH) displacement. In Fig. 2.15 the effects on the steady state shape of increasing the eccentricity at fixed consistency ratio are illustrated. It is interesting to note that increasing the eccentricity has a different effect in an HL displacement than an LH displacement. In the former the density difference promotes slumping along the bottom of the well and an increase in eccentricity actually counters this effect by constricting the area. Increasing eccentricity acts to elongate the interface for the LH displacement. The effects on the steady state shape of increasing the consistency ratio at fixed eccentricity are shown in Fig. 2.16 for e = 0.1. In this case the consistency ratio threshold for the HL displacement is κ1/κ2 = 0.8388, while for the the LH displace- ment it is κ1/κ2 = 1.2959. In both cases the consistency ratios explored are above the threshold values and we can observe that increasing the consistency ratio tends to reduce the extension of the interface along the annulus. As the threshold value is approached the steady states elongate towards infinity, see e.g. Fig. 2.16b. 74 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 0 0.5 1 1.5 2 0 1 Φi x(Φi)a) −0.6 −0.4 −0.2 0 0 1 Φi x(Φi)b) Figure 2.15: The effects on the steady state shape of increasing the eccentricity at fixed consistency ratio for 2 Newtonian fluids: a) HL displacement κ1/κ2 = 1; b) LH displacement κ1/κ2 = 2.02. Eccentricities: e = 0.05 ©, 0.1 , 0.15 ♦, 0.2 ▽; horizontal well α/κ2 = 0. Power law fluids We investigate the effects of shear-thinning on steady displacements only for dis- placements in strictly horizontal wells, α = 0. Figures 2.17 & 2.18 show the effects of varying power law indices on the threshold consistency ratio. Figure 2.17 fixes n1 = 0.5 and explores the effect of increasing eccentricity for different n2. For HL- displacements the threshold is either decreasing with e or has a minimum, (see also the Newtonian case in Fig. 2.14), i.e. a little eccentricity helps the displacement countering the effects of buoyancy. For LH-displacements the consistency ratio increases with e. Increasing n2, the power law index of the displaced fluid in both cases increases the consistency ratio threshold. Essentially the displaced fluid is becoming more viscous. In general, comparing Fig. 2.17a & b, the LH displacement requires a higher consistency ratio than the HL displacement in order to become steady. The results shown in Fig. 2.18 are qualitatively similar with respect to variations in e. Now, as n1 is increased, (for fixed n2), the consistency ratio threshold drops. Again this can simply be interpreted as an increase in viscosity ratio. The shape of the steady states is explored in Figs. 2.19 & 2.20. We fix n1 = 0.5, n2 = 1 and explore the effects of changing consistency ratio and eccentricity. 75 2.4. A lubrication model for gravitational spreading with |b̃| ≫ 1 0 0.1 0.2 0.3 0 1 Φi x(Φi)a) −4 −3 −2 −1 0 0 1 Φi x(Φi)b) Figure 2.16: The effects on the steady state shape of increasing the consistency ratio at fixed eccentricity, e = 0.1, for 2 Newtonian fluids: a) HL displacement; b) LH displacement. Consistency ratios: κ1/κ2 = 1.35©, 1.4 , 1.45 ♦, 1.5 ▽; horizontal well, α/κ2 = 0. Essentially, the steady state shape becomes progressively elongated as the threshold is approached. The results for n1 = 1, n2 = 0.5 are analogous. Yield stress fluids For yield stress fluids we cannot get a closed form expression for a critical consistency ratio, as B1 and B2 depend on the modified pressure gradient. Instead, we follow the procedure of Theorem 1. For example, for a HL-displacement at each Φi we find AL(Φi) and AH(Φi) from (2.40). Provided that AL(Φi) − AH(Φi) < α for all Φi ∈ [0, 1] we can construct a steady state shape. The critical condition is when max Φi∈[0,1] {AL(Φi)−AH(Φi)} = α. (2.61) An analogous procedure is followed for LH displacements. Two example computations of AL(Φi) and AH(Φi) are shown in Fig. 2.21. We may observe that in each case the functions AL(Φi) and AH(Φi) are monotone with respect to Φi. For the parameters of Fig. 2.21a it is possible to compute a steady state shape, but for the parameters of Fig. 2.21b it is not. Evidently, computing the threshold in this way is not very efficient. However, Fig. 2.21 shows that the critical 76 2.5. Discussion and summary 0 0.5 0 4 8 e κ1/κ2 0.5 1 8 1300 e κ1/κ2 b)0 1 0.7 1 1.3 1.6 e κ1/κ2 a) Figure 2.17: Conditions required to have a steady state displacement at fixed n1 = 0.5, for 2 power law fluids: a) HL displacement, (2.48); b) LH displacement, (2.56). In each figure the curves are: n2 = 0.2 △, 0.4 ©, 0.6 , 0.8 ♦, 1.0 ▽. The consistency ratio must lie above the curve to have a steady displacement. conditions may be met at an intermediate Φi, and therefore there does not appear to be any shortcut. Fig. 2.22 shows the conditions on τY,1 & τY,2 required in order to have a steady state displacement, for parameters that are fixed otherwise. It is interesting to note that the marginal curves in the (τY,1, τY,2)-plane are nearly linear. Example steady state shapes are shown in Fig. 2.23. 2.5 Discussion and summary We have presented an analysis of displacement flows in horizontal eccentric annuli, in the Hele-Shaw limit, using 2 complementary models. Using a fully 2D model and computational simulation we have studied a range of displacements involving either HL or LH fluid combinations. The role of density differences appears slightly different depending on whether the displacement is HL or LH. In the former case, density differences and eccentricity compete, so that there is often a threshold in the (St∗, e)-plane delineating steady and unsteady displacements. In a LH displacement both eccentricity and buoyancy act to accentuate unsteadiness and stratification. However, in both cases we have seen that these flows may either stratify or displace 77 2.5. Discussion and summary 0 0.6 0 2 4 6 8 e κ1/κ2 0.6 1 8 150 e κ1/κ2 b)0 1 0.4 0.8 1.2 e κ1/κ2 a) Figure 2.18: Conditions required to have a steady state displacement at fixed n2 = 0.5, for 2 power law fluids: a) HL displacement, (2.48); b) LH displacement, (2.56). In each figure the curves are: n1 = 0.2 △, 0.4 ©, 0.6 , 0.8 ♦, 1.0 ▽. The consistency ratio must lie above the curve to have a steady displacement. in steady state. For long computational domains, the 2D model is less effective and more time consuming. This has motivated derivation of a 1D lubrication-style displacement model. Interestingly, in this model we are able to decouple the effects of buoyancy from those of eccentricity and fluid rheology. For this simplified model we were able to give necessary and sufficient conditions for there to be a steady state displacement, and have explored variations in the results with the main dimensionless parameters. In this model, fluid rheology and eccentricity alone determine whether or not there may be a steady state. The buoyancy force translates simply into an axial length- scale for the slumped propagating steady interface, i.e. a larger density difference gives a longer axial extension to the steady state. We have presented the results for Newtonian and power law fluids in terms of the ratio of consistencies of the two fluids: κ1/κ2. This must exceed a certain critical threshold in order for there to be a steady state. Above the threshold, steady state shapes vary significantly with the other rheological parameters. The variation is fairly intuitive, i.e. as the displacing fluid is made more viscous the axial extent of the steady states decreases. Increasing eccentricity generally results in longer steady states. As the critical threshold is approached, the axial extent of the steady state grows towards infinity. 78 2.5. Discussion and summary −5 −4 −3 −2 −1 0 0 1 Φi x(Φi)b) 0 0.1 0.2 0.3 0.4 0 1 Φi x(Φi)a) Figure 2.19: The effects on the steady state shape of increasing the consistency ratio at fixed eccentricity and power law indices, e = 0.1, n1 = 0.5, n2 = 1: a) HL displacement: threshold κ1/κ2 = 1.2637; b) LH displacement: threshold κ1/κ2 = 1.9154. Consistency ratios: κ1/κ2 = 1.95 ©, 2 , 2.05 ♦, 2.1 ▽; horizontal well, α/κ2 = 0. 2.5.1 Steady state stability A question that is unanswered concerns the stability of the steady states. In the 2D simulations, we only observe steady state solutions that are stable, since they are identified via a transient computation. For the lubrication model we have a semi-analytic expression for the steady state shape, but a priori there is no reason to expect that these shapes are stable. Nevertheless, numerical simulations with the lubrication model do indicate that the steady states are stable. An example of one such computation is shown below in Fig. 2.24. We have not been able to develop any simple stability analysis that predicts the stability of the steady states, but it appears that they are generally stable and perhaps globally so. This appears somewhat strange at first glance, in particular when one considers the moving frame streamline pattern, Φ(x, φ), for a steady state displacement, as shown schematically in Fig. 2.25a, (see also earlier results from the 2D simulation). The steady state is a streamline Φ(x, φ) = 0 that connects wide and narrow sides of the annulus. The Hele-Shaw formulation allows a discontinuity in tangential velocity across an interface, and we see that the steady state has fluids 1 and 2 moving counter-current on either side of it. Since the interface is advected by 79 2.5. Discussion and summary −0.8 −0.6 −0.4 −0.2 0 0 1 Φi x(Φi)b) 0 0.5 1 1.5 0 1 Φi x(Φi)a) Figure 2.20: The effects on the steady state shape of increasing the eccentric- ity at fixed consistency ratio and power law indices, n1 = 0.5, n2 = 1: a) HL displacement, κ1/κ2 = 1.6; b) LH displacement, κ1/κ2 = 2.97. Eccentricities: e = 0.05 ©, 0.1 , 0.15 ♦, 0.2 ▽; horizontal well α/κ2 = 0. the velocity field, this configuration intuitively appears to be unstable to any small perturbation of the interface. However, the above conclusion is based on the false assumption that the stream- line field is not also perturbed by the interface perturbation, i.e. the transient inter- face is advected by the velocity field defined by the moving frame streamfunction: Φ = Φ(x, φ, t,Φi(x, t)). From the 2D simulations, we have seen that the approach to the steady state is characterised by the existence of a slowly counter-rotating zone about the interface, see Fig. 2.25b, which contracts as the steady state is ap- proached. With the streamlines and interface as in Fig. 2.25b, we can see that the interface will remain within the circulatory region, with the two endpoints approach- ing points A and B, which are stable, and the interface being suspended between these two points. The steady state streamline field is the limiting case in which such a (stabilising) recirculation zone collapses to a line. The above appears to be the mechanism by which steady states are stable. This type of stability is likely to be nonlinear, but it is unclear if it is global. 80 2.5. Discussion and summary a) 0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 Φi A H ,A L b) 0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 Φi A H ,A L Figure 2.21: Examples of AL(Φi) and AH(Φi) for parameters τY,1 = 0.4, κ1 = 0.1, n1 = 0.5, τY,2 = 0, κ2 = 1, n2 = 1, α = 0: a) e = 0.1, steady state; b) e = 0.3, no steady state. In each figure AH(Φi) corresponds to the broken line. 2.5.2 Industrial implications From the industrial perspective, the results of the chapter are slightly counter- intuitive but have largely positive implications. They also fill a large gap in industrial understanding of these flows, where there is essentially no published analysis of the displacement mechanics. The 3 key contributions of the work are as follows. 1. Eccentricity and density differences play the dominant roles in vertical annulus displacements. It has long been thought that in horizontal annuli density differences must be detrimental to a good displacement. We see that this is not entirely the case. In fact, (at least in the lubrication limit), whether or not there is steady state displacement in a horizontal well does not appear to be dictated by the density of the fluids! Rheology plays the dominant role in this question, regardless of the density difference. 2. On the other hand, density difference does affect the length of the steady state displacement profile, scaling linearly with density difference (divided by Stokes number). This scaling is evident in results from the 2D computations, (compare e.g. Figs. 2.6-2.8), and is implicit in the scaling adopted for the lubrication model. For 2 Newtonian fluids the Stokes number captures all the effects of pump flow rate, i.e. increasing Q̂ reduces St∗ like 1/Q̂. Thus, density 81 2.5. Discussion and summary 0 0.5 1 1.5 0 2 4 6 τY,2 τY,1 b) 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 τY,2 τY,1 a) Figure 2.22: Conditions required to have a steady state displacement at fixed: n1 = 0.5, n2 = 0.8, e = 0.2: a) HL displacement (2.48); b) LH displacement:(2.56). In each figure the curves are: κ1/κ2 = 0.5©, 1.0 , 1.5 ♦, 2.0▽. The yield stress τY,1 must lie above the curve to have a steady displacement. Horizontal displacement α/κ2 = 0. difference and flow rate act counter to one another in determining the length of the displacement profile. For non-Newtonian fluid displacements, the flow rate affects the dimensionless rheological parameters so that it is harder to assess effects of flow rate. 3. Criteria for the existence of a steady state displacement have been developed and presented as either a critical consistency ratio, or other thresholds. It is relatively simple/quick to compute these criteria and this could easily be computed as part of a design process. Given the apparent stability of these solutions, these are obviously of industrial interest. Apart from the semi-analytical expressions given, which need minimal computing to evaluate, it would be of use to give some clear “rules” that may be followed in industrial design. In this context, the construction leading to Fig. 2.21 is useful. For the HL displacement, AL(Φi) represents the pressure gradient required to move the light fluid at mean speed through a duct consisting only of light fluid in the section φ ∈ [0,Φi]. Similarly, AH(Φi) represents the pressure gradient required to move the heavy fluid at mean speed through a duct consisting only of heavy fluid in the section φ ∈ [1−Φi, 1]. Conditions for which there can be no steady state are 82 2.5. Discussion and summary −0.8 −0.6 −0.4 −0.2 0 0 1 Φi x(Φi)b) 0 0.2 0.4 0.6 0.8 0 1 Φi x(Φi)a) Figure 2.23: The effects on the steady state shape of increasing the yield stress of the displacing fluid. n1 = 0.5, n2 = 0.8, e = 0.2, τY,2 = 0.5: a) HL displacement,α/κ2 = 1.5 (2.48); b) LH displacement,α/κ2 = 3.0 (2.56). In each figure the curves are: τY,1 = 0.0 ©, 0.25 , 0.5 ♦, 0.75 ▽. Horizontal displacement α/κ2 = 0 given (for α = 0) by when AL(Φi) and AH(Φi) intersect. Sufficient conditions for this may be stated in terms of single fluid hydraulic quantities as follows. Theorem 2. For a displacement in a horizontal well, there will be no steady state displacement if either of the following 2 sets of conditions are satisfied. 1. ∆PL,W < ∆PH and ∆PL > ∆PH,N ; 2. ∆PL,W > ∆PH and ∆PL < ∆PH,N . Here ∆PL,W denotes the frictional pressure drop generated when pumping the light (L) fluid through a concentric annulus of the same gap width as the wide (W) side at the imposed mean velocity; ∆PH denotes the frictional pressure drop generated when pumping the heavy (H) fluid through the eccentric annulus as the mean imposed speed; ∆PL denotes the frictional pressure drop generated when pumping the light (L) fluid through the eccentric annulus as the mean imposed speed; ∆PH,N denotes the frictional pressure drop generated when pumping the heavy (H) fluid through a concentric annulus of the same gap width as the narrow (N) side at the imposed mean velocity. Finally, when we do not have a steady state displacement, or when the displace- ment front is steady but elongated, the flow streamlines are approximately pseudo- 83 2.5. Discussion and summary 0 0.5 1 1.5 2 2.5 0 1 x ) i( x ,t ) b) 0 0.5 1 1.5 2 2.5 0 1 x ) i( x ,t ) a) Figure 2.24: Examples of convergence to the steady state profile for the lubrication displacement model. Parameters are: e = 0.2, τY, 1 = 0, τY,2 = 0, n1 = 0.8, n2 = 0.7, α/κ2 = 0. a) HL - displacement, κ1/κ2 = 0.8; b) LH - displacement, κ1/κ2 = 1.8. Both examples have converged after approximately t = 6. In the figures, + is the steady state profile, · the initial condition and the solid lines are transient profiles, evenly spaced in time. Figure 2.25: a) Schematic of the streamlines for a steady state; b) Schematic of the streamlines when close to the steady state. For both figures interface is shown schematically as the thick broken line. 84 2.5. Discussion and summary parallel. The stability of this type of parallel flow, within the Hele-Shaw context, has been recently studied in [4, 3]. The results are too complex to summarise, but it is found that many such parallel flows are linearly unstable. Therefore, although potentially problematic from the industrial perspective, instability and subsequent mixing of the fluids azimuthally may make even these displacements acceptable, i.e. the mixing may prevent development of a long finger on the wide side. Equally, we have not considered situations in which a finger develops that is mechanically un- stable, (i.e. heavy fluid over light fluid). Although possible in our Hele-Shaw model, such configurations are likely to destabilise; (again see [4, 3] for instability of the parallel flow). 85 2.6. Bibliography 2.6 Bibliography [1] S. Bittleston, J. Ferguson, and I. Frigaard. Mud removal and cement placement during primary cementing of an oil well. Journal of Engineering Mathematics, 43:229–253, 2002. [2] I. Ekeland and R. Témam. Convex analysis and variational problems. SIAM, 1976. [3] M. Moyers-González and I. Frigaard. Kinematic instabilities in two-layer ec- centric annular flows, part 1: Newtonian fluids. Journal of Engineering Math- ematics, 62:130–131, 2008. [4] M. Moyers-González and I. Frigaard. Kinematic instabilities in two-layer ec- centric annular flows, part 2: shear thinning and yield stress effects. Journal of Engineering Mathematics, DOI 10.1007/s10665-008-9260-0, 2008. [5] E. Nelson. Well cementing. Schlumberger Educational Services, 1990. [6] M. Payne. Recent advances and emerging technologies for extended reach drilling. S.P.E., 1995. [7] S. Pelipenko and I. Frigaard. On steady state displacements in primary ce- menting of an oil well. Journal of Engineering Mathematics, 48:1–26, 2004. [8] S. Pelipenko and I. Frigaard. Two-dimensional computational simulation of ec- centric annular cementing displacements. IMA Journal of Applied Mathematics, 64:557–583, 2004. [9] S. Pelipenko and I. Frigaard. Visco-plastic fluid displacements in near - vertical narrow eccentric annuli: prediction of travelling wave solutions and interfacial instability. Journal of Fluid Mechanics, 520:343–377, 2004. [10] F. Sabins. Problems in cementing horizontal wells. Journal of Petroleum Tech- nologies, 42:398–400, 1990. [11] S. Zalesak. Fully multidimensional flux corrected transport algorithms for flu- ids. Journal of Computational Physics, 31:335, 1979. 86 Chapter 3 Displacement flows in horizontal, narrow, eccentric annuli with a moving inner cylinder4 3.1 Introduction There is a large scientific literature on flows in an annular geometry in which the walls are either stationary or moving. Some of the main problems studied include: instability and transition in Taylor-Couette type flows; rotational flows and inter- facial 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 appli- cations: in heat exchangers, food processing, fluidized beds, pulp screening, extru- sion, as well as in the domain of oil well construction which forms the motivation for this work. In this chapter we consider the displacement of one Newtonian fluid by another along a long narrow horizontal eccentric annulus. The fluids are considered to be miscible, but the large Péclet number limit is considered, where molecular diffusion acting on its own has an insignificant effect on the flow timescale. Due to the narrowness of the annulus, the classical Hele-Shaw approach is used. If the annulus were concentric and if the fluids were iso-density, this would lead to a displacement 4A version of this chapter has been accepted for publication. Carrasco-Teja, M. and Frigaard, I.A.(2009) Displacement flows in horizontal, narrow, eccentric annuli with a moving inner cylinder Physics of Fluids DOI:10.1063/1.3193712. 87 3.1. Introduction flow mathematically equivalent to a planar Hele-Shaw displacement, (except periodic in the azimuthal direction). In the miscible fluid setting, study of the stability of such displacement flows dates back to Muskat, [16], who dealt with the porous media setting and developed a stability criterion based on consideration of the mobility ratios of the 2 fluids. More detailed linear stability theories of such flows stem from the classical work of Saffman & Taylor, [25, 26]. Generally these have been focused at immiscible fluid displacements and prediction of morphological features of the interfacial instability, although there are also extensions to miscible displacements, e.g. [12, 18]. The reader is referred to [7] for an overview, or to the extensive bibliography5. Such a large part of the Hele-Shaw flow literature concerns viscous fingering that one almost forgets that locally stable interface configuration exist. For misci- ble displacements in planar geometries these correspond to stable two-dimensional fronts, that may degenerate into three-dimensional instabilities at higher flow rates and for adverse viscosity ratios, see e.g. [9, 8]. Two-dimensional studies of miscible displacement flows in plane channels at large Péclet number, Pe, have been carried out by [3, 23, 31]. As Pe→∞, these displacements mimic those of immiscible fluids with low surface tension. At very long time and length-scales we enter the Taylor dispersion regime, but on intermediate scales dispersion is only locally significant. For these regimes a Hele-Shaw approach is adopted as a means of approximating and understanding the bulk fluid motions, which are non-trivial in non-uniform geometries and with significant buoyancy effects. Narrow eccentric annuli are one of these non-uniform geometries. The displace- ment flows that we study occur during the process of primary cementing, in which a steel casing is inserted into a newly drilled oil well and cement is pumped around the outside of the casing, in the tight annular space between casing and rock formation. This space is occupied by drilling fluid which must be removed before the cement is placed, so as to ensure a good bond of the cement with the annular walls and a tight hydraulic seal. The process is described at length in [17]. In the past 20 years there has been a massive increase in the number of horizontal wells constructed world- wide, primarily to increase productivity by aligning the well with the reservoir. The 5A bibliography of over 500 papers, books and articles can be found at: http://www.maths.ox.ac.uk/∼howison/Hele-Shaw/. 88 3.1. Introduction early 1990’s saw a continual pushing of the horizontal extent of wells up to around 10km, see e.g. the detailed description in [19]. The 10km barrier was broken in a number of wells drilled at Wytch Farm, UK, around 2000. The limits of “extreme” extended reach wells are now being pushed into the 15−20km range, but such wells are unusual and do not necessarily bring productivity benefits proportional to their technical challenges. In the present day it is routinely feasible to construct wells with horizontal extensions in the 7− 10km range. Although many of the potential problems of cementing horizontal wells were identified some time ago, (see e.g. [24]), the industrial response has been largely through technological advances, rather than by developing understanding of physical fundamentals that may affect the process. In longer horizontal wells it is increasingly common to have laminar flows, as the high frictional pressures of turbulent flows can lead to fracturing of the surrounding formation. Equally, the steel casing tends to sit eccentrically in the drilled hole, due to its weight. Drilling fluids are usually non-Newtonian and often possess a yield stress, which leads to fluids getting stuck on the lower side of the annulus where the annular gap is smallest. To prevent this it is becoming common to slowly rotate and reciprocate the casing axially. The precise effects of casing movement is however unknown. Developing this understanding is the objective of our study. The underlying idea of using a Hele-Shaw/porous media approach to model these displacements dates back to [11, 10], but was put in the present formulation by Bittleston et al., [1]. The overall approach we take is to adapt the Hele-Shaw model from [1] to the case of casing motion. This model has been analysed in depth for vertical wells by Pelipenko & Frigaard, [20, 21, 22], and for horizontal wells in chapter 2. A time-dependent version of the Hele-Shaw part of the model was developed in [15] and used to investigate interfacial instabilities in [14, 13]. We review the state of knowledge from these fixed casing studies later, in §3.2.2. In this chapter we consider for the first time a moving casing, but restrict atten- tion to Newtonian fluids since we feel that insight into the effects of casing move- ment can be gained by first looking at simpler fluids. This decision is also partly motivated by practical difficulties of computing the non-Newtonian flows, and in particular the model closures in the Hele-Shaw approach. Similar flows do occur in other industrial settings, e.g. in screw extruders and in annular scraped surface heat 89 3.1. Introduction exchangers. However, modeling of these flows with non-Newtonian fluids typically involves assumptions which are not valid here, e.g. in [5] the velocity is primarily in the axial direction. Other than the work considered above, there are a limited number of compu- tational studies of annular displacement flows. Szabo & Hassager, [28, 29], have computed a 3D immiscible displacement flow between 2 Newtonian fluids using the arbitrary Lagrange-Euler formulation. The model shows reasonable agreement with a film-draining model derived for concentric annular displacements and also shows that the displacement efficiency drops significantly with annular eccentricity. 3D ap- proaches have also been taken by other authors, using general purpose CFD codes, e.g. [4, 30]. As with [28, 29], such studies have value in understanding details of the flow near the interface but are of limited use in understanding flows on a larger scale, which is the advantage of the Hele-Shaw approach. A hybrid 2D/3D approach has recently been adopted by [27], in which the Navier-Stokes equations are simplified by ignoring radial velocities and azimuthal pressure gradients, but the model is still resolved in 3D. The authors are able to include casing motion in this approach, but a justification of the assumptions and even details of the final simplified model are not given in [27]. An outline of the chapter is as follows. In §3.2 we present the Hele-Shaw model including casing motion. Section 3.3 investigates the possibility of finding steady traveling wave displacements in these geometries. We are able to show that if the interface is assumed to be close to flat (perpendicular to the axis), then we can find an analytical form for the steady state shape of the interface and the streamfunction, for both concentric and mildly eccentric annuli. Section 3.4 presents computed results that explore the principal effects of casing rotation and reciprocation on displacement. Casing rotation induces a phase shift in the azimuthal position of the interface, which results in the positioning of heavy fluid above light fluid at certain azimuthal positions. This configuration is vulnerable to buoyancy driven fingering, which is observed in the computed results. The instability remains apparently local and over longer time scales we frequently attain stable steady displacements with a diffuse interfacial region. In §3.5 we consider various aspects of stability and instability of these flows. The chapter closes with a brief discussion of the results and summary. 90 3.2. Model outline 3.2 Model outline We consider a laminar displacement flow in a uniform annular space between two cylinders of radii r̂o and r̂i, (r̂o > r̂i), in which the outer cylinder is fixed and the inner cylinder may slowly move, either axially or rotating about its axis. The axial casing speed is denoted ŵC and the angular velocity is Ω̂C . The annulus is assumed to be nearly horizontal, at an angle of inclination to the vertical of β ≈ π/2. The annulus may be eccentric, with eccentricity e ∈ [0, 1), and is narrow in the sense that: δ = r̂o − r̂i π(r̂o + r̂i) ≪ 1, i.e. δ is the aspect ratio of the mean annular gap to the circumference. Throughout the chapter we denote dimensional quantities with the “hat” symbol, ,̂ but shall work primarily with dimensionless variables. The dimensionless geometry of the process is illustrated schematically in Fig. 3.1. Figure 3.1: Schematic of the process geometry: a) primary cementing of a hori- zontal well; b) narrow eccentric annulus; c) annular cross section; d) unwrapped annulus/periodic Hele-Shaw cell. 91 3.2. Model outline The annulus is initially full of fluid 2 and this is displaced by fluid 1, which is pumped at constant flow rate, Q̂. This flow rate defines the mean axial velocity ŵ∗, via Q̂ = π(r̂2o − r̂2i )ŵ∗. The model derivation follows the usual pattern of Hele-Shaw or lubrication flow models. We exploit the narrowness of the gap by scaling azimuthal and axial velocities with ŵ∗; radial velocities with δŵ∗. Azimuthal and axial distances are scaled with 0.5π(r̂o + r̂i), whereas the radial distance is scaled with 0.5(r̂o − r̂i), and is measured from the mean radius of the annulus. The dimensionless azimuthal and axial coordinates, (φ, ξ) are illustrated in Fig. 3.1b: φ ∈ [−1, 1], measuring azimuthally from the top of the annulus, and ξ ∈ [0, Z], measuring axially in the direction of the imposed flow. The dimensionless radial coordinate, y, measures distance from the mean radius, and the annulus walls are at y = ∓H(φ), where H(φ) = 1 + ecosπφ, (3.1) as illustrated in Figs. 3.1c & d. The scaled azimuthal and axial velocities are (v,w), respectively, and the dimensionless velocity of the inner cylinder (or casing) is de- noted (vC , wC). We do not repeat the model derivation, which is adequately described in [1] and is anyway somewhat standard for Newtonian fluids. Having reduced the Navier- Stokes equations to a leading order shear flow, with pressure constant across the narrow annular gap, we average over the gap width, y ∈ [−H,H], to eliminate the radial velocity component and any y-dependence, and “unwrap” the narrow annulus into a 2D domain in the (φ, ξ)-plane, (Fig. 3.1d). We then introduce a stream function, Ψ, to represent the azimuthal and axial gap-averaged velocities, and finally we cross-differentiate to eliminate the pressure. This results in a single linear elliptical equation for the stream function, as follows: ∇ · (S + f) = 0, (3.2) where S = 3κ 2H3 ( ∂Ψ ∂φ −HwC , ∂Ψ ∂ξ +HvC) ) , (3.3) and f = ( ρ cos β St∗ , ρ sin β sinπφ St∗ ) . (3.4) 92 3.2. Model outline The dimensionless parameters in (3.3) & (3.4) are the fluid viscosity, κ, density, ρ, and the Stokes number of the mean flow, St∗, all of which we define and discuss below in §3.2.1. The boundary conditions for the stream function Ψ are Ψ(φ, ξ) = Ψ(φ+ 2, ξ) + 4, (3.5) ∂Ψ ∂ξ (φ, 0) = ∂Ψ ∂ξ (φ,Z) = −HvC ., (3.6) The first of these ensures that the mean axial velocity is equal to 1, as per the scaling. The second ensures that in the far-field the azimuthal flow is simply the Couette component. As usual, a 2D Hele-Shaw model may be expressed either in terms of the stream function or in terms of the pressure. For compatibility with early stud- ies, e.g. [1, 20, 2], where non-Newtonian fluids are considered and the pressure field may be indeterminate, we have chosen the stream function as our principal depen- dent variable. The stream function is related to the gap-averaged azimuthal and axial velocities (v̄, w̄) by: − 1 2H ∂Ψ ∂ξ = v̄ = 1 2H ∫ H −H vdy, 1 2H ∂Ψ ∂φ = w̄ = 1 2H ∫ H −H wdy, (3.7) so that ∂ ∂φ [Hv̄] + ∂ ∂ξ [Hw̄] = 0. In terms of the pressure S can be written as: S = ( −pξ − ρ cos β St∗ , pφ − ρ sinπφ sin β St∗ ) (3.8) The displacement part of the flow we model via a concentration field, again following [1]. The concentration of fluid 1 is denoted, c̄ ∈ [0, 1]. At leading order in the high Péclet number limit, the concentration is simply advected with the mean flow, i.e. molecular diffusion is neglected. ∂ ∂t [Hc̄] + ∂ ∂φ [Hv̄c̄] + ∂ ∂ξ [Hw̄c̄] = 0 (3.9) 93 3.2. Model outline The concentration is periodic in φ and is set equal to 1 at the inflow: c̄(φ+ 2, ξ) = c(φ, ξ) (3.10) c̄(φ, 0) = 1. (3.11) Thus, our model consists of two principal variables, Ψ and c̄, that satisfy (3.2) & (3.9), respectively. The concentration is coupled directly to the stream function, via the velocity field (v̄, w̄), and contains the only time evolution in the model. In (3.2) the coupling of Ψ with c̄ occurs through the physical properties of the fluid, i.e. κ and ρ, which will depend on the concentration c̄ and the properties of the pure fluid phases. 3.2.1 Dimensionless parameters We have introduced the following 9 dimensionless parameters in the model. • Eccentricity: e ∈ [0, 1), which is defined in Fig. 3.1c. • Angle of inclination: β, as illustrated in Fig. 3.1b. For all results here we set β = π/2. • Azimuthal casing speed: vC = 0.5π(r̂o + r̂i)Ω̂C/ŵ∗, and typically |vC | < 1. • Axial casing speed: wC = ŵC/ŵ∗, and typically |wC | < 1. • Fluid j viscosity: κj = κ̂j/maxk=1,2{κ̂k}, so that one of the viscosities is = 1 and the other is ≤ 1, representing the viscosity ratio. Here κ̂j is the dimensional viscosity of fluid j. • Fluid j density: ρj = ρ̂j/maxk=1,2{ρ̂k}, so that one of the densities is = 1 and the other is ≤ 1, representing the density ratio. Here ρ̂j is the dimensional density of fluid j. • Stokes number: St∗ = [4maxk=1,2{κ̂k}ŵ∗]/[maxk=1,2{ρ̂k}ĝ(r̂o − r̂i)2], where ĝ is the gravitational acceleration. Although this may appear formidable, note that we set β = π/2 as we consider only horizontal wells. There is also the above mentioned redundancy in the scaled 94 3.2. Model outline viscosities and densities, so that in reality there are only 6 dimensionless parameters that we consider: e, vC , wC , κj (a single viscosity ratio), ρj (a single density ratio), and St∗. The dimensional physical properties κ and ρ are defined in terms of the concen- tration and the dimensionless pure fluid properties: κk and ρk, for k = 1, 2. For simplicity we have used linear interpolation to define this closure. Theoretically, the concentration is only advected and takes only the initial values of the phases, 0 and 1. Therefore, the closure models for κ and ρ are really only required computationally, where intermediate values are found due to numerical diffusion. In fact, if we consider only the pure fluid properties and assume that the phases are advected without any diffusion/dispersion, the fluid density only enters the picture via buoyancy and always combines with the Stokes number. The effects of buoyancy are characterised by the dimensionless parameter, b̃ = (ρ2 − ρ1)/St∗: b̃ = ρ2 − ρ1 St∗ = (ρ̂2 − ρ̂1)ĝ(r̂o − r̂i)2 4maxk{κ̂k}ŵ∗ , (3.12) clearly representing the ratio of buoyant to viscous forces. Therefore, the flow is governed in its most simple situations by 5 dimensionless parameters: e, vC , wC , κj (a single viscosity ratio), b̃. Of these, the only dimensionless parameter may have a broad range of values is the buoyancy number b̃, which can be small or large, positive or negative. For the most part we consider the effect of constant wC , (indeed rotation is more common than reciprocation), but a time-varying wC(t) can also be considered with care. Derivation of the Hele-Shaw model strictly relies on the changes in wC(t) occurring over a long timescale, so that the flow is pseudo-steady. There may also be situations in which the nonlinear inertial terms are small, but accelerations due to wC(t) are significant, and this requires a modification of the underlying model, as detailed in [15]. In terms of the typical parameter ranges for the casing speed, note that we are focused here on the cementing application, where the casing is moved primarily in order to ensure that any gel in the drilling mud is broken. Large values of wC mean a rapid reciprocation which is impractical over length-scales of 100’s of meters driven mechanically from surface. Note that a value wC > 2 would mean that the Couette component of velocity (driven by pulling the casing) is faster than 95 3.2. Model outline the mean pumping velocity, which is very unlikely. Similarly, we should note that large rotation rates may result in shearing of the casing and/or some component of the drive mechanism, both of which are difficult and expensive to remedy downhole. 3.2.2 Existing results for the stationary casing The inner cylinder or casing is fixed in conventional primary cementing operations. The model derived in [1] has been extensively studied by Pelipenko & Frigaard in the sequence of papers [20, 21, 22], which focus principally at near vertical wells. The dynamics are dominated by the existence (or not) of steady traveling wave so- lutions, 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 [20]. These steady states are in fact found computa- tionally for a much wider range of parameters than those for which it is possible to find analytical solutions, [21], and it is possible to predict the domains of existence of steady and unsteady displacements using a lubrication-style displacement model; see [22]. Another body of work is directed at understanding the stability of these displace- ment 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 in- dustrial 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 [22]. Instead, the approach taken by Moyers-González & Frigaard is to consider the stability of parallel flows in the Hele-Shaw setting; see [14, 13]. 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. 96 3.3. Steady traveling wave displacements: |b̃| ≪ 1 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 in chapter 2. 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 buoy- ancy simply determines the axial length-scale of the interface: whether or not the displacement gives a steady solution depends on the fluid rheologies, annular eccen- tricity 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). 3.3 Steady traveling wave displacements: |b̃| ≪ 1 We first consider the possibility of there being steady traveling wave displacements in the presence of casing movement. The method we use is analytical, namely domain perturbation, i.e. we look for an interface shape that is nearly parallel to the φ-axis, so that the zero-th order domains are rectangular, allowing separable solutions. As with the stationary casing situation, we expect that the domain perturbation method may be unduly restrictive on the parameter space, i.e. we expect that steady states will exist outside of the domain of strict validity of our assumptions. The value of such solutions is three-fold: first, to establish that casing motion does not eliminate this important qualitative feature of these displacement flows; second, to provide a test-bed for validating numerical codes; third, to give some insight into how the dimensionless problem parameters influence the displacement. As in [20], for analytical work is easier to work with the formal large Péclet number limit of the model described in §3.2, in which the fluids are effectively immiscible. Instead of solving (3.9) for the concentration, we assume an interface between the two pure fluid phases and we track the interface using a kinematic condition. Let Ω1 be the domain for the displacing fluid, and Ω2 the domain for the displaced fluid. As there are no density gradients in either pure fluid, ∇ · f = 0, 97 3.3. Steady traveling wave displacements: |b̃| ≪ 1 and equation (3.2) yields ∇ · Sk = 0, (φ, ξ) ∈ Ωk, k = 1, 2, (3.13) Sk = 3κk 2H3 ( ∂Ψ ∂φ −HwC , ∂Ψ ∂ξ +HvC) ) , k = 1, 2. (3.14) The interface is denoted: ξ = h(φ, t), and satisfies the kinematic equation: ∂h ∂t + v̄ ∂h ∂φ = w̄. (3.15) Both pressure and the stream function are continuous at the interface, i.e. [p]21 = 0 and : [Ψ]21 = 0 (3.16) where [q]21 denotes the jump between fluid 1 and 2 of property q across the interface. Assuming the interface is smooth, the tangential derivative of p along the interface is also continuous. Pressure continuity may be expressed in terms of the jump in (Sk · n), which becomes a jump condition on the normal derivative of Ψ [ Sk,ξ + ρk sin β sinπφ St∗ − ( Sk,φ + ρk cos β St∗ ) ∂h ∂φ ]2 1 = 0. (3.17) We shall work in a frame of reference moving with the mean speed of 1, and transform coordinates accordingly: t 7→ t, φ 7→ φ and ξ − t 7→ z. We assume wC is constant. The interface position in the moving frame is z = g(φ, t) = h(φ, t)− t. We assume we are far away from the ends of the annulus, situated in the moving frame at say, z = ±L. We write the stream function as Ψ = Φ + Φ∗ : Φ∗ = 2 ( φ+ e π sin(πφ) ) . (3.18) Note that Φ∗, which is the stream function of the mean flow, satisfies the boundary condition (3.5) but not (3.6). In terms of Φ we have: Sk = 3κk 2H3 ( ∂Ψ ∂φ −H[wC − 2], ∂Ψ ∂ξ +HvC) ) , k = 1, 2. (3.19) 98 3.3. Steady traveling wave displacements: |b̃| ≪ 1 Equation (3.13) still holds, but now subject to boundary conditions: Φk(φ+ 2, z) = Φ(φ, z), (3.20) ∂Φk ∂z (φ,±L) = −HvC . (3.21) The jump conditions at the interface are: [Φ]21 = 0 (3.22)[ 3κk 2H3 ∂Φ ∂φ + 3κk H2 ( 1− wC 2 ) + ρk cos β St∗ ]2 1 ∂g ∂φ = [ 3κk 2H3 ∂Φ ∂z + 3κkvC 2H2 + ρk sinβ sinπφ St∗ ]2 1 . (3.23) The kinematic condition in the moving frame is 2H ∂g ∂t − ∂Φk ∂z ∂g ∂φ = ∂Φk ∂φ , (3.24) and in the case of a steady state, (∂g∂t = 0), this implies that Φ(φ, g(φ, t)) is constant, i.e. the interface is a streamline. 99 3.3. Steady traveling wave displacements: |b̃| ≪ 1 3.3.1 Concentric annuli The simplest case to consider is when the cylinders are concentric, i.e. e = 0. In this case, the interface g(φ) and stream functions, Φk, k = 1, 2, must satisfy: ∆Φk = 0, (φ, z) ∈ Ωk, k = 1, 2, (3.25) Φk(φ+ 2, z) = Φk(φ, z), k = 1, 2, (3.26) ∂Φk ∂z (φ, z = ±L) = −vC (3.27) [Φk] 2 1 = 0. (3.28)[ 3κk 2 ∂Φk ∂φ + 3κk ( 1− wC 2 ) + ρk cos β St∗ ]2 1 ∂g ∂φ = [ 3κk 2 ∂Φk ∂z + 3κkvC 2 + ρk sinβ sinπφ St∗ ]2 1 (3.29) ∂Φ ∂z ∂g ∂φ + ∂Φ ∂φ = 0. (3.30) Here (3.28)-(3.30) are satisfied at the interface z = g. While the form of (3.25)- (3.27) suggests a separable solution, this requires simple domains Ωk, or at least domains that are close to rectangular. We therefore develop a domain perturbation procedure. As z → ±L, we observe that (3.27) implies that Φ ∼ O(1). We assume that Φk(φ, z) = Φk,0 + εΦk,1 + · · · , k = 1, 2, (3.31) where 0 < ε << 1 is a small parameter that we shall identify later. Equations (3.25)-(3.27) have the leading order Couette solution: Φk,0(φ, z) = −vCz, k = 1, 2, (3.32) which also satisfies (3.28), for any interface position. Substituting Φk,0(φ, z) into (3.29) we observe that at leading order the stream function does not appear in this 100 3.3. Steady traveling wave displacements: |b̃| ≪ 1 jump condition. Equation (3.29) is satisfied at leading order by g0(φ): g0(φ) = − b̃ sin β cos πφ π ( 3 (κ2 − κ1) ( 1− wC2 ) + b̃ cos β ) , (3.33) where b̃ = ρ2−ρ1St∗ . The difficulty comes with (3.30), which cannot be satisfied at leading order if |g0(φ)| ∼ O(1). We therefore assume that b̃ sin β( 3 (κ2 − κ1) ( 1− wC2 ) + b̃ cosβ ) = µε (3.34) where µ ∼ O(1). This implies that the interface is nearly-flat. Thus, we complement the assumed perturbation expansion for Φ by an expansion of the following form for g(φ): g(φ) = −εµ cos πφ π + εg1(φ) + ε 2g2(φ) + · · · . (3.35) Note that in a horizontal annulus, the requirement of a near-flat interface will be satisfied only when |b̃| is small since the viscosities κk are of order unity, due to the scaling. Evidently, we may use |b̃| ≪ 1 to define ε, or simply set ε from (3.34) with µ = 1. Alternatively, there may be other small parameters in the problem, e.g. the eccentricity, (see below). The next order solution, Φk,1 and g1(φ), is found from: ∆Φk,1 = 0 (3.36) Φk,1(φ+ 2, z) = Φk,1(φ, z) (3.37) ∂Φk,1 ∂z (φ, z = ±L) = 0 (3.38) [Φk,1] 2 1 = 0. (3.39)[ 3κk ( 1− wC 2 ) + ρk cos β St∗ ]2 1 ∂g1 ∂φ = [ 3κk 2 ∂Φk,1 ∂z ]2 1 (3.40) −∂Φk,0 ∂z ∂g1 ∂φ = ∂Φk,1 ∂φ − vCµ sinπφ. (3.41) Having imposed the condition that the interface perturbation is small, we linearise 101 3.3. Steady traveling wave displacements: |b̃| ≪ 1 the interface jump conditions and kinematic condition onto z = 0 and find a sepa- rable solution. Defining B by: B = 3vC(κ2 + κ1) tanhπL 2 ( 3(κ2 − κ1) ( 1− wC2 ) + b̃ cos β ) , (3.42) we have the following solution: Φ1,1 = vCµ (B sinπφ− cos πφ) π(B2 + 1) coshπ(z + L) coshπL (3.43) Φ2,1 = vCµ (B sinπφ− cos πφ) π(B2 + 1) coshπ(z − L) coshπL (3.44) g1(φ) = 1 vC Φk,1|z=0 + µ π cos πφ. (3.45) For vC = wC = 0 the solutions recover the steady states of [20]. Note that when vC ∼ O(ε), then g1(φ) = 0 and equation (3.33) gives the shape up to O(ε). Thus, for concentric annuli and a nearly-flat interface, a small rotation has no effect on the displacement. Assuming b̃ = O(ǫ) and combining g0 & g1, at leading order the interface position is: g(φ) = − b̃ sin β cos π(φ− φ0) π √ B2 + 1 ( 3 (κ2 − κ1) ( 1− wC2 ) + b̃ cos β ) +O(ε2) (3.46) φ0 = 1 π arccos ( 1√ 1 +B2 ) . (3.47) In the above phase-amplitude form we see that, in the absence of rotation, recipro- cation acts exclusively on the amplitude of g(φ), whereas rotation induces both a change in amplitude and a phase-shift φ0, i.e. the minima and maxima of g(φ) are rotated by an angle πφ0 in a direction determined by the sign of vC . The phase shift φ0 varies between 0 and 1 as |vc| → ∞, implying that the casing rotation can in fact move the maximum in g(φ) from the bottom of the annulus to the top at large rotations. However, also as |vc| → ∞ the amplitude of g(φ) scales as (B2 + 1)−1/2 ∼ |vc|−1, i.e. in this limit the rotation is so strong that the interface is spun around the annulus negating the effects of density difference. Small positive 102 3.3. Steady traveling wave displacements: |b̃| ≪ 1 −1 −0.5 0 0.5 1 0.045 0.05 0.055 0.06 vC |m in φ g (φ ) - m a x φ g (φ )| −1 −0.5 0 0.5 1 −0.22 −0.1 0 0.1 0.22 vC φ o (a) (b) −1 −0.5 0 0.5 1 0.06 0.08 0.1 0.12 0.14 vC −1 −0.5 0 0.5 1 −0.38 −0.2 0 0.2 0.38 vC (c) (d) Figure 3.2: Effects of casing motion on the steady state amplitude, |minφ g(φ) − maxφ g(φ)|, (see figures a & c), and on the phase shift, φ0, (see figures b & d). For a & b, wC = 0 and κ2 = 0.1◦, 0.2 , 0.3 △. For c & d, κ2 = 0.6 and wC = −0.5 ◦, 0 , 0.5 △. Other parameters: e = 0, κ1 = 1, b̃ = −0.2. axial movements wC > 0, will increase the axial length of the steady state, while wC < 0 will decrease it. Figure 3.2 presents typical variations in the length of the interface and the phase shift, for varying displaced fluid viscosity, κ2, and O(1) casing motion. The ampli- tude is an even function of vC and the phase shift an odd function of vC . The interface flattens as |vc| is increased and as the viscosity difference is increased. The largest viscosity difference and smallest rotation rates give the smallest phase shift. Positive wC appears to increase the amplitude and negative wC appears to decrease the amplitude, with similar behaviour for the size of the phase shift. 103 3.3. Steady traveling wave displacements: |b̃| ≪ 1 3.3.2 Small eccentricities We may also derive an analytical solution for the case of slightly eccentric annuli, e≪ 1. In this case we may set ε = e and proceed as above. The field equations are: 3κk 2H3 ∆Φk + 9κk 2H4 eπ sinπφ ∂Φk ∂φ + 6κk H3 eπ sinπφ ( 1− wC 2 ) = 0. (3.48) We use the same perturbation expansion as before, (replacing ǫ with e and requiring g0(φ) = O(e)). The leading order terms are given by (3.32) and (3.33), i.e. the leading order concentric solution. The O(e) solution satisfies ∆Φk,1 + 4π ( 1− wC 2 ) sinπφ = 0 (3.49) Φk,1(φ+ 2, z) = Φk,1(φ, z) (3.50) ∂Φk,1 ∂z (φ, z = ±L) = −vC cos πφ (3.51) [Φk,1] 2 1 = 0 (3.52)[ 3κk ( 1− wC 2 ) + ρk cos β St∗ ]2 1 ∂g1 ∂φ = [ 3κk 2 ( ∂Φk,1 ∂z + vC cos πφ )]2 1 (3.53) −∂Φk,0 ∂z ∂g1 ∂φ = ∂Φk,1 ∂φ − vCµ sinπφ. (3.54) We solve this system using separation of variables, the details of which are straight- forward. The solution is given by: Φ1,1(φ, z) =− vC π sinhπz cosh πL cos πφ+ 4 π ( 1− wC 2 ) sinπφ − (µa1 −Bb1) cosh π(z + L) cos πφ + (µBa1 + b1) cosh π(z + L) sinπφ (3.55) Φ2,1(φ, z) =− vC π sinhπz cosh πL cos πφ+ 4 π ( 1− wC 2 ) sinπφ − (µa1 −Bb1) cosh π(z − L) cos πφ + (µBa1 + b1) cosh π(z − L) sinπφ (3.56) 104 3.3. Steady traveling wave displacements: |b̃| ≪ 1 where a1 = vC π(1 +B2) 1 cosh πL (3.57) b1 = ( − 4 π ( 1− wC 2 ) + 3v2C(κ2 − κ1) ( 1− 1cosh piL ) 2π ( 3(κ2 − κ1) ( 1− wC2 ) + b̃ cosβ ) ) 1 (1 +B2) cosh πL (3.58) Using the kinematic condition (3.54), we get g1(φ) = 1 vC Φk,1(φ, 0) + µ π cos πφ, and so, to leading order, as before: g(φ) = e vC Φk,1(φ, 0) +O(e 2). (3.59) If we substitute for µ and set e = 0 we recover the concentric solution. Equally, when vC = 0, Φk,1, k = 1, 2 are reduced to Φk,1(φ, z) = 4 π ( 1− wC 2 )( 1− cosh(π(z ± L)) coshπL ) sinπφ (3.60) and on using (3.53), to O(e) the interface shape is: g(φ) = g0(φ)−6e π ( 1− wC 2 ) (κ2 + κ1) tanhπL( 3(κ2 − κ1) ( 1− wC2 ) + b̃ cos β ) cos πφ+O(e2). (3.61) which agrees with the stationary casing solution found in [20]. We may also write (3.59) in phase-amplitude form, A cos(π(φ− φ0)), as for the 105 3.4. Computational results concentric case. Here the amplitude A and the phase shift φ0 are defined as follows: A2 = A21 +A 2 2 π2(1 +B2)2 , (3.62) φ0 = −sign(A1) π cos−1 ( |A2| A ) (3.63) A1 = Bb̃ sin β( 3(κ2 − κ1) ( 1− wC2 ) + b̃ cos β ) + 4eB2 vC ( 1− wC 2 ) + eB κ2 − κ1 κ2 + κ1 (3.64) A2 = b̃ sinβ( 3(κ2 − κ1) ( 1− wC2 ) + b̃ cos β ) − 4eB vC ( 1− wC 2 ) + eB2 κ2 − κ1 κ2 + κ1 , (3.65) where the sign of A1 is usually the sign of vC b̃, which in all our examples is −sign(vC). In Fig. 3.3 we plot typical variations in the amplitude and phase shift, for the same parameters as in Fig. 3.2, but with a small eccentricity e = 0.01. We observe that the eccentricity decreases the amplitude of the finger, but has very little effect on the phase shift of the tip of the finger. 3.4 Computational results The condition |b̃| ≪ 1 in the previous section is physically artificial, in the sense that it is motivated by the need to use the domain perturbation method. There is no a priori reason why we should not be able to find steady traveling wave solutions to (3.19)-(3.24) for interfaces that are not flat. In this section we explore the behaviour of solutions computationally, moving away from the assumption |b̃| ≪ 1, and focusing individually on casing rotation and axial motion. Below we will present results drawn from numerical solutions of the model de- scribed in §3.2, in which the interface is represented by the level set c̄ = 0.5. The computational method used is a hybrid finite difference/volume method. We use a staggered mesh, with Ψ represented at the cell nodes and c̄ at the cell centres. Time enters the problem only via the concentration equation (3.9). At each time step, for given values of c̄ we find the stream function Ψ from (3.2), which is a linear Pois- son equation. This is discretised using a 9-point stencil and solved using an SOR 106 3.4. Computational results −1 −0.5 0 0.5 1 0.028 0.03 0.032 0.034 0.036 vC |m in φ g (φ ) - m a x φ g (φ )| −1 −0.5 0 0.5 1 −0.22 −0.1 0 0.1 0.22 vC φ o (a) (b) −1 −0.5 0 0.5 1 0.02 0.04 0.06 0.08 vC −1 −0.5 0 0.5 1 −0.38 −0.3 −0.1 0 0.1 0.3 0.38 vC (c) (d) Figure 3.3: Effects of casing motion on the steady state amplitude, |minφ g(φ) − maxφ g(φ)|, (see figures a & c), and on the phase shift, φ0, (see figures b & d). For a & b, wC = 0 and κ2 = 0.1 ◦, 0.2 , 0.3 △. For c & d, κ2 = 0.6 and wC = −0.5 ◦, 0 , 0.5 △. Other parameters: e = 0.01, β = π/2, κ1 = 1, b̃ = −0.2. method. To advance the concentration field one timestep we use a Flux Corrected Transport (FCT) scheme with limiters, as described in [32]. The treatment of both equations could be improved as we are dealing with a linear Newtonian problem, but our choice is partly dictated by the development of a numerical algorithm for non-Newtonian fluids, in which the linear Poisson solver is embedded iteratively into an Uzawa algorithm as part of an augmented Lagrangian approach; see [21] for a description of this method for the stationary casing situation. We first benchmark the numerical code against the perturbation solutions in §3.3. Two examples of comparison with the concentric annulus steady state solution are given in Figs. 3.4. For both examples we fix parameters κ1 = 1, κ2 = 0.25, b̃ = −0.5, 107 3.4. Computational results (ρ1 = 1, ρ2 = .8, , St ∗ = 0.4). We note that even though |b̃| is not very small, (and also the parameter µε = −2/9), the comparison is good. This suggests that the regime of validity of the analytical results is not strict. The interface is relatively flat in both figures and the error between numerical and analytical solutions is of O(ε2), as expected. If we take the parameters to give a flatter interface the comparison is also good, but we soon approach the limit of the mesh resolution. ξ-t φ −0.08 −0.04 0 0.04 0.08 −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 ξ-t φ −0.08 −0.04 0 0.04 0.08 −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 a) b) Figure 3.4: Comparison of analytical and numerical solutions: a) vC = 0.25, wC = 0; b) vC = 0, wC = 0.5. Other parameters: κ1 = 1.0, κ2 = 0.25, b̃ = −0.5, (ρ1 = 1, ρ2 = .8, St ∗ = 0.4). In Figure 3.5 we compare the numerical solution against the perturbation solu- tion for small e. The comparison is made in a frame of reference moving with the mean pump velocity. For the streamline comparison, we have subtracted off the ro- tational Couette component, HvC . The steady perturbation solution, (3.55), (3.56) & (3.59), is compared against the computed solution at t = 100. The interface is defined as the contour c̄ = 0.5 for the computational results. Evidently there is good agreement between analytic and numerical results. 108 3.4. Computational results ξ-t φ −6 −4 −2 0 2 4 6 −1 −0.5 0 0.5 1 ξ-t φ −6 −4 −2 0 2 4 6 −1 −0.5 0 0.5 1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 Figure 3.5: Comparison of perturbation solution (top) with computational solution (bottom) for parameters: e = 0.05, κ1 = 1, κ2 = 0.75, vC = 0.2, wC = 0, b̃ = −1/6, (ρ1 = 1, ρ2 = .95, St ∗ = 0.3). 3.4.1 Effects of casing rotation In Fig. 3.6 we show a typical evolution of the interface for a pair of fluids, with a heavy viscous fluid displacing a lighter less viscous fluid. The left-hand side of Fig. 3.6 portrays the interface at regularly spaced time intervals (δt = 2.5) through- out the displacement, up until t = 100. On the right-hand side of Fig. 3.6 we plot the streamlines, adjusted for translation, i.e. we plot: Φ(φ, ξ − t, t) = Ψ(φ, ξ, t)− 2 ∫ φ −1 Hdφ. (3.66) In Fig. 3.6a there is no casing rotation and the interface is stable with the interface symmetric about φ = 0. The streamlines are parallel to the annulus in the far-field and recirculate at the interface in a secondary flow, which is counter-current across 109 3.4. Computational results the interface. The magnitude of the secondary flow is governed by the eccentricity, i.e. larger e gives larger |Φ|. These features are found in the numerical solutions in [21], and also shared by the steady state solutions found in §3.3. 0 20 40 60 80 100 −1 −0.5 0 0.5 1 φ −4 −2 0 2 4 −1 0 1 t = 100 −0.05 0 0.05 0 20 40 60 80 100 −1 −0.5 0 0.5 1 φ −4 −2 0 2 4 −1 0 1 −4 −2 0 2 4 0 20 40 60 80 100 −1 −0.5 0 0.5 1 ξ φ −4 −2 0 2 4 −1 0 1 ξ - t −5 0 5 (a) (b) (c) Figure 3.6: Example displacement with increasing casing rotation: a) vC = 0, b) vC = 1.0, c) vC = 1.75. Left panel shows the contour c(φ, ξ, t = j∆t) = 0.5, at time intervals ∆t = 2.5. Right panel shows contours of the stream function Φ(φ, ξ − t, t) at t = 100; the heavy black curve is c(φ, ξ − 100, 100) = 0.5. Other parameters are: κ1 = 1, κ2 = 0.25, e = 0.05, wC = 0, b̃ = −10, (ρ1 = 1, ρ2 = .9, St∗ = 0.01). Figures 3.6b & c present the analogous results for vC = 1 & vC = 1.75, re- spectively. As with the steady states solutions in §3.3, casing rotation breaks the symmetry: the maximal and minimal extensions of the interface are not found ex- actly at the narrow and wide side, but are offset by a phase shift. The phase shift increases with vC . At the same time the axial extension of the interface is reduced as the rotation rate increases. Note that in all the plots, the shape of the interface is 110 3.4. Computational results very close to the shape of the streamline Φ = 0. After a certain time period, small instabilities appear at the interface. The onset occurs earlier at the larger rotation rates. The root cause of the instabilities is buoyancy-driven fingering. The phase shift from symmetry, evident in Figs. 3.6b & c, has the consequence that along seg- ments of the interface close to both narrow and wide sides of the annulus, heavy fluid will lie above light fluid. This mechanically unstable situation is vulnerable to a fingering-type instability. Note that in the far-field the streamlines of Φ are not parallel to the ξ-axis, since here we have only a single fluid and the Couette component of the flow dominates. Figure 3.7 illustrates in finer detail the development of the first instabilities, with the same parameters as for Fig. 3.6c. The upper panel shows the interface from t = 8.5 until t = 30.5, and the lower part of the figure shows the concentration field, close to the interface, at successive time intervals of 0.3. We observe that onset of instability is accompanied by strong secondary flows which advect fluid of inter- mediate concentration away from the interface where ∂g∂φ > 0, (occurring primarily for φ ∈ (0, 1)). This suggests that the secondary flows are primarily rotation driven, although we also expect recirculatory currents associated with the eccentricity, as with stationary casing displacements. Although theoretically there is no diffusion in our model, there is always a small amount of numerical diffusion and it is necessary to prescribe intermediate values of the fluid properties for intermediate c̄, via inter- polation, (here linear). The net effect of numerical diffusion and interpolation, in the absence of (physically driven) secondary flows appears to be fairly minor; (see for example the interface in Fig. 3.7 at t = 3, where ∂g∂φ < 0). However, unquestionably the secondary flows strongly influence behaviour at the interface. At later times in Figure 3.7 we see lighter fluid penetrating the heavy fluid layer, close to the narrow side and a small finger forms. The finger breaks off as it elongates. This process of dispersive mixing/fingering and breakup is not confined to the value c̄ = 0.5, which is an essentially arbitrary choice for the interface. Small scale non-uniformities on the mesh scale are eventually smoothed out by (numerical) diffusion, resulting in a relatively smooth diffuse interfacial region. Thus, on larger scales and over longer times, e.g. Fig. 3.6c, the local instabilities seem to disappear. For the parameters of Figs. 3.6 & 3.7, we have b̃ = −10, so that buoyancy effects were dominant even at the larger rotation rates. Figure 3.8 illustrates the effects of 111 3.4. Computational results 10 15 20 25 30 −1 0 1 ξ φ −1 0 1 −1 0 1 t = 3 φ −1 0 1 −1 0 1 t = 3.3 φ −1 0 1 −1 0 1 t = 3.6 φ −1 0 1 −1 0 1 t = 3.9 φ −1 0 1 −1 0 1 t = 4.2 ξ - t φ −1 0 1 −1 0 1 t = 4.5 −1 0 1 −1 0 1 t = 4.8 −1 0 1 −1 0 1 t = 5.1 −1 0 1 −1 0 1 t = 5.4 −1 0 1 −1 0 1 t = 5.7 ξ - t −1 0 1 −1 0 1 t = 6 −1 0 1 −1 0 1 t = 6.3 −1 0 1 −1 0 1 t = 6.6 −1 0 1 −1 0 1 t = 6.9 −1 0 1 −1 0 1 t = 7.2 ξ - t −1 0 1 −1 0 1 t = 7.5 −1 0 1 −1 0 1 t = 7.8 −1 0 1 −1 0 1 t = 8.1 −1 0 1 −1 0 1 t = 8.4 −1 0 1 −1 0 1 t = 8.7 ξ - t −1 0 1 −1 0 1 t = 9 −1 0 1 −1 0 1 t = 9.3 −1 0 1 −1 0 1 t = 9.6 −1 0 1 −1 0 1 t = 9.9 −1 0 1 −1 0 1 t = 10.2 ξ - t −1 0 1 −1 0 1 t = 10.5 −1 0 1 −1 0 1 t = 10.8 −1 0 1 −1 0 1 t = 11.1 −1 0 1 −1 0 1 t = 11.4 −1 0 1 −1 0 1 t = 11.7 ξ - t 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Figure 3.7: Detailed evolution of the instability for the parameters of Fig. 3.6c. Top panel shows c(φ, ξ, 0.5 + j∆t) = 0.5 with ∆t = 1, (dark line contour is for t = 8.5). Bottom panel shows contour plots of c(φ, ξ − t, t), with c(φ, ξ − t, t) = .5 marked by the solid line. rotation on displacements with a more modest buoyancy number, b̃ = −1, achieved by increasing St∗, (physically this means pumping faster). For this example we have also reduced the viscosity ratio and consider a concentric annulus. To illustrate the effects of rotation on the displacement in a more global sense, both spatially and in terms of other values of c(φ, ξ, t), we average c̄(φ, ξ, t) with respect to φ and present the evolution of c̄(ξ − t, t) = 1 2 ∫ 1 −1 c̄(φ, ξ − t, t) dφ, (3.67) in the form of a spatiotemporal plot as the displacement progresses. The parame- 112 3.4. Computational results ters have been chosen so that in the absence of rotation the interface slumps and elongates under the effects of buoyancy, (Fig. 3.8a). We observe that for sufficiently large vC the interfacial region is severely limited at long times, (Figs. 3.8c & d). This partly answers the question of whether or not rotation may be used to sta- bilize an otherwise unstable displacement. This is evidently possible, but requires dimensionless rotation rates of the order of |b̃|. −1 0 1 0 5 10 15 20 t −1 0 1 0 5 10 15 20 t −1 0 1 0 5 10 15 20 t −1 0 1 0 5 10 15 20 ξ - t t (a) (b) (c) (d) Figure 3.8: Spatiotemporal plot of c̄(ξ − t, t) for: a) vC = 0; b) vC = 0.5; c) vC = 1; d) vC = 2; shading legend is 1 = , 0 = . Other model parameters are: κ1 = 1, κ2 = 0.8, e = 0, wC = 0, b̃ = −1, (ρ1 = 1, ρ2 = .9, St∗ = 0.1). Since |b̃| & 1 typically, (i.e. buoyancy is always significant in the industrial setting), to have a significant effect of this type requires that the rotational speed of the casing is of the same order as the mean fluid velocity. Note that in the dimensional setting the parameter |b̃| increases with decreasing pump speed, as does |vC |. Thus, this type of rotational stabilization requires fast casing rotation as 113 3.4. Computational results opposed to slow flow. In terms of the dimensional model parameters, we have: |b̃| |vC | = |ρ̂2 − ρ̂1|ĝd̂2 maxk{κ̂k}|v̂C | , (3.68) where v̂C is the dimensional rotational speed of the casing (=rotation rate × mean radius). This balance can therefore be interpreted as the ratio of buoyant stress to viscous stress due to the rotational Couette component of the velocity. Large rotation rates increase frictional torques which can lead to shearing of the casing. Thus, there is some risk involved with use of this technique. To summarise the observed effects of casing rotation on displacement, two effects are evident. First, it appears that casing rotation generally reduces the length of annulus over which intermediate concentrations are found. In some cases this appears to result in concentration fields where the diffuse region spreads very slowly at large times. Secondly we have observed a form of rotational stabilization in our results at sufficiently large rotation rates, in comparison to the buoyancy number. We note that the process by which stabilization occurs involves the growth of local instabilities coupled to dispersive mixing driven by secondary flows in the interfacial regime, i.e. these displacements are only stable in a non-local sense at longer times. 3.4.2 Effects of constant axial casing motion Industrially, axial motions of the casing are imposed for the same reasons as for rotation. We start by looking at the effects of constant axial motion. Figure 3.9 shows a spatiotemporal plot of the azimuthally averaged concentration, c̄, plotted in a moving frame, for wC = ±1, 0. We see that when the axial motion is against the flow, the interface length is somewhat reduced, but when the axial motion is in the flow direction, the interface length increases significantly. Figure 3.10 shows a further spatiotemporal plot of the azimuthally averaged concentration. Now instead of constant axial motion we impose a (more realistic) reciprocating motion: wC(t) = w0 sin( 2πt tr ). Figure 3.10a shows baseline results for a stationary casing. Figures 3.10b-d show results for unit amplitude, (w0 = 1), and decreasing period of oscillation, tr. For 114 3.4. Computational results −2 −1 0 1 2 0 5 10 15 20 t −2 −1 0 1 2 0 5 10 15 20 t −2 −1 0 1 2 0 5 10 15 20 ξ - t t (a) (b) (c) Figure 3.9: Spatiotemporal plot of c̄(ξ − t, t) for: a) wC = −1; b) wC = 0; c) wC = 1; shading legend is 1 = , 0 = . Other model parameters are: κ1 = 1, κ2 = 0.65, e = 0, vC = 0, b̃ = −3, (ρ1 = 1, ρ2 = .7, St∗ = 0.1). large tr the net effect of the reciprocation is barely discernible from the stationary casing situation: compare Figs. 3.10a & b. As the period of oscillation is reduced, we observe synchronisation of of the spreading concentration with casing movement: the concentration spreads and contracts periodically. Note that the spreading and contraction observed underlines the fact that this is advective dispersion, i.e. pure diffusive mixing would not be reversible. Also we note that the mean axial extent of the diffuse region does not appear to be changed much by the reciprocation. In Fig. 3.11 we show the evolution of the interface over time intervals ∆t = 0.5 for reciprocation time periods, tr ≥ 1. A range of oscillations of the interface are observable in this parameter range. We note that there is no discernible local fingering instability during reciprocation. Unlike rotation, the axial motion does not result in symmetry breaking or in mechanically unstable configurations that lead to fingering. Figure 3.12 shows more detail of the stream function and interface oscillations for the parameters of Fig. 3.11c, taken at times 20, 20+ tr/4, 20 + tr/2, 115 3.4. Computational results Figure 3.10: Spatiotemporal plot of c̄(ξ − t, t) for wC(t) = w0 sin ( 2pit tr ) : a) w0 = 0; b) w0 = 1, tr = 20; c) w0 = 1, tr = 10; d) w0 = 1, tr = 2; shading legend is 1 = , 0 = . Other model parameters are: κ1 = 1, κ2 = 0.3, e = 0.2, vC = 0, b̃ = −20, (ρ1 = 1, ρ2 = .8, St∗ = 0.01). 20 + 3tr/4. At 20 + tr the solution repeats, as is suggested by the pattern of Fig. 3.11c. It is clear that various types of interesting synchronisation may occur as the frequency of reciprocation is varied, but less clear if there is any practical consequence of these motions. 3.4.3 Combined reciprocation and rotation It is apparent that the reciprocation on its own does not have much effect on a dis- placement. This is not to say that casing reciprocation does not produce secondary flows, but rather that these secondary flows are symmetric and do not apparently lead to mechanically unstable interface configurations. By combining reciprocation with rotation, we might therefore take advantage of the secondary flows from re- ciprocation, in promoting dispersion, but use the rotational motion to promote an 116 3.4. Computational results 0 10 20 30 −1 0 1 φ −3 −1.5 0 1.5 3 −1 0 1 t = 30 −0.2 0 0.2 (a) (b) (c) (d) 0 10 20 30 −1 0 1 φ −3 −1.5 0 1.5 3 −1 0 1 −0.2 0 0.2 0 10 20 30 −1 0 1 φ −3 −1.5 0 1.5 3 −1 0 1 −0.2 0 0.2 0 10 20 30 −1 0 1 ξ φ −3 −1.5 0 1.5 3 −1 0 1 ξ - t −0.2 0 0.2 Figure 3.11: Effects of casing reciprocation wC(t) = w0 sin ( 2pit tr ) : a) w0 = 0; b) w0 = 1, tr = 20; c) w0 = 1, tr = 10; d) w0 = 1, tr = 2. Other model parameters are: κ1 = 1, κ2 = 0.3, e = 0.2, vC = 0, b̃ = −20, (ρ1 = 1, ρ2 = .8, St∗ = 0.01). Left panel shows c(φ, ξ, t = j∆t) = 0.5 for time intervals ∆t = 0.5. Right panel shows stream function Φ and interface (solid line) at t = 30. unstable interface configuration. Figure 3.13 shows the streamlines and interface at t = 20 for 4 different displacements. Figure 3.13a has no casing movement and the interface elongates in a symmetric manner. On adding reciprocation, (Fig. 3.13b), a second oscillation of the interface is formed, but no apparent instability. The length of the interface is similar to the fixed casing. With a small amount of rotation, the interface becomes unstable and the length decreases, (Fig. 3.13c). A further increase in rotation again diminishes the length of interface, (Fig. 3.13d). Here by length of the interface we simply mean the axial extent of the contour c̄ = 0.5. 117 3.5. Stability and instability −3 −1.5 0 1.5 3 −1 0 1 t =20 φ −0.2 0 0.2 −3 −1.5 0 1.5 3 −1 0 1 t =22.5 φ −0.2 0 0.2 −3 −1.5 0 1.5 3 −1 0 1 t =25 φ −0.2 0 0.2 −3 −1.5 0 1.5 3 −1 0 1 t =27.5 ξ-t φ −0.2 0 0.2 Figure 3.12: The stream function Φ and interface, plotted over 1 period of recip- rocation for the parameters of Fig. 3.11c. At t = 30 the solution is the same as at t = 20 (top panel). 3.5 Stability and instability Here we consider two aspects of stability and instability of these flows. First, Hele- Shaw displacements are vulnerable to viscous fingering type instabilities, which we have not shown so far. In §3.5.1 we show that such instabilities are present in our model and explore the effects of casing movement on such fingering. From the industrial perspective, such instabilities are not common since one typically uses a more viscous fluid as the displacing fluid. Secondly, we have seen that buoyancy driven local fingering instabilities occur in the presence of casing rotation. We advance a simplistic analytical prediction of this phenomenon in §3.5.2. 118 3.5. Stability and instability −3 −2 −1 0 1 2 3 −1 0 1 φ t = 20 −0.2 0 0.2 (a) (b) (c) (d) −3 −2 −1 0 1 2 3 −1 0 1 φ −0.2 0 0.2 −3 −2 −1 0 1 2 3 −1 0 1 φ −2 0 2 −3 −2 −1 0 1 2 3 −1 0 1 φ ξ - t −5 0 5 Figure 3.13: Contours of the stream function Φ with combined rotation and recipro- cation: a) w0 = vC = 0; b) w0 = 1, tr = 0.4, vC = 0; c) w0 = 1, tr = 0.4, vC = 0.5; d) w0 = 1, tr = 0.4, vC = 1. Other model parameters are: κ1 = 1, κ2 = 0.3, e = 0.2, vC = 0, b̃ = −2, (ρ1 = 1, ρ2 = .8, St∗ = 0.1) 3.5.1 Viscous fingering and casing motion In Hele-Shaw (and porous media) miscible displacement flows we typically expect to observe viscous fingering for adverse viscosity ratios, i.e. less viscous fluid displacing more viscous. In the absence of any density difference and with e = 0, the annular geometry simply unwraps into a φ-periodic uniform Hele-Shaw cell. Figure 3.14a shows typical evolution of the flow in such a situation, with κ1 = 0.3, κ2 = 1. Unsurprisingly, the interface becomes locally unstable due to viscous fingering. In Figs. 3.14b-d we explore the effects of slow casing reciprocation and casing rotation. The former appears to slightly retard the onset of instability whereas casing rotation appears to bring forward the onset. 119 3.5. Stability and instability 0 2 4 6 8 10 −1 0 1 φ −2 0 2 −1 0 1 t = 8 −0.04 −0.02 0 0.02 0.04 0 2 4 6 8 10 −1 0 1 φ −2 0 2 −1 0 1 t = 8.5 −0.05 0 0.05 0 2 4 6 8 10 −1 0 1 φ −2 0 2 −1 0 1 t = 3.5 −2 0 2 0 2 4 6 8 10 −1 0 1 ξ φ −2 0 2 −1 0 1 t = 4 ξ−t −2 0 2 (a) (b) (c) (d) Figure 3.14: Isodensity displacements in a concentric annulus with an adverse vis- cosity ratio: a) vC = 0, wC = 0; b) vC = 0, wC = sin ( 2pit 10 ) ; c) vC = 0.5, wC = 0; d) vC = 1, wC = 0. Other model parameters are: κ1 = 0.3, κ2 = 1, e = 0, b̃ = 0, (ρ1 = 1, ρ2 = 1, St ∗ = 0.1). Left panel shows evolution of the interface in a fixed frame of reference, at successive time intervals ∆t = 0.5. Right panel shows the interface and contours of the stream function Φ at the last shown interface from the left panel. When the annulus axisymmetry is broken, there is a tendency for the fluids to move faster on the wide side of the annulus and there is no underlying steady dis- placement. As the eccentricity is increased this tendency increases and the front advances unsteadily in the form of a large finger. Although related to viscous fin- gering, as this is essentially driven by differences in mobility at different azimuthal positions, the conditions governing onset of this type of unsteady interfacial motion are non-local, as discussed in [22]. Such non-local fingers may also develop their own local/secondary viscous fingers, and an example of this is shown in Fig. 3.15a. Fig. 3.15b & c show the effects of imposing casing rotation. A small amount of 120 3.5. Stability and instability rotation causes weak secondary flows that seem to lead to asymmetrical fingering. At larger |vc|, as rotation becomes dominant, we see that the initial unsteady finger itself is reduced in length, as fluid is moved around the annulus. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 −1 −0.5 0 0.5 1 φ −6 −4 −2 0 2 4 6 −1 0 1 t = 10.5 −0.2 0 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 −1 −0.5 0 0.5 1 φ −6 −4 −2 0 2 4 6 −1 0 1 −2 −1 0 1 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 −1 −0.5 0 0.5 1 ξ φ −6 −4 −2 0 2 4 6 −1 0 1 ξ - t −5 0 5 (a) (b) (c) Figure 3.15: Isodensity displacements in a mildly eccentric annulus with an adverse viscosity ratio: a) vC = 0; b) vC = 0.25; c) vC = 1. Other model parameters are: κ1 = 0.3, κ2 = 1, e = 0.2, b̃ = 0, (ρ1 = 1, ρ2 = 1, St ∗ = 0.1). Left panel shows evolution of the interface in a fixed frame of reference, at successive time intervals ∆t = 1. Right panel shows the interface and contours of the stream function Φ at t = 10. 3.5.2 A criterion for buoyancy driven fingering As commented before, the majority of industrial displacements do not involve an adverse viscosity ratio and, as we have seen, in the absence of casing rotation do not appear to suffer from local instabilities. On the other hand, since casing rota- tion is becoming a common practice, we would like some prediction of the type of 121 3.5. Stability and instability buouyancy-driven fingering that we have observed. Our observation is that these in- stabilities arise essentially due to the phase shift in the steady-state interface, which results in segments of the interface where heavy fluid lies above light fluid. We develop below a Muskat-type analysis of this interface configuration, [16], by assuming a small finger that protrudes normal to the interface; see Fig. 3.16. We note that in a Hele-Shaw displacement, it is possible for there to be a jump in the tangential velocity across the interface. Thus, a second possible avenue for instability is a two-layer shear flow type instability. However, since we have not observed any clear evidence of these instabilities, we focus for now on fingering. bottom top bottom Heavy Light vC g(φ) vf vf Figure 3.16: Schematic of interfacial fingering in an unwrapped annulus, in regions where heavy fluid lies on top of lighter fluid. For definiteness, we fix on a heavy fluid displacing a light fluid, with the finger of heavy fluid 1 protruding into the lighter fluid 2, as indicated in Fig. 3.16. In the Muskat approach one considers that the pressure within the thin finger is given by the pressure field in the surrounding fluid, and this translates into a velocity of the fluid within the finger. If the finger velocity exceeds the normal velocity of the interface the flow is deemed unstable. 122 3.5. Stability and instability We assume that the interface is propagating as a steady state at unit speed in the axial direction, i.e. z = g(φ) in the moving frame. The tangential and normal vectors are given by t = ( 1, ∂g ∂φ ) vn, n = ( −∂g ∂φ , 1 ) vn : vn = ( 1 + ∂g ∂φ 2)− 12 . (3.69) Note that vn is the speed of the interface in the normal direction. The pressure within the thin finger of fluid 1, say pf , is approximated by that in fluid 2, the gradient of which is given by: −∂pf ∂n = −∂p2 ∂n = [S2 + f2] · t (3.70) = 3κ2 H2 ( vn + ( −wC 2 , vC 2 ) · t ) + ρ2 St∗ ( cos β + ∂g ∂φ sin β sinπφ ) vn. (3.71) This pressure gradient drives fluid 1 in the normal direction according to: 2κ1 H2 ( vf + ( −wC 2 , vC 2 ) · t ) = −∂pf ∂n − ρ1 St∗ ( cos β + ∂g ∂φ sin β sinπφ ) vn. (3.72) Substituting (3.71) and rearranging yields expression (3.73). Fs(φ) = vf − vn vn = ( κ2 κ1 − 1 )( 1− wC 2 ) + H2 3κ1 b̃ cos β + (( κ2 κ1 − 1 ) vC 2 + H2 3κ1 b̃ sinβ sinπφ ) ∂g ∂φ (3.73) If vf > vn at some position, we classify the interface as unstable. Thus, our stability criterion is: min φ∈[−1,1] Fs(φ) ≤ 0. (3.74) Let us consider the different terms in the expression (3.73) above. The first term is the classical stabilising effect of a positive viscosity ratio, κ1 > κ2. We have not considered strong axial casing motions, wC > 2, but the implication is that in such motions the effect of the viscosity ratio is reversed. For a steady interface motion, 123 3.6. Summary and discussion note that wC > 2 implies that the Couette component of the velocity is stronger than the mean imposed flow from pumping, all along the interface, and thus an adverse pressure gradient results pushing the fluids backwards against the viscous drag from the axial casing motion. However, assuming wC < 2, this first term is stabilising for κ1 > κ2. The second term is zero in a perfectly horizontal annulus, and small when the annulus is close to horizontal. For a heavy-light fluid displacement, b̃ < 0, so this term is mildly stabilising if the displacement is uphill β < π/2 and mildly destabilising otherwise. The third term is interesting. For vC > 0 we have seen a phase shift in the steady profile as shown schematically in Fig. 3.16. Thus, in the mechanically unsta- ble segment of the interface close to φ = −1, where ∂g∂φ > 0 rotation is stabilising, (assuming κ1 > κ2). As we continue along the interface, where ∂g ∂φ < 0, the inter- face is destabilised, and this will include the mechanically unstable segment of the interface close to φ = 0. The fourth term in (3.73) is the destabilisation due to buoyancy. For b̃ < 0 we observe that this term will be positive in the two mechanically unstable segments of the interface. Since the third term is destabilising only in the uppermost of these segments, this criterion suggests that the segment close to φ = 0, in the wide side of the annulus is the least stable. Although in many situations we have observed instability first in this segment, this is not always the case in our numerical results. In order to extract quantitative information from (3.73) it is necessary to specify the steady state shape. In our analysis the only solutions we have are for the con- centric and weakly eccentric annuli considered in §3.3. However, although we may substitute from the solutions (3.33) or (3.59), note that the strict domain of validity leading to these solutions required that | ∂g∂φ | ≪ 1, which in a horizontal annulus means that |b̃| ≪ 1 and that the first term in (3.73) will be order 1. Therefore, for such situations we see that the other terms in (3.73) are necessarily small. 3.6 Summary and discussion In this study we have shown that steady traveling wave displacements may occur, as for the situation with static walls; see [2, 20]. This has been confirmed both 124 3.6. Summary and discussion analytically, via a perturbation method for small buoyancy numbers and when the annulus is near to concentric, and also computationally. From the perturbation method results we have seen explicitly that rotation reduces the extension of the interface in the axial direction and also results in an azimuthal phase shift of the steady shape away from a symmetrical profile. For larger buoyancy numbers computational solution has shown that the phase shift results in the positioning of heavy fluid over light fluid along segments of the interface. When the axial extension of the interface is sufficiently large this leads to a local buoyancy driven fingering instability. In terms of the concentration field, local fingering advances via complex secondary flows close to the front that are controlled by both casing rotation and annular eccentricity. Over longer times and on smaller length-scales the local fingering is replaced by steady propagation of a diffuse interfacial region that spreads slowly due to dispersion. In some cases this appears to result in concentration fields where the diffuse re- gion grows very slowly at large times. Here large times are dictated by the temporal limits of our computations, (typically t ∼ 100), which are vulnerable to numerical diffusion effects at large t. In our model molecular diffusion effects are absent at leading order and thus only numerical diffusion is active in smoothing small scale variations. This makes it hard to distinguish whether or not at long times the under- lying physical result would be a bounded diffuse region approached asymptotically or one that grows slowly via dispersion. From a practical perspective, since the growth rates of the diffuse region are small, this makes little difference. Slow steady axial motion of the annulus walls on its own is apparently less interesting. There is no breaking of the symmetry of the interface and hence no instability. However, axial wall motion does generate secondary flows which may combine with those from cylinder rotation resulting in enhanced dispersive effects. For larger axial motions, i.e. where the Couette velocity component exceeds the mean flow velocity, our stability analysis suggests that instability may result, essentially from a reversal of the pressure gradient. We have not studied this regime. Large steady motions of the casing are not physically possible in cementing. If periodic reciprocation is studied in place of steady axial casing motion, we observe interesting synchronisation patterns as the frequency is varied. We have seen that dispersive mixing results from secondary flows that combine 125 3.6. Summary and discussion both rotation and recirculation effects. These secondary flows are too complex to study analytically, although it is clear that |vC | and e are the appropriate scaling parameters for the 2 respective effects. Although a quantitative computational study could be carried out, other effects also need to be taken into account for such a study to be effective. Firstly, it should not be overlooked that the underlying Hele-Shaw approach averages across the annular gap. Dispersion (and other) effects on the gap scale are thus ignored in this approach. Although we have derived (3.9) under the reasonable assumption of negligible molecular effects, neglect of gap-scale dispersion requires more careful examination. In fact we lack a simple estimate for the gap-scale dispersivity. Parametrically we are always outside the Taylor dispersion regime in a typical process situation, but there is still a wide range of conditions. This makes rigorous inclusion or neglect of gap-scale dispersion difficult. Secondly, although numerical diffusion is handled reasonably well by the FCT scheme, other comparable numerical schemes have been developed in the past 20 years. To make a broad quantitative study of the dispersion it would be wise to at least benchmark against other numerical methods. Similarly, the form of interpolation of the physical properties of the fluids, coupled with the numerical solver, may have some effect on the degree of dispersion. For these reasons, we postpone such a study. On the other hand, we are perhaps too pessimistic about the value of such a study. Other researchers, e.g. [33], have found that the details of the concentration field are not dominant in complex flows of this type, but rather it is the bulk characteristics of the pressure field that governs the flow. In terms of our future directions, our study of flow stability here is somewhat preliminary. In this chapter we have focused primarily on displacements where a stable advancing front or diffuse frontal region results. We have considered fairly modest eccentricities and buoyancy numbers, so as to explore casing motion effects. When eccentricity and/or buoyancy is more dominant, the interface tends to elon- gate and become pseudo-parallel. We may expect different types of instability to result other than fingering, as studied e.g. in [14, 13]. The steady state analysis and our stability criterion developed in §3.5.2 both consider the situation with a clean interface, whereas the computations use a concentration field. This disparity of modeling approaches needs addressing. However, with a discontinuity in tangential velocity across the interface due to secondary flows, computation using an interface 126 3.6. Summary and discussion tracking method is challenging. A second option therefore is to develop the stability theory within the concentration-dependent fluid framework, as done for example by [6]. This would allow incorporation of gap dispersion effects via an anisotropic diffusion term, and appears to be a promising direction. A final direction that we are actively investigating is the inclusion of non- Newtonian fluid rheologies, as in the stationary casing model of [1]. The challenge here is primarily computational, i.e. how to include these model closures accurately but in a way that still allows computation in reasonable times. 127 3.7. Bibliography 3.7 Bibliography [1] S. Bittleston, J. Ferguson, and I. Frigaard. Mud removal and cement placement during primary cementing of an oil well. 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In this situation we were able to find conditions on the dimensionless rheological parameters, annulus eccentricity and (small) angle of inclination from horizontal, under which a steady traveling wave displacement front could exist. Such displacements correspond to a 100% efficient displacement. In other situations the displacement is less efficient: the interface is found to elongate progressively during the displacement. When the displacement fluids have a yield stress, a more severe situation arises in which a fluid may get stuck on the narrow side of the annulus (see [9]), in which case the displacement is never complete. The principal motivation for the study of chapter 2 came from 6A version of this chapter has been submitted for publication. Carrasco-Teja, M. and Frigaard, I.A. Non-Newtonian fluid displacements in horizontal narrow eccentric annuli: Effects of slow mo- tion of the inner cylinder (July 2009) 132 4.1. Introduction the industrial process of primary cementing, which involves displacement of non- Newtonian fluids along narrow eccentric annuli. The process is described at length in [13]. See Fig. 4.1 later for a schematic of the process. In the industrial process, the fluids are drilling muds, spacer fluids, washes and cement slurries, which have a broad range of rheological properties. The annular space is formed by the borehole wall and the outer wall of a steel casing that is inserted into the newly drilled borehole. The casing slumps downwards under grav- ity, so that the annuli are eccentric. This space is occupied by drilling fluid which must be removed before the cement is placed, so as to ensure a good bond of the cement with the annular walls and a tight hydraulic seal. Drilling fluids are usually non-Newtonian and often possess a yield stress, which can lead to fluids getting stuck on the narrow side of the annulus. In the past 20 years there has been a mas- sive increase in the number of horizontal wells constructed worldwide, primarily to increase productivity by aligning the well with the reservoir. The early 1990’s saw a continual pushing of the horizontal extent of wells up to around 10km, see e.g. the detailed description in [15]. The 10km barrier was broken in a number of wells drilled at Wytch Farm, UK, around 2000. The limits of “extreme” extended reach wells are now being pushed into the 15−20km range, but such wells are unusual and do not necessarily bring productivity benefits proportional to their technical chal- lenges. In the present day it is routinely feasible to construct wells with horizontal extensions in the 7− 10km range. Although many of the potential problems of cementing horizontal wells were identified some time ago, (see e.g. [19]), the industrial response has been largely through technological advances, rather than by developing understanding of physical fundamentals that may affect the process. In longer horizontal wells it is increasingly common to have laminar flows, both due to smaller annular gap sizes and to the increased risk of high frictional pressures fracturing the surrounding formation at high flow rates. In the absence of a density difference between fluids, displacements in horizontal annuli are the same as in vertical annuli. However, drilling fluids are typically 100− 600kg/m3 lighter than cement slurries, and a chemically compatible spacer fluid designed to have intermediate density and rheological properties typi- cally separates these two fluids. This means that significant density differences are always present in the cementing of horizontal wells. 133 4.1. Introduction In chapter 2 we answered a number of questions, of both fluid mechanic interest and of practical industrial relevance. Which dimensionless groups govern whether or not the flow will become stratified? With a density difference, is it possible to have steady, traveling wave, displacement fronts? What are the effects of an increased flow rate on a horizontal displacement? In this chapter, which is still motivated by primary cementing displacements, we consider motion of the inner cylinder in the annulus. In the industrial setting it is becoming common to slowly rotate and/or reciprocate the inner cylinder, which is called the casing. The precise effects of casing movement is unknown, but the underlying reasoning is to forcibly shear the drilling mud, so as to ease displacement on the narrow side of the annulus. This objective is probably achieved: the walls of the annulus are fairly rough, so that slip is unlikely, and therefore imposing relative motion must yield the fluid. What is unknown however, are the wider effects of casing motion on these displacement flows. Developing this understanding is the objective of our study. The flows that we consider are laminar and the annuli considered have annu- lar gaps that are narrow with respect to both circumferential and axial length- scales. Thus, a Hele-Shaw modelling approach is appropriate and we adapt such an approach to the case of casing motion. The underlying idea of using a Hele- Shaw/porous media approach to model these displacements dates back to [7, 8], but was put in the present formulation by [2]. We first restrict the model of [2] to a uni- form annular section of constant inclination, eccentricity and radii, then incorporate both rotation and axial motion of the inner cylinder. The principal difficulty in developing the Hele-Shaw approach in this direction is computational. At its heart, Hele-Shaw (and related) methods rely on being able to solve a reduced 2D shear-flow, in the plane of the flow, then averaging the velocity across the narrow gap. The closure relation developed in this way relates the gap- averaged velocity field to the modified pressure gradient, (i.e. Darcy‘s law in the porous media context), and is used to eliminate either the pressure or the stream function from the governing equations. In the absence of wall motion, this closure relation can be expressed analytically, even for non-Newtonian fluids of the types commonly used. With a moving wall, the underlying shear flow is a 2D planar Poiseuille-Couette flow, for which the closure relations are only calculable analytically in the case of 134 4.1. Introduction Newtonian fluids. For Newtonian fluids therefore, development of a displacement model with wall motion is quite straightforward, leading to an elliptic linear 2D partial differential equation for the stream function. Although rheological aspects are missing, at least some characteristics of displacements with casing motion can be understood by a Newtonian model, and this we have developed in chapter 3. For non-Newtonian fluids, we formulate the underlying displacement model in this chapter. However, to solve a fully 2D displacement flow problem for such fluids is not attempted. The field equation for the stream function is still elliptic, but now is nonlinear, and to resolve the nonlinearity at each point in space requires numerical solution of the underlying local planar Poiseuille-Couette flow solution. Iteration is needed both for this local closure and for the elliptic field equation. The computational task is thus formidable. The results from chapter 3, on Newtonian fluid displacements are interesting. First of all, it has been possible to find steady state traveling wave solutions for concentric and mildly eccentric annuli, in the presence of casing motion. These solutions are developed via a domain perturbation method, that relies on the leading order interface being almost perpendicular to the direction of gravity. This occurs only when the buoyancy number of the flow is small. The buoyancy number, b̃, reflects the ratio of static pressure difference, (over the scale of the annular gap), to the principal viscous stresses. Although convenient to have such analytical solutions, small buoyancy number is relatively unusual in cementing. In chapter 3 we have shown numerically that in fact steady displacements also occur far away from the strict domain of validity of the assumptions needed for analytical solution. Casing rotation reduces the extension of the interface in the axial direction, and also results in an azimuthal phase shift of the steady shape away from a symmetrical profile. The phase shift results in the positioning of heavy fluid over light fluid along segments of the interface. When the axial extension of the interface is sufficiently large this leads to a local buoyancy driven fingering instability. A simple theory is developed in chapter 3 for this type of fingering. Over longer times, the local fingering is replaced by steady propagation of a diffuse interfacial region that may spread slowly due to dispersion. Slow axial motion of the annulus walls on its own is apparently less interesting. There is no breaking of the symmetry of the interface and hence no instability. However, axial wall motion does generate secondary flows 135 4.1. Introduction which may combine with those from inner cylinder rotation resulting in enhanced dispersive effects. The numerical regimes explored in chapter 3 are restricted to small and O(1) buoyancy number. The large buoyancy number regime was studied in chapter 2, but for a stationary casing (annulus with fixed walls). At large |b̃|, buoyancy effects dominate and the interface slumps under gravity as it advances. The slump distance scales with |b̃| ≫ 1, which prompted the development of a lubrication style displacement model in chapter 2. This approach not only leads to analytical results, (discussed earlier), but also is sensible computationally in avoiding computing over long-thin domains. Later, in the second part of this chapter, we develop a similar lubrication model approach, but now including motion of the inner cylinder. Other than the work reviewed above, the model of [2] has been extensively stud- ied in [16, 17] and [18], for near vertical annuli. As with horizontal displacements, the dynamics are dominated by the existence (or not) of steady traveling wave so- lutions, i.e. for certain parameter values the displacement front advects along the annulus at the mean pumping speed. Analytic solutions to the displacement prob- lem, exhibiting the steady traveling wave behaviour, are constructed by [16] and investigated numerically by [17]. It is also possible to predict the domains of exis- tence of steady and unsteady displacements using a lubrication-style displacement model; see [18]. A time-dependent version of the Hele-Shaw part of the model was developed in [12] and used to investigate interfacial instabilities in [11, 10]. There are also a limited number of computational studies of annular displacement flows, which do not follow the Hele-Shaw approach. Szabo & Hassager, [21, 22], have computed a 3D immiscible displacement flow between 2 Newtonian fluids using the arbitrary Lagrange-Euler formulation. The model shows reasonable agreement with a film-draining model derived for concentric annular displacements and also shows that the displacement efficiency drops significantly with annular eccentricity. 3D approaches have also been taken by other authors, using general purpose CFD codes, e.g. [3, 6, 14, 23]. As with [21, 22], such studies have value in understanding details of the flow near the interface but are of limited use in understanding flows on a larger scale, which is the advantage of the Hele-Shaw approach. A hybrid 2D/3D approach has recently been adopted by [20], in which the Navier-Stokes equations are simplified by ignoring radial velocities and azimuthal pressure gradients, but 136 4.2. Model outline the model is still resolved in 3D. The authors are able to include casing motion in this approach, but a justification of the assumptions and even details of the final simplified model are not given in [20]. An outline of the chapter is as follows. In §4.2 we present the Hele-Shaw model including casing motion. Two formulations of the model are given: one based on a concentration equation and one with a clean interface (more suitable for analytical work). We also discuss the closure problem, in §4.2.1 and later in more depth in appendix B. Displacements at large buoyancy number are considered in §4.3. We first use the Newtonian displacement model from chapter 3 to demonstrate that long interfaces, of length ∼ O(|b̃|), also occur in the presence of casing motion, i.e. as for the stationary casing situation in chapter 2. We then derive a lubrication- style displacement model for the interfacial region of such flows, which results in a quasi-linear advection-diffusion equation for the interface position(s). We simplify the analysis using a perturbation procedure, which shows that the leading order interface position is symmetric, with a rotational asymmetric “shift” occurring at first order. In §4.4 we address the key question of whether or not steady traveling wave solutions are possible. We provide conditions that are necessary and sufficient for the lubrication model to have steady solutions, and explore solutions for typical parameter ranges in Newtonian, power law and Bingham fluid models. The chapter closes with a brief discussion of the results in §4.5. 4.2 Model outline The model derivation that we adopt follows closely that in [2], see also [16] and chapters 2 and 3. We consider displacement flows in a nearly-horizontal, narrow eccentric annulus, in which the inner cylinder may slowly translate and/or rotate. We use a coordinate system (y, φ, ξ), as illustrated in Fig. 4.1: y measures radial dis- tance from the centreline of the annular gap, φ ∈ [−1, 1] denotes a scaled azimuthal coordinate, and ξ measures distance along the annulus. The geometry is described by the eccentricity, e, angle of inclination to the vertical, β, and annular gap width, 2H(φ) ≈ 2(1 + e cos πφ), 137 4.2. Model outline i.e. y ∈ [−H(φ),H(φ)]. The inner cylinder is rotating at a dimensionless speed vC , and reciprocating with a velocity wC . The underlying model we shall derive is a Hele-Shaw style model, with now the added complication that the inner wall of the eccentric annular Hele-Shaw cell is moving. As before, we work with 2 formulations for the model: (i) a fluid concentration -based formulation, which is effective for computational simulation; (ii) an inter- face tracking formulation which is more convenient for analytical work. The fluid concentration formulation is considered in the large Péclet number limit, in which diffusion/dispersion is neglected, and the two models are formally equivalent. Figure 4.1: Schematic of the process geometry: a) primary cementing of a hori- zontal well; b) narrow eccentric annulus; c) annular cross section; d) unwrapped annulus/periodic Hele-Shaw cell. 4.2.1 Fluid concentration formulation Scaling and model reduction follows the steps in [2] and chapter 2. The velocity components in (y, φ, ξ) directions are denoted (u, v,w), and to leading order the 138 4.2. Model outline mass-conservation equation is ∂u ∂y + ∂v ∂φ + ∂w ∂ξ = 0. (4.1) To eliminate the radial velocity, we average across the gap width, using conditions of no-slip at the annulus walls to get ∂ ∂φ [Hv̄] + ∂ ∂ξ [Hw̄] = 0, (4.2) where (v,w) are the averaged velocity components in the (φ, ξ)-directions, i.e. v̄ = 1 2H ∫ H −H v dy, w̄ = 1 2H ∫ H −H w dy. (4.3) The two key assumptions made in this modeling approach are: (i) that the fluid concentration c is uniform across the narrow annular gap; (ii) that diffu- sion/dispersion may be neglected. The reader is referred to [2] for a discussion of these assumptions. Making these assumptions allows us to derive the following leading order equation for the gap-averaged fluid concentration c: ∂ ∂t [Hc] + ∂ ∂φ [Hv c] + ∂ ∂ξ [Hw c] = 0. (4.4) Note that under the assumptions made, c = c, but for consistency with our previous work we use the notation c. The fluids are modeled as Herschel-Bulkley fluids. The dimensionless density, consistency, power law index and yield stress of pure fluid k are denoted respectively by: ρk, κk, nk and τk,Y , for k = 1, 2. In the concentration- based formulation, fluid properties at intermediate concentrations are modeled via closure expressions involving the above pure fluid properties and c, i.e. we write sim- ply ρ(c), κ(c), n(c) and τY (c) for these closures. For simple displacements, (e.g. pure fluid 1 displaces pure fluid 2 at given flow rate with an initially sharp interface), intermediate concentrations only arise local to the interface, due to numerical dif- fusion/dispersion effects. We have not found that the precise closure relationships used have a significant effect on such displacements and consequently have used only linear interpolation of the fluid properties. 139 4.2. Model outline Only the gap-averaged velocity and concentration appear in equations (4.2) and (4.4), which are valid in the 2D domain (φ, ξ) ∈ [−1, 1] × [0, L], where typically L ≫ 1. To provide closure of this system we derive the leading order momentum equations, using the typical scaling arguments of Hele-Shaw models. The pressure does not vary across the annular gap and the reduced momentum equations in (φ, ξ)-directions are: ∂τφy ∂y = −Gφ (4.5) ∂τξy ∂y = −Gξ (4.6) where G = (Gφ, Gξ) is the modified pressure gradient field, given by: G = ( −pφ + ρ sin β sinπφ St∗ ,−pξ − ρ cos β St∗ ) . (4.7) Here St∗ is the Stokes number for the flow, which is defined below in §4.2.3, where we also summarise the scaling used and other dimensionless groups. The scaled leading order constitutive model is τφy = η ∂v ∂y ⇐⇒ τ > τY , (4.8) τξy = η ∂w ∂y ⇐⇒ τ > τY , (4.9) γ̇ = 0⇐⇒ τ ≤ τY , (4.10) where τ = ( τ2φy + τ 2 ξy ) 1 2 , the leading order rate of strain 2nd invariant is γ̇ = ( ∂v ∂y 2 + ∂w ∂y 2) 12 , and the effective viscosity is: η = κγ̇n−1 + τY γ̇ . (4.11) 140 4.2. Model outline The remainder of the model derivation proceeds along standard lines. First we define the gap averaged velocity in terms of a stream function Ψ: 2Hw̄ = ∂Ψ ∂φ , 2Hv̄ = −∂Ψ ∂ξ . (4.12) Second we derive the closure relationship between G and the gap-averaged velocity u = (v,w), (or equivalently ∇Ψ). Thirdly, we cross differentiate G to eliminate the pressure, which leads to an elliptic problem for the stream function Ψ: ∇ · S = −∇ · f (4.13) where S = (Gξ ,−Gφ) and where the buoyancy terms manifest in the term f = ρ(c) ( cos β St∗ , sinβ sinπφ St∗ ) . (4.14) Boundary conditions for (4.13) are: Ψ(φ+ 2, ξ) = Ψ(φ, ξ) + 4 (4.15) ∂Ψ ∂ξ = −HvC , at ξ = 0, L. (4.16) The first of these ensures that the mean axial velocity is equal to 1, which follows from our choice of scaling. The second ensures that in the far-field, away from the interface, the azimuthal flow is given by the Couette component due to the moving casing. Remarks on the formulation and closure relations As is usual in this kind of 2D model, there is the option of eliminating either the pressure from the system and working with the stream function or vice versa. When the casing is moving, the fluids are sheared and the mapping from G to u is one to one. Thus either formulation could be used. However, for a stationary casing, the mapping from u to G is not uniquely defined at u = 0 for yield stress fluids. Physically, a finite range of modified pressure gradients G fail to mobilise a yield stress fluid, i.e. whenever |G| ≤ τY /H we have u = 0. This has led, in [2] and our 141 4.2. Model outline other work, to the adoption of the stream function formulation. For consistency of approach we retain the stream function formulation here for the moving casing. The main difference between this and the stationary casing models that we have worked with before is in the closure relation, between G and u, or equivalently between ∇Ψ and S. This closure is derived by solving the simplified momentum equations, (4.5) & (4.6), at each (φ, ξ), subject to boundary conditions: (v,w) = (vC , wC) at y = −H(φ), (v,w) = (0, 0) at y = H(φ). For a Newtonian fluid we derive straightforwardly the linear relations: G = 3κ H2 ( u− 1 2 (vc, wc) ) , (4.17) S = 3κ 2H3 (∇Ψ−H(wc,−vc)) , (4.18) separating Poiseuille and Couette effects. For the non-Newtonian fluids that we consider there is no fully analytical solution available if the casing velocity is non-zero. For stationary casing, there is an implicit relationship between ∇Ψ and S, that is derived in [2]. For a moving casing, the closure may be computed reasonably quickly and it is also possible to derive certain qualitative results. These results are derived fully in appendix B, but here we mention just 3 results that are useful later. 1. For fixed (vc, wc), the constant vector G uniquely defines u, (or equivalently S uniquely defines ∇Ψ). 2. If ∇Ψ1 is defined by S1 and ∇Ψ2 is defined by S2, the following monotonicity result holds: [∇Ψ1 −∇Ψ2] · [S1 − S2] ≥ 0. (4.19) 3. Defining the j-th moment of the mobility, mj by mj = ∫ H −H ( y H )j 1 η dy, (4.20) 142 4.2. Model outline The following relationships are satisfied: u = 1 2 (vc, wc) ( 1 + m1 m0 ) +H2 ( m2 − m 2 1 m0 ) G, (4.21) ∇Ψ = H(wc,−vc) ( 1 + m1 m0 ) + 2H3 ( m2 − m 2 1 m0 ) S. (4.22) The first of these results (a), ensures that we have a closure to compute. The second of these (b), can be exploited to ensure uniqueness of the stream function solution to (4.13). It also allows us to compute the closure in different ways. In fact, provided that (vc, wc) is non-zero the fluid is sheared and the inequality in (4.19) is strict, (as can be verified numerically). This means that for example, we may do any of: (i) specify ∇Ψ and compute S; (ii) specify S and compute ∇Ψ; (iii) specify both Ψφ & Sξ, and compute Ψξ & Sφ; (iv) specify both Ψξ & Sφ, and compute Ψφ & Sξ. This allows for considerable flexibility, that we shall later exploit. The third of the properties given above (c), shows that, although the mobility moments mj do need to be computed numerically, there is a well-defined splitting into Couette and Poiseuille components directly analogous to (4.17) & (4.18). Note that in the case of a Newtonian fluid, m1 = 0, m2 = 1/(3κ), and (4.17) & (4.18) are recovered. In the case of a pure Couette flow the stress is constant across the annular gap and hence m1 = 0. Finally, it is worth noting that the expression (m2 −m21/m0) > 0, from the Cauchy-Schwarz inequality, except in the degenerate case where mj = 0, ∀j, which corresponds to the fluid being unyielded throughout the channel and is not possible for non-zero (vc, wc). 4.2.2 Interface tracking formulation In this formulation the domain is divided into two fluid domains: Ω1 for the dis- placing fluid 1, and Ω2 for the displaced fluid 2, in each of which (4.13) is replaced by: ∇ · S1 = 0, (φ, ξ) ∈ Ω1, (4.23) ∇ · S2 = 0, (φ, ξ) ∈ Ω2, (4.24) 143 4.2. Model outline with Sk defined exactly as S is defined for the concentration dependent formulation, but with properties ρ1, τ1,Y , κ1, n1 in fluid 1 and ρ2, τ2,Y , κ2, n2 in fluid 2. Note that for constant density ρk, we have that ∇ · fk = 0, hence the right-hand side of (4.23) & (4.24). If the interface is denoted by φ = φi(ξ, t), this satisfies the kinematic condition: ∂φi ∂t + w̄ ∂φi ∂ξ = v̄, (4.25) which essentially replaces the concentration advection equation. The leading order continuity conditions at the interface are that the stream function Ψ and the pressure p are continuous across the interface. Assuming sufficient regularity of the interface, the former condition assures that the normal velocity, the derivative of Ψ along the interface, is well defined at the interface. Pressure continuity is expressed in terms of defining the jump in Sk across the interface: [( Sk,ξ ∂φi ∂ξ − Sk,φ ) + ( ρk sin β sinπφ St∗ ∂φi ∂ξ − ρk cos β St∗ )]2 1 = 0. (4.26) Effectively this is: [(Sk + fk) · n]21 = 0. 4.2.3 Summary of scaling and dimensionless numbers All parameters and variables used above are dimensionless, and our results will be presented in terms of dimensionless quantities. Throughout the chapter we denote dimensional quantities with the “hat” symbol, .̂ Our dimensionless model contains the following 13 dimensionless parameters. • Eccentricity: e ∈ [0, 1), which is defined in Fig. 4.1c. • Angle of inclination: β, as illustrated in Fig. 4.1b. • Azimuthal casing speed: vC = 0.5π(r̂o + r̂i)Ω̂C/ŵ∗, where r̂o and r̂i are the outer and inner radii of the annulus, Ω̂C is the angular velocity of the inner cylinder, and ŵ∗ is the mean velocity, defined in terms of the imposed flow rate, Q̂, by Q̂ = π(r̂2o − r̂2i )ŵ∗. Note that typically |vC | < 1. • Axial casing speed: wC = ŵC/ŵ∗, and typically |wC | < 1. 144 4.2. Model outline • Fluid j power-law index: nj. Typically the fluids are shear thinning, so nj ≤ 1 • Fluid j yield stress: τj,Y = τ̂j,Y /τ̂∗, where τ̂∗ = maxk=1,2{τ̂k,Y + κ̂k[ˆ̇γ∗]nk}, is a viscous stress scale defined using ˆ̇γ∗ = 2ŵ∗/(r̂o − r̂i), as a representative strain rate. Note that by definition, τj,Y ≤ 1. • Fluid j consistency: κj = κ̂j [ˆ̇γ∗]nk/τ̂∗}. Note that by definition, κj ≤ 1. • Fluid j density: ρj = ρ̂j/maxk=1,2{ρ̂k}, so that one of the densities is = 1 and the other is ≤ 1, representing the density ratio. Here ρ̂j is the dimensional density of fluid j. • Stokes number: St∗ = [2τ̂∗]/[maxk=1,2{ρ̂k}ĝ(r̂o − r̂i)], where ĝ is the gravita- tional acceleration. This is a formidable set of parameters, but there are some restrictions in range, as indicated. First, we shall consider inclinations that are close to β = π/2. Secondly, we shall always consider that the heavier fluid displaces the lighter fluid, i.e. ρ1 = 1, ρ2 < 1. Thirdly, if we consider only the pure fluid properties and assume that the phases are advected without any diffusion/dispersion, the fluid density only enters the picture via buoyancy and always combines with the Stokes number. The effects of buoyancy are characterised by the dimensionless parameter, b̃ = (ρ2 − ρ1)/St∗: b̃ = ρ2 − ρ1 St∗ = (ρ̂2 − ρ̂1)ĝ(r̂o − r̂i) 2τ̂∗ , (4.27) clearly representing the ratio of buoyant to viscous forces. Note that since the displacing fluid is assumed heavier here, b̃ < 0. In chapter 3 we have considered |b̃| = O(1), mostly numerically, and for Newtonian fluids only. Here we shall explicitly consider the limit of |b̃| ≫ 1, in which we also allow deviations from a horizontal annulus, with angles |β−π/2| ∼ O(1/|b̃|). Thus, the main dimensionless parameters governing this limit are the eccentricity e, the dimensionless casing speeds, vC and wC , and the 6 rheological parameters (all bounded above by 1). The dimensionless casing speeds are of O(1) partly for practical reasons. The casing is moved primarily in order to ensure that any gel in the drilling mud is broken. Large values of wC mean a rapid reciprocation which is impractical over length-scales of 100’s of meters driven mechanically from surface. Note that a value 145 4.3. Displacements at high buoyancy numbers wC > 2 would mean that the Couette component of velocity (driven by pulling the casing) is faster than the mean pumping velocity, which is very unlikely. Similarly, we should note that large rotation rates may result in shearing of the casing and/or some component of the drive mechanism, both of which are difficult and expensive to remedy downhole. From the mathematical perspective, in deriving the Hele-Shaw model we would require that the casing speeds are of the same order as the mean flow velocity. In this chapter we assume that the casing speeds are constant, but this is not strictly necessary. Suitably slow variations in time would not affect the model derivation, (and as the well is finite in length the axial speed must vary in time, i.e. the casing is reciprocated). To recover dimensional quantities from the reduced model, the axial and az- imuthal velocities have been scaled with the mean flow velocity, ŵ∗, axial and az- imuthal lengths with the half-circumference, 0.5π(r̂o + r̂i), and time with 0.5π(r̂o + r̂i)/ŵ ∗. 4.3 Displacements at high buoyancy numbers The model described in the previous section is derived for laminar displacement flows in horizontal annuli with arbitrary casing motion and rheological parameters. As discussed in §4.2.1 the main complexity of the model comes in the closure between ∇Ψ and S. This complexity arises due to casing motion, which imposes a Couette component and means that the modified pressure gradient is no longer parallel to the gap-averaged velocity field. In chapter 3 the Newtonian version has been solved computationally, which simply consists of a linear elliptic equation for the stream function, coupled to (4.4) for the concentration. For non-Newtonian fluids we could adapt the Augmented Lagrangian approach used in [17] and chapter 2, but having to evaluate the underlying model closures numerically makes this task extremely slow numerically. Therefore, we instead consider a limiting set of flows that allows further model simplification, i.e. the limit |b̃| ≫ 1. 146 4.3. Displacements at high buoyancy numbers 4.3.1 Example Newtonian simulations for |b̃| ≫ 1 In fixed casing flows at large |b̃|, the authors have observed that the displacement front typically elongates along the annulus for a distance of O(|b̃|). The elongated front may either propagate as a steady traveling wave, moving at the mean dis- placement velocity, or may spread further if the rheological parameters do not allow for a stable steady traveling wave solution. We explore whether this situation also occurs in the presence of casing motion, using the model and computational method detailed in chapter 3, for Newtonian fluid displacements. Figure 4.2 presents results from Newtonian computations with a 4:1 viscosity ratio and small eccentricity at b̃ = −50. Three simulations are presented at increas- ing casing rotation rates, with no axial casing velocity. The parameters are chosen so that at vC = 0 there exists a steady traveling wave solution, according to the conditions developed in chapter 2, and this is confirmed numerically. We observe that for modest increases in vC there is also a steady solution. Two interesting observations can be made from Fig. 4.2. First, it appears that the axial length of the interface does not change with the rotation rate. This is in contrast to many of the simulation results presented in chapter 3, where buoyancy was not dominant, i.e. |b̃| ∼ O(1). Secondly, we observe that the orientation of the streamlines changes. In each simulation, the interfacial region appears to be char- acterised by streamlines that are near-parallel to the annulus axis, but just outside this region the streamlines quickly become more angular. The angular behaviour is easily understood as it is simply associated with the Couette component from the rotation. We observe that the angle the streamlines make with the ξ-axis increases as the rotation rate is increased. In Fig. 4.3 we present a second set of simulation results. This time we have a 1:4 viscosity ratio, small eccentricity again, and have reduced |b̃| slightly. For these parameters the fixed casing simulation is unsteady, as is captured in Fig. 4.3a. We observe that the interface elongates progressively along the narrow side of the annulus, under the action of buoyancy. As the casing rotation rate is increased from zero, the displacement remains unsteady. In chapter 3, for more modest |b̃| we observed that O(1) casing rotation could in fact stabilise an otherwise unsteady displacement, but for |b̃| ≫ 1 this does not appear to occur. We also observe in the 147 4.3. Displacements at high buoyancy numbers 28 33 38 43 −1 0 1 ξ φ −5 0 5 (c) 28 33 38 43 −1 0 1 φ −4 −2 0 2 4 (b) 28 33 38 43 −1 0 1 t = 35 φ −4 −2 0 2 4 (a) Figure 4.2: Streamlines and interface, (c̄(φ, ξ) = 0.5), at t = 35 from the 2D numer- ical simulation: a) vC = .1; b) vC = 0.3; c) vC = 0.5. Other model parameters are: κ1 = 1, κ2 = 0.25, e = 0.1, wC = 0, b̃ = −50, (ρ1 = 1, ρ2 = .9, St∗ = 0.002). right hand panel of Fig. 4.3 the same streamline behaviour as in Fig. 4.2. 4.3.2 Derivation of the lubrication displacement model As we have seen, under conditions of dominant buoyancy and with modest casing motions, it is possible to have stable steady displacements with interfaces that be- come long. We consider the case |b̃| ≫ 1 and define ǫ = 1/|b̃| << 1. As in chapter 2 we adopt the interface tracking formulation described in §4.2.2. For simplicity we shall assume throughout that the displacing fluid 1 is heavier than the displaced fluid 2, and that the displacement front slumps towards the bottom of the annulus, φ = ±1. We also assume as in chapter 2 that the annulus is close to horizontal in 148 4.3. Displacements at high buoyancy numbers 0 2 4 6 8 10 12 14 16 18 −1 −0.5 0 0.5 1 ξ φ 0 10 20 30 40 50 −1 0 1 ξ −5 0 5 (c) 0 2 4 6 8 10 12 14 16 18 −1 −0.5 0 0.5 1 ξ φ 0 10 20 30 40 50 −1 0 1 −5 0 5 (b) 0 2 4 6 8 10 12 14 16 18 −1 −0.5 0 0.5 1 ξ φ 0 10 20 30 40 50 −1 0 1 t = 20 −5 0 5 (a) Figure 4.3: Examples of an unsteady displacement. Left: Interface (c̄(φ, ξ) = 0.5) shown at t = 1, 2, 3...15. Right: Streamlines and interface shown at t = 20: a) vC = 0; b) vC = 0.1; c) vC = 0.3. Other model parameters are: κ1 = 0.25, κ2 = 1, e = 0.1, wC = 0, b̃ = −10, (ρ1 = 1, ρ2 = .9, St∗ = 0.01). the sense that cos β = αǫ with α = O(1). The main complication introduced by the casing rotation is that the closure relation between Sk and ∇Ψ in each fluid is not easily specified for non-zero (vc, wc). Thus, the way to reduce the equations further is not immediately apparent. One might consider working with either Sk or ∇Ψ. Our choice of variable is dictated by the example results of Figs. 4.2 & 4.3, which have shown that long-thin interfaces are characterised by near-parallel streamlines within the interfacial region, (but not 149 4.3. Displacements at high buoyancy numbers outside in the single phase regions). Thus, we assume the following scaling (i) ∣∣∣∣∂Ψ∂ξ ∣∣∣∣ = O(ǫ), (ii) ∣∣∣∣∂Ψ∂φ ∣∣∣∣ = O(1), (iii) ∣∣∣∣∂φi∂ξ ∣∣∣∣ = O(ǫ) (4.28) From (4.22) we observe that (4.28)(i) implies |Sk,ξ| = O(vC) +O(ǫ), and (4.28)(ii) implies |Sk,φ| = O(wC) +O(1). We re-scale axial distances and time as follows: z = ǫξ, t̃ = ǫt, w̄ =W , v̄ = ǫV . The interface position is now φ = φi(z, t̃) and the kinematic equation: 2H ∂φi ∂t̃ + ∂Ψ ∂φ ∂φi ∂z + ∂Ψ ∂z = 0, (4.29) where, 2HV = −Ψz, 2HW = Ψφ. The scales on S may be used to derive the pressure scaling: −pξ − ρk cosβ St∗ = −ǫpz − α ρk|ρ2 − ρ1| = Sk,φ = O(wC) +O(1). This suggests that we introduce P = ǫp as a rescaled pressure, i.e. Sk,φ = −Pz − α ρk|ρ2 − ρ1| . Using (4.22) and the scaling of Sk,ξ we have: 1 ǫ [ Pφ − ρk|ρ2 − ρ1| sinπφ+O(ǫ 2) ] = pφ − ρ sin β sinπφ St∗ = Sk,ξ = vc 1 2H2 [ m1 +m0 m0m2 −m21 ] k +O(ǫ). (4.30) We suppose that the interface has slumped towards the narrow side of the an- nulus, φ = ±1, due to buoyancy effects: see Fig. 4.4. At any fixed z, as we increase φ ∈ [−1, 1] we will cross the interface exactly twice, (thus our use of φ = φi(z, t̃) is not precise, as the function is double valued). We shall denote the two crossing positions by φ−i and φ + i , with ∂φ−i ∂z > 0, ∂φ+i ∂z < 0, and assume the narrow side is occupied by the heavier fluid 1. If instead the narrow side is occupied by light fluid, similar expressions hold. 150 4.3. Displacements at high buoyancy numbers We integrate (4.30) with respect to φ at fixed z, within the interfacial zone, giving the following expressions, valid to O(ǫ2): P (φ, z, t) =P (−1, z, t) − ρ1 π|ρ2 − ρ1| [cos πφ+ 1] +ǫvc ∫ φ −1 1 2H2 [ m1 +m0 m0m2 −m21 ] 1 dφ, φ ∈ [−1, φ−i ] (4.31) P (φ, z, t) =P (−1, z, t) − ρ1 π|ρ2 − ρ1| − cos πφ−i π − ρ2 π|ρ2 − ρ1| cos πφ +ǫvc ∫ φ−i −1 1 2H2 [ m1 +m0 m0m2 −m21 ] 1 dφ+ ǫvc ∫ φ φ−i 1 2H2 [ m1 +m0 m0m2 −m21 ] 2 dφ, φ ∈ [φ−i , φ+i ] (4.32) P (φ, z, t) = P (−1, z, t) − ρ1 π|ρ2 − ρ1| [cosπφ+ 1] + cosπφ+i − cos πφ−i π +ǫvc ∫ φ−i −1 1 2H2 [ m1 +m0 m0m2 −m21 ] 1 dφ+ ǫvc ∫ φ+i φ−i 1 2H2 [ m1 +m0 m0m2 −m21 ] 2 dφ +ǫvc ∫ φ φ+i 1 2H2 [ m1 +m0 m0m2 −m21 ] 1 dφ, φ ∈ [φ+i , 1] (4.33) We now impose periodicity of the pressure, i.e. P (−1, z, t) = P (1, z, t), which leads directly to the condition: 0 = cos πφ+i − cos πφ−i π + ǫvc ∫ φ−i −1 1 2H2 [ m1 +m0 m0m2 −m21 ] 1 dφ + ǫvc ∫ φ+i φ−i 1 2H2 [ m1 +m0 m0m2 −m21 ] 2 dφ+ ǫvc ∫ 1 φ+i 1 2H2 [ m1 +m0 m0m2 −m21 ] 1 dφ. (4.34) Only the first term above is of order 1, and thus we have the leading order condi- tion that cos πφ+i = cos πφ − i which implies that φ − i = −φ+i . This has the important conclusion that at leading order the interface will be symmetric about wide and narrow sides of the annulus. We may readily see that in the above expression the effect of casing rotation comes in at first order on the symmetry of the interface. This is not to say that casing rotation can not affect the shape of the interface at 151 4.3. Displacements at high buoyancy numbers Figure 4.4: Schematic of the asymmetric interface leading order. Perturbation expansion Having determined the orders of magnitude of the various terms and some basic properties of the solution above, we now proceed in a formal perturbation expansion: Ψ ∼Ψ0 + ǫΨ1 + ǫ2Ψ2 + ... P ∼P0 + ǫP1 + ǫ2P2 + ... Sk ∼Sk,0 + ǫSk,1 + ǫ2Sk,2 + ... φ±i ∼φ±i,0 + ǫφ±i,1 + ǫ2φ±i,2 + ... Above we have established that φ−i,0 = −φ+i,0, and we shall simply write φ+i,0 = φi for the leading order symmetric interface. The leading order pressure is given by the even function P0(−φ, z, t) = P0(φ, z, t): P0(φ, z, t) = P (1, z, t) − ρ1(1 + cos πφ) π|ρ2 − ρ1| + cos πφ− cosπφi π , φ ∈ [0, φi], P (1, z, t) − ρ1(1 + cos πφ) π|ρ2 − ρ1| , φ ∈ [φi, 1]. (4.35) 152 4.3. Displacements at high buoyancy numbers The equations governing Sk are (4.23) & (4.24), which to leading order give: ∂Sk,0,φ ∂φ =0, (4.36) ∂Sk,1,φ ∂φ =− ∂Sk,0,z ∂z . (4.37) Thus, Sk,0,φ is independent of φ in each layer and using the leading order pressure: S1,φ =− Pz(1, z, t) − α ρ1|ρ2 − ρ1| (4.38) S2,φ =− Pz(1, z, t) − α ρ2|ρ2 − ρ1| − sinπφi ∂φi ∂z S2,φ =S1,φ + α− sinπφi ∂φi ∂z . (4.39) To proceed further we need to fix the closure relation between Sk and ∇Ψ. As discussed in §4.2.1 and described in detail in appendix B, the closure relations may be formulated and computed in many different ways. For the lubrication model, the most convenient is to specify both Ψξ & Sφ, and compute Ψφ & Sξ. Therefore, we assume the following relationships hold: Ψφ =Fk,1(Sk,φ,Ψξ), (4.40) Sk,ξ =Fk,2(Sk,φ,Ψξ), (4.41) where Fk,1 and Fk,2 are sufficiently smooth for what follows. Note that Ψξ ∼ ǫΨ0,z+ O(ǫ2), and thus via a Taylor expansion we have: Ψ0,φ =Fk,1(Sk,0,φ, 0), (4.42) Sk,0,z =Fk,2(Sk,0,φ, 0), (4.43) Ψ1,φ =Sk,1,φ ∂Fk,1 ∂Sk,φ (Sk,0,φ, 0) + Ψ0,z ∂Fk,1 ∂Ψξ (Sk,0,φ, 0). (4.44) As in [18] and chapter 2, the leading order stream function and pressure gradi- ent are now determined via consideration of the global mass conservation, i.e. the 153 4.3. Displacements at high buoyancy numbers periodic boundary condition on Ψ 4 = Ψ(1, z, t̃)−Ψ(−1, z, t̃) = ∫ 1 −1 Ψφ dφ. (4.45) On expanding (4.45) with respect to ǫ, and exploiting the symmetry of the leading order interface, the zero-th order expression is: 2 = ∫ 1 0 Ψ0,φ dφ = ∫ φi 0 F2,1(S2,0,φ, 0) dφ+ ∫ 1 φi F1,1(S1,0,φ, 0) dφ, = ∫ φi 0 F2,1(A, 0) dφ+ ∫ 1 φi F1,1(A− b, 0) dφ, (4.46) b = α− sinπφi∂φi ∂z , (4.47) A = S2,0,φ. (4.48) We note that the functions Fk,1(Sk,φ,Ψξ) increase strictly monotonically with respect to the first argument and thus A is uniquely determined at each z. Fur- thermore, we may observe that the dependency on (z, t̃) enters only via φi and ∂φi ∂z . For a given symmetric leading order interface, φi(z, t̃), let us assume that we have computed A(z, t̃) from (4.46) at each z. Thus, we have Sk,0,φ determined in each fluid and also Ψ0, via integration, e.g. Ψ0 = ∫ φ 0 F2,1(S2,0,φ, 0) dφ, φ ∈ [0, φi]. Therefore Ψ0 = Ψ0(φ, φi(z, t̃), ∂φi ∂z (z, t̃)). From symmetry considerations, we may infer that Ψ0 is an odd function of φ. Using (4.43) we may also compute Sk,0,z. Defining Φi(φi) = ∫ φi 0 H dφ, as the volumetric position of the leading order interface, the leading order kinematic equation is ∂Φi ∂t̃ + ∂q ∂z (Φi,Φi,z) = 0. (4.49) 154 4.3. Displacements at high buoyancy numbers where q ( Φi(z, t̃),Φi,z(z, t̃) ) = 1 2 Ψ0(φ = φi(z, t̃), φi(z, t̃), ∂φi ∂z (z, t̃)). This completes the leading order solution. First order perturbation To understand the asymmetry induced by casing rotation, we need to consider the first order perturbation. To start with we may consider the first order terms in (4.45), which are found to be: 0 =φ−i,1[Ψ0,φ(φ −,− i,0 )−Ψ0,φ(φ−,+i,0 )]− φ+i,1[Ψ0,φ(φ+,+i,0 )−Ψ0,φ(φ−,−i,0 )] + ∫ φ−i,0 −1 Ψ1,φ dφ+ ∫ φ+i,0 φ−i,0 Ψ1,φ dφ+ ∫ 1 φ+i,0 Ψ1,φ dφ, (4.50) where e.g. φ−,−i,0 is the limit as φ − i,0 approached from below, etc. Using the fact that Ψ0,φ is an even function and substituting from (4.44) we have: 0 =[φ−i,1 − φ+i,1][Ψ0,φ(φ+,+i,0 )−Ψ0,φ(φ−,−i,0 )] + ∫ φ−i,0 −1 S1,1,φ ∂F1,1 ∂S1,φ (S1,0,φ, 0) + Ψ0,z ∂F1,1 ∂Ψξ (S1,0,φ, 0) dφ + ∫ φ+i,0 φ−i,0 S2,1,φ ∂F2,1 ∂S2,φ (S2,0,φ, 0) + Ψ0,z ∂F2,1 ∂Ψξ (S2,0,φ, 0) dφ + ∫ 1 φ+i,0 S1,1,φ ∂F1,1 ∂S1,φ (S1,0,φ, 0) + Ψ0,z ∂F1,1 ∂Ψξ (S1,0,φ, 0) dφ, (4.51) We may observe that Sk,0,φ is an even function of φ and the partial derivatives of Fk,1 featured above are also even. From (4.37) we see that Sk,1,φ is an odd function of φ and equally we have that Ψ0,z is an odd function of φ. Combining this we find that all the integrals above vanish, leaving us with: [φ+i,1 − φ−i,1][Ψ0,φ(φ+,+i,0 )−Ψ0,φ(φ−,−i,0 )] =0 (4.52) Note that [Ψ0,φ(φ +,+ i,0 ) − Ψ0,φ(φ−,−i,0 )] is proportional to the jump in tangential ve- locities across the interface, which will in general be non-zero due to different fluid 155 4.3. Displacements at high buoyancy numbers rheological properties. Therefore, we have that: φ+i,1 = φ − i,1. (4.53) Thus, the first order perturbation of the interface gives the asymmetry due to rota- tion. To quantify the degree of asymmetry we enforce the condition of periodicity of the pressure with respect to φ, i.e. we expand (4.34) to first order in ǫ, using (4.53) and the fact that the mobility moments mj and leading order interface are even: φ+i,1 sinπφi = vc [∫ φi 0 1 2H2 [ m1 +m0 m0m2 −m21 ] 2 dφ+ ∫ 1 φi 1 2H2 [ m1 +m0 m0m2 −m21 ] 1 dφ ] . (4.54) Since the integrand is positive definite, we see that the symmetric interface is per- turbed by O(ǫ) in the direction of vc, as above. Note that typically we would not compute the momentsmj as part of the solution procedure, so the above is given for qualitative understanding. In developing the zero-th order solution, we will have evaluated the leading order Sk,0,z from (4.43). This can be used to compute φ+i,1 = φ − i,1: φ+i,1 sinπφi = ∫ φi 0 S2,0,z dφ+ ∫ 1 φi S1,0,z dφ. (4.55) Although we may proceed further to construct the 1st order stream function per- turbation and remainder of the 1st order solution, our principal interest lies with behaviour of the leading order interface and prediction of the asymmetry above. 4.3.3 Computational verification and comments Figure (4.5) shows two comparisons between numerical solutions of the full 2D problem, for Newtonian fluids, and the lubrication model. We have presented both the leading order symmetric profile and the first order asymmetric correction super- imposed. The agreement is evidently good, and the asymmetric shift is verified to be small, i.e. O(1/|b̃|), as expected. An interesting feature of the flows we have modeled is the transition from the 156 4.3. Displacements at high buoyancy numbers 33 38 43 −1 0 1 t = 35 φ −6 −4 −2 0 2 4 6 (a) 33 38 43 −1 0 1 ξ φ −15 −10 −5 0 5 10 15 (b) Figure 4.5: Comparison between the 2D numerical solution, (interface is the contour c̄(φ, ξ) = 0.5, in black), and the lubrication displacement model interface, (leading order symmetric solution in solid white; 1st order asymmetric shift - white dots): a) vC = .1; b) vC = 0.5. Other model parameters are: κ1 = 1, κ2 = 0.25, e = 0.1, wC = 0, b̃ = −50, (ρ1 = 1, ρ2 = .9, St∗ = 0.002). interfacial regime to the pure fluid regimes away from the interface. In the pure fluid regimes, the streamfunction adopts a solution in which Sξ = 0, so that the azimuthal velocity matches the Couette shear component. It can be verified that in a pure fluid, ∇ · f = 0, so that the field equations for the streamfunction are satisfied when Sφ is independent of φ; Sφ is found iteratively by increasing Sφ until (4.45) is satisfied. It is interesting to note that, for example, if the 2 fluids were both Newtonian or were both power law, but with the same power law index, then the streamfunction far upstream and downstream of the interfacial would be identical to within a constant! Nevertheless, the streamlines are distorted in the interfacial region. 157 4.4. Steady traveling wave solutions We should observe that some kind of matching region is necessary between the interfacial and far-field regions. In terms of our perturbation method, there is a transition layer between Ψξ ∼ ǫ in the interfacial region to Sξ ∼ ǫ in the pure fluid regions. The classical lubrication assumptions on the interface being long and thin, i.e. equation (4.28)(iii), break down in the vicinity of the leading and trailing edge. For example, the expression (4.55) for the azimuthal shift in interface position becomes apparently singular at φi = 0, 1. This is a characteristic of the breakdown of a regular perturbation procedure. We have not looked at the matching problem, although it is probably tractable. The same type of breakdown of the modeling assumptions occurs in most lubrication-style displacement (or thin film spreading) models. Finally, note that since the 1st order asymmetric shift is dependent only on the zero-th order solution, via (4.55), one could use the lubrication model as a relatively quick simulation model for the full process, i.e. solving (4.49). The leading order model is symmetric and it can be verified that (4.49) approaches the fixed casing limit, (equation (4.13) in chapter 2, as the casing speeds approach zero. Also the first order asymmetry vanishes, since Sk,0,z → 0 as vC → 0. We have not followed this computational route, but instead below we analyse (4.49) directly in order to ascertain if there exist steady travelling wave solutions. 4.4 Steady traveling wave solutions As in chapter 2 the key feature that we wish to predict is whether or not steady traveling wave solutions exist, and to characterise the effects of casing motion. In computing the flux function, note that formally we have: q ( Φi(z, t̃),Φi,z(z, t̃) ) = q ( Φi(z, t̃), b ) , b = α− sinπφi∂φi ∂z , Φi(φi) = ∫ φi 0 H dφ, and note that the only difference in leading order model between here and chapter 2 is in the detail of computing the closures. In particular we find that q increases 158 4.4. Steady traveling wave solutions monotonically with respect to b, and consequently (4.49) can be written as: ∂Φi ∂t̃ + ∂q ∂Φi (Φi, b) ∂Φi ∂z = ∂q ∂b (Φi, b) ∂ ∂z [sinπφi ∂φi ∂z ], (4.56) from which we see that the coefficient multiplying Φi,zz will be positive. There- fore (4.49) is a quasi-linear advection-diffusion equation. We expect spreading of the interface relative to some advective motion of mean speed 1, (due to scaling considerations). We seek a steady traveling wave solution, shifting to a moving frame, moving at a unit speed, x = z − t. We assume that the steady profile will be monotone and symmetric. Thus Φi(x), (or equivalently x(Φi)), will satisfy x′(0) = 0, x′′(0) > 0 x′(1) = 0, x′′(1) < 0 x′(Φi) > 0 Φi ∈ (0, 1). (4.57) The function Φi(x) must satisfy the following equation q ( Φi, b(Φi,Φ ′ i) )− Φi = 0, (4.58) which may also be interpreted as an algebraic equation for x(Φi). Furthermore, by exactly the same methods as in chapter 2, we may prove the following. Theorem 3. The condition that q(Φi, α) ≥ Φi for all Φi ∈ [0, 1] (4.59) is a necessary and sufficient condition for the existence of a steady-state traveling wave solution to (4.49), that also satisfies the conditions (4.57) 4.4.1 Newtonian fluids Newtonian fluids form an important special case. We may note that all mobility moments mj are constant: mj = 1/(jκk) in fluid k. This means that the leading order solution may be calculated semi-implicitly. The flux function q(Φi, b) is given 159 4.4. Steady traveling wave solutions by: q(Φi, b) = IL κ2 κ1 IH + IL + wc 2 [ Φi − ILκ2 κ1 IH + IL ] + κ2 κ1 IL b 3κ1 IH + IL (4.60) where IL = ∫ φi 0 H(φ)3dφ, IH = ∫ 1 φi H(φ)3dφ. The condition to have a steady state for two Newtonian fluids is therefore given by κ1/κ2 > ΦiIH (1− Φi)IL − αIH 3 ( 1− wC2 ) (1− Φi)κ2 , ∀Φi ∈ [01]. (4.61) Note that for a completely horizontal well α = 0, the axial velocity doesn’t affect this ratio. Apart from the wc term, this expression is identical with that for the stationary casing flows in chapter 2. We may straightforwardly compute the critical viscosity ratio necessary to satisfy the condition (4.59) at different wc and α/κ2. It is interesting that only the axial casing velocity affects the leading order solution. Casing rotation vc introduces a first order asymmetry of the interface, i.e. the interface positions are at: φ = −φi + ǫφ−i,1, φ = φi + ǫφ+i,1, φ+i,1 = φ−i,1, where φ+i,1 sinπφi = 3vc 2 [ κ2 ∫ φi 0 1 H2 dφ+ κ1 ∫ 1 φi 1 H2 dφ ] . (4.62) This explains the apparently identical axial extensions of the interfaces shown earlier in Fig. 4.2, computed from the 2D model. In the left panel of Fig. 4.6 we explore the variation in critical viscosity ratio with the annular eccentricity, for different inclination parameters α, (recall that α < 1 means that the heavy fluid is displacing the lighter fluid downhill). When displacing downhill, (Fig. 4.6a), a positive axial casing velocity requires a larger viscosity ratio to achieve a steady displacement than does a negative axial casing velocity. When displacing uphill, (Fig. 4.6c), this trend is reversed. For a perfectly horizontal annulus there is no effect of the casing motion. We observe a decrease in 160 4.4. Steady traveling wave solutions viscosity ratio for increasing eccentricity, which parallels the fixed casing results in chapter 2. Physically, buoyancy causes the interface to slump towards the bottom of the annulus, which is a destabilising effect. A small amount of eccentricity counters the slumping, by making it harder to flow on the narrow side. At large enough eccentricities the critical viscosity ratio increases, as it becomes increasingly difficult to flow on the narrow side, and we therefore see a minimum in the viscosity ratio needed for a steady state displacement, e.g. see Fig. 4.6a. 0 0.2 0.4 0.6 0.9 1 1.1 1.2 1.3 e κ 1 /κ 2 −0.8 −0.6 −0.4 −0.2 0 −1 −0.5 0 0.5 1 φ i x(φi) 0 0.2 0.4 0.6 0.7 0.8 0.9 e κ 1 /κ 2 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 −1 −0.5 0 0.5 1 φ i x(φi) 0 0.2 0.4 0.6 0.2 0.4 0.6 e κ 1 /κ 2 −0.2 −0.15 −0.1 −0.05 0 −1 −0.5 0 0.5 1 φ i x(φi) (a) (b) (c) Figure 4.6: Effects of small inclination from horizontal on steady state displacement solutions for two Newtonian fluids: Left panel, critical viscosity ratio above which we have a steady state displacement solution; Right panel, sample steady state shapes (solid lines) and with 1st order asymmetric corrections (dotted lines), (parameters are κ1/κ2 = 1.5, e = 0.1); a) α = −1; b) α = 0; c) α = 1. In each figure we plot wC = −0.5 2, 0 △, 0.5 ◦. Other model parameters are: vC = 0.5, b̃ = −50. The right panel of Fig. 4.6 shows examples of the steady state shapes, at small eccentricity, e = 0.1, and with κ1/κ2 = 1.5. A positive axial velocity tends to 161 4.4. Steady traveling wave solutions elongate the steady state profile as does a downhill displacement. The asymmetric first order perturbation is shown with the dotted profile, but only in the central part of the interfacial region since we have seen that the perturbation procedure breaks down at the ends of this region. Figure 4.7 shows the effects of increasing casing rotation rates: both positive and negative. The basic effect is of course to accentuate the asymmetry in the leading order. As we increase the viscosity ratio, steady state shapes become shorter, (see Fig. 4.7a-c). Moderate increases in eccentricity also result in a shorter steady state, (see Fig. 4.7d-f). −0.2 −0.15 −0.1 −0.05 0 −1 −0.5 0 0.5 1 φ i x(φi) −0.2 −0.15 −0.1 −0.05 0 −1 −0.5 0 0.5 1 φ i x(φi) −0.15 −0.1 −0.05 0 −1 −0.5 0 0.5 1 φ i x(φi) −0.4 −0.3 −0.2 −0.1 0 −1 −0.5 0 0.5 1 φ i x(φi) −0.2 −0.15 −0.1 −0.05 0 −1 −0.5 0 0.5 1 φ i x(φi) −0.15 −0.1 −0.05 0 −1 −0.5 0 0.5 1 φ i x(φi) (a) (b) (c) (d) (e) (f) Figure 4.7: Steady state shapes for two Newtonian fluids with varying casing rotat- ing speeds. Right panel: a) κ1/κ2 = 1.5; b) κ1/κ2 = 1.7; c) κ1/κ2 = 1.9. For each plot e = 0.05, and interfaces are plotted at vC = 0 (−) , 1, (· · · ), 2 (+) . Left panel: d)e = 0; e)e = 0.2; f) e = 0.4. For each plot κ1/κ2 = 1.2, and interfaces are plotted at vC = 0 (−) ,−1, (· · · ),−2 (+) . Other model parameters are: wC = 0.3, b̃ = −20. 162 4.4. Steady traveling wave solutions 4.4.2 Power law fluids For power law fluids, we are unable to compute directly the flux function q(Φi, b), as the closure relation is not specified algebraically. However, we may demonstrate that the conditions under which a steady state exists can be expressed in terms of a consistency ratio. To see this, consider first the closure problem for a power law fluid. We have, (see also appendix B): ∂ ∂y τ = −G, τ = (τφy, τξy) = κγ̇n−1 d dy u. The boundary conditions for u = (v,w) are u = uc at y = −H and u = 0 at y = H. On dividing through by κ we observe that the local velocity u depends only on: n, H, uc and G/κ = (Gφ/κ,Gξ/κ). Therefore, the closure expression (4.40), we may observe that Fk,1(Sk,φ, 0) = Fk,1(Sk,φ/κk;nk,H,uc), and we know that Fk,1 increases monotonically with its first argument. Suppose now at any Φi we define AL/κ2 and AH/κ1 by: Φi = 1 2 ∫ φi 0 FL,1(AL/κ2;n2,H,uc) dφ, 1− Φi = 1 2 ∫ 1 φi FH,1(AH/κ1;n1,H,uc) dφ. Following the procedure in chapter 2, these integrals increase monotonically with respect to Ak/κk, and hence for fixed Φi, nk, e and uc we may compute both Ak/κk iteratively. We may therefore write: AL/κ2 = fL(Φi, n2, e,uc); AH/κ1 = fH(1−Φi, n1, e,uc). We now define b(Φi) = AL−AH = fL(Φi, n2, e,uc)κ2−fH(1−Φi, n1, e,uc)κ1. Also we have that b = α− sinπφi ∂φi∂z along the steady state, and require that ∂φi∂z > 0 for 163 4.4. Steady traveling wave solutions 0.2 0.4 0.6 0.8 1 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 n1 κ 1 /κ 2 −2 −1.5 −1 −0.5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 φ i x(φi) 0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1 1.1 n2 κ 1 /κ 2 −4 −3 −2 −1 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 φ i x(φi) (a) (b) Figure 4.8: Left panel shows effects of power law indices on the critical consistency ratio necessary to have a steady state for two power law fluids: a) n2 = 0.2, variations with n1; b) n1 = 0.66, variations with n2. Other model parameters: wC = 0, α = 0, e = 0.2, and we plot vC = 0 , 0.5 △, 1 ◦. Right panel shows sample steady state shapes (solid lines) and asymmetric shifts (dotted lines) for b̃ = −20: a) κ1/κ2 = 1.45, n1 = 0.2, n2 = 0.2; b) κ1/κ2 = 0.6, n1 = .66, n2 = .2. 164 4.4. Steady traveling wave solutions φi ∈ [0, 1]. This gives us the condition: κ1/κ2 > fL(Φi, n2, e,uc)− α/κ2 fH(1− Φi, n1, e,uc) , ∀Φi ∈ [0, 1], (4.63) in order for there to be a steady traveling wave displacement; compare with (2.47) in chapter 2. Figure 4.8 explores variations in the critical consistency ratio, κ1/κ2, in a hor- izontal annulus with no axial motion of the inner cylinder. We explore the effects of varying n1 and n2 at small e = 0.2, for different vC . Note here that the casing rotation rate does enter into the conditions for there to be a steady state, unlike the Newtonian fluid case. We observe that the critical consistency ratio decreases with n1 and increases with n2. These effects are expected on physical grounds as they simply correspond to the displacing fluid becoming more (or less) viscous than the displaced fluid. More surprising is the relative insensitivity of the critical consistency ratio to the casing rotation rate. The shear thinning effects of casing rotation manifest more in the shape of the interface, as shown in the right panel of Fig. 4.8. For identical power law indices, (Fig. 4.8a), as the rotation rate increases the effective viscosity drops and it appears that the interface elongates. In Fig. 4.8b the power law index of the displacing fluid is larger than that of the displaced fluid. As the rotation rate increases the effective viscosity of the displaced fluid decreases more rapidly than that of the displacing fluid. The consequence is that the interface shortens with rotation rate, which is perhaps counterintuitive. Figure 4.9 explores the effects of axial casing velocity for two power law fluids. In the left panel we explore variations in the critical consistency ratio, κ1/κ2, with power law indices. The results are qualitatively similar to Fig. 4.8 in that increasing n1 decreases the critical consistency ratio, whereas increasing n2 increases the critical consistency ratio. The effects of reciprocation wC do however appear to be more pronounced than those of changing vC . It also appears that when n1 > n2 a positive wC reduces the critical κ1/κ2, but for n1 < n2 this effect is reversed. The crossover is however approximately at n1 = n2. In the right-hand panel of Fig. 4.9 we observe that the interface length elongates with wC . 165 4.4. Steady traveling wave solutions 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 n1 κ 1 /κ 2 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 φ i x(φi) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 n2 κ 1 /κ 2 −1.5 −1 −0.5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 φ i x(φi) (a) (b) Figure 4.9: Left panel shows effects of power law indices on the critical consistency ratio necessary to have a steady state for two power law fluids: a) n2 = 0.4, variations with n1; b) n1 = 0.8, variations with n2. Other model parameters: vC = −1, α = 0, e = 0.3, and we plot wC = 0 , 0.5 △, 1 ◦. Right panel shows sample steady state shapes (solid lines) and asymmetric shifts (dotted lines) for b̃ = −10 and κ1/κ2 = 0.6: a) n1 = 0.9, n2 = 0.4; b) n1 = .8, n2 = .3. 4.4.3 Bingham fluids For any yield stress fluids, we can follow the procedure of the above section, to at least determine the functional dependency of the conditions for there to be steady state solutions. The closure expression (4.40) is: Fk,1(Sk,φ, 0) = Fk,1(Sk,φ/κk;Bk, nk,H,uc), 166 4.4. Steady traveling wave solutions with Bk = τk,Y /κk. We define AL/κ2 and AH/κ1 by: Φi = 1 2 ∫ φi 0 FL,1(AL/κ2;B2, n2,H,uc) dφ, 1− Φi = 1 2 ∫ 1 φi FH,1(AH/κ1;B1, n1,H,uc) dφ. Again these integrals increase monotonically with respect to Ak/κk, and there- fore may be inverted: AL/κ2 = fL(Φi, B2, n2, e,uc); AH/κ1 = fH(1− Φi, B1, n1, e,uc). (4.64) resulting again in the condition: κ1/κ2 > fL(Φi, B2, n2, e,uc)− α/κ2 fH(1− Φi, B1, n1, e,uc) , ∀Φi ∈ [0, 1], (4.65) in order for there to be a steady traveling wave displacement. For Bingham fluids in particular, we set n1 = n2 = 1. Figure 4.10 shows typical variations in critical consistency ratio for 2 Bingham fluids, as the eccentricity or either of the yield stresses is increased. The variation with e is qualitatively as before, with Newtonian fluids, and the qualitative effects of changing the yield stress of either fluid are predictable in terms of their effect on the fluid viscosity, i.e. increasing the yield stress also increases the effective viscosity. Hence increasing τ1,Y reduces the critical κ1/κ2 and increasing τ2,Y increases the critical κ1/κ2. As before we observe that the critical conditions are not affected much by changes in vC . The right panel of Fig. 4.10 shows example steady state profiles. In Fig. 4.10a & b casing rotation appears to elongate the interface, perhaps via shear-thinning effects. In Fig. 4.10c, where the displacing fluid yield stress is smaller than that of the displaced fluid, casing rotation results in a shorter interface. Figure 4.11 shows analogous results to Fig. 4.10, but for casing reciprocation. We observe again that casing reciprocation has an apparently larger effect than casing rotation on the conditions for there to be a steady states. However, the effects of wC on length of the steady states appears to be more subdued than those of vC . We again find the trend of elongation of the steady state profile with increasing wC . 167 4.4. Steady traveling wave solutions 0 0.2 0.4 0.6 0.5 0.6 0.7 0.8 e κ 1 /κ 2 −1 −0.8 −0.6 −0.4 −0.2 0 −1 −0.5 0 0.5 1 φ i x(φi) 0 0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 τ1,Y κ 1 /κ 2 −1 −0.8 −0.6 −0.4 −0.2 0 −1 −0.5 0 0.5 1 φ i x(φi) 0 0.2 0.4 0.6 0.8 1 0.6 0.7 0.8 0.9 τ2,Y κ 1 /κ 2 −1 −0.8 −0.6 −0.4 −0.2 0 −1 −0.5 0 0.5 1 φ i x(φi) (a) (b) (c) Figure 4.10: Left panel, critical consistency ratio above which we have a steady state displacement solution for two Bingham fluids: a) κ1/κ2 vs e, with τ1,Y = 0.2, τ2,Y = 0.1; b) κ1/κ2 vs τ1,Y with e = 0.2, τ2,Y = 0.1; c) κ1/κ2 vs τ2,Y with e = 0.2, τ1,Y = 0.5. Other model parameters are: wC = 0, α = 0, and in each figure we plot vC = 0 , 0.5 △, 1 ◦, Right panel shows sample steady state shapes (solid lines) and with 1st order asymmetric corrections (dotted lines): a) κ1/κ2 = 0.72, τ1,Y = 0.2, τ2,Y = 0.1; b) κ1/κ2 = 0.75, τ1,Y = 0.2, τ2,Y = .1; c) κ1/κ2 = 0.9, τ1,Y = 0.5, τ2,Y = 0.9. In each figure we plot vC = −0.5 2, 0 △, 0.5 ◦. Other model parameters are: e = 0.2, b̃ = −20. 168 4.4. Steady traveling wave solutions 0 0.2 0.4 0.6 0.6 0.7 0.8 0.9 1 e κ 1 /κ 2 −3 −2.5 −2 −1.5 −1 −0.5 0 −1 −0.5 0 0.5 1 φ i x(φi) 0 0.2 0.4 0.6 0.8 1 0.6 0.7 0.8 0.9 τ1,Y κ 1 /κ 2 −0.4 −0.3 −0.2 −0.1 0 −1 −0.5 0 0.5 1 φ i x(φi) 0 0.2 0.4 0.6 0.8 1 0.6 0.7 0.8 0.9 τ2,Y κ 1 /κ 2 −1 −0.8 −0.6 −0.4 −0.2 0 −1 −0.5 0 0.5 1 φ i x(φi) (a) (b) (c) Figure 4.11: Left panel, critical consistency ratio above which we have a steady state displacement solution: a) κ1/κ2 vs e, with τ1,Y = 0.5, τ2,Y = 0.5; b) κ1/κ2 vs τ1,Y with e = 0.2, τ2,Y = 0.1; c) κ1/κ2 vs τ2,Y with e = 0.2, τ1,Y = 0.5. Other model parameters are: vC = −0.5, α = 0, and in each figure we plot wC = −0.5 , 0 △ , 0.5 ◦. Right panel shows sample steady state shapes (solid lines) and with 1st order asymmetric corrections (dotted lines): a) κ1/κ2 = 0.83, τ1,Y = 0.5, τ2,Y = 0.5, e = 0.1; b) κ1/κ2 = 0.95, τ1,Y = 0.8, τ2,Y = .1, e = 0.2; c) κ1/κ2 = 0.85, τ1,Y = 0.5, τ2,Y = 0.8, e = 0.2. In each figure we plot wC = −0.5 2, 0 △, 0.5 ◦. Other model parameters are: b̃ = −100. 169 4.5. Discussion and conclusions 4.5 Discussion and conclusions In this chapter we have formulated a Hele-Shaw model for displacement flows along narrow eccentric annuli, in the case where the inner cylinder is moving, and where the fluids are shear-thinning with a yield stress. The resulting quasilinear elliptic PDE for the stream function has not been solved, primarily due to the complexity of the closure relations between flow rate and modified pressure gradient, in the presence of wall motion. This complexity translates into a heavy computational burden. Part of the contribution of this chapter is in characterising the closure problem, deriving qualitative results satisfied by the closure functionals, and in proposing an efficient algorithm with which to solve the closure problem computationally. These results are detailed in appendix B, and are important not only for the results in this chapter but also if one wishes to analyse and solve the fully 2D problem in the future. The main results of the chapter have been in developing a lubrication/thin-film style displacement model and in its analysis. This model focuses on the limit of large |b̃|, in which buoyancy-driven slumping is prevalent. We have been able to show that in this situation, for sufficiently large ratio of displacing fluid consistency to displaced fluid consistency, a steady traveling wave displacement front can be found. The length of the steady interface at leading order varies with all the dimensionless model parameters. An interesting observation is that due to shear-thinning effects, the steady interface can be either longer or shorter in the presence of casing rotation than without. This depends on whether or not the displacing fluid is more or less shear-thinning, as well as on the bulk ratio of effective viscosities. Viewed in a wider context, the large |b̃| results here complement those in chapter 2 for the fixed inner cylinder, suggesting that motion of the inner cylinder does not drastically change the underlying dynamics of the limit, |b̃| → ∞. The results in chapter 3, although only for Newtonian fluids, showed that for |b̃| ≪ 1 it was possible to extend the steady state results of [16] to the case of casing motion. i.e. again the underlying dynamics appear unchanged. However, we should not conclude that motion of the inner cylinder is unimpor- tant. First of all, for many of the flows computed in chapter 3, where |vC | ∼ |b̃|, 170 4.5. Discussion and conclusions localised interfacial instabilities led to mixing over short timescales, and over longer times to displacements that had diffuse interfacial region but nearly steady. Here too, if we consider sufficiently large vC , we may expect that local instabilities ap- pear, and the lubrication model becomes invalid. As an example of this, we present in Figs. 4.12 & 4.13 results of a Newtonian fluid displacement for successively large vC . We observe that the lubrication model prediction becomes progressively poor for vC > 1 and eventually we observe a significant shortening of the interface. 30.5 34.5 38.5 42.5 −1 0 1 ξ φ −50 0 50 (e) 30.5 34.5 38.5 42.5 −1 0 1 t = 35 φ −10 0 10 (a) 30.5 34.5 38.5 42.5 −1 0 1 φ −20 0 20 (b) 30.5 34.5 38.5 42.5 −1 0 1 φ −20 0 20 (c) 30.5 34.5 38.5 42.5 −1 0 1 φ −40 −20 0 20 40 (d) Figure 4.12: The effects of increasing rotation on interfacial stability for 2 Newtonian fluids; comparison of predicted steady states with numerical computation: a) vC = 0.5; b) vC = 1; c) vC = 1.5; d) vC = 2; e) vC = 2.5. Other parameters are: e = 0.1, κ1 = 1, κ2 = 0.25, α = 0, wC = 0, b̃ = −50, (ρ1 = 1, ρ2 = .9, St∗ = 0.002). See Fig. 4.5 for an explanation of the various curves. Thus, although the semi-analytical solutions in chapter 3 and here, are valuable in defining what happens at the limits of large and small buoyancy number, at intermediate buoyancy number, much of the complexity of the displacements does 171 4.5. Discussion and conclusions 28 30 32 34 36 38 40 42 44 −1 0 1 t = 35 φ −4 −2 0 2 4 (a) 28 30 32 34 36 38 40 42 44 −1 0 1 φ −5 0 5 (b) 28 30 32 34 36 38 40 42 44 −1 0 1 φ −5 0 5 (c) 28 30 32 34 36 38 40 42 44 −1 0 1 φ ξ −10 0 10 (d) Figure 4.13: Close-up of the interfacial region in Fig. 4.12, with parameters: a) vC = 0.5; b) vC = 1.5; c) vC = 2.5; d) vC = 5; (other parameters as before). See Fig. 4.5 for an explanation of the various curves. depend on transient 2D dynamics, for which computational solution is needed. In terms of practical consequences, perhaps the most interesting observation is that conditions for steady state displacements appear to be relatively unaffected by casing rotation. This implies that process design using the simpler fixed casing model of chapter 2 may be adequate, simply using casing rotation to ensure that the fluids are yielded/mobilised on the narrow side of the annulus. From an industrial perspective it may be felt desirable to categorise the effective- ness of the displacement flows in terms of a displacement efficiency. This has been considered but is perhaps an oversimplification that may become misleading. If we consider parameters for which we have a steady state, with axial length of O(1), this is clearly the best situation, with 100% efficiency. However, beyond that things 172 4.5. Discussion and conclusions are less clear. A typical cemented section has dimensionless axial length L of order 102 − 104. Therefore, although the the asymptotic results here give conditions for steady displacement, the axial lengths of these steady states are of order |b̃| ≫ 1. If |b̃| becomes of the same order as L, then whether or not the steady state is “useful” becomes an operational or economical question, e.g. (i) is it operationally feasible to pump more of a particular fluid, to ensure that the steady state transits the entire annulus; (ii) in the case that this is possible, are the additional costs of the fluids acceptable. We may just note that the additional volumes needed scale with |b̃| and material costs are a significant part of the cost of a cementing job. Taking this further, if |b̃| exceeds or is comparable to the axial length of the annulus, we must address the question of timescale to achieve the steady state, i.e. if the time required is longer than the residence time in the annulus a steady displacement may not be markedly different from an unsteady displacement. Secondly, once we begin to consider large |b̃| flows and stratification, questions of stability, mixing and en- trainment become important in affecting the efficiency of displacement. Although some progress has been made in this direction, (see [10, 11]), there is still much to be done. Finally, although it must be acknowledged that the cementing application con- sidered is a rather specialised flow, we should consider the wider fluid mechanical context of the work and its other applications. Firstly, the underlying yield stress fluid models, in the Hele-Shaw context, have a porous media analogue in so-called non-Darcy flows with limiting pressure gradient, i.e. these are nonlinear filtration problems in which yield stress effect is replaced by a critical pressure gradient that must be exceeded. This critical pressure gradient may be either a property of the porous media (e.g. argillaceous soils) or of the fluids flowing in the porous media, (e.g. heavy oils). The study of these flows was first carried out by the late V.M. En- tov, in the 1970’s and subsequently, and is summarised in the two texts: [5] and [1]. In this context, the annular geometry that we consider corresponds to a spatially periodic anisotropy in the porous media, and with a spatially periodic conservative (gravitational) force field imposed via f . For the Couette component in our flows the porous media analogy is unclear to us. We comment that for porous media studies, it is common to consider geometries in which the displacement fronts are planar of axisymmetric, and the study is focused 173 4.5. Discussion and conclusions on e.g. local fingering instabilities. Here, the anisotropy due to eccentricity, the casing motion and the complicated effects of buoyancy and rheology, mean that the underlying steady displacement flow is itself challenging enough to find. We have not even touched on questions of instability and fingering. Secondly, in the wider application context, we mention that similar Poiseuille- Couette flows occur in annular screw extruders, with narrow gaps and polymeric liquids. Similarly, rotating annular heat exchangers are used in the food industry, where one of the cylinders is often fitted with a scraper system. Here the closure laws are simplified by the assumptions of slow rotation and a dominant axial Poiseuille flow component; see e.g. [4]. The closure laws that we have developed below in appendix B are general to these flows and the algorithms suggested have wider utility. 174 4.6. Bibliography 4.6 Bibliography [1] G. Barenblatt, V. Entov, and V. Ryzhik. Theory of fluid flows through natural rocks. Theory and Applications of Transport in Porous Media, 3, 1990. [2] S. Bittleston, J. Ferguson, and I. Frigaard. Mud removal and cement placement during primary cementing of an oil well. Journal of Engineering Mathematics, 43:229–253, 2002. [3] E. Dutra, M. Naccache, P. Souza-Mendes, C. Souto, A. Martins, and C. de Mi- randa. Analysis of interface between Newtonian and non-Newtonian fluids in- side annular eccentric tubes. SPE paper 59335, 2004. [4] A. Fitt and C. Please. Asymptotic analysis of the flow of shear-thinning food- stuffs in annular scraped heat exchangers. Journal of Engineering Mathematics, 39:345–366, 2001. [5] R. Goldstein and V. Entov. Qualitative methods in continuum mechanics. Aca- demic Press., 1989. [6] J. Jakobsen, N. Sterri, A. Saasen, and B. Aas. Displacement in eccentric annuli during primary cementing in deviated wells. SPE paper 21686, 1991. [7] J. F. M.A. Tehrani and S. Bittleston. Laminar displacement in annuli: a com- bined experimental and theoretical study. SPE paper 24569, 1992. [8] M. Martin, M. Latil, and P. Vetter. Mud displacement by slurry during primary cementing jobs - Predicting optimum conditions. SPE paper 7590, 1978. [9] R. McLean, C. Manry, and W. Whitaker. Displacement mechanics in primary cementing. SPE paper 1488, 1966. [10] M. Moyers-González and I. Frigaard. Kinematic instabilities in two-layer ec- centric annular flows, part 1: Newtonian fluids. Journal of Engineering Math- ematics, 62:130–131, 2008. [11] M. Moyers-González and I. Frigaard. Kinematic instabilities in two-layer ec- centric annular flows, part 2: shear thinning and yield stress effects. Journal of Engineering Mathematics, DOI 10.1007/s10665-008-9260-0, 2008. 175 4.6. Bibliography [12] M. Moyers-González, I. Frigaard, O. Scherzer, and T.-P. Sai. Transient effects in oilfields cementing flows: Qualitatively behaviour. European Journal of Applied Mathematics, 18:477–512, 2007. [13] E. Nelson and D. Guillot. Well cementing, 2nd Edition. Schlumberger Educa- tional Services, 2006. [14] Q. Nguyen, T. Deawwanich, N. Tonmukayakul, M. Savery, and W. Chin. Flow visualization and numerical simulation of viscoplastic fluid displacements in eccentric annuli. AIP Conf. Proc. XVth International Congress on Rheology: The Society of Rheology 80th Annual Meeting, Monterey, Calif. USA, July 7, 2008, 1027:279–281, 2008. [15] M. Payne. Recent advances and emerging technologies for extended reach drilling. S.P.E., 1995. [16] S. Pelipenko and I. Frigaard. On steady state displacements in primary ce- menting of an oil well. Journal of Engineering Mathematics, 48:1–26, 2004. [17] S. Pelipenko and I. Frigaard. Two-dimensional computational simulation of ec- centric annular cementing displacements. IMA Journal of Applied Mathematics, 64:557–583, 2004. [18] S. Pelipenko and I. Frigaard. Visco-plastic fluid displacements in near - vertical narrow eccentric annuli: prediction of travelling wave solutions and interfacial instability. Journal of Fluid Mechanics, 520:343–377, 2004. [19] F. Sabins. Problems in cementing horizontal wells. Journal of Petroleum Tech- nologies, 42:398–400, 1990. [20] M. Savery, R. Darbe, and W. Chin. Modeling fluid interfaces during cementing using a 3d mud displacement simulator. SPE paper 18513, 2007. [21] P. Szabo and O. Hassager. Simulation of free surfaces in 3-d with the arbi- trary Lagrange-Euler method. International Journal for Numerical Methods in Engineering, 38:717–734, 1995. 176 4.6. Bibliography [22] P. Szabo and O. Hassager. Displacement of one Newtonian fluid by another: density effects in axial annular flow. International Journal of Multiphase Flow, 23(1):113–129, 1997. [23] E. Vefring, K. Bjorkevoll, S. Hansen, N. Sterri, O. Saevareid, B. Aas, and A. Merlo. Optimization of displacement efficiency during primary cementing. SPE paper 39009, 1997. 177 Chapter 5 Conclusions The main goal of this thesis was to model and analyse laminar displacement flows of viscoplastic fluids in nearly-horizontal annuli. The motivation being the primary cementing of highly deviated oil wells, which is idealised as the displacement of one fluid by another in a narrow, nearly-horizontal, eccentric annulus. In the preceding 3 chapters we modelled and analysed these displacements for different scenarios: • Displacement flow of Herschel-Bulkley fluids in an annulus with fixed walls, for both O(1) and large buoyancy number (Chapter 2). • Displacement flow of Newtonian fluids in an annulus with a moving inner wall. This was studied numerically for O(1) and small buoyancy number (Chapter 3). • Displacement flow of Herschel-Bulkley fluids in an annulus with a moving inner wall, in the limit of large buoyancy number using asymptotic methods, (Chapter 4). We used a 2D Hele-Shaw approach in all of the above cases. We have provided valuable analytic and computational tools to design a cementing job in horizontal wells. Overall, we have provided a number of valuable insights into the process design. In this chapter we summarise first the scientific contributions, then the indus- trial implications. We conclude with a critical look at our methodology and make recommendations for future work. 178 5.1. Scientific contributions 5.1 Scientific contributions The scientific advances and understanding are in three main categories: steady travelling waves, stability of the flow and the computational advances. 5.1.1 Steady travelling waves From the perspective of achieving an efficient displacement, the single most impor- tant feature to understand is whether or not there are steady travelling displace- ments. In all the scenarios of the previous three chapters we were able to find steady travelling waves under various (sometimes unexpected) conditions. These results were very surprising as intuitively one may think that with a horizontal an- nulus, the heavier fluid would usually travel faster in the bottom (narrow) side of the annulus than the lighter one, or that rotation would move the fluid around so fast that the interface would always be changing. Using numerical solutions and asymptotic methods we proved this intuition wrong. In Chapter 2 we analysed the displacement when the inner pipe is fixed. Using a fully 2D model and computational simulation we studied a range of displacements involving either a heavier fluid pushing a lighter one or vice versa. The role of density differences appeared slightly different in each case. When the displacing fluid is heavier, density differences and eccentricity compete, so that there is often a threshold in the (St∗, e)-plane delineating steady and unsteady displacements. If the displacing fluid is lighter, both eccentricity and buoyancy act to accentuate unsteadiness and stratification. However, in both cases we observed that these flows may either stratify or displace in steady state, according to different choices of the fluid rheology. In general, making the displacing fluid increasingly viscous led towards a steadier displacement. In a long computational domain, 2D simulation is less effective and more time consuming to compute, so we derived a 1D lubrication model. This model allowed us to find necessary and sufficient conditions for there to be a steady state displacement, and to explore variations in these conditions with the main dimensionless param- eters. In this approach, fluid rheology and eccentricity alone determine whether or not there may be a steady state. The buoyancy force translates simply into an axial length-scale for the slumped propagating steady interface, i.e. a larger density 179 5.1. Scientific contributions difference gives a longer axial extension to the steady state. We presented the results for Newtonian and power law fluids in terms of the ratio of consistencies of the two fluids: κ1/κ2. This ratio must exceed a certain critical threshold in order for there to be a steady state. Above the threshold, steady state shapes vary significantly with the other rheological parameters. The variation was fairly intuitive, i.e. as the displacing fluid is made more viscous the axial extent of the steady states decreases. Increasing eccentricity generally resulted in shorter steady states when the displaced fluid is heavier, and vice versa. As the critical threshold was approached, the axial extent of the steady state grew towards infinity. In Chapter 3 we showed that steady traveling wave displacements may also occur when rotation is present or with constant axial motion, but not with reciprocation. In the case of reciprocation, there was no steady state, but the interface moved in synchronisation with the reciprocation. The existence of the steady state was confirmed both analytically, via a perturbation method for small buoyancy numbers and eccentricities, and also computationally. From the perturbation method results we explicitly saw that rotation reduces the extension of the interface in the axial direction and also results in an azimuthal phase shift of the steady shape away from a symmetrical profile. It is important to note that we also found a different type of (pseudo) steady state numerically. This was found when rotation induced local instabilities and mixing at O(1) buoyancy numbers. We describe these further in the next section. In Chapter 4 we considered the case when the inner cylinder is moving, and where the fluids are shear-thinning with a yield stress. The main results are the development of a lubrication/thin-film style displacement model and in its analysis. This model focuses on the limit of large |b̃|, in which buoyancy-driven slumping is prevalent. We showed that in this situation, for sufficiently large ratio of displacing fluid consistency to displaced fluid consistency, a steady traveling wave displacement front can be found. The length of the steady interface at leading order varies with all the dimensionless model parameters. An interesting observation is that due to shear- thinning effects, the steady interface can be either longer or shorter in the presence of casing rotation than without. This depends on whether or not the displacing fluid is more or less shear-thinning, as well as on the bulk ratio of effective viscosities. 180 5.1. Scientific contributions The large |b̃| results of Chapter 4 complement those in Chapter 2 for the fixed inner cylinder, suggesting that slow motion of the inner cylinder does not drastically change the underlying dynamics of the limit, |b̃| → ∞. The results in Chapter 3, although only for Newtonian fluids, showed that for |b̃| ≪ 1 it was possible to extend the steady state results to the case of casing motion. For the O(1) buoyancy regime a full numerical solution is needed. We have observed steady state displacements in this regime, but also local instabilities and mixing in the presence of significant casing rotation. Largely, this domain still requires exploration and characterisation. 5.1.2 Stability of the flows At various points in the thesis, we have considered stability of the flows. How ever, before reviewing these results, we should first mention that, in most cases, (and in particular without casing motion), the steady travelling wave solution appears computationally to be globally stable. To be more precise, for 2D numerical simula- tions, we converge to the steady state for a wide range of physically sensible initial conditions and this appears to be the only attracting structure. Equally, when we have derived the lubrication displacement model, which is an advection-diffusion equation, it is clear from the construction of our proofs that the steady states solutions are unique, when they exist. From the diffusive nature of the lubrication model, we suspect that these steady states are also nonlinearly stable. When we have solved the transient lubrication model, we have again found that we have convergence for a wide range of initial conditions. However, we have not attempted to analyse the stability directly. In view of the above, our consideration of instability has been confined to situations when there is casing rotation. In Chapter 3 we were able to develop a simple analysis to predict buoyancy driven instabilities of the steady states. For nearly-flat interfaces (small buoyancy numbers), this was done with a Muskat-type fingering analysis. For larger buoyancy numbers, computational solution showed that the phase shift caused by rotation resulted in positioning of heavier fluid on top of lighter fluid along some segments of the interface. When the axial extension of the interface and the rotation were sufficiently large, this led to a local buoyancy driven fingering instability. In terms of the concentration field, local fingering advanced via complex secondary flows close to 181 5.1. Scientific contributions the front that are controlled by both casing rotation and annular eccentricity. Over longer times and on smaller length-scales the local fingering was replaced by steady propagation of a diffuse interfacial region that spreads slowly due to dispersion. These are effectively pseudo-steady displacements. The Muskat-type analysis could be applied to the displacements studied in Chap- ter 4, even though we only considered small rotations compared to the elongation of the interface. This work remains to be done. In Chapter 2, where the casing is fixed, we assumed a priori a symmetric flow, and a mechanically stable configuration (Figure 2.2). Therefore, buoyancy driven fingering is not expected. It remains an open question whether the steady state travelling waves will be- come unstable due to secondary flows and dispersion. The Hele-Shaw formulation allows a discontinuity in tangential velocity across an interface, and in the simu- lations in Chapters 2 and 3 we have observed that the steady state had fluids 1 and 2 moving counter-current on either side of the interface. Since the interface is advected by the velocity field, this configuration intuitively appeared to be unstable to any small perturbation of the interface. In some cases this can result in concen- tration fields where the diffuse region grows slowly at large times. In our model molecular diffusion effects are absent at leading order and thus only numerical diffu- sion is active in smoothing small scale variations. This makes it hard to distinguish whether or not at long times the underlying physical result would be a bounded diffuse region approached asymptotically or one that grows slowly via dispersion. Slow axial movement and reciprocation might contribute to this type of dispersive instability. 5.1.3 Computational advances A main contribution of this work is the characterisation of the closure problem for the flow of viscoplastic fluids in a moving annulus, deriving qualitative results satisfied by the closure functionals, and in proposing an efficient algorithm with which to solve the closure problem computationally. These results are important not only for the results already presented, but also if one wishes to analyse and solve the fully 2D problem in the future. The second main computational advance in the thesis is the extension of the 182 5.2. Industrial perspective model to periodic domains and with wall motion. Although this is only for New- tonian fluids, this gives considerable insight into the effects of casing rotation and reciprocation in the range of O(1) buoyancy numbers. Not only there are no reliable models available to the industry that study these effects during displacement, but we have seen that the O(1) buoyancy number range is hard to study without 2D simulation. Therefore, although the computational novelty here is minimal, in terms of changes required to the algorithm, the utility of the model output is high. 5.2 Industrial perspective From the oil industry perspective, the results are sometimes counter-intuitive but have largely positive implications. They also fill a large gap in industrial understand- ing of horizontal primary cementing displacement flows, where there is essentially no published analysis of the displacement mechanics. The key contributions of the work are as follows. 1. Eccentricity and density differences play the dominant roles in vertical annulus displacements, (see e.g. [8]). It has long been thought that in horizontal annuli density differences must be detrimental to a good displacement. We see that this is not entirely the case. In fact, (at least in the lubrication limit), whether or not there is steady state displacement in a horizontal well does not appear to be dictated by the density difference between the fluids. Rheology plus eccentricity play the dominant role, regardless of the density difference. 2. On the other hand, density difference does affect the length of the steady state displacement profile, scaling linearly with density difference (divided by Stokes number). This scaling is evident in results in Chapter 2 from the 2D computations, and is implicit in the scaling adopted for the lubrication model in Chapters 2 and 4. For 2 Newtonian fluids the Stokes number captures all the effects of pump flow rate, i.e. increasing Q̂ reduces St∗ like 1/Q̂. Thus, density difference and flow rate act counter to one another in determining the length of the displacement profile. For non-Newtonian fluid displacements, the flow rate affects the dimensionless rheological parameters so that it is harder to assess effects of flow rate. However, in broad terms, provided that the fluid 183 5.2. Industrial perspective rheology plus the eccentricity allow a steady state, it appears that increasing the flow rate will reduce the axial length of the interface. Density differences also affect the extent of the phase shift when rotation is present. 3. Criteria for the existence of a steady state displacement have been developed and presented as either a critical consistency ratio, or as other thresholds, for all the displacements studied in this thesis. These criteria are relatively simple to compute and could easily be computed as part of a design process. Given the apparent stability of the solutions, these are obviously of industrial interest. 4. Conditions for steady state displacements appear to be relatively unaffected by casing rotation. This implies that process design using the simpler fixed casing model of Chapter 2 may be adequate, simply using casing rotation to ensure that the fluids are mobilised on the narrow side of the annulus. 5. Although it seems that rotation does not significantly affect the conditions for a steady state displacement in the high buoyancy number regime (chapter 4), it does affect the axial length of the interface. Furthermore, in some of the simulations of chapter 3 we have seen that strong rotation at moderate buoy- ancy number can “stabilise” an interface that would otherwise be unsteady (see Fig. 3.15). This stabilisation is via local instability and mixing that results in an interfacial region in place of the sharp interface. The diffusive region does continue to spread dispersively and diffusively, but at a rate that is slow with respect to the advective timescale of the mean flow. 6. The balance between buoyancy and rotation is given by |vC | / ∣∣∣b̃∣∣∣, which in dimensional terms is: |vC |∣∣∣b̃∣∣∣ = 2π(r̂o + r̂i)maxk{κ̂k}Ω̂C (r̂o − r̂i)2 |ρ̂2 − ρ̂1| ĝ , (5.1) with the dimensionless parameters defined in §3.2.1. In broad terms we can think of three regimes: (i) |vC | ≪ ∣∣∣b̃∣∣∣, when the displacements are very similar to those with fixed 184 5.3. Other contributions casing (chapter 2). (ii) |vC | ∼ ∣∣∣b̃∣∣∣, in this regime rotation is strong enough to move the interface adequately in order to promote local instability and mixing. (iii) |vC | ≫ ∣∣∣b̃∣∣∣, when we appear to have stabilisation via mixing of the fluid interface around the annulus (see 5 above). With a view to the large rotation rates required from (5.1), this regime is not practically feasible for most current operations. 7. Slow steady axial motion or reciprocation of the annulus walls on its own appears less interesting as there is no breaking of the symmetry of the interface and hence no instability. However, axial wall motion or reciprocation does generate secondary flows which may combine with those from cylinder rotation resulting in enhanced dispersive effects. 5.3 Other contributions In the wider application context, we mention that similar Poiseuille-Couette flows occur in annular screw extruders, with narrow gaps and polymeric liquids. Similarly, rotating annular heat exchangers are used in the food industry, where one of the cylinders is often fitted with a scraper system. Here the closure laws are simplified by the assumptions of slow rotation and a dominant axial Poiseuille flow component (see e.g. [2]). The closure laws that we have developed in §B are general to these flows and the algorithms suggested therefore have wider utility. Finally, although it must be acknowledged that the cementing application con- sidered is a rather specialised flow, we should consider the wider fluid mechanical context of the work and its other applications. Firstly, the underlying yield stress fluid models, in the Hele-Shaw context, have a porous media analogue in so-called non-Darcy flows with limiting pressure gradient, i.e. these are nonlinear filtration problems in which yield stress effect is replaced by a critical pressure gradient that must be exceeded. This critical pressure gradient may be either a property of the porous media (e.g. argillaceous soils) or of the fluids flowing in the porous media, (e.g. heavy oils). The study of these flows was first carried out by Entov, in the 1970’s and subsequently, and it is summarised in the two texts: [1] and [3]. In this 185 5.4. Critique of the methodology and future work context, the annular geometry that we consider corresponds to a spatially periodic anisotropy in the porous media, and with a spatially periodic conservative (grav- itational) force field imposed via f . For the Couette component in our flows the porous media analogy is unclear to us. 5.4 Critique of the methodology and future work 5.4.1 Hele-Shaw model We have mentioned earlier that the Hele-Shaw model is an effective and accurate tool to model flows in narrow channels. It is obviously faster to compute than a 3D model, and it is usually possible to do some analytical work with it, so as to develop qualitative physical understanding. These are the principal advantages of this approach. Nevertheless, it should not be overlooked that the underlying Hele-Shaw ap- proach averages across the annular gap. Dispersion (and other) effects on the gap scale are thus ignored in this approach. Although we have derived the concentra- tion equation (3.9) under the reasonable assumption of negligible molecular effects, it requires more careful examination. We lack a simple estimate for the gap-scale dispersivity. We have seen that dispersive mixing results from secondary flows that combine both rotation and recirculation effects. These secondary flows are too complex to study analytically. In laboratory experiments for vertical displacements presented in Appendix C, it is apparent that dispersion should be taken into account in the model. This requires further work and analysis. 5.4.2 Lubrication modelling approach The lubrication approach used in Chapter 2 and 4 proved to be a powerful tool to find steady travelling wave solutions and conditions for their existence. Nevertheless, the assumption of a clean interface with no mixing, might lead to an incorrect classification of the displacement. In Chapter 4, for the moving annulus case, it appears as if the motion of the inner cylinder is unimportant. But for many of the flows computed, where |vC | ∼ |b̃|, localised interfacial instabilities led to mixing over short timescales. For longer times it led to displacements that had diffuse 186 5.4. Critique of the methodology and future work interfacial region but nearly steady. Here too, if we consider sufficiently large vC , we may expect that local instabilities appear, and the lubrication model becomes invalid. We have observed in the Newtonian simulations that the lubrication model prediction becomes progressively poor for an increase in rotation. the main point here is to note that the lubrication model is valid at long times, as well as over long length-scales. If, however, the interface destabilises before it has stretched, the assumptions of the lubrication model are never reached. 5.4.3 Numerical diffusion and dispersion If we are to improve our physical representation of dispersion and diffusion during the displacement, we must first understand better the degree of numerical diffu- sion/dispersion present, and be able to control it. Although numerical diffusion is handled reasonably well by the FCT scheme §A.2, other comparable numerical schemes have been developed in the past 20-30 years. To make a broad quantitative study of the dispersion it would be wise to at least benchmark against other numer- ical methods. Similarly, the form of interpolation of the physical properties of the fluids, coupled with the numerical solver, may have some effect on the degree of dis- persion, as they are active essentially only in the interfacial region. A more detailed study of these effects and benchmarking of other numerical algorithms would be a sensible future research. 5.4.4 Stability In terms of our future directions, our study of flow stability here is somewhat pre- liminary. In this work we have focused primarily on displacements where a stable advancing front or diffuse frontal region results. We have considered fairly modest eccentricities and buoyancy numbers, so as to explore casing motion effects. When eccentricity and/or buoyancy is more dominant, the interface tends to elongate and become pseudo-parallel. We may expect different types of instability to result other than fingering, as studied e.g. in [7, 6]. The steady state analysis and our stabil- ity criterion developed in §3.5.2 both consider the situation with a clean interface, whereas the computations use a concentration field. This disparity of modeling approaches needs addressing. However, with a discontinuity in tangential velocity 187 5.4. Critique of the methodology and future work across the interface due to secondary flows, computation using an interface tracking method is challenging. A second option therefore is to develop the stability theory within the concentration-dependent fluid framework, as done for example by [4]. This would allow incorporation of gap dispersion effects via an anisotropic diffusion term, and appears to be a promising direction. Another limitation of the model that we have used concerns the accuracy of the model when localised fingering instabilities occur, e.g. Figures 3.14 and 3.15 in Chapter 3. Really the model that we have used is aimed at predicting global features of the flow. These global features lead to interface configurations that we have found to be locally unstable. Although we can have confidence in the global features of the flow and hence the prediction of instability, whether or not the small scale evolution of the instability is accurate is questionable. For example, in Figure 3.7, we have shown a detailed example in which rotation of the casing moves the fluids such that heavy fluid lies above light fluid, and we then observe buoyant fingering. The physical causes of this fingering are within the scope of the model, but the local fingering may not evolve exactly as computed. In classical analyses of fingering in porous media displacements, e.g. [5], the growth rates of linear instabilities are inversely proportional to the wavelength, so that short wavelengths are the least stable. Conventionally therefore we expect that the short wavelengths are the dominant modes, but due to the infinite growth rate the short wavelength problem is ill-posed. There are different mechanisms that are used to overcome the ill-posedness. Adding interfacial tension or diffusion can be used to control the growth of the instabilities. In our case, including diffusive terms in the concentration equation (3.9), though small, might play a significant role in modulating the growth of the short wavelength perturbations. Possibly in our present simulations, numerical diffusion is fulfilling this role. 5.4.5 Computational methods The resulting quasilinear elliptic PDE for the stream function in Chapter 4 has not been solved, primarily due to the complexity of the closure relations between flow rate and modified pressure gradient, in the presence of wall motion. This complexity translates into a heavy computational burden that causes the approach to be very 188 5.4. Critique of the methodology and future work slow. This remains an open question and task for future work. 189 5.5. Bibliography 5.5 Bibliography [1] G. Barenblatt, V. Entov, and V. Ryzhik. Theory of fluid flows through natural rocks. Theory and Applications of Transport in Porous Media, 3, 1990. [2] A. Fitt and C. Please. Asymptotic analysis of the flow of shear-thinning food- stuffs in annular scraped heat exchangers. Journal of Engineering Mathematics, 39:345–366, 2001. [3] R. Goldstein and V. Entov. Qualitative methods in continuum mechanics. Aca- demic Press., 1989. [4] N. Goyal, H. Pichler, and E. Meiburg. Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability. Journal of Fluid Mechanics, 584:357– 372, 2007. [5] D. Joseph. Change of type and loss of evolution in the flow of viscoelastic fluids. Journal of Non-Newtonian Fluid Mechanics, 20:117–141, 1986. [6] M. Moyers-González and I. Frigaard. Kinematic instabilities in two-layer eccen- tric annular flows, part 1: Newtonian fluids. Journal of Engineering Mathemat- ics, 62:130–131, 2008. [7] M. Moyers-González and I. Frigaard. Kinematic instabilities in two-layer eccen- tric annular flows, part 2: shear thinning and yield stress effects. Journal of Engineering Mathematics, DOI 10.1007/s10665-008-9260-0, 2008. [8] S. Pelipenko and I. Frigaard. Visco-plastic fluid displacements in near - vertical narrow eccentric annuli: prediction of travelling wave solutions and interfacial instability. Journal of Fluid Mechanics, 520:343–377, 2004. 190 Appendix A Computational methods In this appendix we outline the two main numerical algorithms we have used to solve the 2D Hele-Shaw displacement model, for both, the fixed casing model with viscoplastic fluids, and the moving casing model with Newtonian fluids. In broad terms, the solution of (2.4) uses the augmented Lagrangian algorithm in chapter 4 of [1]. The advection equation for the concentration field (2.2) is solved using the Flux Corrected Transport scheme described in [3]. We first describe in detail both alogrithms for the fixed casing flow, and then, in §A.4 we outline differences needed for the moving casing model. The convergence of these methods used for the same system of equations has been studied by Pelipenko in [2]. For further details, we refer the reader to this work. A.1 Augmented Lagrangian method The augmented Lagrangian method uses a weak formulation of (2.2) to compute the solution. This is a good choice to solve viscoplastic fluid flows as the unyielded regions are included implicitly in the algorithm without any further modification. In order to apply the method to our problem, we need the weak formulation of the field equation (2.4). Let Ψ∗ be any stream function that satisfies the boundary conditions (2.5,2.6). Then Ψ can be expressed as Ψ = Ψ∗ + u, u ∈ V0, (A.1) where V0 is the subset of W 1,1+1/m that contains functions that satisfies u = 0 at the boundaries. Note that Ψ∗ can be determined by taking a linear combination of stream functions at ξ = 0 and ξ = Z. So in case of two fluids we can take Ψ∗ = cΨ0 + (1 − c)ΨZ , where Ψa is the stream function at ξ = a and c is the 191 A.1. Augmented Lagrangian method concentration with c = 1 for the displacing fluid, and c = 0 for the displaced fluid. Then the weak formulation of (2.4) is given by min v∈V0 {J(v)}, (A.2) where J(v) = F (∇v) +G(v), and F (q) = F0(q) + F1(q), q ∈ H, (A.3) F0(q) = ∫ Ω 1 2 ∫ |∇Ψ∗+q|2 0 χ(s1/2) s1/2 ds dΩ, q ∈ H (A.4) F1(q) = ∫ Ω τY H |∇Ψ∗ + q| dΩ, q ∈ H (A.5) G(v) = − ∫ Ω fv dΩ, v ∈ V0. (A.6) In [2] it is shown that F0(q) is strictly convex and differentiable, F1(q) is convex and continuous and G(v) is convex and continuous. Moreover, the minimisation problem has a unique solution in V0, with H = L 1+1/m(Ω)× L1+1/m(Ω). We define a Lagrangian functional L associated with (A.1), by L(v, q, µ) = F (q) +G(v)+ < µ,∇v − q >, and for r ≥ 0 its associated augmented Lagrangian Lr by Lr(v, q, µ) = L(v, q, µ) + r 2 |∇v − q|2. (A.7) where < ·, · > is the inner product defined < u, v >= ∫Ω u·vdΩ and |u| its associated norm. Following the results in chapter 4 of [1], it is true that {u, p, λ} is a saddle point of L if and only if it is a saddle point of Lr, ∀r > 0. Moreover such u is a solution of (A.1) and p = ∇u. Thus, the solution to (2.4) is equivalent to finding the saddle point of Lr which is accomplished by using an algorithm of Uzawa type. The resulting algorithm is the following. Start with an initial λ0 ∈ H. For the 192 A.1. Augmented Lagrangian method nth step λn is known, so we find un, pn, λn+1 by solving the minimisation Lr(un, pn, λn) ≤ L∇(v, q, λn), ∀{v, q} ∈ V0 ×H, {un, pn} ∈ V0 ×H (A.8) and the equation λn+1 = λn + ρn(∇un − pn), ρn > 0. (A.9) The problem (A.8) is equivalent to solving two coupled variational inequalities: G(v) −G(un)+ < λn,∇(v − un) > +r < ∇un − pn,∇(v − un) >≥ 0, ∀v, un ∈ V0, (A.10) and F (q)− F (pn)− < λn, q − pn > +r < pn −∇un, q − pn >≥ 0, ∀q, pn ∈ H. (A.11) The main complication of this algorithm is that it requires the solution of two coupled variational inequalities at each step. To overcome this, we uncouple them to obtain the following modified algorithm. Let p0, λ1 ∈ H (A.12) be given. Then for each step pn−1, λn are known, so we solve for un, pn and λn+1 such that G(v) −G(un) + < λn,∇(v − un) > + r < ∇un − pn−1,∇(v − un) >≥ 0, ∀v, un ∈ V0,(A.13) F (q)− F (pn)− < λn, q − pn > +r < pn −∇un, q − pn >≥ 0, ∀q, pn ∈ H. (A.14) λn+1 = λn + ρn < ∇un − pn >, ρn > 0. (A.15) In fact the first step (A.13) corresponds to minimizing Lr(v, pn−1, λn) with respect to v to get un. The second step (A.14) corresponds to minimizing Lr(un, q, λn) with 193 A.1. Augmented Lagrangian method respect to q to obtain pn. When we apply this algorithm to our particular problem with F and G as in (A.1 - A.6) above, we find that G(v) −G(un)+ < λn,∇(v − un) > +r < ∇un − pn−1,∇(v − un) > = ∫ Ω −f(v − un) + λn · ∇(v − un) + r(∇un − pn) · ∇(v − un) dΩ, (A.16) Integrating by parts the second and third terms we obtain G(v) −G(un)+ < λn,∇(v − un) > +r < ∇un − pn−1,∇(v − un) > = ∫ Ω (−f −∇λn − r∇ · (∇un − pn)) (v − un) dΩ. (A.17) Thus the first step of the algorithm (A.13) is to solve ∫ Ω (f +∇λn + r∇ · (∇un − pn))(un − v) dΩ ≥ 0, v, un ∈ V0. (A.18) This equation is satisfied by the solution of r∇ · ∇un = r∇ · pn −∇λn − f, un ∈ V0, (A.19) which is a Poisson equation on Ω, with the right hand side updated at each iteration. To find pn, we may minimise Lr(un, q, λn) with respect to q, i.e, when pn is given locally by pn = inf q {1 2 ∫ |∇Ψ∗+q|2 0 χ(s1/2) s1/2 ds+ τY H |∇Ψ∗ + q| + 1 2 |∇Ψ∗ + q|2 − (λn + r∇un + r∇Ψ∗) · (q +∇Ψ∗) } , (A.20) i.e., it is minimised when (λn + r∇un + r∇Ψ∗) is parallel to q +∇Ψ∗. Thus, if we define (q +∇Ψ∗) = θ (λn + r∇un + r∇Ψ∗) , (A.21) and x = |λn + r∇un + r∇Ψ∗| (A.22) 194 A.1. Augmented Lagrangian method then we need to minimise M(θ) = 1 2 ∫ (θx)2 0 χ(s1/2) s1/2 ds+ τY H |θ|x+ 1 2 (θx)2 − θx (A.23) with respect to x. Now, if x ≤ τYH then θ = 0 minimizes M , with pn = −∇Ψ∗. If x > τYH then we set ∂M ∂θ = 0, so χ(θx) + τY H + rθx− x = 0 (A.24) which can be solved numerically for θ, giving pn = θ (λn + r∇un + r∇Ψ∗)−∇Ψ∗. (A.25) In summary, given initial guesses p0, λ1 ∈ H, (A.26) at the nth step we find un, pn, λn+1 from r∇ · ∇un = r∇ · pn −∇λn − f, un ∈ V0, (A.27) pn = { −∇Ψ∗ if x ≤ τYH θ(λn + r∇un + r∇Ψ∗)−∇Ψ∗ if x > τYH (A.28) λn+1 = λn + ρn(∇un − pn), ρn > 0. (A.29) where x = |λn + r∇un + r∇Ψ∗|, (A.30) and χ(θx) + τY H + rθx− x = 0. (A.31) The relation between the modified pressure gradient χ and the modulus of the 195 A.2. Flux Corrected Transport scheme stream function |Ψ| is given from the constitutive relationship (equation (2.9)), i.e. |∇Ψ| = 0 χ ≤ 0, Hm+2 κm(m+ 2) χm+1 (χ+ τY /H) 2 [ χ+ (m+ 2)τY (m+ 1)H ] χ > 0. (A.32) The result of solution of the algorithm is a sequence of functions un converging to u ∈ V0 which gives the stream function Ψ = Ψ∗ + u. Also we obtain pn ∈ L1+1/m(Ω) × L1+1/m(Ω) converging to p = ∇u, from which the components of the velocity vector to be used in the concentration-advance equation can be easily calculated. A.2 Flux Corrected Transport scheme The concentration equation (2.2), i.e., ∂ ∂t [Hc̄] + ∂ ∂φ [Hv̄ c̄] + ∂ ∂ξ [Hw̄ c̄] = 0, (A.33) is a hyperbolic conservation law and there are many different methods to find its numerical solution. In the displacement flow problem we have there are large con- centration gradients at the interface. Thus, we need a time advance scheme that captures the propagation of the interface between the fluids. The Flux Corrected Transport (FCT) scheme [3] is a reasonable choice as it effectively minimises both the numerical diffusion and dispersion by hybridising upwind low-order schemes and higher order schemes, while limiting the total numerical flux functions. We used an FCT scheme with donor-cell upwind discretisation for the low order scheme and central difference discretisation for high order scheme. A brief outline of the FCT scheme used is as follows. Let rewrite equation (A.33) as ∂U ∂t + ∂F ∂φ + ∂G ∂ξ = 0, (A.34) where U = Hc̄, F = Hv̄ c̄, G = Hw̄ c̄. (A.35) First, low order fluxes FL and GL and high order fluxes FH and GH are com- 196 A.3. Discretisation and solution puted using a donor-cell upwind and central difference discretisations correspond- ingly. From those we compute antidiffusive fluxes Ai = FH − FL, Aj = GH −GL and the low-order time advance solution U td from the low-order fluxes FL and GL. Secondly, we limit the antidiffusive fluxes by setting AiC = Ai Ci and AiC = Ai Cj. Here the limiting functions 0 ≤ Ci, Cj ≤ 1 are chosen to limit U(t + ∆t) so that it does not exceed some maximum Umax(t) or fall below some minimum Umin(t). In turn we choose Umax(t) to inhibit the overshoots in c̄ as the maximum value of U(t) over the neighboring cells and analogously for Umin(t). Lastly, we apply the limited antidiffusive fluxes to the low order time advanced solution to get the final time advanced solution U(t+∆t) = U td − 1 ∆V [ AiC −A(i−1)C +AjC −A(j−1)C ] , (A.36) where ∆V = ∆φ∆ξ. A.3 Discretisation and solution We descretise the equations by applying a finite volume discretisation with a stag- gered regular rectangular mesh, i.e, we divide the domain Ω into Ni × Nj control volumes with each control volume being a rectangle of width ∆φ = 1/Ni in the φ-direction and height ∆ξ = 1/Nj in the ξ-direction. Denoting by CVi,j the control volume with its lower left-hand corner at coordinates (i∆φ, j∆ξ) we define the dis- cretised functions unh, Ψ n h at the corners of CVi,j and the discretised functions λ n h, p n h and concentration ch at its center (Figure A.1). This allows simplified calculation of gradients arising in the above equations, as well as smaller spatial support of the numerical schemes for the chosen second order of accuracy in each of the numerical calculations of the flux between the control volumes. We let the discretised concentration function ch be defined at the centre of each control volume and take values between 0 and 1 with ch = 1 in the displacing fluid and ch = 0 in the displaced fluid. The density and rheological parameters are then also defined at the centers of the control volumes using linear closure laws. So for example ρhi,j = c h i,jρ1 + (1− chi,j)ρ2. At the start of each run we first determine the homogenised stream function Ψ∗h 197 A.3. Discretisation and solution u u u u e (i∆φ, j∆ξ) ((i+ 1/2)∆φ, (j + 1/2)∆ξ) u - unh, Ψ ∗ h e - λnh, p n h, c h, ∇Ψ∗h 6 - ξ φ Figure A.1: Schematic picture of the control volume CVi,j. by setting Ψ∗h = c hΨh0 + (1 − ch)ΨhZ . Here Ψh0 and ΨhZ are the discretised stream functions at ξ = 0 and ξ = Z correspondingly. These in turn are determined from the boundary conditions which give the modified pressure gradient G = χ+ τY /H constant in the φ-direction at the top and bottom of the well. Thus, χ at ξ = 0 and ξ = Z can be calculated by numerically inverting the expression for the total rate of flow through the well. As the flow is symmetric in the azimuthal direction, we use two “mirror” ghost cells to the left of φ = 0 and to the right of φ = 1. At each iteration of the algorithm we first solve the unh advance equation (A.19). We apply the standard point Gauss-Seidel method with second-order accurate spa- tial fluxes to solve this modified Poisson’s equation. This seems especially appropri- ate as unh converges after some iterations of the algorithm, so taking u h n−1 as starting point of Gauss-Seidel iterations each time improves the convergence time. The same is true after applying the time-advance scheme, the final value of unh at the previous time-step will be the starting point for Gauss-Seidel iterations. Over-relaxation was also used to improve the speed of convergence further in individual cases. The second step in the algorithm is the advance of pnh. We achieve this by first calculating x from (A.30). If the fluid is yielded at that point we perform the numerical inversion in (A.31) to determine θ, and thus, pnh. Note that (A.31) can be differentiated analytically and a Newton-Raphson method used in its inversion 198 A.4. Moving casing algorithms converges in few iterations to the desired tolerance. A.4 Moving casing algorithms The method described in the previous three sections has to be slightly modified to fit the moving casing model of chapter 3: (i) We model the entire circumference of the annulus, since we may no longer assume that the flow is symmetric at wide and narrow sides, i.e., (φ, ξ) ∈ [−1, 1] × [0, Z]. Therefore, we impose periodic boundary conditions in the azimuthal direction.. (ii) The stream function is homogeneised using Ψ∗M = 2H(φ+ e π sinπφ)−HvC (A.37) in order to satisfy the boundary conditions (3.5,3.6). For computing the stream function, we have in fact retained the augmented Lagrangian formulation even though the problem is linear for Newtonian fluids. The algorithm converges fast in the Newtonian case, but this is admittedly inefficient. The main reason for retaining the augmented Lagrangian method was in order to make subsequent extension of the 2D moving casing model to non-Newtonian fluids more transparent. 199 A.5. Bibliography A.5 Bibliography [1] R. Glowinski. Numerical methods for nonlinear variational problems. Spriger- Verlag, 1983. [2] S. Pelipenko. Displacement flow of non-Newtonian fluids in an eccentric annulus. Master’s thesis, UBC, 2001. [3] S. Zalesak. Fully multidimensional flux corrected transport algorithms for fluids. Journal of Computational Physics, 31:335, 1979. 200 Appendix B Analysis and computations of the closure of the moving casing model for viscoplastic fluids In this appendix we address the closure problem between applied pressure gradients and gap-averaged velocity field for a planar 2D shear flow of an Herschel-Bulkley fluid in a plane channel of width 2H with the wall at y = −H translating with speed uc. Mathematically we solve: ∂ ∂y τ = −G, (B.1) where τ = (τφy, τξy), G = (Gφ, Gξ) and with τ = η(γ̇) d dy u⇐⇒ τ > τY , (B.2) γ̇ = 0⇐⇒ τ ≤ τY . (B.3) Here we have u = (v,w), γ̇ = [( dv dy )2 + ( dw dy )2] 12 , τ = [τ2φy + τ 2 ξy] 1/2, and η = κγ̇n−1 + τY γ̇ . The boundary conditions for u are u = uc at y = −H and u = 0 at y = H. Integrating (B.1) gives us: τ = τ 0 − yG, for some unknown stresses, τ 0, at 201 Appendix B. Closure of the moving casing model y = 0. We may note that the velocity gradient is given by: d dy u = 1 η(γ̇) τ = 1 η(γ̇) [τ 0 − yG]. (B.4) From integrating by parts, we have: uc =− ∫ H −H d dy u dy = G ∫ H −H y η(γ̇) dy − τ 0 ∫ H −H 1 η(γ̇) dy (B.5) 2Hū = ∫ H −H u dy = Huc − ∫ H −H y d dy u dy =Huc +G ∫ H −H y2 η(γ̇) dy − τ 0 ∫ H −H y η(γ̇) dy (B.6) or alternatively uc =Hm1G−m0τ 0, (B.7) 2Hū =Huc +H 2m2G−Hm1τ 0, (B.8) for the mobility moments mj : mj = ∫ H −H ( y H )j 1 η dy, (B.9) Rearranging these expressions to eliminate τ 0 leads to (4.21) & (4.22), which reflects the split into Couette and Poiseuille components of the flow. Because the effective viscosity and strain rate appearing in the mobility moments depend on the solution, (4.21) & (4.22) are in fact implicit nonlinear relationships. We turn now to issues of solvability and developing the qualitative understanding of these relationships. 202 Appendix B. Closure of the moving casing model Monotonicity results For any u and v ∈W 1,1+n(−H,H)×W 1,1+n(−H,H), define the functionals a(u,v), j(u) and inner product 〈u,v〉, by: a(u,v) = ∫ H −H ∣∣∣∣dudy ∣∣∣∣ n−1 du dy · dv dy dy, (B.10) j(u) = ∫ H −H ∣∣∣∣dudy ∣∣∣∣ dy, (B.11) 〈u,v〉 = ∫ H −H u · v dy, (B.12) and let V be the space: V = {v ∈W 1,1+n(−H,H)×W 1,1+n(−H,H) : v(−H) = uc, v(H) = 0} . Using standard methods, e.g. in [2], we may characterise the closure problem as the following minimisation problem: min v∈V κ n+ 1 a(v,v) + τY j(v)− 〈G · v〉. (B.13) The above functional is strictly convex and the minimisation consequently has a unique solution; see e.g. [1]. The solution u also satisfies the following variational inequality κa(u,v − u) + τY [j(v)− j(u)] ≥ 〈G · (v − u)〉, u ∈ V, ∀v ∈ V. (B.14) For fixed casing velocity u, consider two modified pressure gradients, G1 & G2, with velocity solutions u1 & u2, respectively. Evidently the solution space V is identical for u1 and u2, so that each solution may be used as a test function for the other. Summing the above variational inequalities for u1 and u2 gives: 〈(G1 −G2) · (u1 − u2)〉 ≥ κ[a(u2,u2 − u1) + a(u1,u1 − u2)] ≥ 0, (B.15) with the last inequality coming from convexity of a(v,v). Since G1 & G2 are 203 Appendix B. Closure of the moving casing model constant, we have (G1 −G2) · (ū1 − ū2) ≥ 0. (B.16) In fact, since a(v,v) is strictly convex, we have strict inequality above except if u1 = u2. Suppose that u1 = u2 6= 0. We note that the mobility moments mj are uniquely defined by γ̇ and hence by u. Thus, in the identity (4.21) we have the same mobility moments for u1 and u2. This implies that either G1 = G2 or the coefficient m2 −m21/m0 = 0. However, from the Cauchy-Schwarz inequality we have that m21 ≥ m0m2 with equal- ity only if the integrands of m0 and m2 are linearly dependent. However, we observe this can only happen if the integrands are identically zero, i.e. if the effective viscos- ity is infinite and the fluid unyielded everywhere. But if the casing velocity uc 6= 0, then the fluid cannot be unyielded everywhere. Therefore, we see that the following is true. • If uc 6= 0 then u1 = u2 if and only if G1 = G2. The inequality (B.16) is strict unless u1 = u2, or equivalently, if and only if G1 = G2. • If uc = 0 then u1 = u2 6= 0 if and only if G1 = G2. The inequality (B.16) is strict unless u1 = u2 6= 0, or equivalently, if and only if G1 = G2. Alterna- tively, if u1 = u2 and G1 6= G2, then u1 = u2 = 0. We note that the latter case corresponds to the fixed casing situation, when indeed different G may lead to u = 0, provided that |G| ≤ τY /H. Computation of the closures The inequality (B.16) and above comments on monotonicity establish the feasibility of computing the closure. In the case that uc 6= 0 there is a one-to-one mapping between u and G, and (B.16) implies that by increasing say Gφ we increase uφ and vice versa. Therefore, if we have a forward solver, (computing u from G), we may use this to iterate monotonically towards finding the correct G that satisfies some constraint ū = ū∗. 204 Appendix B. Closure of the moving casing model For the purpose of this paper, the type of closure we wish to compute is one in which both ūφ and Gξ are fixed. Although this could be computed via an outer iteration, as outlined above, one still needs to compute the forward solver. Instead we outline here a method in which the constraints are satisfied within the framework of the forward solver. Suppose then that ūφ = ū ∗ φ and Gξ = G ∗ ξ . We use Gφ as a Lagrange multiplier for the constraint, ūφ = ū ∗ φ, by minimising: min v, Gφ κ n+ 1 a(v,v) + τY j(v)− 〈Gφ(vφ − ū∗φ)〉 − 〈G∗ξvξ〉. (B.17) Again we know that there is a unique solution to this minimisation. To cope with the non-differentiability of the above functional, we relax dvdy 7→ q ∈ U = {L1+n(−H,H) × L1+n(−H,H)} and replace the above minimisation with the fol- lowing saddle point problem in the classical way, e.g. [2]: max s min v, Gφ, q L(v, Gφ,q, s), (B.18) L(v, Gφ,q, s) = κ n+ 1 〈|q|n+1〉+ τY 〈|q|〉 − 〈Gφ(vφ − ū∗φ)〉 − 〈G∗ξvξ〉 + r 2 〈 ( dv dy − q ) , ( dv dy − q ) 〉+ 〈s, ( dv dy − q ) 〉. (B.19) The saddle point problem (B.19) can be solved iteratively using an Uzawa-type algorithm, sequentially determining the optimality for v, Gφ,q and s. This proce- dure converges under fairly non-restrictive conditions. However, for the problem at hand it does not take full advantage of the known solution structure. Firstly, it is known that in the converged solution: s → τ and since τ satisfies the reduced momentum equations, both components are linear in y. Thus, in place of s we impose: s = τ̃ 0 − yG̃, where G̃ξ = G ∗ ξ , and will iterate to find τ̃ 0 and G̃φ at each step. Evidently, on convergence τ̃ 0 → τ 0 and G̃φ → Gφ. Secondly, the problem for v is linear and we might hope to implement the constraint ūφ = ū ∗ φ directly within the solver, thus determining Gφ directly at each iterate. Lastly, we note that for the closure 205 Appendix B. Closure of the moving casing model problem, we have no direct interest in u, but only in the averaged value ū. Therefore, it appears that there is some wasted effort in computing u at each iterate. We now use these observations, in deriving an improved Uzawa-type algorithm for the saddle point problem. First let us suppose that an initial guess for τ̃ 0, G̃φ and q is available at step k, say τ̃ 0 = τ̃ k 0 , G̃φ = G̃ k φ and q = p k. The optimality condition for v, which defines uk+1, is simply: r d2 dy2 uk+1 = r d dy pk − (Gk+1φ , G∗ξ)− d dy sk = r d dy pk − (Gk+1φ − G̃kφ, 0). Integrating twice and using the boundary conditions at y = ±H, we find: d dy uk+1 = − 1 2H uc + p k − p̄k − y r (Gk+1φ − G̃kφ, 0) uk+1 = H − y 2H uc + ∫ y −H pk(ỹ)− p̄k dỹ − y 2 −H2 2r (Gk+1φ − G̃kφ, 0) ūk+1 = 1 2 uc − 1 2H ∫ H −H ỹ[pk(ỹ)− p̄k] dỹ + H 2 3r (Gk+1φ − G̃kφ, 0), where we have used the notation, e.g. p̄k = 1 2H ∫ H −H pk(ỹ) dỹ. Therefore, at the (k + 1)st iterate we set: Gk+1φ =G̃ k φ + 3r H2 [ ū∗φ − 1 2 vc + 1 2H ∫ H −H ỹ[pkφ(ỹ)− p̄kφ] dỹ ] , (B.20) ūk+1ξ = 1 2 wc − 1 2H ∫ H −H ỹ[pkξ (ỹ)− p̄kξ ] dỹ, (B.21) so that the imposed constraint on ūφ is automatically satisfied by this choice of Gk+1φ . The optimality condition for q, which defines pk+1, consists of minimising over 206 Appendix B. Closure of the moving casing model q the following functional, K(q): K(q) = κ n+ 1 |q|n+1 + r 2 |q|2 + τY |q| − ck · q, ck = r d dy uk+1 + sk = r[pk − p̄k]− r 2H uc − y(Gk+1φ , G∗ξ) + τ̃ k0 , Note that ck = ck(y), so that the minimisation of K(q) is carried out for y ∈ [−H,H]. We have the following solution: pk+1 = 0, ⇔ |ck| ≤ τY , (B.22) pk+1 = |pk+1| |ck| c k, ⇔ |ck| > τY , (B.23) κ|pk+1|n + r|pk+1| = |ck| − τY . (B.24) This last equation requires computational solution if n 6= 1, but for the case of a Bingham fluid we see that pk+1(y) is specified as a simple algebraic function at each iterate. Finally, we update τ̃ k0 and G̃ k φ by setting: G̃k+1φ = G k+1 φ , (B.25) τ̃ k+10 = τ̃ k 0 − ρ [ p̄k+1 + 1 2H uc ] . (B.26) The last of these comes from the usual update for s, i.e. sk+1 = sk + ρ [ d dy uk+1 − pk+1 ] . On substituting for the linear form of s we observe that τ̃ 0 is the mean value of s. Substituting from the expression for ddyu k+1 and averaging over [−H,H] leads to the above projection for τ̃ k0 . The parameters r and ρ above are the usual numerical parameters of the Uzawa algorithm. In order to implement this algorithm we use the Newtonian solution to 207 Appendix B. Closure of the moving casing model give the initial iterate, k = 1. After some algebra this gives: τ̃ 10 = − κ 2H uc, G̃ 1 φ = 3κ H2 [ ū∗φ − vc 2 ] , p1 = − 1 2H uc − y κ (G̃1φ, G ∗ ξ). Comments There are alternatives to the above closure algorithm. However, it is worth noting that for the non-Newtonian fluids, all require some form of numerical integration across the interval [−H,H] as well as some iteration to find the solution. In this sense we can expect that computational times will all be comparable. The advantage of the above approach is that we have not computed the velocity field pointwise in [−H,H] and that the constraints on the mean velocity and modified pressure gradient could be easily incorporated into the algorithm. 208 B.1. Bibliography B.1 Bibliography [1] I. Ekeland and R. Témam. Convex analysis and variational problems. SIAM, 1976. [2] R. Glowinski. Numerical methods for nonlinear variational problems. Spriger- Verlag, 1983. 209 Appendix C An experimental study of displacement flow phenomena in narrow vertical eccentric annuli7 C.1 Introduction This chapter presents 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 an- nulus; 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, fluidized beds, pulp screening, extrusion and oil well construction, which motivates the chapter. Primary cementing is described at length in the recent text by [30]. 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 7A version of this chapter has been accepted for publication. Makelmohammadi, S., Carrasco- Teja, M., Storey, S. Frigaard, I.A. andMartinez, D.M. (2009) An experimental study of displacement flow phenomena in narrow vertical eccentric annuli Journal of Fluid Mechanics 210 C.1. Introduction D r i l l n e w s t a g e o f w e l l R e m o v e d r i l l p i p e I n s e r t s t e e l c a s i n g P u m p s p a c e r f l u i d P u m p l e a d & t a i l s l u r r y D i s p l a c e m u d i n a n n u l u s E n d o f o p e r a t i o n Figure C.1: Schematic of the primary cementing process, showing the various stages (left to right) in cementing a new casing. full of drilling fluid (or other fluids), which must be removed during the cement placement. The primary cementing process proceeds as follows, see Fig. 2.12. 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 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 Fig. 2.1, and the cement is allowed to set. 211 C.1. Introduction 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 rel- atively 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 Lebouché, e.g. [29, 32, 33], those by Nouri and Whitelaw, e.g. [34, 35, 36], and the extensive studies of Escudier and co-workers, e.g. [9, 10, 11, 13]. Globally these studies consider fluids similar to those we use here: CMC, Xanthan, Car- bopol, etc., which are the most commonly used fluids for this type of experiment, and cover 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 computa- tional results. There are also numerous computational studies for these flows and some analytical solutions. The reader is referred to [12], (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: [24, 23, 44]. 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 8 212 C.1. Introduction 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 [24]. 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, 31, 45]. Our initial interest in annular displacement flows came from revisiting the stud- ies 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, buoy- ancy number, Péclet 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 ex- perimentally 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 Péclet 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 [25] and [24]. This model has been analysed in depth by Pelipenko & Frigaard in the sequence of papers [37, 38, 39], 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, 213 C.1. Introduction 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 [37]. These steady states are in fact found computationally for a much wider range of parameters than those for which it is possible to find analytical solutions, [38], and it is possible to approximately predict steady and unsteady displacements using a lubrication-style displacement model; see [39]. Another body of work is directed at understanding the stability of these displace- ment 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 in- dustrial 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 [39]. Instead, the approach taken by Moyers-González & Frigaard is to consider the stability of parallel flows in the Hele-Shaw setting; see [28, 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 buoy- ancy simply determines the axial length-scale of the interface: whether or not the displacement gives a steady solution depends on the fluid rheologies, annular eccen- tricity 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 214 C.1. Introduction in the case of yield stress fluids). Our very recent work in this 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 §C.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. [48] 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 a review, we refer the reader to [39] 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 the chapter is as follows. In §C.2 following, we describe the ex- perimental setup. The first sequence of results is given in §C.3, where we classify 215 C.2. Experimental methodology the displacements as either steady or unsteady, in each of our 6 series of experi- ments. Section C.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. C.2 Experimental methodology A schematic of the experimental setup is given in Fig. C.2. The annulus dimensions are r̂o =1.91 cm, r̂i =1.27 cm, and a length of 188.3 cm. The outer pipe is con- structed from acrylic tubing with a wall thickness of 12.7 mm. The inner pipe is an aluminium pipe with wall thickness of 1.58mm. 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 and an outlet pump that drains the fluid for recircu- lation or waste disposal. 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 single-phase AC motor. Variable speed operation is achieved through a variable frequency driver. 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 directed to the control computer. The flowmeter accuracy was cross-checked in a simple calibration experiment by mea- suring the mass of fluid pumped over a fixed time interval. Finally, a scale measures the weight 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 216 C.2. Experimental methodology Figure C.2: Schematic of the experimental setup 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, from the bottom of the annulus to the top, (see Fig. C.2). The 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. C.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 Fig. C.3. Each image thus consists of a reflected image from the side of the annulus and a non-reflected 217 C.2. Experimental methodology Inner pipe Outer pipe Displacing Fluid Fish Tank Optical Ray Path Optical Ray Path CCD Camera Mirror Figure C.3: Schematic of the optical set-up. 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 lo- cations 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 back- ground 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 circumfer- ential line increases from zero 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 218 C.2. Experimental methodology 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 ∆t̂j, 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, say ŵ∗ = Q̂/π(r̂2o − r̂2i ), the idealised mean residence time is denoted ∆t̂p: ∆t̂p = ∆ŷ ŵ∗ , (C.1) where ∆ŷ =700 mm, is the vertical distance between upper and lower measurement locations. Throughout the chapter we shall use the “hat” notation, i.e. ·̂, to denote variables that are dimensional. The piston-like residence time is used to scale the individual residence times, resulting in: ∆tj = ∆t̂j ∆t̂p , j = 1, 2, ..., 130. (C.2) In a steady displacement, where the interface travels in a steady manner, the nor- malized ∆tj should give the same constant value for each pixel j, (see Fig. C.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 Fig. C.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. 219 C.2. Experimental methodology 1 × 2 × 3 × × × × x y 1 1 × 3 × 2 × ! y x × × × 3 × 1 × 2 × 1 1 × 3 × 2 × ! Figure C.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. C.2.2 Experimental design and process related issues As explained in §C.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 displace- ments in parameter ranges having some overlap with 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 [39]. 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 ver- tical 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, (in- 220 C.2. Experimental methodology cluding the Bingham, power law and Newtonian models as sub-cases). This model contains 3 parameters: the yield stress, τ̂k,Y , the consistency, κ̂k, and the (shear- thinning or) power law index, nk, (where k = 1, 2 denotes the fluid). The fluid densities are denoted ρ̂k. 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 = r̂o − r̂i π(r̂o + r̂i) . (C.3) Azimuthal and axial distances are scaled with 0.5π(r̂o+ r̂i), whereas radial distance from the annular centreline is scaled with 0.5(r̂o − r̂i). For a velocity scale we take the mean flow velocity, ŵ∗: ŵ∗ = Q̂∗ π(r̂2o − r̂2i ) . (C.4) Axial and azimuthal velocity components are scaled with ŵ∗, and radial velocity with δŵ∗. A representative shear rate is ˆ̇γ∗ = 2ŵ∗/(r̂o − r̂i), which is used for the viscous stress scale τ̂∗: τ̂∗ = max k=1,2 [τ̂k,Y + κ̂k(ˆ̇γ ∗)nk ]. (C.5) The viscosity scale is, µ̂∗ = τ̂∗/ˆ̇γ∗, and finally, densities are scaled with the maximum density: ρ̂∗ = maxk=1,2{ρ̂k}. 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(r̂o − r̂i)ŵ∗ρ̂∗ µ̂∗ , (C.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 de- flections 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 221 C.2. Experimental methodology 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 indus- trial 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: t̂v = ρ̂∗(r̂o − r̂i)2 4µ̂∗ (C.7) and the length-scale is thus: ŵ∗t̂v ≪ 0.5π(r̂o+ r̂i) ≈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 fixed boundaries in the azimuthal direction to generate boundary layer flows. Therefore, we expect flow developments to take place on the shortest length-scale of the Hele-Shaw cell, i.e. the azimuthal length-scale: 0.5π(r̂o + r̂i). 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(r̂o + r̂i). 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∗ = τ̂∗ 0.5ρ̂∗ĝ(r̂o − r̂i) . (C.8) The Stokes number can vary in the range 0.1-100 in the field setting, but the dimen- sionless density difference (ρ2− ρ1) is also small. In terms of dimensional quantities 222 C.2. Experimental methodology the buoyancy parameter b is defined by: b = 0.5[ρ̂2 − ρ̂1]ĝ(r̂o − r̂i) τ̂∗ , (C.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 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: κk = κ̂k(ˆ̇γ ∗)nk τ̂∗ , τk,Y = τ̂k,Y τ̂∗ . (C.10) 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. 223 C.2. Experimental methodology C.2.3 Selection of fluids Although ideally one would like to be able to select fluids to match given dimension- less 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 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 prop- erties 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 23oC. The Xanthan measurements were highly re- peatable, with an error of less than 2% between successive measurements. The data were fitted to a power law model: τ̂ = κ̂ˆ̇γn, (C.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 224 C.2. Experimental methodology 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. τ̂ = τ̂Y + κ̂ˆ̇γ n. (C.12) 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 τ̂Y 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, κ̂, 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 20-30oC, with only a small (7%) decrease in viscosity observed over this range. The actual lab temperature variation range is less than 5oC, 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 225 C.3. Experimental results sterilization with Chlorine at a concentration of 100ppm, and by keeping the fluid as cool as possible. On the other hand, Carbopol spontaneously ages due to a slow and irreversible change in the micro structure; 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. C.2.4 Experimental plan We aimed to study the effects on the displacement flows of variations in the dimen- sionless governing parameters of the Hele-Shaw model. Six series of experiments with Newtonian and non-Newtonian fluids were performed. Each experimental se- ries 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 Table C.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 C.1 summarizes the experimental conditions for each series of experiments. C.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. 226 C.3. Experimental results Series Flow Rate (l/min) e |b| κ1/κ2 1 0.17 - 0.72 0 - 0.5 0.9 - 3.7 1 2 0.17 - 0.72 0 - 0.5 1.5 - 6.3 3.1 3 0.17 - 0.72 0 - 0.5 0.5 - 2.2 13.1 4 0.17 - 0.72 0 - 0.5 1.2 - 2.2 1.45 5 0.17 - 0.72 0 - 0.5 0 1.6 6 0.10 - 2.16 0 - 0.5 0.7 - 1.4 0.37 - 0.46 Table C.1: Experimental condition for the different series. C.3.1 Illustrations of typical displacements We commence by showing in Fig. C.5 typical images from both a steady and un- steady 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 Fig. C.5a translates axially while remaining constant, whereas that in Fig. C.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. Displacing Fluid (1) Displaced Fluid (2) Series Fluid Additive ρ̂1 κ̂1 n1 Fluid ρ̂2 τ̂Y κ̂2 n2 (kg/m3) (Pa.sn) (kg/m3) (Pa) (Pa.sn) 1 78% Gly 0.4% Salt 1200 0.049 1 78% Gly 1192 - 0.049 1 2 88% Gly - 1234 0.15 1 78% Gly 1192 - 0.049 1 3 98% Gly - 1254 0.64 1 78% Gly 1192 - 0.049 1 4 0.3% Xan 30% Sug 1118 1.45 0.38 0.38% Xan 1000 - 1.00 0.38 5 0.8% Xan - 1000 3.6 0.3 0.1% Car 1000 2 2.2 0.3 6 0.3% Xan 33% Sug 1048 0.32 0.51 0.07% Car 1000 0.6 0.91 0.36 Table C.2: Fluids and properties in the experimental series. 227 C.3. Experimental results y = 1 0 0 0 m my = 3 0 0 m m 0 s 2 s 6 s 4 s F r o n tS i d e 1 1 2 s 1 1 4 s 1 1 8 s 1 1 6 s F r o n tS i d e 0 s 4 s 1 2 s 8 s 4 6 s 5 0 s 5 8 s 5 4 s F r o n tS i d e F r o n tS i d e y = 1 0 0 0 m my = 3 0 0 m m Figure C.5: Examples of steady and unsteady displacements: a) e = 0.5, Q̂ =0.17 l/min, fluid properties from series 3; b) e = 0.2, Q̂ =0.34 l/min, 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. To quantify more precisely this notion of steady/unsteady displacement and to measure the amount of dispersion, we construct spatio-temporal plots at the up- stream and downstream locations, using the normalised darkness intensity obtained via the method explained in §C.2.1. These are shown in Figs. C.6 & C.7, for the experiments of Fig. C.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. In Fig. C.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 t̂ =20 s. This effect comes from opening of the gate valve, which entrains fluid 1 into fluid 228 C.3. Experimental results 2 as can be observed in the images of Fig. C.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 Fig. C.6, the spatiotemporal plot for the unsteady displacement, Fig. C.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. C.3.2 Parametric results: Newtonian fluids We now present the results of our different series of experiments, in particular fo- cusing on the question of whether the displacement may be considered 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 §C.2.1 we compute the scaled residence times ∆tj at each pixel value j, i.e. the time taken for the inter- face 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 Fig. C.8 for the displacements of Fig. C.5a & b. C.3.3 Parametric results: Newtonian fluids We now present the results of our different series of experiments, in particular fo- cusing on the question of whether the displacement may be considered 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 §C.2.1 we compute the scaled residence times ∆tj at each pixel value j, i.e. the time taken for the inter- face 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 Fig. C.8 for the displacements of Fig. C.5a & b. 229 C.3. Experimental results 0 20 40 60 80 100 120 Azimuthal Distance (pixel) 60 50 40 30 20 10 0 Ti m e (se c) a) Fluid 2 Fluid 1 Ti m e (se c) 200 180 160 140 120 0 20 40 60 80 100 120 Azimuthal Distance (pixel) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Fluid 1 Concentration b) Figure C.6: Spatio-temporal for the displacement of Figure C.5a: a) Upstream location; b) Downstream location. Ti m e (se c) 50 40 30 20 10 0 0 20 40 60 80 100 120 Azimuthal Distance (pixel) Fluid 1 Fluid 2 a) 95 85 75 65 55 Ti m e (se c) 0 20 40 60 80 100 120 Azimuthal Distance (pixel) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Fluid 1 Concentration b) Figure C.7: Spatio-temporal for the displacement of Figure C.5b: a) Upstream location; b) Downstream location. 230 C.3. Experimental results 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 20 40 60 80 100 120 140 Fr eq ue nc y tj Experiment Theory a) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 20 40 60 80 100 120 140 b) Fr eq ue nc y tj Experiment Theory Figure C.8: Residence time distribution for the displacements of Figs. C.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. Evidently, the standard deviation of the residence time distribution, σ∆t, is much smaller for the displacement of Fig. C.5a than for that of Fig. C.5b. Perhaps a less obvious effect is that the mean residence time, µ∆t, of the displacement of Fig. C.5a is significantly larger than that of Fig. C.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 dis- counts 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 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. C.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 231 C.3. Experimental results range of experimental |b|, appears non-physical in Fig. C.9c, but otherwise is at least consistent with our physical intuition. By comparison between Figs. C.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|. a) e |b | 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 3 3.5 0.10 0.20 0.30 0.40 0.50 b) e |b | 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 c) e |b | 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 1 2 0.00 0.02 0.04 0.06 0.08 0.10 t/ t |b| e = 0.2 d) Figure C.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. We have also classified each experimental displacement as either steady (squares) or unsteady (triangles). Figure C.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 232 C.3. Experimental results 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 (Fig. C.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 (Fig. C.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. C.3.4 Parametric results: non-Newtonian fluids Three series of experiments were conducted with non-Newtonian fluids using Xan- than gum and Carbopol 940 solutions. In the first series of non-Newtonian experi- ments, Fig. C.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 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 C.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 dis- placing fluid had a larger consistency value, but no yield stress. Steady displacement 233 C.4. Secondary flows and dispersion 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 concen- tric annuli; see Fig. C.10b. Although the displacing fluid 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. C.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. C.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 dis- placement 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 [16, 15, 22, 20, 21, 19, 40, 42], and computationally by [1, 5, 14, 41, 46, 47]. 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 234 C.4. Secondary flows and dispersion a) e |b | 0 0.1 0.2 0.3 0.4 0.5 1 1.2 1.4 1.6 1.8 2 2.2 0.02 0.03 0.04 0.05 0.06 b) e |b | 0.2 0.3 0.4 0.5 0.6 0.8 1 1.2 1.4 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 Figure C.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). and possibly by buoyancy b < 0, both of which act via modifying the velocity field of the underlying Poiseuille flow, close to the interface. Amongst locally stable dis- placements, dispersive effects are most prominent when the two fluids are identical. An example of such a displacement is shown in Fig. C.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, (Fig. C.11a). As the displacement advances, in Fig. C.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 235 C.4. Secondary flows and dispersion (a) (b) (c) (d) Figure C.11: Dispersive finger in the displacement flow of two identical Newtonian fluids: κ̂1 = κ̂2 =0.31Pa.s: a) t̂ =0s; b) t̂ =4.1s; c) t̂ =16s; d) t̂ =32s; The annular eccentricity is e =0.25 and the aspect ratio δ =0.084. can be gained from a lubrication/thin-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 multi-dimensional 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. C.9a-c), the mean scaled residence times are of the order of 0.95 ≈ 1/1.05. 236 C.4. Secondary flows and dispersion C.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, 37, 44]. 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 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 brand) 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: ρ̂1 =1398 kg/m 3 and ρ̂2 =1363 kg/m 3. The viscosities were κ̂1 =6.24 Pa.s and κ̂2 =0.76 Pa.s. In the 3 experiments shown, the flow rate Q̂ was increased: Q̂ = 3× 10−6m3/s, 6× 10−6m3/s, 12× 10−6m3/s. The annulus had 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 [38], shown in Fig. C.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 237 C.4. Secondary flows and dispersion 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 Fig. C.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 with the eccentricity, as suggested by the argument outlined above. Figure C.13 shows the experimental results corresponding to Fig. C.12a-c. 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 Fig. C.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 starts, this entrained fluid is 238 C.4. Secondary flows and dispersion a) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 b) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 c) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 d) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 e) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 f) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 g) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 h) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 i) ξ − t φ 0 0.2 0.4 0.6 0.8 1 4.5 5 5.5 6 6.5 Figure C.12: Moving frame streamlines and interface (heavy line) computed from the model of [38], 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). 239 C.4. Secondary flows and dispersion (a) (b) (c) W N Figure C.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. visible and is advected downstream. We show an example of this, (an expanded version of Figure C.5a), below in Fig. C.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. C.4.3 Combined effects: spikes and tails Although we have introduced dispersive effects above, by considering gap-scale 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 sec- 240 C.4. Secondary flows and dispersion Figure C.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 ρ̂1 = 1254kg/m 3, ρ̂2 = 1192kg/m 3; κ̂1 = 0.64Pa.s, κ̂2 = 0.049Pa.s; n1 = n2 = 1. ondary 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. An example of this is shown in Fig. C.15, (see also Fig. C.13). These images are from series 4, at the highest eccentricity, e = 0.5, and show the phenomenon at increasing flow rates, for Figs. C.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 Fig. C.16. The spike is 241 C.4. Secondary flows and dispersion Figure C.15: Spikes on the wide side in non-Newtonian displacements from series 4, at e = 0.5; a) Q̂ =0.17 l/min; b) Q̂ =0.34 l/min; c) Q̂ = 0.72 l/min. slightly longer and more visible at higher flow rates. It was observed in almost all Newtonian experiments, but not in non-Newtonian experiments, see e.g. Fig. C.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 [40, 42] in capillary tubes and by [19, 20, 22] in plane channel/Hele-Shaw geometries. Petitjeans & Maxworthy relate the occurrence to a transition in the pattern of streamlines, as suggested by [43]. Lajeunesse and co-workers 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 the ‘U’ shape indicates that the narrowest part is moving at a steady speed, over some range of azimuthal angles. These two 242 C.4. Secondary flows and dispersion Figure C.16: Spikes on the narrow side in Newtonian displacements from series 3, at e = 0.5; a) Q̂ =0.17 l/min; b) Q̂ =0.34 l/min; c) Q̂ = 0.72 l/min. features are illustrated in Figure C.17. In the lubrication model developed by [39], 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. [44] 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. C.4.4 Quantifying dispersion In order to quantify the amount of dispersion a number of measures are possible. Following the procedure described in §C.2.1, suppose we consider a 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 Fig. C.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 t̂1 is the time at which the front of interface just passed the upstream detection line and t̂2 is the time at which the back of interface passed 243 C.4. Secondary flows and dispersion (a) (b) Figure C.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. this line, t̂2 − t̂1 is the amount of time that corresponds to the dispersion at the lower (upstream) measurement locations, i.e. a saturation time. Similarly, t̂4 − t̂3 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, ∆t̂p, leads to the a normalized growth rate of the saturation time, at pixel j, say ∆tD,j: ∆tD,j = [(t̂4 − t̂3)− (t̂2 − t̂1)]j ∆t̂p . (C.13) Examining the distribution of ∆tD,j allows us to compare the effects of dispersion at different points around the annulus. Figure C.19 shows the effect of eccentricity on the saturation time growth rate, in the displacements of series 3. In this figure, 244 C.4. Secondary flows and dispersion D i s p e r s i o n a t l o w e r p o s i t i o n D i s p e r s i o n a t u p p e r p o s i t i o n 0 . 0 5 0 . 9 5 1 2 3 4 Figure C.18: Interface detection from the saturation curve. 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 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 Fig. C.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 variations 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. C.4.5 Other interesting phenomena In addition to the dispersion phenomena reported, several other effects were ob- served. 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 C.21 shows a 245 C.4. Secondary flows and dispersion 0 10 20 30 40 50 60 70 80 90 100 110 120 130 0.0 0.2 0.4 0.6 0.8 1.0 t D , j Azimuthal Distance (pixel) e = 0 e = 0.2 e = 0.5 Figure C.19: Effect of eccentricity on the saturation time growth rate, ∆tD,j, in series 3, at Q̂ = 0.72 l/min. 0 10 20 30 40 50 60 70 80 90 100 110 120 130 0.0 0.2 0.4 0.6 0.8 1.0 t D , j Azimuthal Distance (pixel) Q = 0.17 L/min Q = 0.34 L/min Q = 0.52 L/min Q = 0.72 L/min a) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 0.0 0.2 0.4 0.6 0.8 1.0 Azimuthal Distance (pixel) t D , j Q = 0.17 L/min Q = 0.34 L/min Q = 0.52 L/min Q = 0.72 L/min b) Figure C.20: Effect of flow rate on the saturation time growth rate, ∆tD,j, at e = 0.5 in: a) series 2; b) series 3. 246 C.5. Discussion and conclusions 0.6mm Figure C.21: Drainage wall layer observed during a Newtonian displacement. 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 visi- ble 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 was common to find a channel of Carbopol left behind on the narrow side of the annulus. This phenomenon was first highlighted by [26] in the cementing context as a potential process problem. An example is shown in Fig. C.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. C.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, 30]. This same scenario has also been extensively studied in the context of a Hele-Shaw style displacement 247 C.5. Discussion and conclusions Figure C.22: Static channel on the narrow side for an experiment in series 5: e = 0.5, Q̂ =0.34 l/min. Images are shown at: 10s, 45s and 75s after opening of the gate valve model, by [2, 37, 38, 39]. 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) steadiness 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 248 C.5. Discussion and conclusions 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. 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 gap-scale 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 σ∆t/µ∆t ≪ 1, over a broad range of parameters, only increasing sharply at some threshold value: see e.g. Fig. C.9d. This suggests that the net result of the dispersion does not change the dynamics of the underlying large- 249 C.5. Discussion and conclusions scale 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. [48]. 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 un- steady 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 repeti- tive time-dependent simulations. Not least, this is because as the frontier is ap- proached, growth rates approach zero, requiring very long times to infer stability or instability from simulation results. In [39] 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 Fig. C.23 we compare the predictions from this model with the experimental results. The re- gions that are predicted to be unsteady by the model in [39] 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 present Hele-Shaw model, and then study its effects on the displacement. This will allow a more detailed comparison with the results presented here. 250 C.6. Bibliography 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 a) e |b | Steady Unsteady 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 b) e |b | Steady 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 c) e |b | Steady 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 e |b | d) Steady Unsteady Figure C.23: Comparison of the classified steady and unsteady experiments with the lubrication model predictions from [39]: a) Series 1; b) Series 2; c) Series 3; d) Series 4. C.6 Bibliography [1] M. Allouche, I. Frigaard, and G. Sona. Static wall layers in the displacement of two visco-plastic fluids in a plane channel. Journal of Fluids Mechanics, 424:243–277, 2000. [2] S. Bittleston, J. Ferguson, and I. Frigaard. Mud removal and cement placement during primary cementing of an oil well. Journal of Engineering Mathematics, 43:229–253, 2002. [3] M. Carrasco-Teja and I. Frigaard. Displacement flows in horizontal, narrow, 251 C.6. Bibliography eccentric annuli with a moving inner cylinder. Physics of Fluids, submitted, 2009. [4] M. Carrasco-Teja, I. Frigaard, B. Seymour, and S. Storey. Viscoplastic fluid dis- placements in horizontal narrow eccentric annuli: stratification and travelling waves solutions. Journal of Fluid Mechanics, 605:293–327, 2008. [5] C. Chen and E. Meiburg. Miscible displacements in capillary tubes. Part 2. Numerical simulations. Journal of Fluid Mechanics, 326:57–90, 1996. [6] M. Cloitre, R. Borrega, and L. Leibler. Rheological aging and rejuvenation in microgel pastes. Physical Review Letters, 85(22):4819–4822, 2000. [7] M. Couturier, D. Guillot, H. Hendriks, and F. Callet. Design rules and associ- ated spacer properties for optimal mud removal in eccentric annuli. SPE paper 21594, 1990. [8] E. Dutra, M. Naccache, P. Souza-Mendes, C. Souto, A. Martins, and C. de Mi- randa. Analysis of interface between Newtonian and non-Newtonian fluids in- side annular eccentric tubes. SPE paper 59335, 2004. [9] M. Escudier and I. Gouldson. Concentric annular flow with centrebody rotation of a Newtonian and a shear-thinning liquid. International Journal of Heat and Fluid Flow, 16:156–162, 1995. [10] M. Escudier, I. Gouldson, and D. Jones. Flow of shear-thinning fluids in a concentric annulus. Experiments in Fluids, 18:225–238, 1995. [11] M. Escudier, I. Gouldson, P. Oliveira, and F. Pihno. Effects of inner cylinder rotation on laminar flow of a Newtonian fluid through an eccentrica annulus. International Journal of Heat and Fluid Flow, 21:92–113, 2000. [12] M. Escudier, P. Oliveira, and F. Pihno. Fully developed laminar flow of purely viscous non-Newtonian liquids through annuli, including effects of eccentricity and inner-cylinder rotation. International Journal of Heat and Fluid Flow, 23:52–73, 2002. 252 C.6. Bibliography [13] M. Escudier, P. Oliveira, and F. Pinho. Fully developed laminar flow of non- Newtonian liquids through annuli: comparison of numerical calculations with experiments. International Journal of Heat and Fluid Flow, 23:52–73, 2002. [14] I. Frigaard, O. Scherzer, and G. Sona. Uniqueness and non-uniqueness in the steady displacement of two viscoplastic fluids. ZAMM - Journal of Applied Mathematics and Mechanics, 81:99–118, 2001. [15] C. Gabard. Etude de la stabilité de films liquides sur les parois d’une conduite verticale lors de l’ecoulement de fluides miscibles non-newtoniens. PhD thesis, Universite Pierre et Marie Curie, 2001. [16] C. Gabard and J. Hulin. Miscible displacements of non-Newtonian fluids in a vertical tube. The European Physical Journal E, 11:231–241, 2003. [17] G. Homsy. Viscous fingering in porous media. Journal of Fluid Mechanics, 19:271–311, 1987. [18] J. Jakobsen, N. Sterri, A. Saasen, and B. Aas. Displacement in eccentric annuli during primary cementing in deviated wells. SPE paper 21686, 1991. [19] E. Lajeunesse. Déplacement et instabilités de fluides miscibles et immiscibles en cellules de Hele-Shaw. PhD thesis, Universite Pierre et Marie Curie, 1999. [20] E. Lajeunesse, J. Martin, N. Rakotomalala, and D. Salin. 3d instability of miscible displacements in a Hele-Shaw cell. Physical Review Letters, 79:5254– 5257, 1997. [21] E. Lajeunesse, J. Martin, N. Rakotomalala, and D. Salin. The threshold of the instability in miscible displacements in a Hele-Shaw cell at high rates. Physics of Fluids, 13:799–801, 2001. [22] E. Lajeunesse, J. Martin, N. Rakotomalala, D. Salin, and Y. Yortsos. Miscible displacement in a Hele-Shaw cell at high rates. Journal of Fluid Mechanics, 398:299–319, 1999. [23] P. Long. Experimental studies of fluid-fluid displacement in annuli. PhD thesis, Cambridge University, 91. 253 C.6. Bibliography [24] J. F. M.A. Tehrani and S. Bittleston. Laminar displacement in annuli: a com- bined experimental and theoretical study. SPE paper 24569, 1992. [25] M. Martin, M. Latil, and P. Vetter. Mud displacement by slurry during primary cementing jobs - Predicting optimum conditions. SPE paper 7590, 1978. [26] R. McLean, C. Manry, and W. Whitaker. Displacement mechanics in primary cementing. SPE paper 1488, 1966. [27] M. Moyers-González and I. Frigaard. Kinematic instabilities in two-layer ec- centric annular flows, part 1: Newtonian fluids. Journal of Engineering Math- ematics, 62:130–131, 2008. [28] M. Moyers-González and I. Frigaard. Kinematic instabilities in two-layer ec- centric annular flows, part 2: shear thinning and yield stress effects. Journal of Engineering Mathematics, DOI 10.1007/s10665-008-9260-0, 2008. [29] M. Naimi, R. Devienne, and M. Lebouché. Étude dynamique et thermique de lecoulement de Couette-Taylor-Poiseuille; cas dun fluide présentant un seuil découlement. International Journal of Heat and Mass Transfer, 33:381–391, 1990. [30] E. Nelson and D. Guillot. Well cementing, 2nd Edition. Schlumberger Educa- tional Services, 2006. [31] Q. Nguyen, T. Deawwanich, N. Tonmukayakul, M. Savery, and W. Chin. Flow visualization and numerical simulation of viscoplastic fluid displacements in eccentric annuli. AIP Conf. Proc. XVth International Congress on Rheology: The Society of Rheology 80th Annual Meeting, Monterey, Calif. USA, July 7, 2008, 1027:279–281, 2008. [32] M. Nouar, C. Desaubry, and H. Zenaidi. Numerical and experimental investi- gation of thermal convection for a thermodependent Herschel-Bulkley fluid in an annular duct with rotating inner cylinder. European Journal of Mechanics B, 17:875–900, 1998. 254 C.6. Bibliography [33] M. Nouar, R. Devienne, and M. Lebouché. Convection thermique pour lécoulement de Couette avec debit axial: cas dun fluide pseudoplastique. In- ternational Journal of Heat and Mass Transfer, 30:639–647, 1987. [34] J. Nouri, H. Umur, and J. Whitelaw. Flow of Newtonian and non-Newtonian fluids in concentric and eccentric annuli. Journal of Fluid Mechanics, 253:617– 641, 1993. [35] J. Nouri and J. Whitelaw. Flow of Newtonian and non-Newtonian fluids in an concentric annulus with rotation of the inner cylinder. Journal of Fluids Engineering, 116:821–827, 1994. [36] J. Nouri and J. Whitelaw. Flow of Newtonian and non-Newtonian fluids in an eccentric annulus with rotation of the inner cylinder. International Journal of Heat and Fluid Flow, 18:236–246, 1997. [37] S. Pelipenko and I. Frigaard. On steady state displacements in primary ce- menting of an oil well. Journal of Engineering Mathematics, 48:1–26, 2004. [38] S. Pelipenko and I. Frigaard. Two-dimensional computational simulation of ec- centric annular cementing displacements. IMA Journal of Applied Mathematics, 64:557–583, 2004. [39] S. Pelipenko and I. Frigaard. Visco-plastic fluid displacements in near - vertical narrow eccentric annuli: prediction of travelling wave solutions and interfacial instability. Journal of Fluid Mechanics, 520:343–377, 2004. [40] P. Petitjeans and T. Maxworthy. Miscible displacements in capillary tubes. part 1. Experiments. Journal of Fluid Mechanics, 326:37–56, 1996. [41] N. Rakotomalala, D. Salin, and P. Watzky. Miscible displacement between two parallel plates: BGK lattice gas simulations. Journal of Fluid Mechanics, 338:277–297, 1997. [42] J. Scoffoni, E. Lajeunesse, and G. Homsy. Interfacial instabilities during dis- placements of two miscible fluids in a vertical pipe. Physics of Fluids, 13(3):553– 556, 2001. 255 C.6. Bibliography [43] G. Taylor. Deposition of a viscous fluid on the wall of a tube. Journal of Fluid Mechanics, 10:161, 1961. [44] M. Tehrani, S. Blittleston, and P. Long. Flow instabilities during annular displacement of one non-Newtonian fluid by another. Experiments in Fluids, 14:246–256, 1993. [45] E. Vefring, K. Bjorkevoll, S. Hansen, N. Sterri, O. Saevareid, B. Aas, and A. Merlo. Optimization of displacement efficiency during primary cementing. SPE paper 39009, 1997. [46] Z. Yang and Y. Yortsos. Asymptotic solutions of miscible displacements in geometries of large aspect ratio. Physics of Fluids, 9:286–298, 1997. [47] J. Zhang and I. Frigaard. Dispersion effects in the miscible displacement of two fluids in a duct of large aspect ratio. Journal of Fluid Mechanics, 549:225–251, 2006. [48] W. Zimmerman and G. Homsy. Nonlinear viscous fingering in miscible dis- placement with anisotropic dispersion. Physics of Fluids A, 3:1859–1872, 1991. 256 Appendix D List of publications 1. Carrasco-Teja, M. and Frigaard, I.A. (2009) Non-Newtonian fluid displace- ments in horizontal narrow eccentric annuli: Effects of slow motion of the inner cylinder Submitted July 2009 . 2. Makelmohammadi, S., Carrasco-Teja, M., Storey, S. Frigaard, I.A. and Mar- tinez, D.M. (2009) An experimental study of displacement flow phenomena in narrow vertical eccentric annuli J Fluid Mech Accepted November 2009. 3. Carrasco-Teja, M. and Frigaard, I.A.(2009) Displacement flows in horizon- tal, narrow, eccentric annuli with a moving inner cylinder Physics of Fluids DOI:10.1063/1.3193712. 4. Carrasco-Teja, M., Frigaard, I.A., Seymour, B.R. and Storey S.(2008) Visco- plastic fluid displacements in horizontal narrow eccentric annuli: stratification and traveling wave solutions, J Fluid Mech 605:293-327. 5. Carrasco-Teja, M., Frigaard, I.A. and Seymour, B.R. (2008) Cementing Hor- izontal Wells: Complete zonal isolation without casing rotation SPE paper 114955, Presented at the CIPC/SPE Gas Technology Symposium 2008: Cal- gary, Alberta, Canada, 16-19 June 2008. 6. Lewis, G., Frigaard, I.A., Huang, H., Myers, T., Westbrook, R. and Carrasco- Teja, M.(2005) Simple models for an injection molding system Canadian Ap- plied Maths Quarterly 12:4. 257
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Primary cementing of a highly deviated oil well Carrasco-Teja, Mariana 2010
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Title | Primary cementing of a highly deviated oil well |
Creator |
Carrasco-Teja, Mariana |
Publisher | University of British Columbia |
Date Issued | 2010 |
Description | In this thesis we study laminar displacement flows of one ﬂuid by another in a horizontal annulus. The study comes from the primary cementing of highly deviated oil and gas wells. Highly deviated wells are those in which part of the well bore is nearly horizontal. Primary cementing is a critical process in the construction of a well. The objective is to provide zonal isolation, i.e., a hydraulic seal between the well and the surrounding rock. This is essential to protect the environment and increase the productivity of the well. Therefore, an understanding of the process is indispensable. We model primary cementing displacement flows using a Hele-Shaw approach, and provide simple scientific tools to improve the design of cementing jobs. The contribution of the thesis comes in three parts. Firstly, we analyse the displacement of one viscoplastic fluid by another in a near-horizontal eccentric annulus with a fixed inner pipe. We present examples that illustrate the differences between vertical and horizontal displacements. We then derive a 1D lubrication model which gives analytical conditions that predict when the flow will stratify, according to the fluid properties and the annulus geometry. Secondly, we derive a 2D displacement model for Newtonian fluids which includes rotation and reciprocation of the inner cylinder. This is a common practice in the industry and not well understood. Using an asymptotic approach, we find steady-state traveling wave solutions for nearly- flat interfaces. Then we use numerical simulations to understand the flow dynamics for more elongated interfaces. In particular, we show that casing rotation can lead to local instabilities and mixing, which can shorten the length of the interface. Finally, we generalise this moving casing model to viscoplastic ﬂuids. Using a lubrication- type model we explore the effects of casing motion, again deriving conditions for there to be steady solutions. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-04 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0068718 |
URI | http://hdl.handle.net/2429/17466 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
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Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2010-05 |
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UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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