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Annular displacement flows in turbulent and mixed flow regimes Maleki Zamenjani, Amir 2019

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Annular displacement flows in turbulent and mixed flowregimesbyAmir Maleki ZamenjaniBSc. Mechanical Engineering, Sharif University of Technology, 2011MSc. Mechanical Engineering, The University of British Columbia, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)November 2018© Amir Maleki Zamenjani, 2018The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:Annular displacement flows in turbulent and mixed flow regimessubmitted by Amir Maleki in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Mechanical EngineeringExamining Committee:Ian A. Frigaard, Mechanical Engineering and MathematicsSupervisorColin Macdonald, MathematicsSupervisory Committee MemberAnthony Wachs, Chemical Engineering and MathematicsSupervisory Committee MemberBernard Laval, Civil EngineeringUniversity ExaminerAnthony Peirce, MathematicsUniversity ExaminerAdditional Supervisory Committee Members:Dana Grecov, Mechanical EngineeringSupervisory Committee MemberiiAbstractThis thesis presents a comprehensive, yet practical, two-dimensional model for thedisplacement of viscoplastic fluids in eccentric annuli in laminar, turbulent andmixed flow regimes. The motivations originally stem from primary cementing ofoil and gas wells, as well as other types of wells such as those in Carbon Captureand Storage applications. During primary cementing, cement slurries are placedin an annular region between a steel casing and a wellbore to provide mechani-cal stability and hydraulic isolation. Several complications may arise due to theeccentricity of the annular region, as well as the viscoplastic nature of the fluidsinvolved.The existing 2D and 3D models of primary cementing assume the flow is lam-inar, while in practice, turbulent and more importantly, mixed flow regimes arecommon. In this thesis, we fill this gap in knowledge. More specifically, we ex-pand the laminar model of Bittleston et al. [24] and develop a new formulation thatincludes turbulent and mixed flow regimes. This new formulation considers scalingbased on the disparity of length-scales, which allows a narrow-gap averaging ap-proach to be effective. With respect to the momentum equations, the leading-orderequations correspond to a turbulent shear flow in the direction of the modified pres-sure gradient. With respect to the mass transport equations that model the miscibledisplacement, to leading-order turbulence effectively mixes the fluids. Changes inconcentrations within the annular gap arise due to the combined effects of advec-tion with the mean flow, anisotropic Taylor dispersion (along the streamlines) andturbulent diffusivity.This new extension allows us to understand the process of cementing moredeeply, and resolve several questions that have been left unanswered for manyiiiyears. In particular, we show that many simple statements/rules that are oftenemployed in industry do not stand up to serious analysis. Instead, modelling ap-proaches such as the one developed here can incorporate specific features of wellsin the simulations, and therefore, yield more accurate predictions.ivLay SummaryPrimary cementing is a process by which cement slurries (dry cement + water)are placed outside of a metal pipe through which oil and gas are extracted. Whencured, the cement will provide mechanical stability and a hydraulic seal. Failureof primary cementing operations can have severe economical and environmentalconsequences, which can range from leakage of gas (in varying degrees), throughto rarer but more extreme well control incidents. In this study, we derive a mathe-matical model that accounts for several complexities of this process. For example,cement slurries behave like solid materials when the force they experience is belowsome threshold. Above that threshold, they behave like liquids. Further compli-cations may arise from the varying and complex annular geometry between thewellbore and the metal pipe. Having derived our cementing model and solvedit numerically, we then study a wide range of practically interesting cementingscenarios in order to understand limitations in current practice and make recom-mendations. The results of this study have immediate implications for oil and gasproducers, as well as government regulators.vPrefaceThe research presented in the current thesis is conducted by the author, AmirMaleki, under the supervision of Professor Ian Frigaard. The following papersare published or in preparation for publication:• A. Maleki and I. Frigaard, “Axial dispersion in turbulent flows of yield stressfluids”, J. Non-Newt. Fluid Mech. 235 (2016) , 1-19Both authors had equal contributions in this paper, both in terms of generat-ing the contents as well as writing the draft.• A. Maleki, and I. Frigaard, “Primary Cementing of oil and gas wells in tur-bulent and mixed regime”, J. Eng. Math. (2017), 107, 201-230.A. Maleki primarily generated the content and prepared the initial draft. I.Frigaard supervised model derivation, and assisted with writing the paper.• N. Hanachi, A. Maleki and I. Frigaard, “A model for foamed cementing ofoil and gas wells”, J. Eng. Math. (2018). Hanachi was the principal contributor for this paper. A. Maleki assistedin deriving the model and developing the computational codes. I. Frigaardsupervised model derivation, and assisted with writing the paper.• A. Renteria, A. Maleki, I. Frgaard, B. Lund, A. Taghipour and J. Ytrehus,“Effects of irregularity on the displacement flows in primary cementing ofhighly deviated wells”, J. Petrol. Sci. Eng. (2019), 172, 662-680.A. Renteria was the principal contributor for this paper. A. Maleki assistedin running the annular displacement code. I. Frigaard supervised the project,viand assisted with writing the paper. B. Lund, A. Taghipour and J. Ytrehusperformed the experiments and aided with data analysis.• A. Maleki, I. Frgaard, “Tracking fluid interfaces in primary cementing ofsurface casing”, Phys. Fluids (2018), 30, 093104.A. Maleki was the principal contributor for this study. I. Frigaard supervisedmodel derivation, and assisted with writing the paper.• A. Maleki, I. Frgaard, “Turbulent displacement flows in primary cementingof oil and gas wells”, Phys. Fluids (accepted).A. Maleki was the principal contributor for this study. I. Frigaard supervisedmodel derivation, and assisted with writing the paper.• A. Maleki, I. Frgaard, “Comparing laminar and turbulent primary cementingflows”, J. Petrol. Sci. Eng. (under review).A. Maleki was the principal contributor for this study. I. Frigaard supervisedmodel derivation, and assisted with writing the paper.Several parts of this study have been disseminated in forms of oral or posterpresentation at the following conferences:• 37th International Conference on Ocean, Offshore & Arctic Engineering,Madrid, Spain.• 24th IInternational Conference on Theoretical and Applied Mathematics (IC-TAM), Montreal, 2016.• The XVIIth International Congress on Rheology (ICR2016), Kyoto, 2016.A. Maleki has also benefited from a summer undergraduate student, N. Heim,who worked on turning the existing code into a standalone software.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 What is primary cementing? . . . . . . . . . . . . . . . . . . . . 21.2 Why is primary cementing important? . . . . . . . . . . . . . . . 31.3 Why do primary cementing operations fail? . . . . . . . . . . . . 41.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Flows of a single yield stress fluid in annular geometry . . 8viii1.4.2 Experimental works on the displacement flows of yieldstress fluids . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.3 Rule-based systems of primary cementing . . . . . . . . . 111.4.4 2D and 3D models of primary cementing . . . . . . . . . 141.5 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Cementing Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1 Pipe flow hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.1 Choice of dimensionless groups . . . . . . . . . . . . . . 252.1.2 Flow regimes . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Plane channel flows . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Turbulent transition in generalized Newtonian fluids . . . . . . . . 312.4 Dispersion and diffusion of passive scalars . . . . . . . . . . . . . 372.4.1 Velocity profiles in turbulent pipe flows . . . . . . . . . . 392.4.2 Diffusivity and dispersivity in turbulent pipe flows . . . . 502.4.3 Velocity profile in turbulent channel flows . . . . . . . . . 602.4.4 Diffusivity and dispersivity in turbulent channel flows . . 622.5 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . 643 Primary cementing modelling . . . . . . . . . . . . . . . . . . . . . . 663.1 Dimensional governing equations . . . . . . . . . . . . . . . . . 663.2 Scaling and simplification . . . . . . . . . . . . . . . . . . . . . . 693.3 Narrow gap approximation . . . . . . . . . . . . . . . . . . . . . 733.4 Mass transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.4.1 Dispersion effects . . . . . . . . . . . . . . . . . . . . . . 793.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 843.6 One-dimensional closures . . . . . . . . . . . . . . . . . . . . . . 873.6.1 Hydraulic closure . . . . . . . . . . . . . . . . . . . . . . 873.6.2 Dispersion and diffusion closures . . . . . . . . . . . . . 883.7 Model summary and conclusions . . . . . . . . . . . . . . . . . . 89ix4 Computational Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 914.1 Discretization and variable storage . . . . . . . . . . . . . . . . . 914.2 Solving for Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2.1 Variational inequality and weak solution . . . . . . . . . . 934.2.2 Stream function and pressure potential functionals . . . . 944.2.3 Existence and uniqueness . . . . . . . . . . . . . . . . . 974.2.4 Computational algorithm . . . . . . . . . . . . . . . . . . 984.3 Solving for ck . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3.1 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.2 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.3 Displacement example . . . . . . . . . . . . . . . . . . . 1164.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 Fully turbulent displacement flows . . . . . . . . . . . . . . . . . . . 1225.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . 1225.2 Rheology is not very relevant . . . . . . . . . . . . . . . . . . . . 1235.3 Turbulence vs buoyancy . . . . . . . . . . . . . . . . . . . . . . 1315.4 Is turbulence necessarily good? . . . . . . . . . . . . . . . . . . . 1385.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456 Mixed regimes displacement flows . . . . . . . . . . . . . . . . . . . 1486.1 Surface casing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.2 Production casing . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.3 Removing the preflush . . . . . . . . . . . . . . . . . . . . . . . 1636.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667 Using washes for primary cementing . . . . . . . . . . . . . . . . . . 1687.1 Displacement parameters . . . . . . . . . . . . . . . . . . . . . . 1697.2 Laminar wash . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.3 Turbulent wash . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.4 Actual contact time . . . . . . . . . . . . . . . . . . . . . . . . . 1777.5 Low viscous wash . . . . . . . . . . . . . . . . . . . . . . . . . . 1797.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182x8 Tracking displacement interface using suspending particles . . . . . 1858.1 A simplified view of annular displacements and interface tracking 1868.1.1 Annular cementing fluid mechanics . . . . . . . . . . . . 1868.1.2 Particle migration . . . . . . . . . . . . . . . . . . . . . . 1878.1.3 Concentric annulus (wˆ = wˆi) . . . . . . . . . . . . . . . . 1908.1.4 Eccentric annulus (wˆ 6= wˆi) . . . . . . . . . . . . . . . . . 1928.2 Two-dimensional model . . . . . . . . . . . . . . . . . . . . . . 1948.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1958.3.1 Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . 1968.3.2 Non-Newtonian fluids . . . . . . . . . . . . . . . . . . . 2038.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . 2089 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 2129.1 Contributions of the individual chapters . . . . . . . . . . . . . . 2139.2 Industrial implications . . . . . . . . . . . . . . . . . . . . . . . 2179.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . 222Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225A OGRE Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248A.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . 248A.2 Data input and output . . . . . . . . . . . . . . . . . . . . . . . . 249A.3 Data storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252A.4 Elliptic equation solver . . . . . . . . . . . . . . . . . . . . . . . 253A.4.1 Slice model implementation . . . . . . . . . . . . . . . . 253A.4.2 Augmented Lagrangian implementation . . . . . . . . . . 254A.5 Hydraulic module . . . . . . . . . . . . . . . . . . . . . . . . . . 256A.6 Hyperbolic equation solver . . . . . . . . . . . . . . . . . . . . . 260B A model for foamed cementing of oil & gas wells . . . . . . . . . . . 262B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263B.2 Foamed cement hydraulics . . . . . . . . . . . . . . . . . . . . . 265B.2.1 Description of foamed cements . . . . . . . . . . . . . . 267xiB.2.2 Foam rheology . . . . . . . . . . . . . . . . . . . . . . . 269B.2.3 Wall shear stress . . . . . . . . . . . . . . . . . . . . . . 270B.2.4 One-dimensional model . . . . . . . . . . . . . . . . . . 270B.2.5 Analytical Model . . . . . . . . . . . . . . . . . . . . . . 278B.2.6 Mechanical stability limitation . . . . . . . . . . . . . . . 280B.2.7 Positive flow and U-tubing . . . . . . . . . . . . . . . . . 282B.3 Modeling annular displacement flows . . . . . . . . . . . . . . . 282B.3.1 Scaled and simplified model . . . . . . . . . . . . . . . . 283B.3.2 Compressible Hele-Shaw model . . . . . . . . . . . . . . 289B.3.3 Closure model for τw . . . . . . . . . . . . . . . . . . . . 290B.3.4 Computational method . . . . . . . . . . . . . . . . . . . 291B.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292B.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299C Effects of geometrical irregularities on displacement flows . . . . . . 304C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305C.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307C.2.1 Simulation model . . . . . . . . . . . . . . . . . . . . . . 308C.2.2 Experimental description . . . . . . . . . . . . . . . . . . 308C.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 313C.3.1 Experimental results . . . . . . . . . . . . . . . . . . . . 315C.3.2 Simulation results . . . . . . . . . . . . . . . . . . . . . 320C.3.3 Parametric studies . . . . . . . . . . . . . . . . . . . . . 329C.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333C.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338D Rheological and geometric effects in cementing of irregularly shapedwells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348D.2 Displacements in uniform wells . . . . . . . . . . . . . . . . . . 352D.3 Displacements in irregular wells . . . . . . . . . . . . . . . . . . 353D.4 A helical displacement . . . . . . . . . . . . . . . . . . . . . . . 357D.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357xiiList of TablesTable 4.1 L2-norm of error for the three methods in problem 1 (computedsolution - analytical solution) calculated at t = 1. . . . . . . . . 111Table 4.2 L2-norm of error for the three methods in problem 2 (computedsolution - analytical solution) calculated at t = 1. . . . . . . . . 112Table 6.1 Range of density and rheological parameters as well as pumpingrates for the mud, preflush and cement slurry. Red readings arein SI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Table 6.2 Candidate preflush fluids for displacement in the surface casing. 154Table 6.3 Candidate preflushes for displacement in the production casing. 162Table C.1 Parameter range of our experimental study . . . . . . . . . . . 310Table C.2 Rheology of the fluids used in the concentric experiments. . . . 311Table C.3 Rheology of the fluids used in the eccentric experiments (e=0.42).311xiiiList of FiguresFigure 1.1 Schematic view of successive steps of primary cementing . . . 2Figure 2.1 a) n′(n,rY ) for n = 0.1, 0.2, ... 0.9, 1; b) E(n,rY ) for n =0.1, 0.2, ... 0.9, 1. . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.2 Example of the hydraulic quantities for pipe flow . . . . . . . 30Figure 2.3 Example of the hydraulic quantities for channel flow . . . . . 32Figure 2.4 Critical ReMR as a function rY and n . . . . . . . . . . . . . . 36Figure 2.5 The wall-layer scaling parameter ψ . . . . . . . . . . . . . . 44Figure 2.6 The critical layer thickness yc . . . . . . . . . . . . . . . . . 47Figure 2.7 Example velocity profiles in wall coordinate (W+(y+/y+c )) . . 49Figure 2.8 Comparison of Newtonian velocity profile in the wall layer . . 50Figure 2.9 Example profiles of Dt(r) . . . . . . . . . . . . . . . . . . . 53Figure 2.10 Examples of D¯t . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 2.11 Examples of DT and comparison with experimental data . . . 58Figure 2.12 Examples of D¯t as a function of wall shear stress . . . . . . . 59Figure 2.13 Critical wall layer thickness xc . . . . . . . . . . . . . . . . . 61Figure 2.14 Example velocity profiles in wall coordinate (W+(x+/x+c )) forchannel flow . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 2.15 Examples of DT for channel flow . . . . . . . . . . . . . . . 64Figure 3.1 Geometrical parameters of primary cementing . . . . . . . . . 67Figure 3.2 a) |S|(|∇aΨ|); b) |∇aΨ|(|S|) . . . . . . . . . . . . . . . . . . 78Figure 3.3 The closure Hw(Rep), showing asymptotic behaviour . . . . . 88xivFigure 3.4 Variation of D¯1D(blue lines), DT,1D(black lines) and D∗T,1D(redlines) with wall shear stress for n = 0.2,0.4,0.6,0.8 and 1. . . 89Figure 4.1 Staggered mesh . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 4.2 a) |S|(|∇aΨ|); b) |∇aΨ|(|S|) . . . . . . . . . . . . . . . . . . 95Figure 4.3 Contour of c in benchmark Problem 2 at t = 9 . . . . . . . . . 113Figure 4.4 Effect of diffusion and dispersion in turbulent displacement inan eccentric annulus: e = 0.5 . . . . . . . . . . . . . . . . . . 117Figure 4.5 accuracy of the advection-diffusion solver . . . . . . . . . . . 119Figure 4.6 Same as Figure 4.5a, except the displaced and displacing fluidproperties are changed to (4.56) and (4.57), respectively. . . . 120Figure 5.1 Effect of rheological parameter in fully turbulent displacementflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Figure 5.2 Same as Figure 5.1 a, except the flow rate is reduced to Qˆ =0.00071 m3/s (= 0.273 bbl/min), ˆ¯W = 0.025 m/s . . . . . . . 126Figure 5.3 Contour of a) axial velocity and b) wall shear stress corre-sponding to the displacement example shown in Figure 5.1 att = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Figure 5.4 Profile of wall shear stress corresponding to the displacementexample of Figure 5.1 . . . . . . . . . . . . . . . . . . . . . 128Figure 5.5 Profile of wall shear stress corresponding to the displacementexample of Figure 5.1 . . . . . . . . . . . . . . . . . . . . . 128Figure 5.6 Effect of rheological parameter in fully turbulent displacementflows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Figure 5.7 Illustration of the buoyancy effect in steady/unsteady displace-ment flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Figure 5.8 Competition between turbulent stresses and Buoyancy stresses 135Figure 5.9 Profile of differential velocity (wW −wN) along the annulus,corresponding to the simulations in Figure 5.8 . . . . . . . . . 136Figure 5.10 Contour of wall shear stress corresponding to the displacementexample shown in Figure 5.8 . . . . . . . . . . . . . . . . . . 137Figure 5.11 Competition of turbulent stresses vs Buyancy stresses . . . . . 139xvFigure 5.12 Profile of differential velocity (wW −wN) along the annulus,corresponding to the simulations in Figure 5.11 . . . . . . . . 140Figure 5.13 Illustration of too turbulent displacement regime . . . . . . . 141Figure 5.14 Contour of wall shear stress (τw) corresponding to the displace-ment example shown in Figure 5.13 . . . . . . . . . . . . . . 142Figure 5.15 Steady vs unsteady displacement for different values of eccen-tricity e and Richardson number Ri. . . . . . . . . . . . . . . 144Figure 5.16 Axial velocity profile, corresponding to the simulations in Fig-ure 5.13b-c . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Figure 6.1 Typical profile of standoff along an annulus (standoff is 1− e) 150Figure 6.2 Effect of flow regime in a largely eccentric surface casing;cases A1, A2 and A3 . . . . . . . . . . . . . . . . . . . . . . 156Figure 6.3 Effect of flow regime in a largely eccentric surface casing casesB and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Figure 6.4 Effect of flow regime in a largely eccentric surface casing casesAp and Bp . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Figure 6.5 Displacement efficiency as a function of time in a highly ec-centric annulus . . . . . . . . . . . . . . . . . . . . . . . . . 159Figure 6.6 Displacement efficiency as a function of time in a moderatelyeccentric annulus . . . . . . . . . . . . . . . . . . . . . . . . 160Figure 6.7 Narrow side displacement efficiency (ηN) vs time (t) in a mod-erately eccentric annulus . . . . . . . . . . . . . . . . . . . . 161Figure 6.8 Narrow side displacement efficiency (ηN) vs time (t) in pro-duction casing . . . . . . . . . . . . . . . . . . . . . . . . . 163Figure 6.9 Effect of flow regime in removing the spacer in a largely ec-centric surface casing; cases A1, A2, A3, B and C . . . . . . . 164Figure 6.10 Displacement efficiency as a function of time; removing preflush165Figure 7.1 Laminar pre-flush displacement in a nearly concentric annuluswith no density difference . . . . . . . . . . . . . . . . . . . 171Figure 7.2 Laminar pre-flush displacement in a moderately eccentric an-nulus with no density difference . . . . . . . . . . . . . . . . 172xviFigure 7.3 Laminar pre-flush displacement in a highly eccentric annuluswith no density difference . . . . . . . . . . . . . . . . . . . 173Figure 7.4 Laminar pre-flush displacement in a moderately eccentric an-nulus with moderate negative density difference . . . . . . . . 174Figure 7.5 Laminar pre-flush displacement in a moderately eccentric an-nulus with large negative density difference . . . . . . . . . . 175Figure 7.6 Turbulent pre-flush displacement in a nearly concentric annu-lus with no density difference . . . . . . . . . . . . . . . . . 176Figure 7.7 Turbulent pre-flush displacement in a moderately eccentric an-nulus with no density difference . . . . . . . . . . . . . . . . 177Figure 7.8 Turbulent pre-flush displacement in a moderately eccentric an-nulus with moderate negative density difference . . . . . . . . 178Figure 7.9 Actual contact time (in seconds) for laminar preflush. . . . . . 179Figure 7.10 Actual contact time vs actual turbulent contact time; effect ofdensity difference . . . . . . . . . . . . . . . . . . . . . . . . 180Figure 7.11 Actual contact time vs actual turbulent contact time; effect ofeccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Figure 7.12 Displacement with a low viscous lightweight preflush; smallflow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Figure 7.13 Displacement with a low viscous lightweight preflush; largeflow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Figure 7.14 Actual contact time vs actual turbulent contact time (bottomrow) for a displacement with a low viscous lightweight preflush 183Figure 8.1 Schematic of our 1D simplified displacement model. . . . . . 188Figure 8.2 Phase plane for the system of (8.7) when ∆w = 0 . . . . . . . 191Figure 8.3 Phase plane and sample solutions for the system (8.7) . . . . . 193Figure 8.4 Snapshot of the displacement with particles shown with greencircles; large buoyancy number . . . . . . . . . . . . . . . . . 197Figure 8.5 Snapshot of the displacement with particles shown with greencircles; moderate buoyancy number . . . . . . . . . . . . . . 199Figure 8.6 Snapshot of the displacement with particles shown with greencircles; small buoyancy number . . . . . . . . . . . . . . . . 199xviiFigure 8.7 Snapshot of the displacement with particles shown with greencircles; Effect of position of particle at releasing time . . . . . 201Figure 8.8 Snapshot of the displacement with particles shown with greencircles; effect of eccentricity . . . . . . . . . . . . . . . . . . 202Figure 8.9 Snapshot of the displacement with particles shown with greencircles; effect of shear-thinning a) Same as Figure 8.8b, exceptn2 = 0.5; b) Same as Figure 8.5b, except n2 = 0.5. . . . . . . 204Figure 8.10 Snapshot of the displacement with particles shown with greencircles; effect of yield stress . . . . . . . . . . . . . . . . . . 206Figure 8.11 Snapshot of the displacement with particles shown with greencircles; unsteady displacement . . . . . . . . . . . . . . . . . 207Figure 8.12 Snapshot of the displacement with particles shown with greencircles; when pump is shut down . . . . . . . . . . . . . . . . 208Figure B.1 a,b) Schematic of the primary cementing. c,d) Unwrapping theannulus into a channel of varying width . . . . . . . . . . . . 266Figure B.2 Results from 1D displacement model with parameters from ex-ample (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276Figure B.3 Results from 1D displacement model with parameters from ex-ample (ii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Figure B.4 Results from 1D displacement model with parameters from ex-ample (iii) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Figure B.5 Results from the analytical model for the parameters of exam-ple (i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280Figure B.6 Results from one-dimensional simulation in example (i). a)ρˆ computed when friction is included. b) ∆ρˆ = ρˆwith friction−ρˆno friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280Figure B.7 Density of the foam at the interface, for different pˆc = pˆa usingthe analytical model . . . . . . . . . . . . . . . . . . . . . . 281xviiiFigure B.8 2D simulation of displacement of Example (i) with no eccen-tricity in the annulus. a) Snapshots of mud mass fraction Y1.White lines show mass streamlines. b) From left to right, con-tours of mud mass fraction, density, quality, axial velocity andpressure at time tˆ = 1991 s. Only the annulus is shown, withhere ξˆbh = 0 and ξˆtz = 500m. . . . . . . . . . . . . . . . . . . 293Figure B.9 Same as Fig B.8, except the well has eccentricity e= 0.1. Wideand narrow sides are denoted with W and N. . . . . . . . . . . 296Figure B.10 Same as Fig B.8, except the well has eccentricity of e = 0.5.Wide and narrow sides are denoted with W and N. . . . . . . 297Figure B.11 Incompressible displacement with cement properties ρˆ = 1524kg/m3, n = 1, κˆ = 0.055 Pa.s and τˆY = 7.94 Pa, (identical tothe foamed slurry properties at ξˆ = 150 m and tˆ = 926 s inFigure B.8): a) e = 0; b) e = 0.1 and c) e = 0.5. . . . . . . . . 298Figure B.12 Same as Figure B.8, except a back pressure of pˆa = 20 atm isenforced on top of annulus. . . . . . . . . . . . . . . . . . . . 300Figure B.13 Same as Figure B.8, except a back pressure of pˆa = 40 atm isenforced on top of annulus. . . . . . . . . . . . . . . . . . . . 301Figure C.1 Schematic of highly deviated wellbore with an irregular en-larged section. . . . . . . . . . . . . . . . . . . . . . . . . . 306Figure C.2 a) Flow loop schematic; b) Enlarged washout section. . . . . . 309Figure C.3 Repeatability of the setup . . . . . . . . . . . . . . . . . . . . 310Figure C.4 position of conductivity probes . . . . . . . . . . . . . . . . . 312Figure C.5 Example of the conductivity signal recorded in the annulus;case H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313Figure C.6 Example of the conductivity signal recorded in the annulus;case C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314Figure C.7 Normalized local axial velocity in half annulus from the exper-imental data; concentric cases . . . . . . . . . . . . . . . . . 317Figure C.8 Normalized local axial velocity in half annulus from the exper-imental data; eccentric cases . . . . . . . . . . . . . . . . . . 319Figure C.9 Displacement simulation for case G2 . . . . . . . . . . . . . 321xixFigure C.10 Displacement simulation for case H2 . . . . . . . . . . . . . 322Figure C.11 Normalized local axial velocity in half annulus from simula-tions; concentric cases . . . . . . . . . . . . . . . . . . . . . 323Figure C.12 Normalized local axial velocity in half annulus from simula-tions; eccentric cases . . . . . . . . . . . . . . . . . . . . . . 324Figure C.13 Displacement simulation for case A2 . . . . . . . . . . . . . 325Figure C.14 Conductivity signal recorded in the annulus . . . . . . . . . . 326Figure C.15 Displacement simulation in a concentric annulus with moder-ately increased yield stress; 90 degrees inclined . . . . . . . . 331Figure C.16 Displacement simulation in a concentric annulus with moder-ately increased yield stress; 80 degrees inclined . . . . . . . . 332Figure C.17 Displacement simulation in a concentric annulus with moder-ately increased yield stress; 70 degrees inclined . . . . . . . . 333Figure C.18 Displacement simulation in a concentric annulus with moder-ately increased yield stress; 60 degrees inclined . . . . . . . . 334Figure C.19 Displacement simulation in a concentric annulus with largelyincreased yield stress; 90 degrees inclined . . . . . . . . . . . 335Figure C.20 Concentration colourmaps and streamlines of simulation forthe concentric annulus with largely increased yield stress; 80degrees inclined . . . . . . . . . . . . . . . . . . . . . . . . . 336Figure C.21 Displacement simulation in a concentric annulus with largelyincreased yield stress; 70 degrees inclined . . . . . . . . . . . 337Figure C.22 Displacement simulation in a concentric annulus with largelyincreased yield stress; 60 degrees inclined . . . . . . . . . . . 338Figure C.23 Displacement simulation in an eccentric annulus with moder-ately increased yield stress; 90 degrees inclined . . . . . . . . 339Figure C.24 Displacement simulation in an eccentric annulus with moder-ately increased yield stress; 80 degrees inclined . . . . . . . . 340Figure C.25 Displacement simulation in an eccentric annulus with moder-ately increased yield stress; 70 degrees inclined . . . . . . . . 341Figure C.26 Displacement simulation in an eccentric annulus with moder-ately increased yield stress; 60 degrees inclined . . . . . . . . 342xxFigure C.27 Displacement simulation in an eccentric annulus with largelyincreased yield stress; 90 degrees inclined . . . . . . . . . . . 343Figure C.28 Displacement simulation in an eccentric annulus with largelyincreased yield stress; 80 degrees inclined . . . . . . . . . . . 344Figure C.29 Displacement simulation in an eccentric annulus with largelyincreased yield stress; 70 degrees inclined . . . . . . . . . . . 345Figure C.30 Displacement simulation in an eccentric annulus with largelyincreased yield stress; 60 degrees inclined . . . . . . . . . . . 346Figure D.1 Uniform eccentric wellbore unwrapped into a Hell-Shaw cell.Schematic from Pelipenko and Frigaard [180]. . . . . . . . . 349Figure D.2 Progression of two uniform eccentric annular displacements . 350Figure D.3 Mud displacement for uniform wells . . . . . . . . . . . . . . 352Figure D.4 Eccentricity vs. depth: proposed shape for irregular wells; heree = 0.3 is the uniform value. . . . . . . . . . . . . . . . . . . 354Figure D.5 Mud displacement in the irregular annulus with e = 0.3±0.2 . 355Figure D.6 Mud displacement in the irregular annulus with e = 0.6±0.2 . 355Figure D.7 Mud displacement for irregular well with helical geometry . . 356xxiNomenclatureADM,BDM: constants of log-law (turbulent velocity profile)Bu: Buoyancy numberck: concentration of fluid kDˆ: Pipe diameterDˆe: eddy diffusivityDˆt : turbulent diffusivityD¯: averaged dimensionless turbulent diffusivityDˆT : Taylor dispersivityDˆ∗T : secondary dispersivity (originated by geometry variations)dˆ: dimensional mean gap widthE: Axillary variable defined by κˆp ˆ˙γL/τˆwes: local direction of streamlinese: eccentricityFr: Froude numberf f : Fanning friction factorH: dimensionless mean gap widthHe: Hedström numberHw: Hedström number based on wall shear stressHw,1: Hw when flow is not laminar anymoreHw,2: Hw when flow is fully turbulentn: power-law indexn′: flow dependent power-law index defined by (2.3)Pe: Péclet numberQˆ0: Representative flow ratexxiiReMR: Metzner-Reed Reynolds numberRep: power-law Reynolds numberRe1: Metzner-Reed Reynolds number when flow is not laminar anymoreRe2: Metzner-Reed Reynolds number when flow is fully turbulentRi: Richardson numberrˆ: dimensional radial coordinaterˆi: inner radius of annulusrˆo: outer radius of annulusrˆa: mean radius of annulus (local)rˆa,0: axially averaged mean radius of annulus (global)ra: dimensionless mean radius of annulus (local)rY : ratio of yield stress to wall shear stressrALG,ρALG: augmented Lagrangian parameterstˆ: timetˆACT : contact timeˆ¯W0: mean velocity (global scale)ˆ¯WL: mean velocity in a laminar flowWˆ∗: friction velocityy: local annular gap coordinatey+ and x+: dimensionless wall coordinate defined based on friction velocityzˆ: axial coordinateβ : inclination angle from verticalµˆe f f : effective viscosityˆ˙γL: laminar strain rate at the wallˆ˙γN : Newtonian strain rate at the wallˆ˙γ∗: nominal strain rateρˆ: densityκˆ: consistencyκˆp: power-law consistencyκˆ ′: flow-dependent consistency defined by (2.3)ψ: wall layer scaling parameterΨ: stream functionθ : azimuthal coordinatexxiiiφ : scaled azimuthal coordinate (θ/pi)τˆ: dimensionless time (only in chapter 8)τˆY : yield stressτˆw: wall shear stressδ : mean aspect ratio of annulus (local)δ0: axially averaged mean aspect ratio of annulus (global)η : displacement efficiencyηN : narrow side displacement efficiencyξ : dimensioneless coordinate axis along the well (measuring upward)ξˆ : coordinate axis along the well (measuring downward)ξˆbh: bottom hole axial coordinatexxivGlossaryThe following technical terms are used throughout the thesis:• Bottom plug: A mechanical device to isolate the spacer from the cementslurry, while moving downwards inside the casing. When the cement slurryhas reached to the bottom of well, the plug is ruptured by pressure.• Casing: A metallic pipe inserted in, and cemented to, the borehole. Wellboreequipments, such as production packers and blowout preventers are installedon the casing. The casing provides mechanical stability and hydraulic isola-tion and controls pressure.• Cement slurry: A mixture of cement powder, water and several additives.The additives modify the physical properties of the cement such as its vis-cosity, yield stress, thickening time, etc.• Centralizer: A device that keeps the casing or liner at the center of the well-bore. Centralizers are fitted to the casing and typically have bowsprings tokeep the casing at the center.• Conditioning (Mud Conditioning): For the sake of a better displacement,mud properties are changed by adding additives and circulating the mudaround the flowpath before cementing. This process is called mud condi-tioning. Often water or dispersants are added to the mud to change its den-sity and rheology. It is recommended to circulate the mud before and afterremoval of drill pipe.• Dogleg: Places in the wellbore where the trajectory of the wellbore changesrapidly, faster than anticipated or desired.xxv• Drilling mud: Drilling mud is muddy liquid that is left after drilling. It is pri-marily made up of drilling fluid and formation cuttings. Drilling fluid is usedduring drilling to facilitate drilling. Drilling fluid suspends and carries thedrill cuttings and keeps the drill bit cool. In addition, it stops the formationfluids from invading the wellbore.• Drillpipe: Tubular steel pipes are fitted together to form a drillpipe. Thedrillpipe is connected to the rig surface from one end and to the drill bit atthe other end. Drilling fluid is pumped through the drillpipe during drilling.• Eccentricity: In the region between two parallel cylinders, eccentricity is ameasure of offset of the two cylinders’ axes. When the two cylinders areco-axial, the eccentricity is zero (stand off is 100%). The other extreme iswhen the inner cylinder touches the outer cylinder. In this case eccentricityis 1 (stand off is 0%). See §3.1 for the precise definition.• Frictional pressure: The pressure losses due to the wall shear stresses arecalled frictional pressure losses.• Gel strength: A property of drilling mud, working analogously to a yieldstress, which increases when the mud is static until it reaches a plateauafter 10-60 minutes. The gel strength has to be broken for the mud toflow. Gel strengths occur in drilling fluids due to the presence of electri-cally charged molecules and clay particles (Drilling Fluid Reference Manual,Baker Hughes, 2006).• Liner: A liner is a casing that extends downwards from just above the previ-ous casing.• Preflush: Primary cementing typically starts with conditioning the mud andthen proceeds to pumping a sequence of the so called preflushes inside thecasing. Preflush can be a spacer or a wash. Preflushes are intended to breakthe static gelation in the mud, clean the wellbore from the cuttings and pro-vide a buffer layer between the mud and cement slurry.• Spacer: Spacers are heavier preflushes with viscosifying constituents. Be-cause of their larger density and viscosity, spacers are more often pumpedxxviin laminar regime. Spacers are placed to separate cement slurry from thedrilling mud. Often cement slurries and the drilling mud are chemically in-compatible, meaning that their contact can damage highly tuned propertiesof cement slurry (such as thickening time) [167]. Furthermore, spacers rhe-ological parameters can be designed carefully to aid the displacement of thedrilling mud [232].• Wash: Washes may be water-based or oil-based. Rheologically, they aregenerally Newtonian fluid solutions (e.g. water). They are designed to washthe walls of the annulus free from residual fluids (and any remaining solids),to leave the annulus water-wet for the cement slurry. In addition, they shouldbreak any static gelation of the mud, mobilizing the mud in general. The lowviscosity of these fluids allows them to be pumped in turbulent flow regimes.Most common wash is water with a range of chemical additives.• Wellbore: A hole that is drilled to extract natural resources such as oil andgas.• Yield stress fluid: Yield stress fluids, also call viscoplastic fluids, are a cat-egory of non-Newtonian fluid which exhibit yield stress behavior; i.e. theydo not flow, unless they are sufficiently stressed.xxviiAcknowledgmentsToday, I am grateful and fortunate to have been surrounded by many wonderfulindividuals who made this day possible; the day that I write the acknowledgementof my PhD thesis.I have reserved three big “Thank you!” for five outstanding individuals. Thefirst one is for my mentor and supervisor, Professor Ian A. Frigaard. Ian is an ex-emplary supervisor: knowledgeable yet humble, insightful, passionate but patient,friendly, considerate and funny. Ian is committed to his students’ growth and de-velopment. I feel truly privileged to have worked under his supervision. “Thankyou Ian!”My second “Thank you!” goes to Dr. Sarah Hormozi. Sarah is a source of en-couragement and commitment and a true example of “hard work pays off”. Thereis no more confident version of me than when I just have talked to Sarah. I am trulyindebted to her for her constant support and wise advices. “Thank you Sarah!”My last “Thank you!” belongs to, of course, my family: Hamid, Afsaneh andAmin, my father, mother and brother. Without their sacrifices, I could never bestanding here at this exciting moment of my life. Although my heart aches thatthey are not here today to celebrate this very moment, I feel their presence in myeveryday life. “I love you.”I would like to seize this opportunity to thank two of my mathematics teachersin highschool, Mr. Akbarzadeh and Mr. Askarinejad, who introduced me to theamazing world of mathematics. I remember I kept both of them in the class duringtheir breaks, asking them (most likely) stupid questions, and they open heartedlyanswered. Fast forward, I am a PhD student at UBC. At UBC, the most inspiringcourses I took were those with Professor Michael Ward. I thank Michael for hisxxviiiamazing lectures, and his fruitful office hours.During my PhD a lesson that I learned, not quite easily though, was that re-search does not work until it does. One of the activities that kept me going on the"not working" days of research was teaching. I would like to thank Drs. JosephTopornycky, Shaya Golparian and Jonathan Verrett and Professor Richard Ansteeand the Centre for Teaching, Learning and Technology for helping me to grow andprosper as a teacher.Dr. Peterson, the author of “12 Rules for life, an antidote to chaos” writesin his book, as one of the rules of life: “Make friend with people who want thebest for you”. Completion of this thesis would not be possible without the supportand love of Sharareh, Morteza, Parisa, Shakiba, Alireza, Yves, Sulul, Mahdiar,Ida, Behrooz, Bana, Ehsan, Hamed, Nazanin, Majid, Ata, Rosie, Nikoo, Giordano,Analise, Babak, Kata, Hedieh, Mehdi, Ali, Ali, Elizabeth, Vahid, Elizabeth, Lau-ren, Marjan and Alondra.I acknowledge the funding that I received from the Natural Sciences and Engi-neering Research Council of Canada, British Columbia Oil and gas Commission,Schlumberger and Sintef Petroleum.Last but certainly not least, I would like to show my sincere respect to the giantsof calculus: Mr. Isaac Newton and Mr. Gottfried Leibniz. Had they not dedicatedtheir lives to mathematics, human would live in a dark calculus-less world, a worldI would never want to live in. Needless to mention, there was no thesis then for mefor which I want to write the acknowledgement chapter.xxixDedicationI dedicate this thesis to myself for not giving up, for believing in my own abilities,and for my perseverance. I am proud of myself.xxxChapter 1IntroductionIn this work we study the displacement of one fluid by another in a narrow, eccen-tric annulus in laminar, turbulent and mixed flow regimes. The study is motivatedby the processes involved in the primary cementing of oil and gas wells. The mainfocus of this work is to understand the primary cementing from a fluid mechanicperspective, identify the current industrial misunderstandings, and propose meansto improve the efficiency of this critical operation.In the following sections, we briefly review the primary cementing and differ-ent procedures that it entails and explain why this operation plays a critical role,not only economically (in terms of oil and gas production), but also environmen-tally. We continue the chapter by outlining the types of fluids that are common inthe primary cementing operations, and explain how their physical properties mayinfluence the outcome of a cement job. Subsequently, we proceed to reviewing theexisting literature on the displacement of yield stress fluids. In doing so, we brieflytouch on displacement flows in pipe geometry, in Hele-Shaw cells and in porousmedia, but the main focus remains on those in annular or plane channel geome-tries. Moreover, we review the industrial design rules that have been developedfor successful annular displacements between 1970-1990. These rules are only de-rived for laminar displacement in vertical wells. Finally, we turn our attention tothe more recent developments, including 2D and 3D models of primary cementing.We will then close the chapter by outlining the objectives of the thesis.1Drill new stage of wellRemovedrillpipeInsertsteelcasingPump spacerfluidPump lead & tail slurryDisplacemud in annulusEnd of operationFigure 1.1: Schematic view of successive steps of primary cementing1.1 What is primary cementing?Primary cementing is the process by which oil and gas wells are sealed duringconstruction. The seal is achieved by placing cement into a narrow gap formedbetween the drilled borehole and the outside of a steel casing1 (or liner), that isplaced in the well. The cement not only seals the well hydraulically, preventingfluids from migrating axially along the wellbore between fluid-bearing zones, butalso provides mechanical support, resisting geo-mechanical stresses. Primary ce-menting in the conventional form of pumping and displacement was first used in1910 in shallow wells in California [167]. The process proceeds as follows; seeFigure 1.1. A new section of the well is drilled. The drillpipe is removed from thewellbore, leaving drilling mud inside the wellbore. A steel tube (casing or liner)is inserted into the wellbore, typically leaving an average annular gap of≈ 2-3 cm.The tubing is inserted in sections of length roughly 10 m each, threaded togetherso that cemented sections can extend 100 to 1000 meters. So-called centralizersare fitted to the outside of the tube, to prevent the heavy steel tubing from slumping1Throughout this chapter, several technical terms are highlighted with a bold font, which are fullydefined in the glossary provided on page xxv.2to the lower side of the wellbore. However, even in (nominally) vertical wells itis common that the annulus is eccentric and this is especially true in inclined andhorizontal wells. With the steel casing in place and drilling mud on the inside andoutside, the operation begins. First, the drilling mud is conditioned by circulatingaround the flow path. Next a sequence of fluids are circulated down inside of thecasing and returning up the outside of the annulus. Preflushes (washes or spacerfluids) are followed by one or more cement slurries. The fluid volumes are de-signed so that the cement slurries fill the annular space to be cemented. Drillingmud follows the final cement slurry to be pumped and the operations ends withthe cement slurry held in the annulus (with a valve system). The cement is thenset over a period of many hours. With reference to Figure 1.1, it can be seen thatthe completed well often has a telescopic arrangement of casings and liners. Thus,the operation is repeated more than once on most wells. Typically, annulus innerdiameters can start at anything up to 50 cm and can end as small as 10 cm in theproducing zone.1.2 Why is primary cementing important?The whole process of primary cementing takes place over the span of 1 or 2 days,but its economical and environmental impacts will remain for tens or maybe hun-dreds of years. In Canada, there is an estimate that 5-20% of wells leak to variousdegrees [63], due to the failure of the primary cementing. Similar estimates arereported for other countries too [51]. This has both economical and environmentalconsequences.From an economical point of view, the recovery rates for conventional reser-voirs are typically 20-40%, which means losing even a few percent is expensive. Inaddition, a well with a poor cementing will normally have further complications,such as necessary remedial treatments, which imposes a significant economicalburden [244]. As a result, both industry and regulators have advocated for “Doingit right the first time” [213, 63].More critically, the environmental impacts of oil and gas wells may span sev-eral hundreds of years. That means a leaking well will be a continuous source ofenvironmental damage and safety hazards. Such environmental concerns include3groundwater contamination due to wellbore leakage [38, 12] as well as greenhousegas emission [126]. An example of a massive consequence of a poor cementingjob is the notorious Deep Water Horizon blowout in the Gulf of Mexico.1.3 Why do primary cementing operations fail?To understand the sources of failure of primary cementing, and the subsequentleakage of wells, we begin to consider primary cementing as a fluid mechanicsproblem. The fluids involving in primary cementing are drilling muds, washes,spacers and cement slurries. With the exception of washes being Newtonian, theother fluids mentioned above typically exhibit yield stress behavior; i.e. they donot flow, unless they are sufficiently stressed. These fluids are called yield stress,or interchangeably viscoplastic, fluids.Commonly, the fluids in the oil and gas well cementing are described by the“Herschel-Bulkley” (HB) model. The HB model is defined by the following con-stitutive relation between the stress and strain rate:2τˆi j =(κˆ ˆ˙γn−1+τˆYˆ˙γ)ˆ˙γ i j⇐⇒ τˆ > τˆY (1.1a)ˆ˙γ = 0⇐⇒ τˆ ≤ τˆY (1.1b)Here τˆi j is the deviatoric stress tensor and ˆ˙γi j is the strain rate tensor:ˆ˙γi j =∂ uˆi∂ xˆ j+∂ uˆ j∂ xˆi, (1.2)the uˆ j’s are the components of the velocity field, and τˆ and ˆ˙γ are the second invari-ant of their respective tensor; i.e.ˆ˙γ =√√√√123∑i, j=1| ˆ˙γ2i j| τˆ =√√√√123∑i, j=1| ˆ˙τ2i j|. (1.3)The three rheological parameters of HB model are the consistency (κˆ), the power-2Throughout this thesis, dimensional variables are denoted with a hat (e.g. τˆ) and dimensionlessvariables without.4law index (n) and the yield stress (τˆY ). If we set τˆY = 0, the HB model reduces to theso called power-law model. The power-law model takes into account that viscosityof some fluids varies with the shear rate. Instead, if we set n = 1, we recoverthe Bingham model. Bingham model is the simplest model for yield stress fluids,which basically assumes the fluid behaves Newtonian (in a sense that the plasticviscosity remains constant), once it is yielded. In addition, there are a handful ofother rheological models, such as Casson model or Carreau model, which are lesscommonly used in the oil and gas industry.Cement slurries also show time-dependent behavior, such as aging or thixotropicbehavior, which is primarily due to the micro-structure of cement particles withinthe suspending fluid [167]. These structures build up or break down which thenresults in change in macro-properties of the fluid. The time-dependent nature ofphysical properties of the fluids is neglected in this thesis.An unsuccessful cement job allows the hydrocarbon to leak. Leakage duringthe primary cementing operation can lead to gas pockets and channels, that com-promise the well. A number of other defects may arise either during the cementingof a well, or afterwards during cement hydration, that allow the well to leak later.The most common fluid-related defects include:• Residual mud channeling: This is where the yield stress of the mud holdsit in place, typically on the narrow side of the annulus, as preflushes andcement slurry by-pass. This is a bulk flow feature predicted well by simplemechanical arguments [156], as well as more sophisticated models [24].• Wet micro-annulus: This is a local mechanical effect, where the displacingfluid does not generate sufficient shear stress to mobilize the mud at thewall. To some extent this is predictable in model flows [9, 85, 247, 254]. Forexample, in a channel of half width Hˆ, where a Newtonian fluid displaces aBingham fluid, Zare et al. [254] suggested that ifBN =τˆY,displacedHˆµˆdisplacingWˆ0< 3, (1.4)the thickness of micro-annulus will gradually diminish, as the displacementprogresses. Here Wˆ0 is displacement mean velocity. At higher values of BN ,5some mud layer may be left behind, depending on the viscosity ratio, densitydifference between the fluids as well as the flow rate. Of course, featuressuch as mud dehydration and uneven wellbores create some complicationswith these results.• Mixing/contamination of the slurry: Mixing (and consequent contamination)occur in different scenarios such as downwards displacement within the cas-ing [8, 223], fluid instabilities in laminar annular flows [181, 162, 163] or inturbulent annular displacement flows. The latter scenario will be extensivelystudied in Chapter 2. In combination with mud channels, micro-annuli oreven mud pockets left behind in irregular wellbores (e.g. washouts), resid-ual drilling fluid can be partially eroded/dispersed in a passing slurry andcontinue to contaminate cement over long lengths.Examples of these features can be found in Watson [244].While the above mechanisms are flow-related, not all wellbore leakage hasa fluid-mechanical cause. The most common cause is formation of dry micro-annulus which is due to cement-to-casing or cement-to-formation bonding defects.De-bonding can happen as a result of cement volumetric shrinkage, downholethermal/hydraulic/mechanical stresses or lack of casing and formation roughness[167]. The interested reader is also referred to Watson and Bachu [245, 246], whichanalyze different leakage factors for significant data sets.From a fluid mechanics perspective, one of the main operational questions iswhether it is preferable to cement a well in turbulent or laminar flow. To explainthis, displacement flow regime depends on local geometry and fluid properties aswell as the overall imposed flow rate. It is relatively common within the annulusthat one fluid can be fully turbulent (e.g. a chemical wash or low-viscous spacer)while others are laminar. Indeed, as it will be shown later, this also can occur on asingle section of the annulus, e.g. turbulent on the wide side, laminar or even staticon the narrow side. Furthermore, although some fluids can be strongly turbulent,the more viscous fluids (muds, viscous spacers and slurries) are often only weaklyturbulent, transitional or laminar. These flow regimes have become more prevalentin recent decades as extended reach and horizontal wells, require reduced flow ratesto control friction pressures.6The early literature on cementing generally stated that the turbulent displace-ment is more successful than those in laminar in removing the mud during primarycementing [200, 166, 130]. While some recent studies have been less definitiveand warned that certain conditions must be satisfied for turbulent displacement tosucceed [167, 132], there still appears to be a widely accepted perception in thecementing community that turbulent displacement is necessarily superior to lam-inar displacement. See for example Kelessidis et al. [127] and the very recentLavrov and Torsæter [140] and Enayatpour and van Oort [70]. However, the sci-entific evidence to support this appears to be scant. Two papers by Howard andClark [119] and Smith and Ravi [216] are often cited as references, but althoughthese have observed that displacement experiments with higher flow rate led to bet-ter displacement efficiency, they did not compare turbulent and laminar regimes.Similarly, two separate studies by Smith [215] and Haut and Crook [115] suggestthat “as the annular velocity is increased there is no sharp increase in the displace-ment efficiency at the transition from laminar to turbulent flow” and “high flowrates, whether or not the cement is in turbulent, provide better displacement thanplug flow rates”. The other work that is often cited is Brice and Holmes [32], inwhich 26 wells are experimented where “turbulent flow techniques were appliedin primary cementing”. The quality of the cement jobs was then evaluated withdifferent metrics, including cement bond-log and pressure test, all of which give abulk assessment of the cement, and are rather insensitive to small features such asmicro-annuli. In addition, the study does not use laminar displacement in any ofthe wells, so it is not entirely clear if the success of cement jobs can be attributed tothe turbulent flow regime. Moreover, this study was performed in an era before ourunderstanding of laminar displacements evolved. Ideally, we would like to confirmif the above perception is correct or not. Comparing the laminar vs turbulent dis-placement flows is the objective of Chapter 6. Before we state our other objectivesmore explicitly, we review the available literature on primary cementing, and moregenerally annular displacement flows.71.4 Literature reviewPrimary cementing involves the displacement of one yield stress fluid with anotheryield stress fluid. The displacement happens in the pipe (casing) and then in theannulus. The key parameters here are: i) physical properties of the displacingfluid; ii) physical properties of the displaced fluid; iii) local geometry and iv) flowparameters. These dozen parameters form a multi-dimensional parameter spacewhich is practically impossible to fully study. Instead, to understand the underlyingphysics, we may start by looking at simpler problems, e.g. single flows of a yieldstress fluid in an annular geometry, and then build complexity.1.4.1 Flows of a single yield stress fluid in annular geometryDespite its apparent simplicity, analytical solution for flows of a yield stress fluidin eccentric annulus has proved to be controversial, primarily due to the existenceof the yield stress. It is known that a naive application of classical fluid mechanicsapproaches, such as the lubrication approximation or thin film analysis, can leadto solutions that exhibit velocity gradients within unyielded plug regions. This isinconsistent with the yield stress closure model. The inconsistency is called thelubrication paradox, which has led to many confusing and incorrect statements[141]. The lubrication paradox was later resolved in several cases; see for exampleBalmforth and Craster [14], Frigaard and Ryan [83], Maleki et al. [152] . In partic-ular, Walton and Bittleston [240] analytically solved the laminar flow of a Binghamfluid in an eccentric annulus. They adopted a systematic perturbation technique andcleverly defined a plug-like region where stress exceeds yield stress, but only by avery small amount and called it pseudo-plug. They also showed that true plugs canexist on both narrow and wide sides of the annulus. The prediction of Walton andBittleston [240] was later confirmed numerically by Szabo and Hassager [218].Another approach in analyzing the flow in an eccentric annulus is to use theso-called slot approximation, where the curvature of the annulus is neglected, andthe annulus is modeled as a slot of variable height; see for example Iyoho andAzar [121], Fordham et al. [76], Bittleston and Hassager [23]. When the annulus isconcentric, the flow is axisymmetric and reduces to 1D. Several studies have con-sidered such a configuration under different conditions. Recent examples consider8these flows with wall slip [125, 172] or with linear [142] and rotational [18] motionof the inner cylinder.1.4.2 Experimental works on the displacement flows of yield stressfluidsWe now move to displacement flows, where one fluid displaces another fluid. Inaddition to primary cementing, displacement flow problems appear in other indus-trial applications such as drilling, mining and water treatment [34, 46, 151] as wellas biomedical [120] and geophysical [210] applications.In the context of primary cementing, there exists two distinctly different cat-egories of displacement flows: i) displacement in the casing (pipe) where mostlythe flow is downward and a lighter fluid displaces a heavier fluid. ii) displacementin the annulus where the flow is upward and generally a heavier fluid displaces alighter fluid. In both cases, the main objective of any analysis is to predict displace-ment efficiency, and to provide an estimate for the mixing.Downward displacement flows in pipes have been extensively studied by Frigaardand coworkers [9, 221, 222, 223, 219, 6, 7, 8] as well as by Gabard and Hulin [86].In particular, Taghavi et al. [223] classified the near horizontal displacement flowof Newtonian iso-viscous fluids as viscous and inertial. The former is character-ized by lack of interfacial instabilities whereas the latter is characterized by theexistence of interfacial instabilities and mixing. A third class of fully-diffusive dis-placement was then proposed in Alba et al. [8] to include displacement flows ininclined pipes. This new class denotes displacement flows where the fluids com-pletely mix across the pipe. More recently, Etrati and Frigaard [73] extend thisanalysis to displacement flow with different viscosities for both near horizontaland near vertical orientations. They report that viscosifying the less dense fluidtends to significantly destabilize the flow. More interestingly, Hasnain et al. [114]experimentally investigated pipe displacement for immiscible fluids, to mimic dis-placement of oil-based mud with water-based slurries. They found that immisci-bility can significantly enhance the efficiency of the displacement.A slightly different class of displacement flows is exchange flows where thereis no imposed flow. Exchange flows in pipe have been extensively studied by Hulin9and coworkers [204, 205, 206, 207, 208] as well as by Kerswell [129], Beckett et al.[17], Varges et al. [238]. These groups found that the relative magnitude of viscousand inertial velocity scales informs of the leading order behavior of the flow.There are relatively few experimental studies in the annulus. Many of exper-imental studies rely on measuring the electrical conductivity of the fluids aroundthe annulus at multiple axial locations. The conductivity values can then be trans-lated into the concentration of each fluid. Typically, salt is added to one of thefluids to increase the conductivity jump across the interface. Other studies takepictures/movies during the process of displacement. Early experimental studiessuch as those in McLean et al. [156], Lockyear and Hibbert [143] considered easydisplacement scenarios; i.e. no mud with extreme rheological parameters. Moreinterestingly, Tehrani et al. [231] conducted several annular displacement exper-iments with non-Newtonian fluids. They found eccentricity as the most criticalparameter in determining the displacement efficiency. Furthermore, they also ob-serve that a thin layer of fluid may be left behind due to development of interfacialinstabilities. More recently, Malekmohammadi et al. [153] conducted a similarstudy and elucidated the roles of eccentricity, viscosity, density as well as flowrate in annular displacement of Newtonian fluids. In another work, Deawwanich[56] analyzed the role of inner cylinder rotation and reported significant displace-ment improvement in the presence of casing rotation, specially when the annulusis highly eccentric.Another relevant family of displacement flows is the displacement of a yieldstress fluid with air, which is common in injection molding or oil recovery applica-tions. de Souza Mendes et al. [54] conducted a series of displacement experimentswhere air pushes Carbopol in a capillary tube. They identified a critical flow ratebelow which Carbopol is perfectly displaced. Above the critical flow rate how-ever, liquid film start to deposit on the wall tube. Similar behavior was observed byPoslinski et al. [185] who reported that the deposition layer thickness can approachas high as 35% of the pipe radius at high flow rates of gas.Finally, flows in porous media and flows in Hele-Shaw cells resemble simi-larities with those in annular displacement. In particular, we will show that our2D governing equations are similar to those in porous media as well as Hele-Shaw cells. Fluid displacement in porous media has been studied for several10years. The early literature is summarized in Barenblatt et al. [15]. More recentworks concern the stability of miscible displacement and development of viscosity-driven and density-driven instabilities [224, 118, 154, 39]. Displacement flowsin Hele-Shaw cells is rather more recent. A representative list of examples is:[175, 176, 199, 47, 139, 74, 13, 239].1.4.3 Rule-based systems of primary cementingMore broadly in the context of primary cementing, there exists an extensive in-dustrial technical literature. Unfortunately, this literature is less relevant/useful forseveral reasons: i) Much of the literature do not concern displacement flows atall. ii) Even those that deal with the displacement flows rarely focus on the fluidmechanics aspects of the problem. iii) Finally, much of the discussions are basedon sparse studies on a dozen or so cement jobs with anecdotal evidence, whichcan hardly be generalized. Nonetheless, some of these studies bear more atten-tion which we will review here. In particular, we look at a number of rule-basedsystems that evolved, from 1970s to early 1990s, to guide laminar mud displace-ment in vertical wells. These studies often lead to conservative and occasionallycontradictory predictions.• McLean et al. [156] suggested the cement slurry should be thicker than themud. Otherwise, the displacement shall be “aided by the motion of the cas-ing or buoyant forces”. To account for the eccentricity, a rule of thumb is tohaveτˆY,mud(1+ e)< τˆY,cement slurry(1− e)where here e is the eccentricity of the well (see §3.1). The study discouragedthinning a cement slurry for the purpose of achieving turbulent displacement,as it would “reduce the efficiency of the displacement” and increase the pos-sibility of viscous fingering.• Smith [214] defined “cementable wellbore” as the one “as nearly gauge aspossible (without washouts), as straight as possible (without severe dog-legs)and stabilized and properly conditioned” with a gap width of 1.5 in (3.8cm). The absolute minimum gap width is 0.75 in (1.9 cm). The author11also suggested “the maximum slurry density to prevent losing circulation tobe 2096 kg/m3 for the production liner, 1797 kg/m3 for the drilling liner,and 1378 kg/m3 to 1498 kg/m3 filler slurry followed by 1893 kg/m3 tail-inslurry for the intermediate string.” In regard to mud condition, the paperrecommended that “cement pumping should not begin until at least 95% ofthe hole volume is being circulated”.• The work of Lockyear and Hibbert [143] and Lockyear et al. [144] concludedthat three main rules should be satisfied:– The wall shear stress has to be large enough to break the gel strengthof the mud. More explicitly,τˆw > τˆghas to be satisfied everywhere around the annulus (and particularly onthe narrow side). Here τˆg is the gel strength of the mud and τˆw is thewall shear stress.– The wall shear stress on the narrow side must exceed the yield stress ofeach fluid (mud, spacer, and cement).– The interface velocity shall be uniform around the annulus (we willlater call this the steady flow condition). To quantify this conditionmore precisely, the ratio ofwˆnarrowwˆwidemust be calculated at the interface. Ideally the ratio ought to be 1.However, the problem is that the interface velocities on the wide andnarrow sides cannot be determined using any 1D analysis.The three criteria discussed above were then implemented in simulation soft-ware by Ryan et al. [197].• A number of rules were developed by Couturler et al. [48] and then listedmore explicitly in Brady et al. [31] and Theron et al. [232]. These rules are12known as the Effective Laminar Flow (ELF) rule system, which has someoverlap with those of Lockyear and Hibbert [143] and Lockyear et al. [144].ELF states:– The displacing fluid must be at least 10% heavier than the displacedfluid:ρˆdisplacing > 1.1ρˆdisplacedwhere here ρˆ is the density of each fluid.– The frictional pressure gradient exerted by the displacing fluid shouldbe at least 20% larger than that of displaced fluid.∂ pˆ f∂ zˆ displacing> 1.2∂ pˆ f∂ zˆ displacedhere ∂ pˆ f /∂ zˆ is the frictional pressure gradient along the well axis.– The shear stress on the narrow side of the annulus should exceed theyield stress of the displaced fluid. To formulate this, notice that theshear stress depends on the total pressure gradient. There are two fac-tors contributing to wall shear stress: frictional pressure gradient andbuoyancy. Thus,∂ pˆ f∂ zˆ displacing+(ρˆdisplacing− ρˆdisplaced)gˆcosβ > 2τˆY,displaceddˆmin.Here gˆ is the gravitational acceleration, β is the inclination angle mea-sured from vertical, and dˆmin is the gap width on the narrow side.– The displacing fluid speed on the wide side must be equal or smallerthan the displaced fluid speed on the narrow side (steady flow condi-tion).wˆnarrow ≥ wˆwide.Again the last rule which is often called differential velocity criteriamay not be so useful, because 1D models cannot estimate wide andnarrow side velocities at the interface.131.4.4 2D and 3D models of primary cementingSince the 1990s, the industry has been able to access model-based simulators.Model based simulators are now actively used in cementing case studies, wherethey compare favourably with post-placement logging of the wells [173, 26, 96,101]. Model-based simulations have been shown to improve the rule-based sys-tems. In particular, Pelipenko and Frigaard [181] reviewed these rules and com-pared them with predictions of their model-based simulations. They concludedthat there is a general agreement between the model-based simulations and rule-based systems, although the rule-based systems are often too conservative. Herewe review the most recent model-based simulators.A 2D model for annular displacement was first introduced by Bittleston et al.[24]. The key assumption of the model is that the annular gap is narrow comparedto the mean circumference. This assumption has several implications: i) It allowsto average the radial profile of velocity and fluid concentration, and therefore, re-duces the problem to 2D. ii) It justifies neglect of the local curvature of the annuluswhich then allows to unwrap the annulus into a channel of varying width. iii) Itallows us to assume the flow is locally a shear-flow. Since the flow is locally ashear-flow, 1D closures are derived and employed in the model. This model wasanalyzed numerically in Pelipenko and Frigaard [179, 180]. In particular, [180]provided a computational algorithm that is guaranteed to converge and does notrequire any viscosity regularization of the yield stress fluid. Using this compu-tational framework, the model has been extensively studied for analyzing nearlyvertical annuli [181], nearly horizontal annuli [37], as well as exchange flows [81].A new generation of 2D models for primary cementing is introduced by Tardyand Bittleston [226]. Although the model derivation is somewhat different, it isessentially based on the same principles of Bittleston et al. [24]. In particular, itassumes a narrow gap and employs a lubrication-type approximation. However,Tardy and Bittleston [226] work with the pressure formulation and require the flowlaw. More recently, Tardy [225] modified the earlier formulation and reconstructedthe radial velocity profiles. To this end, instead of integrating across the entiregap width, the gap width is divided into several smaller sections and integration isperformed across each small section. The new formulation provides an economical14extension of the previous models to 3D, where the velocity field is still updatedby solving a 2D Poisson equation, and the radial variations can be computed aposteriori.Apart from the above two families of models, other 2D cementing models existin the literature. However, these models fail to properly model the yield stress ofthe mud or cement. For example, Carvalho and coworkers’ models [90, 91, 52] aregenerally derived for Newtonian fluids and then extended to non-Newtonian fluidsvia an effective viscosity. The Lattice-Boltzman based model of Zhao et al. [256]also assumes the fluids are power-law. In addition to the above openly accessiblemodels, many service companies, such as Schlumberger and Halliburton, have in-house simulators that are mostly confidential and unfortunately, make few detailspublicly accessible.The simulators above are all based on lubrication-type 2D models of primarycementing. In fact, 3D simulation of primary cementing is generally not economi-cal, because of the large aspect ratio of the problem. Recall the typical gap width ofthe annulus is 2 cm, but the length of a cemented section is up to 1000 m. Resolv-ing such a skewed geometry numerically requires a large number of computationalcells (based on resolving the gap scale) and might need to run for hour-long pump-ing schedules to complete some operations. Given also the additional iterationsrequired for most non-Newtonian fluid models, this scale of computation is stillpractically infeasible even with multi-processor machines and parallel codes. Inaddition to lower computational cost, 2D models are effective at reproducing largescale process features. However, there are some limitations in using 2D models:i) These models are based on shear-flows, which relies on the gap being narrowwith respect to the annular circumference.ii) These models neglect inertial effects under the assumption that the aspectratio times the Reynolds number is small, which is not always the case. Thus,inertial effects are not accurately represented.iii) Averaging across the gap means that these models predict only the meanconcentrations of fluids and not the transverse distribution.iv) These models are based on the flows being fully developed locally, i.e. rela-15tive to the gap width.Putting these into perspective, 3D simulations can be useful to understand particu-lar features of displacement flows, for example those in a washout where the flowis inherently 3D.Unfortunately, several such 3D analyses in the literature lack a deep physicalunderstanding. For example, many of these studies do not consider or properlymodel rheological effects (see Savery et al. [201], Zulqarnain [258], Chen et al.[40], Gomes et al. [92]). Some other 3D analysis investigates features that canbe more readily studied using a 2D model. For example, Shadravan et al. [209]concluded that rheological hierarchy is critical in horizontal wells in preventingintermixing and viscous fingering. Even worse, some other studies have led toerroneous conclusions. For instance, Dutra et al. [64] concluded that “The influ-ence of eccentricity on the interface shape is rather small”, while it is now widelyaccepted (and will be demonstrated later) that eccentricity greatly influences theshape of interface. Another example of a false conclusion can be found in thework by Savery et al. [201], where the authors argued that for water-based muds,“the [molecular] diffusion term likely will have a great impact” compared to theconvective term. This statement is easily debunked, because a simple dimensionalanalysis shows that in laminar regimes Pe 1, suggesting that molecular diffusiondoes not play any significant role in the mixing. Perhaps more interestingly, thereare a number of controversial conclusions in Zulqarnain [258]. In particular, thisstudy argued that• “Fresh water will be the most effective means of displacing mud and de-taching the adhered mud layer to walls.” On the same note, Aranha et al.[11] concluded: “In the case of vertical wells with good centralization, theabsence of hierarchy for rheology and specific weight between the drillingfluid and washer did not lead [to] the formation of interface mixing becausethe displacement happened in turbulent regime”. Our results in Chapter 7 aswell as the work of Guillot et al. [99] will disprove these statements.• In horizontal displacement “There are no significant changes observed whenthe spacer viscosity was changed.” This contradicts several previous studies(e.g. see Shadravan et al. [209]).16• “For vertical wells the final cement fraction slightly decreases with increas-ing displacement rate for spacer having density less than cement, while forthe spacer density equal to cement the opposite is true”. We will show thisis not necessarily true.Nonetheless, several other studies can be found that elucidate the 3D dynamicsof cementing. Gomes et al. [92] looked at the 3D flow patterns near a liner hangerand suggested design modifications that improve flow recirculation. More recently,Kragset et al. [135] have investigated the flows in a nearly horizontal well with awashout. In particular, they explore the competition between buoyancy and eccen-tricity in driving the displacement on the wide or narrow side of the annulus. Theresults of this study confirms that the flow is 3D inside a washout, emphasizingthe weakness of 2D models in modeling such scenarios. Furthermore, they showdisplacement inside the irregular section is facilitated by lowering the flow ratesand increasing the density difference. In the absence of washout, the general ob-servations agree well with those of the 2D model in Carrasco-Teja et al. [37] andthe results of Appendix C.1.5 Thesis objectivesThe literature review presented above concerned entirely laminar displacementflows, with the exception of the work of de Araujo et al. [52] in which the authorsmodel turbulent displacement of two Newtonian fluids in annular geometry. Whilethis study has shed light to our understanding of turbulent displacement flows, fullyturbulent flows are relatively uncommon in primary cementing. It is more likely tohave one fluid fully turbulent (e.g. a chemical wash or low-viscosity spacer), whileothers are laminar. Indeed, as it will be shown later, this also can occur on a singlesection of the annulus, e.g. turbulent on the wide side, laminar or even static onthe narrow side. To the best of our knowledge, there is no study that systematicallyinvestigated turbulent displacement flows in the annulus and accounted for regimetransition. The aim of the present study is to fill this gap in knowledge. Morespecifically, we aim to derive a 2D model for the turbulent displacement of yieldstress fluids in annular geometry. Although the main target is turbulent flows, the17model has to be consistent with the previous laminar models, e.g. Bittleston et al.[24], so that mixed flow regimes can be modeled. In doing so, we may modify theexisting model to account for (i) Turbulent stresses and (ii) Diffusive and dispersivemixing.Evidently, a model for turbulent stresses is needed. This seems particularlychallenging, because, as shown by a number of studies [171, 80, 43, 169], allcomponents of turbulent stress have similar magnitude (the fluctuating velocityis inherently three-dimensional). This is in contrast to laminar stresses in narrowgeometries in which the normal components of stress can be neglected. In addition,although diffusion exists in laminar displacement, it is molecular, and therefore itcan be legitimately neglected. Turbulent diffusion however is significantly largerand therefore, has to be taken into account. Even more importantly, as it will beshown later, weakly turbulent displacement flows fall into the Taylor dispersionregime, where the mixing by dispersion is at least one order of magnitude largerthan that by turbulent diffusion. Therefore, a key aspect of the thesis is to modelboth turbulent diffusion and Taylor dispersion in turbulent and transitional regimes.Finally, the model has to be implemented using robust and fast algorithms. Of par-ticular concern is to properly model the yield stress fluid to capture the unyieldedimmobile regions.Our goal is to simulate displacement scenarios that are found in real cementingoperations. We focus particularly on the types of primary cementing job performedin British Columbia, Canada. Thanks to the data collected from British ColumbiaOil and Gas Commission, we look to provide insight about the following cementingpractices:1. It is common to start primary cementing with a light wash (usually water).This is believed to provide cleaning and enhance displacement quality. Sci-entific intuition however, does not support this argument. More explicitly, isa low viscous lightweight wash able to remove a heavy viscous mud at all?If not, a large volume of water is being used ineffectively.2. There is a perception that turbulent displacement is always better than lam-inar displacement. Which displacement regime should we generally prefer:laminar or turbulent? This question is particularly suited for numerical stud-18ies such as the present study, as we can rarely find wells with comparablegeometries and geophysical specifications that have been cemented with dif-ferent displacement flow regimes. In other words, no systematic experimen-tal data exists that favour or disfavour laminar displacements over turbulentdisplacements.3. Where adequate pump capacity is present and where there are no risks ofeither an influx nor fracturing the well, how does laminar displacement com-pare with a weak turbulent displacement? What about strong turbulence?4. Is it possible at all to derive a simple and computationally cheap model toquickly check any displacement design?5. Currently, the quality of a cement job is evaluated using Cement Bond-Log(CBL) measurements. These measurements are costly, insensitive and of-ten inaccurate. Can we use simulations to identify new means of post jobevaluation?1.6 Thesis outlineThe content of the thesis, broadly speaking, can be divided into two parts.1. In the first part, which includes Chapters 2-4 we extend the existing laminarmodel of Bittleston et al. [24] to transitional and turbulent regimes and derivethe closure laws needed for the model. We then analyze the model from acomputational point of view. More specifically,• In Chapter 2 we lay down a consistent framework to perform one-dimensional (1D) hydraulic calculations for yield stress shear-thinningmaterials. More specifically, we find the relationship between the meanvelocity and the wall shear stress, as a function of fluid rheology, chan-nel (or pipe) geometry and flow parameters. In addition, we extendthe classical turbulent Taylor dispersion analysis to Herschel-Bulklyfluids. In doing so, we derive an accurate profile of velocity in turbu-lent regimes and give estimates for turbulent diffusivity using Reynoldsanalogy.19• In Chapter 3, we derive a comprehensive two-dimensional (2D) modelfor displacement flows of yield stress fluids in annular geometry. Thenovelty of this new formulation is in inclusion of turbulent and mixedflow regimes, which is absent in all similar developments. The finalmodel consists of a nonlinear elliptic equation for the momentum equa-tions as well as a transport equation which governs the evolution of theconcentration of different fluids. The model is based on narrow-gap ap-proximation, which allows the momentum equations to be simplified tolocally 1D shear-flow hydraulic calculations (as laid down in Chapter2).• In Chapter 4, we analyze our annular model from a computational pointof view. For the momentum equation, we establish a variational formand show that the system has a unique solution. This variational form isthen employed to construct a robust augmented Lagrangian algorithm,particularly suitable for the modeling of yield stress fluids. We will alsoanalyze and compare different algorithms that exist in the literature forsolving our 2D transport equation.2. In the second part, we systematically investigate many of the questions men-tioned above. More specifically,• In Chapter 5, we focus on fully turbulent displacement flows and showhow different parameters such as rheology or buoyancy may or maynot influence the outcome of the cement job.• In Chapter 6, we explore displacement flows in mixed flow regimes; i.e.either when one fluid is in laminar and the other is turbulent or whenone fluid undergoes a regime change around the well. In particular,we are interested to understand if one flow regime is more favorablecompared to the other one.• In Chapter 7, we turn our attention to density unstable displacementflows. In particular, through a set of simulations, we consider the ef-fectiveness of using washes for primary cementing.20• In Chapter 8, we present a feasibility analysis on using particles totrack the displacement front. The main idea is to exploit the densitydifference between successive fluids pumped in order to design a tracerparticle to sit at the interface. Although apparently trivial, such parti-cles must overcome viscous drag and strong secondary flows in orderto reach and remain at the interface. Here we present simplistic modelsthat show such technique can be indeed employed to track the interfacein displacement flows.The main body of the thesis is closed with a list of conclusions in Chapter 9.The thesis is accompanied by four appendices. The details of our software pro-totype are presented in appendix A. Appendix B presents our group’s most recentdevelopment of 2D models to include (weakly compressible) foamed cement slur-ries. This work is the core subject of a master thesis in which A. Maleki has beendirectly involved. Appendices C and D present some additional studies of our 2Dmodel. The analysis is primarily conducted by A. Rentaria, who is another PhDstudent in our group.21Chapter 2Cementing HydraulicsIn this chapter, we lay down a consistent framework within which to perform hy-draulic calculations for shear-thinning yield stress fluids. All fluids involved inconventional cementing operations (drilling muds, washes, spacers, cement slur-ries) are rheologically included in this description, which is widely employed inindustry. To characterize turbulent flows, we adopt the widely popular Dodge-Metzner-Reed approach. Although the focus is on turbulent and transitional flows,the framework is consistent with the laminar regime too. This computationalframework is later utilized in the development of our cementing model in Chapter3.The velocity profile in laminar regimes is directly integrable from the consti-tutive law. For turbulent flows however, hydraulic-style calculations have beenstudied since the 1950’s. In particular, Ryan and Johnson [198] and Hank andcoworkers [105, 106, 107, 110, 111] have focused on shear thinning yield stress flu-ids. Although not universally accepted, the phenomenological method of Dodge-Metzner-Reed [158, 58] is popular in many process industries. In this methoda generalized Reynolds number is defined based on the local power-law parame-ters. Then, a closure relationship is established for the frictional pressure drop as afunction of the generalized Reynolds number, calibrated with available data. TheDodge-Metzner-Reed approach was intended to apply to all generalized Newto-nian fluids. The extension to yield stress fluids can be found in Reed and Pilehvari[188], Pilehvari et al. [182], Founargiotakis et al. [77], as well as internally within22technical literature of many petroleum companies. Tests against experimental dataare described by Guillot and Denis [98]. More recently, comparisons with directnumerical simulation data were made by Rudman et al. [196].As it will be explained later, the original version of Dodge-Metzner-Reed for-mulation loses its simplicity when applied to complex generalized Newtonian flu-ids, such as Herschel-Bulkley fluids. To retain this simplicity, we overhaul and re-construct this formulation here. Hydraulic calculations were historically developedfor pipe flows and then extended to other flows such as those in channel. Althoughchannel flow hydraulics is of main interest in this work, it is more natural to derivethe equations for pipe flows first. We then briefly outline the formulation for chan-nel flows. Finally, we end the chapter by estimating the streamwise dispersion anddiffusion effects in fully turbulent flows. A version of this chapter is published inMaleki and Frigaard [146].2.1 Pipe flow hydraulicsConsider fully developed steady flow of a Herschel-Bulkley fluid along a pipe. Theaxial momentum balance relates the axial gradient of frictional pressure pˆ f to thewall shear stress τˆw, which is then described in terms of the inertial stress scaleρˆ ˆ¯W 20 /2 and (Fanning) friction factor f f :− Dˆ4∂ pˆ f∂ zˆ= τˆw =ρˆ ˆ¯W 202f f , (2.1)where ˆ¯W0 is the mean velocity and ρˆ is the fluid density. Herschel-Bulkley fluidsare defined rheologically by 3 parameters: the yield stress τˆY , the consistency κˆ ,and the power law index n. In the hydraulic calculations that are generally per-formed, the fluid properties: ρˆ , τˆY , κˆ , n, and the pipe diameter Dˆ are known. Theaim is to define the closure relationship between the wall shear-stress τˆw and themean velocity ˆ¯W0 for the different flow regimes.A widely used approach is that of Dodge and Metzner [58] in defining f f asa function of the generalized (Metzner-Reed) Reynolds number and power law in-dex, with an additional dimensionless parameter needed to quantify yield stress ef-23fects. Although we are concerned with turbulent flows, the Metzner-Reed approachrequires the laminar flow relations. The Metzner-Reed generalized Reynolds num-ber is defined:ReMR =8ρˆ ˆ¯W 20κˆ ′( ˆ˙γN)n′(2.2)where the primed variables are:κˆ ′ =τˆw( ˆ˙γL)n′, n′ =dln τˆwdln ˆ˙γL. (2.3)The Newtonian strain rate at the wall is ˆ˙γN and ˆ˙γL is the laminar strain rate:ˆ˙γN =8 ˆ¯W0Dˆ, ˆ˙γL =8 ˆ¯WLDˆ. (2.4)The velocity ˆ¯WL, used to define ˆ˙γL, is the mean velocity that the fluid would have ina laminar flow, driven by the wall shear-stress τˆw. Note that ˆ¯WL and ˆ˙γL are definedby the wall shear stress τˆw across all flow regimes, but will only equal ˆ¯W0 and ˆ˙γNin the case that the flow is laminar.For laminar flows, the Buckingham-Reiner equation can be derived, which is analgebraic equation relating the flow rate to the wall shear stress. The Rabinowitsch-Mooney procedure results in the same expression. For Herschel-Bulkley fluids theresult is:ˆ˙γL =4n3n+1(1− rY )1/n+1[τˆwκˆ]1/n×[(1− rY )2+ 2(3n+1)(1− rY )rY2n+1 +(3n+1)r2Yn+1]. (2.5)Here rY = τˆY/τˆw, which also represents the dimensionless radial position of theyield surface. Combining (2.3) with (B.9) we find:n′ = n(1− rY ) (n+1)(2n+1)+2n(n+1)rY +2n2r2Y(n+1)(2n+1)+3n(n+1)rY +6n2r2Y +6n3r3Y, (2.6)and hence can define ReMR etc. Note that the expression for n′ in Zamora and Bleier[253] is incorrect. Figure 2.1a illustrates the variation of n′ with (n,rY ): increasing24a) rY = He/Hw0 0.2 0.4 0.6 0.8 1n′ = 1n = 0.1b) rY = He/Hw0 0.2 0.4 0.6 0.8 1E(n,rY) = 1n = 0.1Figure 2.1: a) n′(n,rY ) for n = 0.1, 0.2, ... 0.9, 1; b) E(n,rY ) for n =0.1, 0.2, ... 0.9, 1.the yield stress (and hence rY ) reduces n′ and the laminar velocity profiles becomeincreasingly plug-like.The complicated derivation of ReMR has the virtue of ensuring that f f = 16/ReMRin the laminar regime for all generalized Newtonian fluids. The original derivationwas for power law fluids, where n′ = n andˆ˙γL =4n3n+1[τˆwκˆ]1/n⇒ κˆ ′ = κˆ[3n+14n]n. (2.7)Thus, for power law fluids, in all flow regimes, ReMR is explicitly defined in termsof the mean velocity, making it straightforward to work with f f , ReMR and n indefining the mapping between τˆw and ˆ¯W0. The simplicity of the Metzner-Reedformulation however is lost once we move more complex generalized Newtonianfluids and study different flow regimes.2.1.1 Choice of dimensionless groupsFrom dimensional considerations, we expect the relation between τˆw and ˆ¯W0 to beexpressible in terms of n and 3 other dimensionless groups. Although differentexpressions have been used to define f f in terms of n′ & ReMR, when these areexpressed in terms of ˆ¯W0 the definition is typically implicit, which makes thesevariables less appealing for characterising τˆw 7−→ ˆ¯W0. Instead, we feel it is more25convenient to work with a Reynolds number that can be defined explicitly in termsof ˆ¯W0 and that is independent of τˆw. Motivated by (2.7) we use a rescaled consis-tency κˆp, referred to as the power-law consistency, and use n, to define the powerlaw Reynolds number Rep, as follows:Rep =8ρˆ ˆ¯W 20κˆp( ˆ˙γN)n, κˆp = κˆ[3n+14n]n. (2.8)For a power-law fluid, ReMR = Rep, and Rep is always an explicit function of ˆ¯W0.The Buckingham-Reiner equation (B.9) may now be simplified to:κˆp ˆ˙γnLτˆw= E(n,rY ) : (2.9)E(n,rY ) = (1− rY )1+n((1− rY )2+ 2(3n+1)(1− rY )rY2n+1 +(3n+1)r2Yn+1)n,see Figure 2.1b.It is common to represent yield effects with either the Bingham number, with rYor with the Hedström number. The Bingham number involves ˆ¯W0, and rY involvesτˆw. Thus, we select the Hedström number, the definition of which varies in theliterature for n 6= 1. We choose to normalize so that the Hedström number has alinear variation in yield stress and use κˆp for later convenience:He = τˆY(ρˆnDˆ2nκˆ2p)1/(2−n). (2.10)This definition agrees with other common definitions at n = 1.Finally, for a dimensionless group that depends on τˆw, but is independent of ˆ¯W0we mimic the definition of He, replacing yield stress with wall shear stress:Hw = τˆw(ρˆnDˆ2nκˆ2p)1/(2−n), (2.11)noting that rY = He/Hw.Our aim has been to isolate effects of τˆw and ˆ¯W0 from other targeted physi-26cal effects (e.g. τˆY ) in our dimensionless description, achieved with (Rep,Hw,He),which independently represent the effects of increasing ˆ¯W0, τˆw and τˆY . Other vari-ables (such as f f , ReMR and rY ) may be economical for expressing specific analyt-ical or empirical relationships, but their utility has partially eroded with the adventof modern computing power and there is no gain in simplicity once we consideryield stress fluids and different flow regimes. It is also worth mentioning herethat He is a system dependent parameter (depends only on pipe diameter and thefluid properties), whereas Rep and Hw are flow dependent and thus the relationshipbetween them uniquely specifies the problem.2.1.2 Flow regimesAs the flow rate (and wall shear stress) increases the flow changes from lam-inar through a transitional regime to fully turbulent flow. In each regime themapping between τˆw and ˆ¯W0 is to be defined, represented dimensionlessly bythe mapping between Hw and Rep. For laminar flows ˆ˙γN = ˆ˙γL, and from (2.8):ˆ˙γL = [8κˆpRep,Lam/(ρˆDˆ2)]1/(2−n), and on using (2.9) we have:(8Rep,Lam)n/(2−n)Hw= E(n,rY ) = E(n,HeHw). (2.12)This defines Rep,Lam explicitly in terms of Hw and vice-versa if Rep,Lam is specified,solving (2.12) iteratively, e.g. by finding rY ∈ [0,1] to any required precision. Thus,we may readily compute the mapping Rep,Lam←→Hw, after which we may define:f f =2τˆwρˆ ˆ¯W 20=16τˆw8ρˆ ˆ¯W 20=16Rep,LamE(n, HeHw) = 16ReMR,Lam. (2.13)For fully turbulent flows, following Dodge and Metzner [58]:1√f f=4.0(n′)0.75log(ReMR f1−n′/2f )−0.4(n′)1.2. (2.14)We will use (2.14) (and its counterpart for channel flow (2.27)) in the following27analysis wherever needed. Noting that:ˆ˙γN =[Rep8κˆpρˆDˆ2]1/(2−n)and ˆ˙γL =[E(n,HeHw)τˆwκˆp]1/n,after a little algebra we find:ReMR =8n−n′2−n Re2−n′2−np E(n, HeHw)n′/nH1−n′/nw, f f =2Hw82−2n2−nRe22−np. (2.15)Substituting into (2.14) and simplifying leads to:Rep = H1− n2w 24−7n2 4.0(n′)0.75log24− 7n′2 E(n, HeHw) n′nHn′n − n′2w− 0.4(n′)1.22−n(2.16)Again, in the case that Hw is specified (i.e. τˆw), then (2.16) defines Rep explicitly.If instead Rep is specified (i.e. ˆ¯W0), then Hw is found iteratively from (2.16).Note that the 2 nonlinear equations that must be solved in the case that Rep isspecified, (2.12) & (2.16), can be straightforwardly written as monotone functionsof Hw within specified bounds. Such equations can be solved with simple but robustroot-finders such as the bisection method, Ridder’s method, Brent’s method etc..Transitional flows are found for Re1(n′) < ReMR < Re2(n′). The choice ofRe1(n′), Re2(n′) and transition to turbulence are discussed at length in §2.3. Recallthat n′= n′(n,He/Hw), and since ReMR depends on (n,He) and either of Hw or Rep,the critical values Re1 and Re2 can be used to define critical (transitional) values ofeither Hw or f f , e.g. we solve the equation:Re1(n,He/Hw) = ReMR(n,He,Hw), (2.17)by iterating with respect to Hw and using the laminar flow closure expression, thusdefining f f ,1 and Hw,1. Similarly, on solving:Re2(n,He/Hw) = ReMR(n,He,Hw), (2.18)28by iterating with respect to Hw and using the turbulent flow closure expression, wedefine f f ,2 and Hw,2.For representing hydraulic quantities in transitional flows it is common to usesome form of interpolation. Here we choose to interpolate log f f linearly withrespect to logReMR. More explicitly:log f f =log f f ,1[logRe2− logReMR]+ log f f ,2[logReMR− logRe1]logRe2− logRe1 . (2.19)Figure 2.2a illustrates the 3 flow regimes in (ReMR, f f )-space at He = 500 for n =0.2, 0.4, ... 0.8, 1. We observe the usual collapse of data in the laminar regime.For n closer to 1 we see a sharp change in f f at transition, but not for smaller n.The transitional curves are linear in the log-log plot, as shown. Figure 2.2b & cplots Rep and ReMR against (Hw−Hw,1)/(Hw,2−Hw,1) for n= 0.2, 0.4, ... 0.8, 1 atHe = 500, i.e. this is the same data as Figure 2.2a. We see large relative differencebetween Rep and ReMR at smaller values of n and Hw (laminar and transitional),which corresponds to those parameters where n′ is smallest. Qualitatively similarplots are found at other He.2.2 Plane channel flowsA broadly similar analysis to that for the pipe can be performed for a plane channelflow. This flow is often used to locally approximate flow along a narrow annulus.We outline here only the main results, highlighting any differences with the pipeflow. Here we briefly repeat the above computational procedure for channel flows.We consider axial flow in a 2D channel of width 2Hˆ. In order to defineMetzner-Reed and power law Reynolds numbers as well as Hedström numberswe need to replace Dˆ with 2Hˆ and the prefactor 8 with 6; i.e.ReMR =6ρˆ ˆ¯W 20κˆ ′( ˆ˙γN)n′Rep =6ρˆ ˆ¯W 20κˆp( ˆ˙γN)n(2.20)He = τˆY(ρˆn(2Hˆ)2nκˆ2p)1/(2−n)Hw = τˆw(ρˆn(2Hˆ)2nκˆ2p)1/(2−n)(2.21)29a)102 104 10610−410−310−210−1ff = 16/ReMRn = 0.2n = 1ReMRf fb)−1 0 1 2 3102104106Turbulentn = 0.2n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)Re pc)−1 0 1 2 3102104106Turbulentn = 0.2n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)Re MRFigure 2.2: Examples for He = 500 and n = 0.2, 0.4, ... 0.8, 1: a) f f vsReMR; b) Rep against (Hw−Hw,1)/(Hw,2−Hw,1); c) ReMR against (Hw−Hw,1)/(Hw,2−Hw,1). Regimes are denoted: laminar (green), transitional(red), turbulent (black).where κ ′ and n′ are still defined as (2.3) andˆ˙γN =6 ˆ¯W02Hˆ, ˆ˙γL =6 ˆ¯WL2Hˆ. (2.22)The power law consistency is defined similarly to (2.8) as:κˆp = κˆ(2n+13n)n(2.23)30The Rabinowitsch-Mooney procedure applied to the laminar flow results in thefollowing expressions for n′(n,yY ) and E(n,yY ):n′ = n(1− yY ) nyY +n+12n2y2Y +2nyY +n+1(2.24a)E =κˆp ˆ˙γnLτˆw= (1− yY )(n+1)(nn+1yY +1)n(2.24b)where yY = τˆY/τˆw represents the dimensionless (laminar) plug width.In the laminar regime, the mapping from Hw↔ Rep (i.e. τˆw↔ ˆ¯W0) is:(6Rep,Lam)n/(2−n)Hw= E(n,yY ), (2.25)from which we then definef f =2τˆwρˆ ˆ¯W 20=12τˆw6ρˆ ˆ¯W 20=12Rep,LamE(n, HeHw) = 12ReMR,Lam. (2.26)In fully turbulent regime, the Dodge-Metzner relation is1√f f=4.0n′0.75log(ReMR f1− n′2f)− 0.395n′1.2(2.27)which leads to the following equation defining Hw↔ Rep:Rep = H1− n2w 61−n21−n2[4.0n′0.75log(61−n′21−n′2 En′n Hn′n − n′2w)− 0.395n′1.2]2−n. (2.28)As with the laminar flows, this must be solved iteratively if Rep is specified, butis explicit if Hw is specified. In the following section, we will discuss transition toturbulence and the choices of Re1 and Re2.2.3 Turbulent transition in generalized Newtonian fluidsThe question of transition from laminar into fully turbulent flows for yield stressfluids was first considered by Hedström [116] who advocated a criterion based on31a)102 104 10610−410−310−210−1ff = 12/ReMRn = 0.2n = 1ReMRf fb)−1 0 1 2 3102104106Turbulentn = 0.2n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)Re pc)−1 0 1 2 3102104106Turbulentn = 0.2n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)Re MRd)−1 0 1 2 310−410−310−210−1Turbulentn = 0.2n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)f fFigure 2.3: Example of the hydraulic quantities for channel flow He = 200and n = 0.2,0.4, ...,1. a) f f against ReRM. Regimes are denoted:laminar (green), turbulent(black). broken line is extrapolation intotransitional range; b) f f against ReMR Regimes are denoted: lam-inar (green), transitional (red) and turbulent(black); c) Rep against(Hw−Hw,1)/(Hw,2−Hw,1); d) ReMR against (Hw−Hw,1)/(Hw,2−Hw,1);e) f f against (Hw−Hw,1)/(Hw,2−Hw,1)32the point of intersection of the laminar and turbulent friction factor curves (intersec-tion method). A later approach by Metzner-Dodge-Reed was to use f f ≈ 0.0076, asthe transition parameter (expressed equivalently with generalized Reynolds num-ber); Metzner and Reed [158], Dodge and Metzner [58]. This concept was ex-tended to the Bingham model by Govier and Aziz [94] and can be applied toany purely viscous non-Newtonian fluids, assuming transition takes place at f f ≈0.0076. A related approach followed more recently is due to Desouky and Al-Awad[57], which combines the Metzner-Reed and intersection methods.Furthermore, a number of approaches have evolved that balance stabilizingand destabilizing effects on the flow, setting a criterion based on when this balanceexceeds some critical value. Two identical predictions of transitional Reynoldsnumbers have been made by Ryan and Johnson [198] and by Hanks & co-workers[105, 106, 107, 108, 109, 110, 111], although arrived at using different rationale.A slightly different balance approach is advanced by Mishra and Tripathi [159],balancing the mean kinetic energy and the wall shear stress. Güzel et al. [103]have developed another local balance approach that shares similarities. Wilson &Thomas [248, 233, 249] have evolved analyses based on estimates of the viscoussub-layer, postulating that transition depends only on He. Other approaches haveevolved that are industry-specific, e.g. those of Slatter [211], Slatter and Wasp [212]are predominantly developed for mining applications. Pilehvari and co-workershave reviewed available data and many of the existing phenomenological criteria[188, 182] and advocate a type of intersection method using the Metzner-Reedapproach.It is noteworthy that many of the above non-Newtonian approaches have devel-oped from roots 40-60 years old. Over this same period our understanding of New-tonian fluid transition has evolved considerably. Although transitional Re for New-tonian fluids are typically quoted at Re≈ 2100, theoretically pipe flow is subcriticaland believed to be linearly stable at all Re. Indeed, with careful control it has beenpossible to achieve stable laminar pipe flows at Re in the 20,000−−60,000 range;see for example Draad et al. [61], Hof [117]. The transitional Re essentially givesa measure of the quality of the experimental flow loop. Thus, the common engi-neering perspective that “transition” (meaning the end of the laminar regime) willoccur at a given Re is flawed, even for a Newtonian pipe flow.33More detailed experimental studies of transition in non-Newtonian fluids haveappeared e.g. Pinho and Whitelaw [183], Draad et al. [61], Escudier et al. [71],Peixinho et al. [177, 178], Esmael and Nouar [72], Güzel et al. [102]. Coupledto these are a range of theoretical and computational studies, e.g. Frigaard et al.[84], Nouar and Frigaard [170], Rudman et al. [195], Frigaard and Nouar [82], Rud-man et al. [196], Esmael and Nouar [72]. This list covers only those studies focusedat inelastic fluids. Although by comparison to Newtonian fluids, our understandingof transition in shear-thinning yield stress fluids remains limited, it has signifi-cantly evolved in the past 20 years. Some aspects of this understanding can nowbe applied pragmatically to improve the common descriptions of transition.Firstly, despite the existence of stability at elevated Re for Newtonian flows,application requires criteria that are approximately correct for typical hydraulicsettings, i.e. industrial pumps and pipes. In essence, there is a critical flow pa-rameter at which stability of the laminar flow is lost. Secondly, it is observed thattransition occurs over an extended range of Re for shear-thinning yield stress fluids.Thirdly, all this is modulated by realization of experimental factors not all known40-60 years ago, including the following. (i) The initial loss of stability is oftenhard to detect at the pipe centre, but is visible at the walls. (ii) Sharp changes in f fare also not always evident, especially in more strongly shear-thinning fluids. (iii)A wide range of different phenomena are found in transitional flows (e.g. puffs &slugs, coherent structures, flow asymmetry...) and these specific phenomena haverheological dependencies that are not fully explored. However, eventually all flowstransition into full turbulence.In the above context, we advocate an approach that uses 2 critical Re: thesmaller one reflecting loss of stability and the larger reflecting onset of full turbu-lence. Although we dismiss the intersection method (as it predicts only a singletransition), we must recognise that such approaches are in some sense robust asthe friction factor relationships extrapolated are based on data valid over ranges oflaminar and turbulent flow rates, instead of at a particular transition point whichmay be hard to detect at smaller n.There are in fact a number of approaches in usage that adhere to the abovepicture, and this is reasonably common in oilfield application; e.g. Nelson andGuillot [167]. Here we assume 2 critical Reynolds numbers: Re1(n′) < Re2(n′),34depending only on the local power law index n′ = n′(n,He/Hw), and use thesecritical values to delineate laminar, transitional and turbulent flow regimes. Thefirst critical value is given by:Re1(n′) = 3250−1150n′, (2.29)as advocated by Nelson and Guillot [167], i.e. laminar flow for ReMR ≤ Re1(n′).For the second critical Reynolds number, one option is that of Guillot and De-nis [98], Founargiotakis et al. [77], which is algebraically similar to (2.29). Un-fortunately, although well behaved for power law fluids (He = 0), as the yieldstress (He) is increased the Dodge-Metzner expression (2.14) loses monotonicityfor smaller n and eventually ceases to be single valued, as illustrated in Figure 2.4a.Thus, expressions that extend (2.29) algebraically such as that of Guillot and Denis[98], Founargiotakis et al. [77] tend to fail to produce physically realistic transitioncriteria at small n once we have an significant yield stress.Mathematically, at fixed (n,He) the variable n′ varies with ReMR. The expres-sion (2.14) gives f f monotone with respect to ReMR only for fixed n′. This be-haviour does not agree with experimental observation. Consequently if (2.14) rep-resents the frictional behaviour for fully turbulent flows, it is necessary to restrictthe transitional range approximately to those for which (2.14) is well-behaved. Anexpression which does this effectively is the following:Re2(n′) = 1.328529×10(6.00−7.84n′) n′ < 0.31,3000+[1a(n′)] 11+b(n′) −[1a(n′=1)] 11+b(n′=1) n′ ≥ 0.31,(2.30)a(n′) = 0.078504+0.0098085logn′, (2.31)b(n′) = −0.24984+0.059646logn′. (2.32)The dependency of Re1 and Re2 on n and rY = He/Hw is shown in Figs. 2.4b &c. Note that the effects of the yield stress at fixed n are felt wholly through n′. Inparticular full turbulence (ReMR ≥ Re2) is significantly delayed by a strong yieldstress, as is observed experimentally. An example of the flow regimes, plotted asf f vs ReMR and computed using Re1 and Re2 defined above, has been given in35a) ReMR102 104 106f f10-410-310-210-1ff = 16/ReMRn = 0.2n = 1b) rY = He/Hw0 0.2 0.4 0.6 0.8 1Re 120002200240026002800300032003400n = 1n = 0.1c) rY = He/Hw0 0.2 0.4 0.6 0.8 1Re 2×10502468101214n = 1n = 0.1Figure 2.4: a) f f vs ReMR for He = 2000 and n = 0.2, 0.4, 0.6, 0.8, 1.Regimes are denoted: laminar (green), turbulent (black); broken lineis an extrapolation into transitional range using the turbulent closure.b) Re1(n′(n,rY )) for n = 0.1, 0.2, ... 0.9, 1. c) Re2(n′(n,rY )) forn = 0.1, 0.2, ... 0.9, 1.36Figure 2.2a. This figure illustrates the effectiveness of (2.30)-(2.32) in truncatingthe fully turbulent regime, ensuring a single-valued f f .For channel flows (§2.2), the same issues arise with extrapolating the turbulentf f (ReMR) at small n and large He. The criterion (2.30)-(2.32) is replaced by:Re2(n′) = 1.106969×10(6.00−8.19n′) n′ < 0.28,3000+[24a(n′)] 11+b(n′) −[24a(n′=1)] 11+b(n′=1) n′ ≥ 0.28,(2.33)a(n′) = 0.5b(n′)× [0.096045+0.0082711logn′] , (2.34)b(n′) = −0.27103+0.063985logn′. (2.35)The same expression (2.29) is used for Re1.A final comment regarding transitional flows is more pragmatic. On the onehand, applications involve real fluids. Frequently, models such as the Herschel-Bulkley model only describe a limited range of shear rates and are fitted to rhe-ological data from viscometric flows. Although shear-thinning and yield stressaspects may be the dominant rheological behaviours observed over these shear rateranges, invariably the fluids used experimentally have other rheological behavioursdepending on the flow history and fluid micro-structure. As fully turbulent shearflows are characterized by broad ranges of time and length-scales it is unknown ifand how smaller-order rheological features may influence turbulence phenomena.Analytical and computational study of inelastic generalized Newtonian fluid mod-els is one approach that explicitly removes other rheological influences. On theother hand, for industrial application even the use of models such as the Herschel-Bulkley fluid presents problems. Rheological measurements in application arefrequently dictated by industry protocol and standardization. Thus, the fitting ofrheological parameters to application data is an imperfect science, and these errorspropagate into whatever predictions we make.2.4 Dispersion and diffusion of passive scalarsAs mentioned earlier in Chapter 1, the fluids used in primary cementing are drillingfluids, washes, spacer fluids and cement slurries, all of which are characterized37within the industry as shear-thinning yield stress fluids. If water-based, these fluidsare miscible. In turbulent flows they rapidly mix transversely and then disperselongitudinally, presumably driven by the Taylor dispersion mechanism, [229, 230].Although Zhang and Frigaard [255] have considered dispersion of such fluids inlaminar regimes, for laminar flows primary cementing does not typically fall intothe Taylor-regime.Axial dispersion in turbulent flows of Newtonian fluids was initially studied byTaylor [230]. Upon applying the Reynolds analogy to model the turbulent disper-sivity, he then integrated the relative velocity profile across the pipe to calculate theaxial bulk dispersivity. Taylor used tabulated data from the universal distributionof velocity which is known to be valid only at high Reynolds number and thereforehis results significantly deviate from experimental data [234, 67, 113]. Taylor’sanalysis was later revisited by Tichacek et al. [234] and Flint and Eisenklam [75]who utilized experimental velocity profiles to give better estimates. Nonetheless,both of these studies deviate from experimental results at low Reynolds number(Re < 104) mainly because the experimental velocity profile was unable to capturethe wall layer. In another study Ekambara and Joshi [67] estimated the axial dis-persion with a velocity profile obtained computationally using the k− ε model. Acomparison of these approaches with the experimental data can be found in Hartet al. [113].For inelastic non-Newtonian fluids, axial dispersion in laminar [28, 4, 3, 255]and turbulent [241, 137, 220] flows has been studied. In the case of turbulentregimes, Wasan and Dayan [241], Krantz and Wasan [137] studied dispersion ofpower-law fluids using the turbulent velocity profile of Bogue and Metzner [27].Wasan and Dayan [241] predicted the axial dispersion to increase with Reynoldsnumber, contradicting Taylor’s model for dispersion. Krantz and Wasan [137]modified the earlier results by adding a wall layer to the velocity profile. However,the validity of their results is questionable since the velocity scale used appears tobe different from that of Bogue and Metzner [27].As noted by Tichacek et al. [234], Krantz and Wasan [137], Ekambara andJoshi [67], Hart et al. [113], good estimation of the Taylor dispersion demands anaccurate velocity profile. Laminar velocity profiles are integrable from the constitu-tive law. As seen above, the Metzner-Reed generalized Reynolds number provides38an economical description of the hydraulic closure relationship. In particular, theDodge-Metzner-Reed approach is attractive in that the hydraulic calculations (andclosure) are linked to a universal log-law velocity profile, proposed by Dodge andMetzner [58]. Such profiles may be used directly to calculate Taylor dispersioncoefficients. However, two common deficiencies occur: (i) the log-law is not validat the centreline of the pipe/duct; (ii) the log-law must be matched/patched to adifferent velocity approximation close to the wall. Various centreline correctionshave been suggested, including the correction of Reichardt [189] and exponentialcorrection of [27]. Near the wall, Krantz and Wasan [136] argued that Reynoldsstresses decay as the cube of the distance, and therefore suggested that the walllayer effect could be significant. Krantz and Wasan [137] developed the analysisframework to evaluate the wall layer for power-law fluids.In laminar flows, increasing the yield stress tends to flatten the velocity profileand hence reduce Taylor dispersion. In turbulent flows it is generally thought thatthe yield stress has little influence on the velocity profile in the turbulent core, butis known to retard turbulent transition. Equally, since the yield stress contributesto the effective viscosity we might expect that wall-layer effects are significant asthe yield stress increases. Hence the interest in weak turbulence where wall-layersare thicker and occupy a larger proportion of the duct area, also where the velocitychanges are greatest. This study explores the subtlety of this relationship.Here we aim to estimate streamwise dispersion and diffusion effects, focusingon fully turbulent flows. Pragmatically, we are unable to easily model diffusionand dispersion in transitional flow regimes. Furthermore, in the laminar flows ofindustrial interest we are typically far from the laminar Taylor dispersion regime.In fully developed turbulent flows the dominant transport mechanism is invariablyTaylor dispersion, which is modelled straightforwardly once the turbulent diffusiv-ity and velocity profile are known.2.4.1 Velocity profiles in turbulent pipe flowsDodge and Metzner [58] derive the following velocity profile in the turbulent core:Wˆ (rˆ)Wˆ∗= ADM log y˜++BDM, (2.36)39where the friction velocity Wˆ∗ is defined by: Wˆ∗ =√τˆw/ρˆ =√f f /2 ˆ¯W0, andy˜+ = (1− r)n′ RˆnρˆWˆ 2−n∗κˆ= (1− r)n′ f 1−n2f Rep[3n+14n]n 8n−121+n2, (2.37)for r = rˆ/Rˆ. This velocity profile, when averaged across the pipe should give anexpression equivalent to (2.14), thus defining ADM & BDM. More precisely, sincethe dimensionless velocity W (r) has mean value 1, we have:1 = 2∫ 10rW (r) dr (2.38)=√f f22∫ 10r[ADM log((1− r)n′ f 1−n2f Rep[3n+14n]n 8n−121+n2)+BDM]drIn order that (2.38) is equivalent to (2.14), we find:ADM =4.0√2(n′)0.75, (2.39)BDM = −0.4√2(n′)1.2−ADM(log(fn′−n2fRepReMR2n2−4(3+1n)n)− 3n′2ln(10)),(2.40)which can be verified1 with those in Dodge and Metzner [58] for a power law fluid(n′ = n). With a little algebra, the Dodge-Metzner velocity profile is given in termsof r by:W (r) =√f f2[ADM(log[(1− r)n′ f 1−n′2f ReMR]+3n′2ln(10))− 0.4√2(n′)1.2](2.41)Two common deficiencies of such log-law profiles are the centreline behaviour anda correction for the wall layer, as we now describe.1Note that there is an errata to the formula in equation (48) of Dodge and Metzner [58]; thecorrected coefficients in the velocity profile may be found in Dodge and Metzner [59].40Centreline correctionFirstly, it is common to adjust the profile near the pipe centre so that the meanturbulent velocity W (r) has zero gradient at centreline. This correction is purelyempirical and there are many suggested forms in the literature. As these differentcorrections work on the derivative of a smooth velocity profile at the pipe centre,it is hard to differentiate between these expressions by comparing with experimen-tal data, even for Newtonian fluids. The condition that any correction functionc(r, f f ,n′) should satisfy is:ddrc(0, f f ,n′) =√f f2n′ADMln10, (2.42)which ensures that the corrected velocity has zero derivative at the pipe centre.We also expect that the maximum velocity is at the pipe centre (an inequality con-straint on the 2nd derivative of c), and that the correction remains relatively smallfor r ∈ [0,1].We consider the following 2 candidates for the centreline correction functionand proceed our analysis:c1(r, f f ,n′) =√f f2n′ADMln10(0.375e0.04−(r−0.2)20.15)(2.43)as suggested by Bogue and Metzner [27], andc2(r, f f ,n′) =√f f2n′ADMln10r(1− r)2. (2.44)which is a modified version of the one suggested by Reichardt [189]. Althoughthe correction is assumed small relative to the dominant term in (2.41), it still con-tributes to the flow rate. This contribution must be subtracted from the constantBDM, to balance the flow rate of the corrected profile. The corrected dimensionlessturbulent core velocity becomes W (r) =W0(r):W0(r) =√f f2[A0 ln(1− r)+B0+B0,c(r)] (2.45)41A0 =ADMn′ln10(2.46)B0 =−0.4√2(n′)1.2+A0(1n′ln( f 1−n′2f ReMR)+32)(2.47)where B0,c(r) is a zero-mean correction:B0,c(r) = A0(0.375e0.04−(r−0.2)20.15 −0.1581529), associated with (2.43),B0,c(r) = A0(r(1− r)2− 115), associated with (2.44).We shall see in §2.4.2 that these small corrections may have a significant effect onthe turbulent diffusivity. Below, unless otherwise stated, all the figures are basedon the correction function of (2.44).Wall-layer correctionThe second correction concerns the pipe wall, where viscous effects come intoplay. The velocity (2.45) clearly does not satisfy the boundary conditions at r = 1.Equally (2.14) is based on a velocity profile such as (2.45), that ignores the walllayer but conserves the flow rate. These approximations are reasonable in highlyturbulent flows where we expect the wall layers to be very thin. However, in weaklyturbulent flows we expect thicker wall layers to emerge, that may affect both theflow rate and the Taylor dispersion coefficient. To analyse these effects we followthe approach of Krantz and Wasan [136, 137].We first introduce wall coordinates, yˆ = Rˆ− rˆ. Using the wall shear stress wedefine a wall shear rate scale ˆ˙γ∗ to satisfy the constitutive law, i.e.ˆ˙γ∗ =[τˆw− τˆYκˆ]1/n=[τˆwκˆ]1/n[1− rY ]1/n. (2.48)The viscous wall layer length-scale yˆ∗ is then defined using ˆ˙γ∗ and the friction42velocity Wˆ∗, i.e. yˆ∗ = Wˆ∗/ ˆ˙γ∗. The wall layer length and velocity variables are:y+ =yˆyˆ∗, W+(y+) =Wˆ (rˆ)Wˆ∗=W (r)ˆ¯W0Wˆ∗=√2f fW (r). (2.49)The pressure gradient is independent of rˆ and consequently we may integrate theaxial momentum equation with respect to rˆ to give:Rˆ− yˆ2∂ pˆ f∂ zˆ= −ρˆ uˆ′wˆ′+ τˆzr (2.50)−(1− yˆ∗Rˆy+)= −u′w′+(y+)− (1− rY )[(∂W+∂y+)n+rY1− rY]. (2.51)Note here that the Reynolds stress term, uˆ′wˆ′, has been scaled with Wˆ 2∗ . Equation(2.50) is valid across the wall layer and into the turbulent core. Only in the walllayer are we justified in evaluating τˆzr in terms of the mean turbulent velocity usingthe leading order constitutive laws, i.e. because the wall layer is dominated byshear.Within the wall layer we may deduce that u′w′+ → 0 as (y+)3. We expandvelocity profile and Reynolds stress in wall layer as polynomial series in y+:W (y+) = W0+W1y++W2(y+)2+W3(y+)3+W4(y+)4+W5(y+)5 (2.52)u′w′+ = u′w′+3 (y+)3+u′w′+4 (y+)4 (2.53)Upon substituting (2.52) and (2.53) in (2.51) we get:0 = 1−ψy+− (u′w′+3 (y+)3+u′w′+4 (y+)4+ ...)+ rY − (1− rY )×(W+1 +2W+2 y++3W+3 (y+)2+4W+4 (y+)3+5W+5 (y+)4+ ...)n(2.54)where ψ = yˆ∗/Rˆ gives the wall layer scaling, and the various coefficients are con-stants with subscript denoting the power of y+ in the expansions. Equating atsuccessive powers of y+ we find:W+0 = 0, W+1 = 1, W+2 =−ψ2n(1− rY ) , W+3 = (1−n)ψ26n2(1− rY )2 . (2.55)43a)0 1 2 310−1510−1010−5100n = 0.2n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)ψb)0 1 2 310−1510−1010−5100n = 0.2n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)ψc)0 1 2 310−1510−1010−5100n = 0.2n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)ψFigure 2.5: The wall-layer scaling parameter ψ for n = 0.2, 0.4, ... 0.8, 1: a)He = 5; b) He = 100; c) He = 2000. The red part of the curves showsthe transitional regime.These expressions match those in Krantz and Wasan [136] for rY = 0. The scalingparameter ψ is defined2 by:ψ =yˆ∗Rˆ=24/n−1/2(1− rY )1/n(3+1/n)[Rep f1− n2f]−1/n. (2.56)Figure 2.5 plots representative ψ for wall shear stresses just above and belowfull turbulence. We see that ψ < 10−2, and that ψ decreases rapidly with wall2Note a factor of n different in our ψ , compared to Krantz and Wasan [136].44shear stress, particularly for smaller n. Note that Hw,2 1, and therefore He = 5(Figure 2.5a) is close to power law fluid behaviour: ψ is only sensitive to Hw forsmaller n < 0.3. As the yield stress becomes significant (Figure 2.5b & c), we seethat for n≤ 0.5, ψ becomes extremely small. Again this is largely the effect of theyield stress on n′ that we are seeing. Certainly, the very thin wall layers predictedat small n′ are physically unrealistic. Values within the transitional regime that areplotted in Figure 2.5 indicate that choices of other Re2 in place of (2.30) are stilllikely to result in very small ψ at modest n for any significant yield stress.The wall layer ends at r = rc = 1− yc = 1− y+c yˆ∗/Rˆ = 1−ψy+c . This is to befound by matching with the core velocity. First however, on integrating the corevelocity W0(r) across the pipe we deduce that the wall layer perturbs the flow rateby a term of order√f f2 y+c [ψy+c ]. This suggests that the core velocity (2.45) mustitself be corrected to take account of the flux in the wall layer. More explicitly:W (r) =W0(r)+√f f2Bw,c.where we expect Bw,c to scale with the critical layer thickness, yc = [ψy+c ]. Theterm W0(r) also satisfies:2∫ 10rW0(r) dr = 1.Subtracting W0(r) from W (r) and integrating across the pipe, we find:r2c Bw,c =2∫ 1rcr[A0 ln(1− r)+B0+B0,c(r)] dr−2ψ5∑j=1([y+c ]j+1j+1−ψ [y+c ]j+2j+2)W+j=r2c [Bw,core−Bw,wall](2.57)Bw,core =A0[ψy+c ][(2− [ψy+c ]) ln[ψy+c ]−2+[ψy+c ]2](1− [ψy+c ])2+B0[ψy+c ] (2− [ψy+c ])+2B¯0,c[ψy+c ](1− [ψy+c ])2 ,(2.58)45Bw,wall =2ψ(1− [ψy+c ])25∑j=1([y+c ]j+1j+1−ψ [y+c ]j+2j+2)W+j , (2.59)B¯0,c =1ψy+c∫ 1rcrB0,c(r) dr (2.60)The correction term B¯0,c is generally small. The leading order terms come fromA0[ψy+c ] ln[ψy+c ] and the term [ψy+c ] log( f1− n′2f ReMR), contained within B0. Thecorrected core velocity is:W (r) =√f f2[A0 ln(1− r)+B0+B0,c(r)+Bw,core−Bw,wall], (2.61)which we note is defined in terms of y+c and the coefficients of the wall velocity,W+j , which are as yet unknown for j > 3.To find y+c we follow the procedure outlined by Krantz and Wasan [136, 137],using the above core velocity. We match W+(y+) and the first 2 derivatives at theedge of the viscous sublayer: y+ = y+c :A0 ln[ψy+c ]+B0,c|r=1−ψy+c +B0+Bw,core−Bw,wall =5∑j=1[y+c ]jW+j (2.62)A0y+c−ψ dB0,cdr|r=1−ψy+c =5∑j=1j[y+c ]j−1W+j (2.63)− A0(y+c )2+ψ2d2B0,cdr2|r=1−ψy+c =5∑j=1j( j−1)[y+c ] j−2W+j (2.64)The last 2 of these equations are used to express the unknown W+4 and W+5 in termsof y+c :W+4 =1.25A0(y+c )−4−ψ(y+c )−3dB0,cdr|r=1−ψy+c −0.25ψ2(y+c )−2d2B0,cdr2|r=1−ψy+c− (y+c )−33∑j=1j[y+c ]j−1W+j +0.25(y+c )−23∑j=1j( j−1)[y+c ] j−2W+j ,(2.65)46a)0 1 2 300. = 0.4n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)y cb)0 1 2 300. = 0.6n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)y cc)0 1 2 300. = 0.6n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)y cFigure 2.6: The critical layer thickness yc = ψy+c , for n = 0.2, 0.4, ... 0.8, 1:a) He = 5; b) He = 100; c) He = 2000.W+5 =−0.8A0(y+c )−5+0.6ψ(y+c )−4dB0,cdr|r=1−ψy+c +0.2ψ2(y+c )−3d2B0,cdr2|r=1−ψy+c+0.6(y+c )−43∑j=1j[y+c ]j−1W+j −0.2(y+c )−33∑j=1j( j−1)[y+c ] j−2W+j .(2.66)These expressions are substituted into the first equation to give a single nonlinearequation for y+c , which may be solved iteratively. Figure 2.6 shows the results ofthis calculation, in terms of yc = ψy+c , for the same (He,n,Hw) as in Figure 2.5.Although for smaller n< 0.3 the critical layer is insignificant, in full turbulence47we see that the critical layer thickness can be 5-15% of the pipe radius. This radialthickness (at the wall) corresponds to a larger area fraction of the pipe and maysignificantly affect Taylor dispersion, being close to the wall where the velocityvariation is maximal. Note that in the transitional regime, it is to be expectedthat the log-law profile loses validity progressively with decreasing flow rate; forFigure 2.6 we have simply extended the calculations into the transitional regime.Again it is observed that an increasing yield stress (He) reduces the effective powerlaw index and hence reduces the critical layer thickness, so that for significant yieldstresses, we see wall layers at 5-15% of the pipe radius only for n ≥ 0.5. Notehowever, that y+c increases with He, but this is masked by the decrease in ψ .Having found y+c we can evaluate W+4 & W+5 and hence the contributions to theReynolds stresses in the wall layer, u′w′+3 & u′w′+4 :u′w′+3 = −4(n(1− rY )W+4 +ψ32n2−3n+124n2(1− rY )2), (2.67)u′w′+4 = −5(n(1− rY )W+5 −0.8(n−1)ψW+4 +ψ42n3−9n2+10n−3120n3(1− rY )3).(2.68)These expressions are now used to define the turbulent diffusivity within the walllayer.In Figure 2.7 we plot some example velocity profiles, lying just within the fullyturbulent regime: Hw = 1.05Hw,2 (i.e. with wall shear stress 5% larger than that re-quired for full turbulence), for He = 5, 100, 2000, and n = 0.2, 0.4, ... 0.8, 1.The main differences within the wall layer profiles as He is increased are found forsmaller n, which is of course also where the layer thickness is insignificant. Theturbulent core profiles appear to vary only modestly with He, being mostly depen-dent on n. This coincides with both computational and experimental observations,Rudman et al. [196], Güzel et al. [102].There is little data regarding the velocity distribution in the wall-layer for shear-thinning fluids. However, we have compared the velocity profile with Newtonianfluid data from the DNS computations of [250]. Figure 2.8 shows this comparison.We can see that velocity profiles are matched very well, both close to the walland in the core. They deviate at the edge of the wall layer, which is partly to be48a)0 1 2 3 4051015202530n = 0.2n = 1y+/y+cW+(y+/y+ c)0 0.5 100.51rW(r)b)0 1 2 3 405101520253035n = 0.2n = 1y+/y+cW+(y+/y+ c)0 0.5 100.51rW(r)c)0 1 2 3 40510152025303540n = 0.2n = 1y+/y+cW+(y+/y+ c)0 0.5 100.51rW(r)Figure 2.7: Example velocity profiles in wall coordinate (W+(y+/y+c )) forHw = 1.05Hw,2 and n = 0.2, 0.4, ... 0.8, 1. Insets show velocity profilesin global coordinate. The black dots show the case of n = 0.2. Velocityprofiles within the wall layer are marked red. a) He = 5; b) He = 100;c) He = 2000.49a)100 10205101520y+W+(y+)b)100 10205101520y+W+(y+)Figure 2.8: Comparison of Newtonian velocity profile in the wall layer ob-tained in this study (solid lines) with those of [250] (dashed lines). a)Re = 5300 and b) Re = 44000.expected, as we have simply “patched” the wall layer to the core region here.2.4.2 Diffusivity and dispersivity in turbulent pipe flowsWe now follow a classical path towards estimating streamwise spreading of a pas-sive tracer by the turbulent flow via diffusive and dispersive mechanisms, e.g. Tay-lor [230]. The net diffusivity is denoted DˆD = Dˆm+Dˆt , representing molecular andturbulent terms respectively. The turbulent diffusivity Dˆt is usually modelled usingthe Reynolds analogy for the turbulent transport of mass and momentum, and theaxial momentum balance to evaluate the shear stress, i.e.Dˆt =1SctDˆe =1Sct ρˆ∣∣∣∣dWˆdrˆ∣∣∣∣(rˆRˆτˆw−|τˆzr|). (2.69)Here Dˆe and Sct are the eddy diffusivity and the turbulent Schmidt number respec-tively, and on the right-hand side we have the total shear stress minus the meanviscous shear stress.We work primarily with dimensionless diffusivities, scaled by ˆ¯W0Dˆ. In the wall50layer we can evaluate (2.69) directly from our approximate solution:DD(y+) = Dm+ψ2Sct(f f2)1/2 [1−ψy+− rY − (1− rY ) ∣∣∣dW+dy+ ∣∣∣n]∣∣∣dW+dy+ ∣∣∣ (2.70)In the turbulent core the velocity is given by (2.61). The velocity gradient and Dtare continuous at r = rc. However for r < rc, the averaged viscous stress τˆzr in(2.69), is not simply defined by inserting the strain rates of the averaged velocityinto the constitutive law, (it is the average of the shear stress, not the shear stressof the average). In the core we expect that velocity fluctuations will be of sizeWˆ∗ ∼√f f ˆ¯W0, which would be the same size as the strain rate evaluated from themean flow. Since the strain rate tensor is assumed locally isotropic, at most we getan order of magnitude for τˆzr. It is unclear how to approximate this term.Krantz and Wasan [136] argue that there is no theoretically justified form forthe molecular diffusion of vorticity in the turbulent core for the power law fluidsthey consider, so they simply neglect τˆzr. On the other hand this seems at odds withthe significant effects of n on the mean velocity profile and of both (n,He) in af-fecting transition. In Güzel et al. [102] it is shown that full turbulence waits for theaverage Reynolds stresses to exceed the yield stress, i.e. breaking the laminar plug.Thus, at least close to transition and for weak turbulence, there are suggestions thatviscous stresses are still relevant within the core. For simplicity, we assume thatτˆzr vanishes at the centreline (from symmetry) and approximate (2.69) by assum-ing that(rˆτˆw/Rˆ−|τˆzr|)varies linearly with r across the core. We then use valuesas r→ r−c to match with the wall layer:DD(r) = Dm+12Sct(f f2)1/2 rrc1G(r)×[rc− rY − 8Rep[n3n+1]n[G(rc)]n(f f2)n/2−1], (2.71)G(r) =∣∣∣∣− A01− r + ddr B0,c(r)∣∣∣∣= ∣∣∣∣dWdr∣∣∣∣√2f f. (2.72)For numerical robustness, in the case of very small ψ , for r→ r−c we evaluate at51r = 0.99rc. Note also that G(r) vanishes at r = 0, due to the centreline correction.Thus, a Taylor series and l’Hôpital’s rule are used to resolve Dt(r) as r→ 0. In(2.71) Dm is the dimensionless molecular diffusivity, equal to the inverse of thePéclet number, Pe = ˆ¯W0Dˆ/Dˆm 1 (for our flows of interest).Examples of Dt(r) are illustrated in Figure 2.9 for the same parameters as thevelocity profiles in Figure 2.7. We observe that Dt is reduced both by decreasingn and by increasing He. The wall layer variation is characteristically cubic withy+. The variation of Dt(r) is curious. Via the Reynolds analogy (2.69) and our as-sumed linear variation of stresses with r, this variation is clearly related to dividingthrough by the velocity gradient. Since the velocity gradient vanishes as r→ 0, wesee that the variation in Dt(r) close to the pipe centreline is directly related to thechoice of centreline correction function. The first derivative of the correction func-tion is fixed and the size of correction is small. Thus, it is essentially the secondderivative of the correction function that is important!We remark that the existence of a local maximum in the radial profile of tur-bulent diffusivity somewhere away from the centreline is found in the literature;see e.g. Seagrave [203], Koo [133], Travis et al. [235]. It seems the correctionfunction (2.44) captures this feature qualitatively. The more exotic variations inDt(r) that correspond to the correction function (2.43) of Bogue and Metzner [27]are not supported by any computational or experimental data that we have found.Of course, this is not conclusive, but favours (2.44).It may be of concern that the correction function can influence Dt(r) to thisextent. A more rudimentary analyses would simply approximate Dt(r) as constant,perhaps evaluated from the wall layer, (hence vanishing for small n and large He asψ → 0). Alternatively, if we ignore the centreline and wall layer corrections, justusing the Reynolds analogy and the logarithmic velocity profile leads to a variation:Dt(r) ∝ r(1− r). This clearly does not represent either the expected diffusivitybehaviour at the centreline, nor can it resolve any effects of weak turbulence on walllayers as the cubic variation is gone. Practically speaking, although the correctionfunction can influence Dt(r), the effects are primarily in the central part of the pipe,which does not contribute greatly to either the averaged turbulent diffusivity nor tothe Taylor dispersivity.In computing mean dispersive and diffusive transport along the pipe, 3 compo-52a)0 0.2 0.4 0.6 0.8 100.511.522.533.54 x 10−3n = 1rDt(r)b)0 0.2 0.4 0.6 0.8 100.511.522.533.54 x 10−3n = 1rDt(r)c)0 0.2 0.4 0.6 0.8 100.511.522.533.54 x 10−3n = 1rDt(r)Figure 2.9: Example profiles of Dt(r) for Hw = 1.05Hw,2 and n =0.2, 0.4, ... 0.8, 1, with Sct = 1: a) He= 5; b) He= 100; c) He= 2000.Solid and broken lines are associated with centreline corrections (2.44)and (2.43), respectively. Profiles within the critical wall layer aremarked red.53nents contribute. The first component is the molecular diffusivity (Dm), which istypically much smaller than the second component, the radially averaged turbulentdiffusivity Dt :Dt = 2∫ 10rDt(r) dr = 2∫ rc0rDt(r) dr+2ψ∫ y+c0(1−ψy+)Dt(y+) dy+.=1Sct(f f2)1/2 [rc− rY − 8Rep [ n3n+1]n [G(rc)]n( f f2 )n/2−1]rc∫ rc0r2G(r)dr+ψ2Sct(f f2)1/2 ∫ y+c0(1−ψy+)[1−ψy+− rY − (1− rY )∣∣∣dW+dy+ ∣∣∣n]∣∣∣dW+dy+ ∣∣∣ dy+(2.73)Both integrals above must be evaluated numerically. Although potentially timeconsuming, the integrands have been normalised and are well-behaved. Thus, arelatively coarse mesh can be used for the integration with a high order approxi-mation, e.g. Simpson’s rule.The calculation of D¯t is sensitive to the approximation of τˆzr in (2.69) and alsoto the velocity gradient, hence correction function. Examples of the variations inDt are shown in Figure 2.10. We see that Dt is not particularly sensitive to eitherwall shear stress nor He (yield stress) over these ranges. The main variation is withn.The third (and usually dominant) component is the Taylor dispersion coeffi-cient, which is defined as:DT =DˆTˆ¯W0Dˆ=12∫ 10(∫ r0[W (r˜)−1]r˜ dr˜)2rDD(r)dr = Ic(rc)+ψ3I+(y+c ) (2.74)whereIc(rc) =12∫ rc0(∫ r0[W (r˜)−1]r˜ dr˜)2rDD(r)dr54a)0 2 4 6 8 100.60.811. x 10−3n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)D¯tb)0 2 4 6 8 100.511.52 x 10−3n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)D¯tc)0 2 4 6 8 x 10−3n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)D¯tFigure 2.10: Examples of D¯t for n= 0.2, 0.4, ... 0.8, 1, with Sct = 1: a) He=5; b) He = 100; c) He = 2000. Broken and solid lines are associatedwith the centreline correction functions (2.43) and (2.44), respectively.andI+(y+c ) =12∫ y+c0(∫ y+0[√0.5 f fW+(s)−1](1−ψs) ds)2(1−ψy+)DD(y+) dy+Both these terms require numerical integration. However, the integral terms in thenumerator can be evaluated explicitly, which accelerates computation, i.e. only onenumerical integration is needed.55For the core integral Ic(rc), the velocity is given by (2.61), so that we see:∫ r0[W (r˜)−1]r˜ dr˜ =√f f2∫ r0[A0 ln(1− r˜)+B0 +B0,c(r˜)+Bw,core−Bw,wall ]r˜ dr˜=(√f f2[B0 +Bw,core−Bw,wall ]−1)r22+√f f2∫ r0[A0 ln(1− r˜)+B0,c(r˜)]r˜ dr˜ (2.75)Integrating the first term on the right hand side, we have:∫ r0A0r˜ ln(1− r˜) dr˜ = A0 14(2r2 ln(1− r)− (r+2)r−2ln(1− r)) (2.76)For the third term on the right hand side, we have two expressions:∫ r0r˜B0,c(r˜) dr˜ =∫ r0A0r˜(0.375e0.04−(r˜−0.2)20.15 −0.1581529)dr˜=3A01600(2√15pie4/15[erf(10r−2√15)− erf(2√15)])− 45A01600(e(8r−20r2)/3−1)+0.079076A0r2which is associated to the centreline correction function (2.43) and∫ r0r˜B0,c(r˜) dr˜ =∫ r0A0[r˜2(1− r˜)2− 115r˜]dr˜ = A0(r55− r42+r33− r230)which is associate to the centreline correction function (2.44).In the wall layer integral I+(y+c ):∫ y+0[√0.5 f fW+(s)−1](1−ψs) ds = (2.77)√f f25∑j=1W+j(y+) j+1j+1− y++ψ((y+)22−√f f25∑j=1W+j(y+) j+2j+2)Note that the molecular diffusivity contributes very close to the wall in removing alogarithmic singularity from the calculation of I+(y+c ), i.e. the eddy viscosity termsin the denominator vanish cubically and the numerator vanishes quadratically as56y+→ 0.Examples of the variations in DT are shown in Figure 2.11a-c. The main obser-vations that we see are: (i) DT decreases significantly with n; (ii) DT decreases withthe yield stress He; (iii) for larger n we see a very significant rise in DT as the wallshear stress approaches its transitional value. The overall trend of increasing DTwith n and the size of DT are similar to those of Krantz and Wasan [136] for powerlaw fluids. In computing DT we divide by the diffusivity [Dt(r)+Dm] in the inte-grands. Although we have seen significant differences in the turbulent diffusivitiesDt(r) within the core, according to the choice of centreline correction function, thenumerator in the core involves integrals of [W (r)−1], which is of order√ f f , andthese terms scale with r4. Thus, the choice between corrections functions such as(2.43) & (2.44) is not critical insofar as calculating DT is concerned.In the original work on dispersion, Taylor [230] used a coarse approxima-tion of the (universal) velocity distribution taken from available measurementsand performed a numerical integration. This gave DˆT = 10.06Wˆ∗Dˆ/2 and ¯ˆDt =0.052Wˆ∗Dˆ/2, giving DT/D¯t ≈ 193. A comparison of Taylor’s coefficients withours for n = 1 and He = 0 (Newtonian fluid) is shown in Figure 2.11d for increas-ing Reynolds number in the weak turbulent range. Our computed DT is signifi-cantly larger than that of Taylor in the weak turbulent range, but converges as thewall layers thin. The main reason for the difference is (of course) including ouranalysis of the wall layers, where we expect to have a significant contribution toDT for weakly turbulent flows.It is interesting to understand where the main contributions to Dt and DT comefrom. This is explored in Figure 2.12 for He = 10 (although analogous effectsare found at other He). Firstly, Figure 2.12a shows that the contribution of thecore region to Dt is always dominant; typically at least 90%. This explains thelarge differences in Dt according to the corrections functions. On the other hand,Figure 2.12b shows the wall-layer contribution to computing DT . The wall layerscorrespond to regions where [W (r)− 1] is of order 1 and where the diffusivity issmall. In the core, [W (r)− 1] is of order √ f f and the diffusivity is of size D¯t .Thus we see an interesting transition in Figure 2.12b. Where the wall layer is rela-tively thick, it gives the dominant contribution to DT . As n decreases sufficiently,or simply as we move further into the fully turbulent regime, the wall layer scaling57a)0 2 4 6 8 1010−310−210−1100101n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)DTb)0 2 4 6 8 1010−310−210−1100101n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)DTc)0 2 4 6 8 1010−310−210−1100101n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)DTd)104 10510−1100101G.I. TaylorReMRDTFigure 2.11: Examples of DT for n= 0.2, 0.4, ... 0.8, 1, with Sct = 1 & Dm =10−6; a) He = 5; b) He = 100; c) He = 2000. Broken and solid linesare associated with centreline correction functions (2.43) and (2.44),respectively. d) Newtonian fluid (He = 0, n = 1). Broken and solidlines: our results for centreline correction functions (2.44) and (2.43);solid thick line: Taylor’s prediction [230]; black point-line; numericalresults of Ekambara and Joshi [67]; diamonds: experimental resultsof Flint and Eisenklam [75]; filled squares: experimental results ofHart et al. [113]; hollow squares: experimental results of Keyes [131];circles: experimental results of Fowler and Brown [79]. All data aretaken from Hart et al. [113].58a)0 2 4 6 8 100.811. x 10−3n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)D¯tb)0 2 4 6 8 1010−210−1100101n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)DTFigure 2.12: a) D¯t for n= 0.2, 0.4, ... 0.8, 1 and He= 10. Broken lines showcontribution of the core region. b) DT for n = 0.2, 0.4, ... 0.8, 1 andHe = 10. Broken lines show the contribution of the wall layer.parameter ψ becomes extremely small and the wall layer contribution reduces sig-nificantly due to the small thickness of the wall layer. This effect occurs at moremoderate n for larger yield stresses, He. We see a corresponding effect on DT ,which decreases significantly as n decreases at any fixed He. As Hw increases thewall layer effects diminish, but relatively slowly for n≈ 1.Figure 2.11 and the comparison with Taylor [230] and other data in Figure 2.11dindicate clearly the importance of modelling the wall layers in estimating stream-wise dispersion in weakly turbulent flows. Although the effects are significant wemust regard our analysis as approximate. Quantitative values rely on the mean ve-locity profile. Earlier authors, e.g. Taylor [230], Tichacek et al. [234], have usedempirical values of the mean turbulent velocity profile. These profiles are availablefor Newtonian fluid flows, but are lacking for non-Newtonian fluids (for which onemust consider at least some range of dimensionless (n,He/Hw)). Partly this is be-cause experimental studies use real fluids for which rheological models like theHerschel-Bulkley model have limitations at high shear rates. Also experimentalstudies with such fluids in order to accurately measure pointwise velocity valuesare time intensive and often involve a degree of rheological degradation (and/orother rheological effects that deviate from simple model descriptions). Thus, we59are pushed towards expressions such as the log-law, which do arise naturally froma dimensional analysis, but nevertheless need correcting. The matching procedureused to define the wall layer thickness (and hence the velocity coefficients) is de-pendent on the core velocity profile.2.4.3 Velocity profile in turbulent channel flowsFollowing the same procedure explained in §2.4.1, we correct the velocity profilenear the centreline and the wall. To do so, we introduce the wall layer coordinatexˆ = Hˆ − yˆ, x+ = xˆ/xˆ∗ and W+(x+) = Wˆ (yˆ)/Wˆ∗ =√2/ f fW (y), where Wˆ∗ is thefriction velocity andxˆ∗ =Wˆ∗γ˙∗, γ˙∗ =[τˆwκˆ] 1n(1− yY )1nWe eventually find the core velocity profile:W (y) =√f f2[A0 ln(1− y)+B0+B0,c(y)+Bw,core−Bw,wall], (2.78)whereA0 =ADMn′ln10, ADM =4.0√2(n′)0.75(2.79a)B0 = A0(1n′ln(ReMR f 1−n′2)+1)− 0.395√2n′1.2(2.79b)B0,c(y) = A0(y(1− y)2− 112)(2.79c)B0,c =11− yc∫ 1ycB0,c dy (2.79d)Bw,core =ψx+c1−ψx+c[A0(ln(ψx+c )−1)+B0+B0,c](2.79e)Bw,wall =ψ1− [ψx+c ]5∑j=1[x+c ]j+1j+1W+j (2.79f)60a)0 5 10 15 2000. = 0.8n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)xc0 1 2 310−1010−5100(Hw −Hw,1)/(Hw,2 −Hw,1)ψb)0 5 10 15 2000. = 0.8n = 1(Hw −Hw,1)/(Hw,2 −Hw,1)xc0 1 2 310−1010−5100(Hw −Hw,1)/(Hw,2 −Hw,1)ψFigure 2.13: Critical wall layer thickness xc =ψx+c for n= 0.2, 0.4, ... 0.8, 1:a)He = 10; b) He = 400. Inset figures show the wall-layer scalingparameter ψ . The black dot is associated with n = 0.2.Note that the centreline correction function we used here is of form (2.44). Thewall scaling parameter in the channel flow is:ψ =61n .21n− 122+ 1n((1− yY )Rep f 1−n2f)− 1n(2.80)We are left to determine the position of the wall layer thickness (x+c ) using ananalogous matching procedure to that of (2.62-2.64). Examples of the wall layerthickness and scaling parameter ψ are shown below in Figure 2.13 at two values ofHe. We again observe significant wall layers for weakly turbulent flows providedn′ is not too small. The scaling parameter decreases rapidly at small n or as He isincreased significantly.Example velocity profiles are shown in Figure 2.14 for wall shear stresses justabove full turbulence (Hw = 1.05Hw,2), for the same He and n as in Figure 2.13.The trends observed are quite similar to those in the pipe flow.61a)0 1 2 3 40510152025x+/x+cW+(x+/x+ c)n = 0.2n = 10 0.5 100.511.5yW(y)b)0 1 2 3 4051015202530x+/x+cW+(x+/x+ c)n = 0.2n = 10 0.5 100.511.5yW(y)Figure 2.14: Example velocity profiles in wall coordinate (W+(x+/x+c )) forHw = 1.05Hw,2 and n= 0.2, 0.4, ... 0.8, 1. Insets show velocity profilesin global coordinate. The black dots show the case of n= 0.2. Velocityprofiles within the wall layer are marked red. a) He= 10; b) He= 400.2.4.4 Diffusivity and dispersivity in turbulent channel flowsA similar approach as in §2.4 is taken here. Using the Reynolds analogy, the tur-bulent diffusivity Dˆt can be written as:Dˆt =1SctDˆe =1Sct ρˆ∣∣∣∣dWˆdyˆ∣∣∣∣(yˆHˆτˆw−|τˆzy|). (2.81)We scale Dˆt with Wˆ0Hˆ, to give dimensionless expressions in the core and walllayers, as follows:Dt(x+) =ψ2Sct(f f2)1/2 [1−ψx+− yY − (1− yY ) ∣∣∣dW+dx+ ∣∣∣n]∣∣∣dW+dx+ ∣∣∣ , (2.82)andDt(y) =12Sct(f f2)1/2 yyc1G(y)[yc− yY − 6Rep[n2n+1]n[G(yc)]n(f f2)n/2−1],(2.83)62whereG(y) =∣∣∣∣− A01− y + ddyB0,c(y)∣∣∣∣= ∣∣∣∣dWdy∣∣∣∣√2f f.Integrating Dt(y) across the channel gives the average turbulent diffusivity (D¯t),exploiting symmetry:Dt =∫ 10Dt(y) dy =∫ yc0Dt(y) dy+ψ∫ x+c0Dt(x+) dx+.=1Sct(f f2)1/2 [yc− yY − 6Rep [ n2n+1]n [G(yc)]n( f f2 )n/2−1]yc∫ yc0yG(y)dy+ψ2Sct(f f2)1/2 ∫ x+c0[1−ψx+− yY − (1− yY )∣∣∣dW+dx+ ∣∣∣n]∣∣∣dW+dx+ ∣∣∣ dx+(2.84)The Taylor dispersion coefficient is also evaluated straightforwardly from the ve-locity profile and turbulent diffusivity, as below.DT =DˆT2 ˆ¯W0Hˆ=12∫ 10(∫ y0[W (y˜)−1] dy˜)2DD(y)dy = Ic(yc)+ψ3I+(x+c ) (2.85)whereIc(yc) =12∫ yc0(∫ y0[W (y˜)−1] dy˜)2Dt(y)dyandI+(x+c ) =12∫ x+c0(∫ x+0[√0.5 f fW+(s)−1] ds)2DD(x+)dx+.Again the calculations involved in Dt and DT involve a single numerical inte-gration in core and wall layer.The Taylor dispersion term is again dominant in diffusing/dispersing mass ax-ially. Computed examples are shown in Figure 2.15. Again we see the main sen-sitivity is to n although He does decrease the dispersivity slightly (acting through63a)0 2 4 6 8 1010−1100101n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)DTb)0 2 4 6 8 1010−1100101n = 0.2n = 1(Hw −Hw,2)/(Hw,2 −Hw,1)DTFigure 2.15: Examples of DT for channel flow, with n = 0.2, 0.4, ... 0.8, 1,Sct = 1 & Dm = 10−6: a) He = 10; b) He = 400n′). In the weakly turbulent regime we again see a significant increase in DT , as-sociated with the wall layers. This is an O(1) increase in DT , but is a smallereffect than in the pipe flows. The reasons for this difference are largely geometric.First, computed xc are slightly smaller than yc (pipe). Secondly, thick wall layersin the pipe represent a larger area fraction than in the channel flow. We can alsocompare the expressions for the core contributions to the DT integrals, close to thecentrelines: in pipe flow we effectively integrate r3 f f /DD and in channel flow weintegrate y2 f f /DD.2.5 Discussion and SummaryThe aim of this chapter was two-fold: i) To analyze turbulent flows of shear-thinning yield stress fluids in both pipe and channel geometries and lay down a con-sistent procedure for hydraulic calculation of Herschel-Bulkley fluids. This meansfinding the relationship between the mean velocity and the wall shear stress. ii)To extend the classical turbulent Taylor dispersion analysis [230] to shear-thinningyield stress fluids and find estimates of turbulent diffusivity and dispersivity.The chapter can be summarized into the following concluding remarks:• We have explored the effects of the yield stress on turbulent transport of64mass along pipes and plane channels. The yield stress produces competingeffects in the wall layers. The critical layer thickness in wall coordinates isincreased, but the scaling parameter ψ decreases rapidly with n′. Thus, wefind that for very large yield stresses (He) and small n the wall layer thicknessis vanishingly small, indeed unrealistically so. This is however dictated bythe friction factor closure and delayed transition.• For larger n≤ 1 and a wide range of practical He the wall layer thickness canbe over 10% of the pipe radius. Following the procedure of Wasan & Krantzwe have developed approximations to the velocity and turbulent diffusivityin the wall region, and for these parameter ranges we show a significantincrease in the (dominant) Taylor dispersivity in weakly turbulent flows. Inpipe flows this effect can be anO(10) increase, compared to values of highlyturbulent regimes where the wall layers thin. In plane channel flows it is amore modest O(1) increase.• This demonstrates that in weakly turbulent regimes (as found in the primarycementing applications of interest), it is necessary to include the effects ofthe wall layers. Our predictions, when applied to Newtonian fluids in weaklyturbulent regimes, bound above the available data and show similar variationwith Re (see Figure 2.11d). The classical expression of Taylor under-predictsthe same data.• It is shown that for weakly turbulent flows it is necessary to include an anal-ysis of wall layers in studying dispersion. In particular, in pipe flows, weobserve an O(10) increase in Taylor dispersion coefficients, compared tohighly turbulent values. This arises from a combination of large velocityand small turbulent dispersivity, acting over a wall layer that can represent& 20% of the pipe area. In channel flows the wall layer effect is more mod-est, but still represents anO(1) increase in Taylor dispersion coefficient. Thepreceding effects are negated for small power law index, due to rapid reduc-tion of the wall layer, and are observed to reduce modestly as the the yieldstress increases.65Chapter 3Primary cementing modellingThis chapter presents a detailed derivation of a practical two-dimensional modelfor displacement flows of viscoplastic fluids in an annular geometry. The two-dimensional formulation presented here models laminar, turbulent and mixed regimeannular flows. Such mixed regimes, including those in which different regimes ex-ist in the same annular cross-section are relatively common in primary cementing.A version of this chapter is published in Maleki and Frigaard [147].3.1 Dimensional governing equationsA cylindrical coordinate system (rˆ,θ , ξˆ ) is used to describe the well geometry: ξˆmeasures distance along the central axis of the casing rˆ = 0 which is assumed tobe inclined to the vertical with angle β (ξˆ ). The local cross-section of the well,outside the casing, is assumed to be that of an eccentric annulus, with inner radiusrˆi(ξˆ ), equal to the outer radius of the casing and outer radius rˆo(ξˆ ) equal to theinner radius of the hole (or previous casing). At each depth ξˆ , the mean radiusrˆa(ξˆ ) and the mean half-gap width dˆ(ξˆ ) are defined by:rˆa(ξˆ )≡ 12 [rˆo(ξˆ )+ rˆi(ξˆ )], dˆ(ξˆ )≡12[rˆo(ξˆ )− rˆi(ξˆ )]. (3.1)As well as inner and outer radii, the distance between the centres of the two cylin-ders is given by eˆ(ξˆ ), see Figure 3.1. It is assumed that eˆ(ξˆ )< 2dˆ(ξˆ ) (the cylinders66Figure 3.1: Geometrical parameters of primary cementingdo not touch), and the variations in the geometry along ξˆ are slow. For simplicity,it is also assumed that the narrow side of the annulus will be found on the lowerside of the well and that the casing remains stationary. Both these assumptions canbe relaxed with care, e.g. see Carrasco-Teja and Frigaard [36], Tardy and Bittleston[226].The annular displacement is modeled as a concentration dependent multi-fluidflow that is fully turbulent and incompressible. The fluid constituents contribute tothe mixture density ρˆ and to the rheological properties of the mixture. The latter areused in closure expressions that define the deviatoric stress tensor τˆi j. In primarycementing a sequence of K fluids is pumped around the flow path, from the bottomof the annulus to the top. A typical sequence would be: mud, wash, spacer, leadslurry, tail slurry, mud. When circulating, drilling muds, spacer fluids and cementslurries are predominantly shear thinning, nonlinearly viscous and inelastic, oftenalso with a significant yield stress. Washes are Newtonian fluids. In general eachconstituent fluid can be effectively modeled as a Herschel-Bulkley fluid. We shalladdress evolution of the mixture concentrations at length in §3.4, but for now weassume that all fluid properties are approximated effectively by closures that de-pend on the local mean fluid concentrations: both ensemble averaged and averagedacross the annular gap.67We adopt the usual Reynolds decomposition for turbulent flows, into mean andfluctuating parts. Because the flows will be time-varying as the displacement pro-ceeds, the mean part (denoted with an overbar below) is interpreted as an ensembleaverage. The velocity and pressure are uˆ = (uˆ, vˆ, wˆ) and pˆ, respectively. In thelocal coordinate system the Reynolds-decomposed Navier-Stokes system is:∂∂ tˆ[ρˆ ¯ˆu]+ ∇ˆ · [ρˆ ¯ˆu ¯ˆu] = 1rˆ∂∂ rˆ[rˆ ¯ˆτrˆrˆ]+1rˆ∂∂θ¯ˆτrˆθ +∂∂ ξˆτˆrˆξˆ −¯ˆτθθrˆ− ∂¯ˆp∂ rˆ+ ρˆ gˆrˆ +1rˆ∂∂ rˆ[rˆτˆ trˆrˆ]+1rˆ∂∂θτˆ trˆθ +∂∂ ξˆτˆ trˆξˆ −τˆ tθθrˆ, (3.2)∂∂ tˆ[ρˆ ¯ˆv]+ ∇ˆ · [ρˆ ¯ˆv ¯ˆu] = 1rˆ2∂∂ rˆ[rˆ2 ¯ˆτθ rˆ]+1rˆ∂∂θ¯ˆτθθ +∂∂ ξˆ¯ˆτθξˆ −1rˆ∂ ¯ˆp∂θ+ ρˆ gˆθ +1rˆ2∂∂ rˆ[rˆ2τˆ tθ rˆ]+1rˆ∂∂θτˆ tθθ +∂∂ ξˆτˆ tθξˆ (3.3)∂∂ tˆ[ρˆ ¯ˆw]+ ∇ˆ · [ρˆ ¯ˆw ¯ˆu] = 1rˆ∂∂ rˆ[rˆ ¯ˆτξˆ rˆ]+1rˆ∂∂θ¯ˆτξˆ θ +∂∂ ξˆ¯ˆτξˆ ξˆ −∂ ¯ˆp∂ ξˆ+ ρˆ gˆξˆ +1rˆ∂∂ rˆ[rˆτˆ tξˆ rˆ]+1rˆ∂∂θτˆ tξˆ θ +∂∂ ξˆτˆ tξˆ ξˆ , (3.4)0 =1rˆ∂∂ rˆ[rˆ ¯ˆu]+1rˆ∂ ¯ˆv∂θ+∂ ¯ˆw∂ ξˆ. (3.5)The components ¯ˆτi j are those that come from ensemble averages of the viscousstress tensor (which itself is nonlinear). The components τˆ ti j are the turbulent(Reynolds) stresses, resulting from the fluctuating components of the velocity fielduˆ′, defined as:τˆ ti j =−ρˆ uˆ′iuˆ′j. (3.6)The gravitational acceleration, gˆ = (gˆrˆ, gˆθ , gˆξˆ ), is given bygˆrˆ =−gˆsinβ (ξˆ )cosθ , gˆθ = gˆsinβ (ξˆ )sinθ , gˆξˆ =−gˆcosβ (ξˆ ), (3.7)where gˆ = 9.81m/s2.683.2 Scaling and simplificationWe wish to reduce our model to something more tractable than (3.2)-(3.5), byexploiting the aspect ratio of the annulus. The annulus geometry is typically longand narrow, with the typical gap half-width (∼ 1cm) being much smaller than atypical azimuthal distance (∼ 30cm), which in turn is much smaller than a typicallength of the annulus (∼ 500m).Following Bittleston et al. [24], let the mean radius (rˆa,0), the local and globalaspect ratios (δ (ξˆ ) and δ0 respectively) be defined by:rˆa,0 =1Zˆ∫ ξˆtzξˆbhrˆa(ξˆ ) dξˆ , δ (ξˆ ) =dˆ(ξˆ )rˆa(ξˆ ), δ0 =1Zˆ∫ ξˆtzξˆbhδ (ξˆ ) dξˆ .(3.8)where Zˆ is the length of the zone of the well to be cemented, extending upwardsfrom bottom hole, ξˆbh, to the top of the zone, ξˆtz. Scaled axial and azimuthalcoordinates ξ and φ are then:ξ =ξˆ − ξˆbhpi rˆa,0, φ =θpi. (3.9)In each cross-section, we define the local average radius, r = ra(ξ ), and local an-nulus eccentricity, e(ξ ), by:ra(ξ ) =rˆa(ξˆ )rˆa,0, e(ξ ) =eˆ(ξˆ )2dˆ(ξˆ ). (3.10)The centreline of the annular gap is at rˆ = rˆa,0ra(ξ )rc(φ ,ξ ). The radial coordinateis scaled relative to the distance from the centreline of the annulus, as follows:y =rˆ− rˆa,0ra(ξ )rc(φ ,ξ )rˆa,0δ0, (3.11)i.e. y is a local annular gap coordinate.We assume a narrow annulus approximation: δ (ξ )∼ δ0 pi; noting that δ0/pi ,denotes the ratio of radial (gap) length-scales to azimuthal length-scale. To leadingorder in δ0/pi , we have rc(φ ,ξ ) ∼ 1 and find that the inner and outer walls are at69y =∓H(φ ,ξ ), where:H(φ ,ξ ) =δ (ξ )ra(ξ )[1+ e(ξ )cospiφ ]δ0. (3.12)More complex geometries are readily accommodated by specifying any H(φ ,ξ )of O(1) that varies slowly with ξ , e.g. H(φ − φ0(ξ ),ξ ) with H as above, retainsthe eccentric annular shape but shifts the wide-side of the annulus to φ = φ0(ξ ).Helically varying well eccentricity, elliptic cross sections and irregular washoutsare each fairly common deviations away from (3.12).Times are scaled with an advective timescale: pi rˆa,0/ ˆ¯W where ˆ¯W is representa-tive of a mean axial velocity. For simplicity, we assume that the fluids are pumpedfollowing a schedule of pump rates: Qˆpump(tˆ), typically a step function. The pumpschedule is used to define a representative flow rate Qˆ0, e.g. the maximum flowrate. The mean axial velocity is: ˆ¯W = Qˆ0/Aˆ0, where Aˆ0 = 4piδ0[rˆa,0]2 is a typicalcross-sectional area of the annulus.We now consider the relative sizes of the different terms in (3.2)-(3.5), witha view to simplification. Given that δ0/pi  1, we may assume that the domi-nant components of mean velocity will be in the (φ ,ξ )-directions, scaling approx-imately with ˆ¯W . The incompressibility condition (3.5) suggests the radial compo-nent of mean velocity scales with δ0 ˆ¯W/pi . Therefore, we can see that the accelera-tion and inertial terms on the left-hand side of (3.2), (3.3) & (3.4) have respectivesizes:δ0piρˆ ˆ¯W 2pi rˆa,0,ρˆ ˆ¯W 2pi rˆa,0,ρˆ ˆ¯W 2pi rˆa,0.Simplifying the stress terms on the right-hand side of (3.2), (3.3) & (3.4) is lessstraightforward. A number of authors have computed turbulent flows of Newtonianfluids in uniform eccentric annuli and calculated the Reynolds stresses, e.g. Nouriand Whitelaw [171], Freund et al. [80], Chung et al. [43], Nikitin et al. [169].Insofar as we are concerned here, the main point is that each component of τˆ ti j hassimilar magnitude, because the fluctuating velocity is inherently three-dimensional.In considering the viscous stresses, we need to consider two regions separately:the turbulent core of the annular flow and the wall region. In the turbulent core themean viscous stresses ¯ˆτi j, are the average of the viscous stress and not the viscous70stress of the averaged strain rate. Although the strain rate associated with the meanvelocity can be estimated using scaling arguments, the fluctuating velocity alsocontributes to each component of the strain rate and indeed this contribution maybe dominant in the core. Nonlinearity of the constitutive relations together withthe probabilistic nature of the strain rate tensor prevents any easy simplification. Anice discussion may be found in chapters 4 & 5 of Sawko [202]. All that we assumehere is that the viscous stress components in the core all have similar magnitude,say ¯ˆτi j ∼ τˆν ,0 for some viscous stress scale τˆν ,0.Conventionally, the Reynolds stresses in the turbulent core have similar size tothe wall shear stresses (which will vary azimuthally in an eccentric annulus). In thewall region the Reynolds stresses vanish and the viscous stresses increase over athin layer to match the wall shear stress. In this wall region it is possible to estimaterelative sizes of the different shear rates and approximate the flow. Thus, bothturbulent and viscous stresses have roles to play in describing these flows. This isparticularly true in cementing flows which are generally not highly turbulent, withthe exception of low viscosity washes. For now we adopt a stress scale τˆ0 that weapply to mean turbulent and viscous stresses, and discuss the order of magnitudesof dimensionless terms in different flow regimes later.Next we consider the gravitational terms. It is common to exploit density dif-ferences in creating buoyancy forces to aid in displacing the in-situ drilling mud.To capture this aspect we scale all densities with the density ρˆ1 of the first fluidin the pumping sequence, i.e. the in-situ drilling mud, and subtract the hydrostaticpressure from ¯ˆp. It is assumed that the pressure remaining balances the domi-nant stress gradients: ¯ˆp = ¯ˆpbh(tˆ)+ ρˆ1g · x+(piτˆ0/δ0)p¯, where ¯ˆpbh(tˆ) denotes thebottom-hole pressure.We now substitute the above variables into (3.2), (3.3) & (3.4) and dividethrough by the largest dimensional scales, to give the following.O(δ 30pi3ρˆ ˆ¯W 2τˆ0)︸ ︷︷ ︸IT=−∂ p¯∂y+O(δ0piρ−1Fr2)︸ ︷︷ ︸BT+O(δ0pi)︸ ︷︷ ︸ST+O(δ0pi)︸ ︷︷ ︸CT, (3.13)71O(δ0piρˆ ˆ¯W 2τˆ0)︸ ︷︷ ︸IT=− 1ra∂ p¯∂φ+(ρ−1)sinβ sinpiφFr2+∂∂y[τ tφy+ τ¯φy]+O(δ0pi)︸ ︷︷ ︸ST+O(δ0pi)︸ ︷︷ ︸CT(3.14)O(δ0piρˆ ˆ¯W 2τˆ0)︸ ︷︷ ︸IT=−∂ p¯∂ξ− (ρ−1)cosβFr2+∂∂y[τ tξy+ τ¯ξy]+O(δ0pi)︸ ︷︷ ︸ST+O(δ0pi)︸ ︷︷ ︸CT(3.15)0 =∂ u¯∂y+1ra∂ v¯∂φ+∂ w¯∂ξ+O(δ0pi)︸ ︷︷ ︸CT(3.16)where ρ is the scaled density and Fr is a Froude number:Fr =√τˆ0ρˆ1gˆδ0rˆa,0.Our leading order model is the narrow gap limit, δ0/pi → 0, with other param-eters fixed. The different terms that will vanish in this limit are identified by theunder-braces as follows: IT denotes the leading order inertial terms; BT denotesthe next order buoyancy terms; ST denote the next order stress terms (turbulent andviscous); CT denote terms associated with curvature.If the flow is fully turbulent, the stress scale τˆ0 is the wall shear stress, so theratio ρˆ ˆ¯W 2/τˆ0 that appears in the IT under-braces is effectively ∼ 1/ f f , for a rep-resentative turbulent friction factor f f . Although multiplied by δ0/pi and formallyvanishing in the narrow gap limit, these are likely to be the next largest terms.In the case that the flow is laminar, the Reynolds stress terms vanish and the ex-pression ρˆ ˆ¯W 2/τˆ0 becomes effectively the Reynolds number, since in this case τˆ0is correctly interpreted as a viscous stress scale. Taking δ0/pi → 0 leads to thelaminar displacement model of Bittleston et al. [24].The curvature terms arise both due to replacing the r-derivatives with y-derivatives(changing 1/r to 1/ra+O(δ0/pi)), and due to slow variations in ξ as the well tra-jectory changes (assumed of O(δ0/pi)). The next order stress terms are also onlyO(δ0/pi) smaller in the turbulent flow: from scaling using the geometric aspectratio, i.e. the partial derivatives are smaller in the (φ ,ξ )-plane than with respect to72y. If the flow were laminar, then the Reynolds stresses would vanish and scalingarguments can be used to estimate the size of the viscous stresses; the next largeststresses appear at O((δ0/pi)2).To summarise, if we consider the next order terms we see that there is a prolif-eration of terms at order δ0/pi: buoyancy terms, inertial, stress and curvature. Themain point here is that to include the next order of terms in δ0/pi is prohibitivelycomplex. On the other hand, considering the formal narrow gap limit, δ0/pi → 0,although the scaling arguments are different the leading order equations are similarto those of Bittleston et al. [24].3.3 Narrow gap approximationProceeding with the narrow gap approximation, we take δ0/pi → 0 in (3.13) –(3.16). To eliminate u¯ we integrate (3.16) across the gap width, using conditions ofno-slip at the annulus walls:∂∂φ[2H ¯¯v]+∂∂ξ[2raH ¯¯w] = 0, (3.17)where¯¯v(φ ,ξ , t) =12H∫ H−Hv¯ dy, ¯¯w(φ ,ξ , t) =12H∫ H−Hw¯ dy. (3.18)Equation (3.17) is satisfied using a stream function:2raH ¯¯w =∂Ψ∂φ, 2H ¯¯v =−∂Ψ∂ξ. (3.19)For later convenience we introduce 2D annular divergence and gradient operatorsas follows:∇a ·q = 1ra∂qφ∂φ+∂qξ∂ξ, ∇aq =(1ra∂q∂φ,∂q∂ξ).Turning now to the momentum balance, from (3.13) we see the pressure isindependent of y, as is ρ (see below in §3.4). Equations (3.14) & (3.15) may beintegrated across the annular gap, assuming symmetry at y = 0 for this leading73order approximation:(τ tφy+ τ¯φy,τtξy+ τ¯ξy)= y(1ra∂ p¯∂φ− (ρ−1)sinβ sinpiφFr2,∂ p¯∂ξ+(ρ−1)cosβFr2).(3.20)The leading order stresses are only non-zero in the direction of the modified pres-sure gradient. Viewed in the (φ ,ξ )-plane this is a one-dimensional (1D) turbulentshear flow through a channel of width 2H(φ ,ξ ), driven in the direction of:−(1ra∂ p¯∂φ− (ρ−1)sinβ sinpiφFr2,∂ p¯∂ξ+(ρ−1)cosβFr2),which is therefore also the direction of the streamlines, say es:es =( ¯¯v, ¯¯w)√¯¯v20+ ¯¯w20=1|∇aΨ|(−∂Ψ∂ξ,1ra∂Ψ∂φ)The integrated momentum balance (3.20) can now be resolved along the stream-lines, in the es direction:τ tsy+ τ¯sy =−yHτw (3.21)where the dimensionless wall shear stress τw is:τw = H∣∣∣∣( 1ra ∂ p¯∂φ − (ρ−1)sinβ sinpiφFr2 , ∂ p¯∂ξ + (ρ−1)cosβFr2)∣∣∣∣ . (3.22)Combining the above we have:1|∇aΨ|(−∂Ψ∂ξ,1ra∂Ψ∂φ)= −Hτw(1ra∂ p¯∂φ− (ρ−1)sinβ sinpiφFr2,∂ p¯∂ξ+(ρ−1)cosβFr2)(3.23)which can be re-arranged and cross-differentiated to eliminate the pressure:∇a · [S+b] = 0 (3.24)74in whichS =raτw(|∇aΨ|)H|∇aΨ| ∇aΨ and b =ra (ρ−1)Fr2(cosβ ,sinpiφ sinβ ) . (3.25)The term ∇a ·S in (3.24) is a quasilinear elliptic operator on Ψ and the term ∇a ·bprovides a source term that is driven by the spatial gradients of the buoyancy vectorb. Note that (3.24) contains no time derivatives: time enters only via (i) boundarydata, e.g. if the flow rate changes; (ii) through the fluid concentrations, which affectboth fluid rheology and buoyancy.The following remarks are to be noted:i) We have still not fixed the shear stress scale τˆ0, and this is not particularlyimportant as all terms in (3.24) are scaled with τˆ0. Although the main targethere is to model turbulent flows, laminar flows are more prevalent. The scaleρˆ1 ˆ¯W 2 also over-estimates the turbulent stresses (by using the mean velocityand not a friction velocity), and these in turn are larger than the typical lam-inar viscous stresses. In addition, the objective has been to develop a modelcapable of dealing with mixed flow regimes. Consequently, we define a τˆ0 thatis relevant to the laminar viscous stresses:τˆ0 = maxk{τˆk,Y + κˆk ˆ˙γnk0}, ˆ˙γ0 =3 ˆ¯Wδ0rˆa,0, (3.26)with τˆk,Y , κˆk, and nk, respectively the yield stress, consistency and power-law index of fluid k in the pumped sequence. Using this scale we define thefunctions τw(|∇aΨ|) and S, by using the closure expressions in Chapter 2, asoutlined below in §3.6.1.ii) The term S/|∇aΨ| is singular as |∇aΨ| → 0, which is the limit where theyield stress of the fluid is not exceeded at the walls of the channel and τw <τY is indeterminate. In the laminar displacement model of Bittleston et al.[24], Pelipenko and Frigaard [180] the vector S is defined explicitly to reflect75this yielding phenomenon, i.e.S =[raχ(|∇aΨ|)|∇aΨ| +raτYH|∇aΨ|]∇aΨ ⇔ |S|> raτYH (3.27)|∇aΨ| = 0 ⇔ |S| ≤ raτYH (3.28)To connect the model derivation here with that of Bittleston et al. [24], Pelipenkoand Frigaard [180], note that the function χ(|∇aΨ|) is simply:χ(|∇aΨ|) = τw(|∇aΨ|)− τYH , (3.29)where τw is defined implicitly (2.25). Furthermore, provided that |∇aΨ| > 0,we can write:|S|(|∇aΨ|) = raχ(|∇aΨ|)+ raτYH .The function χ(|∇aΨ|) represents the contribution to the modified pressuregradient that is surplus to that needed to yield the fluid locally. It is continuousand strictly monotone. As the flow transitions through regimes, from laminarthrough to turbulent, the gradient of χ is continuous within any flow regime(but discontinuous when the flow transitions between regimes). Recall thatχ(|∇aΨ|) also has a local dependency on (φ ,ξ , t) through the local geometryand fluid concentrations present. However, in general we may represent |S|graphically at any (φ ,ξ , t) as in Figure 3.2a.iii) The function χ(|∇aΨ|) increases strictly monotonically (as does τw(|∇aΨ|)).We can examine the limits of (2.25), both as Hw→ ∞ and as Hw→ He (yieldlimit). For the latter we find:Ren/(2−n)p ∼ [Hw−He]n+1 ⇒ χ ∼ |∇aΨ|n/(n+1);see Figure 3.3b. As Hw→ ∞ we find Ren/(2−n)p ∼Hw, i.e. χ ∼ |∇aΨ|n, reflect-ing the shear-thinning behaviour. These limiting behaviours agree with thosein Pelipenko and Frigaard [180], where the laminar model is analysed in moredepth. The difference here though is that the limit Hw→ ∞ is not physicallyattained in the laminar regime: we transition to turbulent flow.76iv) The function τw(|∇aΨ|) is found to increase monotonically in the turbulentregime. Considering He fixed (the rheology) and taking Hw → ∞, we findyY → 0, n′→ n and E→ 1. Thus, we find thatRep ∼ H1−n2w logHw ⇒ |∇aΨ| ∼√τw [logτw]1/(2−n) ,as τw → ∞. Thus, τw grows slightly less fast than |∇aΨ|2, which would bethe expectation in a fully rough turbulent regime, and the rheological depen-dency on n is minimal (in the exponent of the log term only), as would also beexpected. Thus, we see essentially parallel curves in Figure 3.3a at large Hw,independent of n.v) In deriving (3.24), we cross differentiate to eliminate pressure in favour ofstream function. Alternatively, we may derive the pressure equation directlyby reorganizing (3.21) to eliminate Ψ:0 = ∇a ·[r2a|∇aΨ|(|S|)|S| (∇a p+bp)](3.30)bp =ρ−1Fr2(−sinβ sinpiφ ,cosβ ) (3.31)|S| =∣∣∣∣(−ra ∂ p¯∂ξ − ra(ρ−1)cosβFr2 , ∂ p¯∂φ − ra(ρ−1)sinβ sinpiφFr2)∣∣∣∣ .(3.32)The function |∇aΨ|(|S|) is qualitatively illustrated in Figure 3.2b.3.4 Mass transportWe now turn to transport of the different fluids along the annulus, assuming thatat each time the elliptic problem (3.24) can be solved to give the gap-averagedvelocity field. This suggests a similar model reduction will be appropriate for thedifferent fluids and now proceed to derive this.The concentrations of each individual fluid component ck are modeled by an77a)𝐒𝛻𝑎Ψ𝑟𝑎𝜏𝑌𝐻𝑟𝑎𝜏𝑌𝐻𝛻𝑎Ψ + 𝑟𝑎 0𝛻𝑎Ψ𝜒 𝑠 𝑑𝑠b)𝑟𝑎𝜒 𝛻𝑎Ψ𝑟𝑎𝜏𝑌𝐻𝐒𝛻𝑎ΨFigure 3.2: a) |S|(|∇aΨ|); b) |∇aΨ|(|S|)advection-diffusion equation:∂ck∂ tˆ+1rˆ∂∂ rˆ[rˆuˆck]+1rˆ∂∂θ[vˆck]+∂∂ ξˆ[wˆck] = ∇ˆ · [Dˆk,m∇ˆck], (3.33)where ∑Kk=1 ck = 1 and Dˆk,m represents the molecular diffusivity of species k withinthe mixture. For the turbulent flow, we apply the usual Reynolds decompositionand introduce the closure:− uˆ′c′k = Dˆt∇ˆc¯k. (3.34)where Dˆt is the turbulent diffusivity of species k (assumed the same for eachspecies). Equation (3.33) becomes:∂ c¯k∂ tˆ+1rˆ∂∂ rˆ[rˆ ¯ˆuc¯k]+1rˆ∂∂θ[ ¯ˆvc¯k]+∂∂ ξˆ[ ¯ˆwc¯k] = ∇ˆ · [(Dˆt + Dˆk,m)∇ˆck]. (3.35)We now apply the scaling introduced earlier. We anticipate that the main dif-fusive term is Dˆt which will scale with the local gap width and friction velocity,but here introduce a global scale: δ0rˆa,0 ˆ¯W for the purposes of simplification. The78scaled system is:δ0pi(∂ c¯k∂ t+∂∂y[u¯c¯k]+1ra∂∂φ[v¯c¯k]+∂∂ξ[w¯c¯k])=∂∂y[Dk∂ c¯k∂y](3.36)+(δ0pi)2∇a · [Dk∇ac¯k]with Dk = Dt + 1/Pek and Pek = δ0rˆa,0 ˆ¯W/Dˆk,m, Dt = Dˆt/[δ0rˆa,0 ˆ¯W ]. On the left-hand side of (3.36) we have neglected terms that come from approximating ge-ometry/curvature effects, which are O(δ/pi) smaller than those considered. Onthe right-hand side we have also neglected terms of O(δ/pi) that come from theapproximating the radial diffusion term.Eliminating these curvature/geometry terms only, while retaining the otherterms may appear questionable as a perturbation procedure. However, note thatthe intention is to include the leading order effects of all physically relevant trans-port processes. It is evident that (3.36) represents a singular perturbation, in whichthe leading order concentration will be constant across the annular gap (see below).We thus retain the first order advective component on the left-hand side as this isresponsible both for advection of the mean concentration and dispersive effects,within the plane of the narrow annulus. We also wish to evaluate the balance be-tween turbulent diffusion and dispersion within the (φ ,ξ )-plane and consequentlyretain the diffusive terms in (φ ,ξ ) directions. Lastly, although curvature may effectcross-gap diffusion the leading order effect is included in the order 1 terms.3.4.1 Dispersion effectsWe look for a perturbation approximation to (3.36) in terms of the parameterδ0/pi  1. The velocity is assumed to have the following formu¯ = ¯¯u0+ u˜0+(δ0pi)u˜1+(δ0pi)2u˜2+ .... (3.37)0 =∫ H−Hu˜ j dy, j = 0,1,2... (3.38)79i.e. the velocity (which we recall is anyway ensemble-averaged) is decomposedinto a gap-averaged component ¯¯u0 and successive components at each order thatdescribe the y-variation. We assume that ¯¯u0 =(0, ¯¯v, ¯¯w), as defined in (3.18), and thatu˜0 = 0, i.e. the radial component of velocity only arises at the first order. Similarlywe write:c¯k = c¯k,0+(δ0pi)c¯k,1+(δ0pi)2c¯k,2+ .... (3.39)0 =∫ H−Hc¯k, j dy, j = 1,2... (3.40)Here following Zhang and Frigaard [255], Taghavi and Frigaard [220], we usemethod of multiple timescales suggested by Fowler [78] in which we assume thatthe variables respond on both the advective time t and on a slower timescale T =(δ0/pi) t, where we expect diffusive effects to come into play. Note that no-slipboundary conditions are satisfied by u¯ at the walls, where also the diffusive fluxesof c¯k are zero (Neumann condition).Substituting these expressions into (3.36) we find that at leading order c¯k,0 isindependent of y, as we have already assumed previously in analysing the momen-tum balance. From (3.40) we interpret c¯k,0 as the gap-averaged mean concentration.The first order equations are as follows:(∂ c¯k,0∂ t+∂∂y[u˜0c¯k,0]+∇a · [c¯k,0( ¯¯v+ v˜0, ¯¯w+ w˜0)])=∂∂y[Dk,0∂ c¯k,1∂y](3.41)Integrating across the channel shows that:∂ c¯k,0∂ t+( ¯¯v, ¯¯w) ·∇ac¯k,0 = 0. (3.42)and on substituting back into (3.41) and using the continuity equation:∂∂y[Dk,0∂ c¯k,1∂y]= (v˜0, w˜0) ·∇ac¯k,0, ⇒ ∂ c¯k,1∂y =∇ac¯k,0 ·(∫ y−H(v˜0, w˜0) dy′)Dk,0.(3.43)The expression (3.42) says that on the advective timescale t the leading order con-80centration is simply advected along the streamlines.To understand evolution on the slow timescale T we move to a frame of ref-erence moving along the gap-averaged streamlines. The coordinates (s,n) alignlocally with the directions es and en: tangential and normal to the streamlines,respectively. The gap-averaged speed in the direction of the streamline is denoted¯¯s0 =√¯¯v20+ ¯¯w20 = |∇aΨ|/(2H).To further integrate (3.43), recall that in analyzing the momentum balance in theprevious section we have shown that the leading order turbulent velocity is in thedirection of the pressure gradient. Therefore, the two vectors ( ¯¯v0, ¯¯w0) and (v˜0, w˜0)are parallel and in the direction of es, i.e.( ¯¯v, ¯¯w) ·∇ac¯k,0 = ¯¯s0 ∂ c¯k,0∂ s and (v˜0, w˜0) ·∇ac¯k,0 = s˜0∂ c¯k,0∂ s,where s˜0(y) gives the variation in the mean speed across the narrow gap. Substi-tuting into (3.41) and integrating the first order terms, we get:c¯k,1 = c¯k,1(−H)+ ∂ c¯0∂ s∫ y−H1Dk,0∫ y′−Hs˜0dy′′dy′ = c¯k,1(−H)+ ∂ c¯0∂ s k(y) (3.44)in whichk(y) =∫ y−H1Dk,0∫ y′−Hs˜0dy′′dy′.By construction, c¯k,1(−H) =−k¯ ∂ c¯k,0∂ s where k¯ is the average of k(y) across the gap.Thus, c¯k,1 is expressed in terms of∂ c¯k,0∂ s and the distribution of velocity across thegap. Before proceeding, note that the leading order velocity is based on the narrowchannel approximation, which leads to an even function: s˜0(y) is symmetric abouty = 0. Since also0 =∫ H−Hs˜0(y)dy = 2∫ H0s˜0(y)dy = 2∫ 0−Hs˜0(y)dy,we may write:k(y) =∫ y−H1Dk,0∫ y′0s˜0dy′′dy′,81and note that the integral of s˜0(y) will be an odd function. The function Dk,0(y)is also defined by the leading order velocity and can be assumed to be an evenfunction. The integrand above is therefore also an odd function. From this we mayconclude that k(y) is an even function and that k(−H) = k(H) = 0. Similarly, c¯k,1is an even function of y.At the next order of asymptotic expansion, in the moving frame of referencewe collect terms of O((δ0/pi)2):∂ c¯k,0∂T= −∇a · [c¯k,1(v˜0, w˜0)]+∇a.(Dk,0∇ac¯k,0)−∂ c¯k,1∂ t− ¯¯s0 ∂ c¯k,1∂ s −∂∂y(u˜1c¯k,0)−∇a · [c¯k,0(v˜1, w˜1)]+∂∂y(Dk,0∂ c¯k,2∂y)+∂∂y(Dk,1∂ c¯k,1∂y)+∂∂y(Dk,2∂ c¯k,0∂y). (3.45)We integrate (3.45) across the gap width. The terms in the first line of (3.45) do notvanish. In the second line, the first two terms are linear in quantities that integrateto zero. For the last two terms we use the incompressibility of u˜1:∂∂y(u˜1c¯k,0)+∇a · [c¯k,0(v˜1, w˜1)] = u˜1 ·∇c¯k,0 = ∇ac¯k,0 · (v˜1, w˜1).These terms now integrate to zero across the gap. In the 3rd line the terms vanishas there is no flux through the walls.On substituting from (3.44) we see that the slow time evolution of c¯k,0 in theframe of reference moving along the streamline is governed by:2H∂ c¯k,0∂T= −∫ H−H∂∂ s[s˜0(k(y)− k¯)∂ c¯k,0∂ s]dy+∫ H−H∇a.(Dk,0∇ac¯k,0) dy.(3.46)The first term on the right-hand side of (3.46) is the Taylor dispersion term, whichwe write as follows:82∫ H−H∂∂ s[s˜0(k− k¯)∂ c¯k,0∂ s]dy =∂∂ s[(∫ H−Hs˜0k dy)∂ c¯k,0∂ s]+2∂H∂ s∂ c¯k,0∂ s[s˜0(H)k¯]=− ∂∂ s[2HDT∂ c¯k,0∂ s]+2∂H∂ s∂ c¯k,0∂ s[s˜0(H)k¯](3.47)whereDT =12H∫ H−H1Dk,0(y)(∫ y−Hs˜0(y′)dy′)2dy. (3.48)The second term on the right-hand side of (3.46) reflects the average effect of thediffusivity Dk,0. At y = ±H the turbulent term vanishes, leaving only a negligiblemolecular contribution 1/Pe 1. Therefore, we can write:∫ H−H∇a · (Dk,0∇ac¯k,0) dy = ∇a ·∫ H−H(Dk,0∇ac¯k,0) dy−∇aH ·∇ac¯k,0[Dk,0(H)+Dk,0(−H)]= ∇a · [2HD¯∇ac¯k,0]+O(1/Pe)whereD¯ =12H∫ H−HDk,0 dy.Combining the above expressions with (3.42), reverting back to the singletimescale t and transforming back into the fixed frame of reference, we arrive atthe following equation for the evolution of the leading order concentration:∂ c¯k,0∂ t= −( ¯¯v0, ¯¯w0) ·∇ac¯k,0+ δ0pi(12Hes ·∇a[(2HDT )es ·∇ac¯k,0] (3.49)− k¯s˜0(H)H(es ·∇aH)(es ·∇ac¯k,0)+ 12H∇a · [2HD¯∇ac¯k,0])Equation (3.49) describes how the leading order concentrations of fluid k change.The right-hand side has four terms. Firstly, we have advection with the mean flow.Secondly we have a pure Taylor-dispersion term, which we can see takes the formof an anisotropic diffusivity, i.e. only along the streamlines (in direction es). The83third term results from variations in width of the annulus. The fourth term onthe right-hand side of gives the averaged effect of the diffusivity. In §2.4.4 wehave modelled the velocity profiles for the flow along a uniform plane channel,i.e. ¯¯s0+ s˜0(y) and have used this, together with estimates of the turbulent diffusivity,to compute the Taylor dispersivity DT . In general it is found that D¯DT . In highlyturbulent flows, DT decreases, but still remains two orders of magnitude larger thanD¯. Thus, it is the first two dispersive terms that are of most interest.We may extract the variation of H from the first term and rewrite (3.49) as:∂ c¯k,0∂ t= −( ¯¯v0, ¯¯w0) ·∇ac¯k,0+ δ0pi(es ·∇a[DT es ·∇ac¯k,0]+ 12H∇a · [2HD¯∇ac¯k,0])+δ0pi(es ·∇aH)(es ·∇ac¯k,0)DT −D∗TH, (3.50)whereD∗T = k¯s˜0(H) =−s˜0(H)2H∫ H−HyDk,0(y)∫ y−Hs˜0(y′)dy′dy. (3.51)We see that nominally D∗T has the same size as DT . Note that the last term in(3.50) will vanish when the annulus is concentric. Also we should note that inmost displacement flows through long annuli the streamlines are pseudo-parallelto the ξ -axis for most of the annulus and H generally varies slowly with ξ , sothis last term is mostly insignificant. However, in interfacial regions between twodisplacing fluids we often see azimuthal velocities of a similar size to the axialvelocity. These regions are of course also where the main Taylor dispersion termis active. Local closure expressions for DT , D∗T and D¯, in terms of the expressionsderived in §2.4.4 need to be rescaled as shown in § Boundary conditionsThe elliptic second order equation (3.24) determines the stream function, and hencegap-averaged velocity, at each time. It requires suitable boundary conditions inorder to be solved. The annular domain has been reduced via the scaling to rect-angular domain Ω, representing the unwrapped gap-averaged annulus. At eachtimestep it is necessary to specify suitable boundary conditions on ∂Ω in order to84solve (3.24). We suppose that ∂Ω can be split into ∂ΩΨ and ∂ΩS:Ψ = Ψb, (φ ,ξ ) ∈ ∂ΩΨ, (3.52)S ·n = f , (φ ,ξ ) ∈ ∂ΩS, (3.53)where Ψb and f are specified boundary data. The conditions are explained below.First, in the azimuthal direction if the geometry is fixed so that the narrow sideof the annulus is the lowest side, then one may simplify the model by assumingthat the flow is symmetric in φ along both wide and narrow sides:Ψ(0,ξ , t) = 0, (3.54)Ψ(1,ξ , t) = 2Q(t), (3.55)where Q(t) is the dimensionless flow rate (and Ω = [0,1]× [0,Z], Z being thedimensionless well depth). Note that by rearranging (3.23), S can also be expressedin terms of the pressure gradients as:S =(−ra ∂ p¯∂ξ −ra(ρ−1)cosβFr2,∂ p¯∂φ− ra(ρ−1)sinβ sinpiφFr2,).Thus, the symmetry condition ¯¯v = 0, which gives Sξ = 0, also implies that∂ p¯∂φ= 0.On the other hand, suppose we consider a full annulus, with no symmetry im-posed at wide and narrow sides (Ω= [0,2]× [0,Z]). Then an alternate to (3.54) &(3.55) would be:Ψ(φ +2,ξ , t) =Ψ(φ ,ξ , t)+4Q(t), (3.56)fixing only the total flow rate. In using (3.56), if one wanted to work with thepressure, the pressure would be 2-periodic in φ .Secondly for the end conditions, following Pelipenko and Frigaard [179] wemight expect to impose Dirichlet conditions at the inflow, ξ = 0:Ψ(φ ,0, t) = Ψ0(φ , t), (3.57)85e.g. a uniform inflow velocity can be specified, reflecting the fact of some kindof entry/development region, following local mixing as the fluids enter the annu-lus. Similarly, at the outflow (ξ = Z), it might be appropriate to assume a fullydeveloped flow profile:Ψ(φ ,Z, t) = ΨZ(φ , t). (3.58)The fully developed profile above of course needs to be specified. The naturalway to do this is by neglecting ξ -derivatives in (3.24), due to the length of theannulus, which leads to∂∂φ[Sφ +bφ ] = 0,which in turn implies that ∂ p¯/∂ξ is independent of φ . The fully developed profilewould be found computationally by decreasing ∂ p¯/∂ξ , which increases the axialvelocity at each φ (hence the flowrate), until the net imposed flowrate throughthe exit section is attained. Although this appears a convoluted procedure, it isstraightforward numerically.Alternatively to (3.57) & (3.58), one might impose Neumann conditions∂Ψ∂ξ= 0at each end of the annulus, e.g. as in Pelipenko and Frigaard [181]. Note that theNeumann condition, implies that Sξ = 0, which specifies S · n on the boundarywith outward normal n, i.e. a condition of type (3.53). Depending on the densitygradients at inflow and outflow we might choose to specify S · n in terms of thebuoyancy b ·n.Boundary conditions for (3.50) are generally that c¯k,0 is specified at the inflowto the annulus, either from a pump schedule or from coupling with a predictivemodel of the downwards displacement flow in the casing. Along the sides of theannulus, either a symmetry condition is imposed, i.e.∂ c¯k,0∂φ= 0,or potentially a periodicity condition in the case that the full annulus is resolved86and no symmetry is assumed. At the outflow we generally assume that∂ c¯k,0∂ξ= 0,although in cases of large density differences we may have counter-current flowswith denser fluids entering the annulus and need to specify accordingly.3.6 One-dimensional closuresTo close the model derived above, we need to introduce closure relations to evaluatewall shear stress as well as diffusion and dispersion coefficients.3.6.1 Hydraulic closureIn Chapter 2 we solved (3.21) to give an expression for the leading order turbu-lent mean velocity for a Herschel-Bulkley fluid, based on the phenomenologicalapproach of Dodge-Metzner-Reed [58, 158]. This analysis provides a closure ex-pression for τw = τw(|∇aΨ|;φ ,ξ , t). The local dependency (φ ,ξ ) explicitly reflectsthe local geometric variables and the fluid rheology and density are represented im-plicitly with dependency (φ ,ξ , t) as the fluids are displaced.Following the notation introduced in Chapter 2, the dimensional wall shearstress isτˆw = τˆ0τwand the dimensional mean speed (averaged across the local gap) and the dimen-sional local annular gap are given as:Wˆ0 = ˆ¯W12H|∇aΨ|, 2Hˆ = 2Hδ0rˆa,0recall that ˆ¯W is the velocity scale for the entire annulus. We also assume that,according to the concentrations of the fluids at (φ ,ξ , t) we may construct the di-mensional local mixture density ρˆ and the rheological parameters: τˆY , κˆ , n. Wecan the proceed to find Rep, He and Hw as defined by (2.20) and (2.21).The procedure in Chapter 2 gives a detailed description of the mapping from87a)Hw / =w102 104 106Re1=(2!n)p/jra*j100102104106n = 0:2n = 1b)(Hw !He) / @100 105Re1=(2!n)p/jra*j10-5100n = 0:2 n = 1Figure 3.3: The closure Hw(Rep), showing asymptotic behaviour: a) Hw →∞; b) Hw → He. The closure is plotted for He = 100 and n =0.2, 0.4, 0.6, 0.8, 1: green - laminar; red - transitional; black - tur-bulent.Rep to Hw and vice versa, which is parameterized by (n,He). Observe that Hw ∝ τwand Rep ∝ |∇aΨ|2−n, so the mapping Hw↔Rep defines our closure relation. Figure3.3 shows an example of this mapping for different n at He = 100. The sensitivityto He is not extreme.3.6.2 Dispersion and diffusion closuresSimilar to §3.6.1, we can find the averaged turbulent diffusivity and Taylor disper-sion coefficients (D¯,DT ,D∗T ) using the method introduced in Chapter 2. We firstconstruct the dimensional parameters; e.g.DˆT = DTδ0rˆa ˆ¯W0and then rescale it using the scaling defined in Chapter 2. We eventually find:DT =12|∇aΨ|DT,1D (3.59)where DT,1D is the dispersion coefficient obtained assuming a locally one-dimensionalchannel flow (see §2.4.4). Similar relations can be derived for D¯ and D∗T , i.e. mul-880 2 4 6 8 1010−310−210−1100101102(Hw −Hw,2)/(Hw,2 −Hw,1)D¯t,1D,DT,1D,D∗ T,1DnnnFigure 3.4: Variation of D¯1D(blue lines), DT,1D(black lines) and D∗T,1D(redlines) with wall shear stress for n = 0.2,0.4,0.6,0.8 and 1.tiplying the one-dimensional results from §2.4.4 by |∇aΨ|/2.It is worthwhile to compare the values of D¯1D, DT,1D and D∗T,1D (or equivalentlyD¯, DT and D∗T ). Figure 3.4 plots turbulent diffusivity and dispersion coefficients asa function of wall shear stress for fully turbulent flows. Hw,1 and Hw,2 are defined in§2.2. As Figure 3.4 shows, D¯ is 2 to 3 orders of magnitude smaller than DT . Thisis a typical feature of turbulent Taylor dispersion [230]. In addition, D∗T is almostalways larger than DT . This is interesting, although the results shown later havenot revealed where these terms become important.3.7 Model summary and conclusionsWe derived a practical leading-order model for simulating 2D turbulent and mixedregime displacement flows, as encountered in the process of primary cementing.89Leading order annular displacement flows are governed by the coupled system:∇a · [S+b] = 0 (3.60)12H∇aΨ = ( ¯¯w0,− ¯¯v0) (3.61)∂ c¯k,0∂ t+( ¯¯v0, ¯¯w0) ·∇ac¯k,0 = δ0pi(es ·∇a[DT es ·∇ac¯k,0]+ 12H∇a · [2HD¯∇ac¯k,0])+δ0pi(es ·∇aH)(es ·∇ac¯k,0)DT −D∗TH(3.62)with associated boundary conditions (§3.5) and the one-dimensional closures (§3.6).Compared to the laminar model of Bittleston et al. [24], the turbulent model hastwo main differences: I) The treatment of the momentum equations differs. Unlikethe laminar flow, the turbulent Reynolds stress components all have similar size.The leading-order flow is a turbulent shear flow, but only due to differential scalingof the lengths. At the next order of approximation many more unknowns enter themodel. Having derived the turbulent shear flow the analysis is similar to Bittlestonet al. [24] in developing the field equations for the stream function (or pressure), butwith the closure expressions coming from turbulent-flow hydraulics. II) The sec-ond principal difference comes in the treatment of the fluid concentrations. Firstwe note that the assumed decoupling of averaged concentration from velocity, inthe advective part of (3.62), is more valid here than for the laminar model in Bit-tleston et al. [24]. Second, following our analysis, we find that turbulent flows aregoverned by complex diffusive and dispersive transport processes (absent in thelaminar flows). The largest effect is Taylor dispersion, which diffuses only alongthe streamlines. The gap-averaged turbulent diffusivity acts isotropically but ap-pears to be relevant only in sufficiently long wells. We have also derived terms thatdescribe the influence of annulus gap variations on dispersion, again in the stream-wise direction. For the laminar displacement flow, only molecular diffusion shouldbe present, which results in smaller effects than those here.90Chapter 4Computational AnalysisIn the previous chapter, we showed that the leading order annular displacementflows (across mixed flow regimes) is governed by this coupled system of partialdifferential equations:∇a · [S+b] = 0 (4.1a)12H∇aΨ= ( ¯¯w0,− ¯¯v0) (4.1b)∂ c¯k,0∂ t+( ¯¯v0, ¯¯w0) ·∇ac¯k,0 = δ0pi(es ·∇a[DT es ·∇ac¯k,0]+ 12H∇a · [2HD¯∇ac¯k,0])+δ0pi(es ·∇aH)(es ·∇ac¯k,0)DT −D∗TH(4.1c)with associated boundary conditions (§3.5) and the one-dimensional closures (§3.6).This system of equations is to be solved numerically. In this chapter, we analyzethis system of equations from a computational point of view, discuss the commonnumerical challenges and provide robust algorithms to accurately solve them.4.1 Discretization and variable storageFirst of all, it is important to discuss the choice of discretization. It is well-knownthat the choice of discretization scheme plays a key role in the accuracy and ro-bustness of any numerical algorithm. In particular, features such a checkerboard91pressure field are possible consequences of poor discretization choices. Our choiceof discretization is primarily motivated by the fact that the velocity field is com-puted via differentiating stream functions.We discretize the system of (4.1) on a staggered mesh. The domain (φ ,ξ ) ∈[0,φend ]× [0,ξend ] is divided into nφ × nξ mesh cells. Here φend can take valueof 1 or 2, depending on whether the half or full annulus is modeled. ξend is thedimensionless length of cemented interval. Vertices of the mesh cells are at:(φi,ξ j) = (i∆φ , j∆ξ ),where the mesh cell sizes (∆φ ,∆ξ ) are given by∆φ =1nφand ∆ξ =ξendnξ;i.e. we have uniform structured mesh cells. Notice here, in the case of a fullannulus, the number of mesh cells is 2nφnξ . The stream function is approximatedat the vertices:Ψi, j ≈Ψ(φi,ξ j, t)Naturally, this leads us to store the velocity fields at the edge centres:wi+1/2, j ≈ w(φi+1/2,ξ j, t), and vi, j+1/2 ≈ v(φi,ξ j+1/2, t).This choice enables central differencing to give a second order approximation to themean velocity components. Besides, the velocity components are readily availablefor finite-volume-type discretization of ck at cell centre:ck,i+1/2, j+1/2 ≈ c(k,φi+1/2,ξ j+1/2, t).In reference to Figure 4.1, Ψ,v,w,ck are stored at circles, diamonds, squares andtriangles, respectively. Note, had we chosen to work with the pressure formulation,instead of the stream function, we would also represent p at the cell centres.92Figure 4.1: Staggered mesh4.2 Solving for Ψ4.2.1 Variational inequality and weak solutionEquation (4.1a) is an elliptic second order partial differential equation, in whichtime evolution enters indirectly only via the fluid concentrations (see §3.4) or viaflow rate changes. Our ultimate goal is to use some form of augmented Lagrangianalgorithms, which are suitable for computing the flow of viscoplastic fluids andare guaranteed to converge. Therefore, we begin our analysis by establishing avariational for of (4.1a) and demonstrating certain features of the problem.Consider a rectangle Ω with boundary ∂ΩΨ⋃∂ΩS, on which boundary condi-tions (3.52) & (3.53), respectively are satisfied. We regard any suitably smooth Ψ˜as an admissible stream function provided that (3.52) is satisfied. Similarly, S˜ willbe regarded as admissible provided that:∇a · [S˜+b] = 0,and that (3.53) is satisfied. The following statements are easily proven usingGreen’s theorem in the plane.• For any admissible Ψ˜ & S˜:0 =∫ΩΨ˜∇a ·b−∇aΨ˜ · S˜ dΩ+∫∂ΩΨΨbS˜ ·n ds+∫∂ΩSΨ˜ f ds. (4.2)93• For Ψ & S that solve (3.24) with boundary conditions (3.52) & (3.53):0 =∫ΩΨ∇a ·b−∇aΨ ·S dΩ+∫∂ΩΨΨbS ·n ds+∫∂ΩSΨ f ds. (4.3)• For the solution Ψ & S, and any other admissible Ψ˜:0 =∫Ω[Ψ˜−Ψ]∇a ·b− [∇aΨ˜−∇aΨ] ·S dΩ+∫∂ΩS[Ψ˜−Ψ] f ds. (4.4)• For the solution Ψ & S, and any other admissible S˜:∫Ω∇aΨ · [S˜−S] dΩ=∫∂ΩΨΨb[S˜−S] ·n ds. (4.5)Now we consider the closure relationship defining S. Provided that |∇aΨ|> 0or equivalently |S|> raτY/H, we can write this as:|S|(|∇aΨ|) = raH τw(|∇aΨ|) = raχ(|∇aΨ|)+raτYH. (4.6)The function χ(|∇aΨ|) represents the contribution to the modified pressure gra-dient that is surplus to that needed to yield the fluid locally. It is continuous andstrictly monotone. As the flow transitions through regimes, from laminar throughto turbulent, the gradient of χ is continuous within any flow regime (but discontin-uous when the flow transitions between regimes). Recall that χ(|∇aΨ|) also has alocal dependency on (φ ,ξ , t) through the local geometry and fluid concentrationspresent. However, in general we may represent |S| graphically (at any (φ ,ξ , t)) asin Figure 4.2a.4.2.2 Stream function and pressure potential functionalsThe stream function potential functional J(Ψ˜) is defined as:J(Ψ˜) =∫Ω∫ |∇aΨ˜|0|S|(x) dx− Ψ˜∇a ·b dΩ−∫∂ΩSΨ˜ f ds, (4.7)which has the following property.94a)𝐒𝛻𝑎Ψ𝑟𝑎𝜏𝑌𝐻𝑟𝑎𝜏𝑌𝐻𝛻𝑎Ψ + 𝑟𝑎 0𝛻𝑎Ψ𝜒 𝑠 𝑑𝑠b)𝑟𝑎𝜒 𝛻𝑎Ψ𝑟𝑎𝜏𝑌𝐻𝐒𝛻𝑎ΨFigure 4.2: a) |S|(|∇aΨ|); b) |∇aΨ|(|S|). The shaded areas contribute to thedissipation and potential functions. The shaded areas sum to |S||∇aΨ|,used to establish Lemma 4.2.3.Lemma 4.2.1. The solution Ψ minimizes J(Ψ˜) over all admissible Ψ˜.Proof. We look at:J(Ψ˜)− J(Ψ) =∫Ω(∫ |∇aΨ˜||∇aΨ||S|(x) dx)− (Ψ˜−Ψ)∇a ·b dΩ−∫∂ΩS(Ψ˜−Ψ) f ds=∫Ω∫ |∇aΨ˜||∇aΨ||S|(x) dx dΩ−∫Ω[∇aΨ˜−∇aΨ] ·S(|∇aΨ|) dΩ≥∫Ω∫ |∇aΨ˜||∇aΨ||S|(x) dx dΩ−∫Ω(|∇aΨ˜|− |∇aΨ|)|S|(|∇aΨ|) dΩ=∫Ω∫ |∇aΨ˜||∇aΨ|(|S|(x)−|S|(|∇aΨ|)) dx≥ 0.We have used (4.4) and then the Cauchy-Schwarz inequality above. In this lastexpression note that |S|(x)> |S|(|∇aΨ|) whenever x > |∇aΨ| due to monotonicity.Thus the sign of the integrand changes according to the limits and the integral isalways positive. 2The minimization of J(Ψ˜) can also be expressed as a variational inequality,which is the basis of our augmented Lagrangian method; see §4.2.4. Consideringnow Figure 4.2b, we can define the function |∇aΨ|(|S|) by effectively inverting95|S|(|∇aΨ|), as illustrated, i.e.|∇aΨ|(|S|) = |S|−1(|∇aΨ|), |S|> raτYH ,0, |S| ≤ raτYH .We now define the pressure potential function K(S˜) for any admissible S˜ as follows.K(S˜) =−∫Ω∫ |S˜|raτYH|∇aΨ|(y) dy dΩ+∫∂ΩΨΨbS˜ ·n ds. (4.8)Analogous to Lemma 4.2.1 we have the following.Lemma 4.2.2. The solution S maximizes K(S˜) over all admissible S˜.Proof. We look at:K(S)−K(S˜) =∫Ω∫ |S˜||S||∇aΨ|(y) dy dΩ+∫∂ΩΨΨb[S− S˜] ·n ds=∫Ω∫ |S˜||S||∇aΨ|(y) dy dΩ−∫Ω∇aΨ · [S˜−S] dΩ≥∫Ω∫ |S˜||S||∇aΨ|(y) dy dΩ−∫Ω|∇aΨ|(|S˜|− |S|) dΩ=∫Ω∫ |S˜||S|(|∇aΨ|(y)−|∇aΨ|) dy dΩ≥ 0.Here we have used (4.5) and then the Cauchy-Schwarz inequality. In the last ex-pression note that if |S˜|> |S| then |∇aΨ|(y)> |∇aΨ| due to monotonicity; similarlywhen |S˜|< |S|. Thus the sign of the integrand changes according to the limits andthe integral is always positive. 2Finally, since the shaded areas in Figures. 4.2a & b, sum to give |S||∇aΨ|, wehave: ∫Ω|S||∇aΨ| dΩ=∫Ω∫ |∇aΨ|0|S|(x) dx dΩ+∫Ω∫ |S|raτYH|∇aΨ|(y) dy dΩ,which can be combined with (4.3). In combination with the minimization andmaximization principles above we have the following minimax principle:96Lemma 4.2.3. The solution pair (S,Ψ) satisfy:K(S˜)≤ K(S) = J(Ψ)≤ J(Ψ˜),for all admissible S˜ and Ψ˜.In the porous media context, similar variational principles are used to describenonlinear filtration, see e.g. Barenblatt et al. [15], Spena and Vacca [217]. The first(integral) terms in both J(·) and K(·) are referred to as dissipation potentials. Ina porous media flow, one is often more concerned with determining the pressurefield and a stream function formulation is restrictive in only applying to 2D flows.Thus, typically K(·) is referred to as the primal potential and J(·) as dual potential.Here however, we treat Lemma 4.2.1 as the primal principle as it leads to a uniquestream function (see below). Note that the terminology dissipation results from(4.3) which is essentially a mechanical energy balance, equating the dissipationwithin the system to the work done by buoyancy forces and by the boundary terms.4.2.3 Existence and uniquenessLemma 4.2.1 is the basis of an existence and uniqueness result. Firstly, note thatJ(Ψ˜) can be split as follows:J(Ψ˜) =∫Ωra∫ |∇aΨ˜|0χ(x) dx dΩ+∫ΩraτYH|∇aΨ˜| dΩ−∫ΩΨ˜∇a ·b dΩ−∫∂ΩSΨ˜ f ds,= J0(Ψ˜)+ J1(Ψ˜)−L(Ψ˜). (4.9)The functional J0 is strictly convex as the integrand has second derivative equalto the derivative of χ , which is a strictly monotone function. The functional J1containing the yield stress is convex, but not strictly. Finally, L denotes the linearparts that are in Ψ˜.This problem structure is in a format where standard results may be applied,(e.g. Theorem 2.1 in chapter 5 of Glowinski [89]), to guarantees existence of aunique weak solution. The relevant function space is determined by the behaviour97of J0 as ||Ψ˜|| → ∞. In the fully turbulent regime |∇aΨ| ∼ √τw [logτw]1/(2−n) as|∇aΨ| → ∞ , and therefore alsoχ [logχ]2/(2−n) ∼ |∇aΨ|2.This suggests χ & O(|∇aΨ|2−ε) for any small ε > 0 as |∇aΨ| → ∞, i.e. the logterm is less significant than any power.Proceeding now as in Pelipenko and Frigaard [181] we can infer that Ψ ∈W 1,3−ε(Ω), with further details specific to the boundary conditions to be con-sidered. It is interesting to compare with the results for the purely laminar caseconsidered in Pelipenko and Frigaard [181], where the growth of χ using only thelaminar closure resulted in Ψ ∈W 1,1+nmin(Ω). It appears that the turbulent closureresults in a smoother weak solution and a function space largely independent of therheology.4.2.4 Computational algorithmEquation (4.1a) can be solved using Newton’s-like iterative algorithms; see Tardyand Bittleston [226], Tardy [225]. However these algorithms are not guaranteedto converged; i.e a poor initial guess may result in divergence. Therefore, insteadwe opt to derive and implement an augmented Lagrangian (AL) algorithm. Theadvantages of AL algorithms is two-fold: i) By the convexity of (4.1a), as shownabove, AL algorithm is guaranteed to converge. ii) It is common that inside aneccentric annulus, flow regime varies from turbulent in the wider side to laminar inthe narrow side. More crucially, we may even have mud stuck on the narrow side(= unyielded viscoplastic fluid). AL algorithms can fully resolve the unyieldedregions. This is superior to the e.g. a typical viscosity regularisation techniquescommon in other models, e.g. in Tardy and Bittleston [226], Tardy [225].An alternative simpler approach for solving (4.1a) is the so called “slice model”.In the slice model, we assume the gradients in the axial direction are much smallerthan those in azimuthal direction which then simplifies (4.1a) to a 1D equation.This approximate equation can be straightforwardly solved iteratively. We explainboth algorithms below:98Augmented Lagrangian algorithmWe expand S(∇Ψ) toS =[raχ |∇aΨ|)|∇aΨ| +raτYH|∇aΨ|]∇aΨ (4.10)where χ(∇Ψ) is given by (4.6):χ =τw (|∇aΨ|)− τYH. (4.11)The advantage of using χ instead of χW in the formulation is that the effect of yieldstress can be explicitly seen. We develop the algorithm for both notations.In §3.5, we discussed in length different choices of boundary conditions. Forsimplicity, we adopt the following boundary conditions: For half annulus simula-tions, the boundary conditions are:Ψ(0,ξ , t)= 0, Ψ(1,ξ , t)= 2Q(t), Ψ(φ ,0, t)=Ψ0(φ , t), and Ψ(φ ,1, t)=ΨZ(φ , t).(4.12)For full annulus simulations, the boundary conditions areΨ(φ+2,ξ , t)=Ψ(φ ,ξ , t)+4Q(t), Ψ(φ ,0, t)=Ψ0(φ , t), and Ψ(φ ,1, t)=ΨZ(φ , t)(4.13)We intend to homogenize the boundary conditions, therefore we decomposethe stream function Ψ into homogeneous (Ψ˜) and particular (Ψ∗) parts.Ψ=Ψ∗+ Ψ˜, (4.14)where the particular parts (Ψ∗) satisfies the boundary condition (4.12) or (4.13)and the homogeneous part (Ψ˜) satisfies the homogeneous Dirichlet boundary con-ditions:Ψ˜(0,ξ , t) = Ψ˜(1,ξ , t) = Ψ˜(φ ,0, t) = Ψ˜(φ ,1, t) = 0 half annulus (4.15)Ψ˜(φ ,0, t) = Ψ˜(φ ,1, t) = 0 full annulus (4.16)99The particular part can be constructed through different approaches. In [180],the slice model (§4.2.4) is employed at the bottom and the top of annulus (ξ =0,ξbh) and then the particular solution was constructed via a linear interpolation.Here we choose to use the full slice model to compute a particular solution.For Ψ˜, we use the minimisation problem introduced in Lemma 4.2.1. Lemma4.2.1 showed that the solution of (4.1a) can be found by minimising the functionalintroduced in 4.7. Substituting (4.14), we find that Ψ˜ minimises the followingfunctional:J(ψ˜) =∫Ω∫ |∇aΨ∗+∇aψ˜|0raτw(s)Hds− ψ˜∇.b dΩ (4.17)over all admissible homogeneous solutions ψ˜ . Notice that ∂Ωs = ∅ (an emptyspace) with either choice of boundary conditions. To find the minimiser, we relax|∇ψ˜| → q and augment the functional J(ψ˜) with a Lagrange multiplier term∫ΩT.(∇aψ˜−q) dΩas well as a stabilizer termrALG2∫Ω|∇aψ˜−q|2dΩ.This effectively replaces the minimisation problem with the following saddle pointproblem:minqmaxTJr(ψ˜,q,T). (4.18)where the augmented function Jr(ψ˜,q,T) is defined by:Jr(ψ˜,q,T) =∫Ω∫ |∇aΨ∗+q|0raτw(s)Hds+ Ψ˜∇.b dΩ+ (4.19)rALG2∫Ω|∇aψ˜−q|2dΩ+∫ΩT.(∇aψ˜−q) dΩ. (4.20)The following Uzawa-type algorithm iteratively solves this saddle point prob-lem:1001. Updating ψ˜: solverALG∇2aψ˜ = ∇a.(rALGq−T−b)with the boundary conditions given by (4.15).2. Updating q :q = θM|M| −∇aΨ∗whereM = T+ rALG∇aψ˜+ rALGq, m = |M| (4.21)andraHτw(θ)+θrALG = |M|This equation clearly has a solution between 0 < θ < |M|/rALG which canbe found iteratively.3. Updating T :T = T+ρALG (∇aψ˜−q)The parameters rALG and ρALG are two free parameters in this algorithm. Theconvergence speed depends on the choice of these parameter. However, there isno systematic way to choose the optimum values. A sufficient requirement forconvergence is given in Glowinski [89]:ρALG <1+√52rALG.Recently, more advanced algorithms have been developed where the value of thesefree parameters are automatically chosen; see Treskatis et al. [236, 237].Remark 1: Each iteration of the Uzawa algorithm above requires solving a Pois-son equation for ψ˜ . This may sound computationally expensive. However thediscretization matrix associated to this equation only depends on mesh sizes andmean radius which do not change during the simulation. This effectively meansthe discretization matrix needs to be computed and factorized only once.101Remark 2: The second step of Uzawa algorithm requires finding τw as a functionof θ(= |∇aΨ∗+∇aψ˜|). This operation is computationally heavy, because τw (orequivalently Hw) is only known implicitly as a function of θ (or equivalently Rep).To circumvent this problem, instead of having θ the independent variable, we canthink of τw as the independent variable. i.e. write:raHτw+θ (τw)rALG = |M|This formulation allows us to iterate on the value of τw to compute θ , which is amuch lighter operation because the hydraulic relation is known explicitly.Remark 3: With the above formulation, the effect of yield stress is hidden insidethe second step of the Uzawa algorithm where the hydraulic module is called. Itis not obvious under what condition the flow will be partially unyielded. However,notice that inside the hydraulic module, the first condition investigated is whetherHe < Hw. If this conditions is not satisfied, then the fluid is unyielded. To seethe role of yield stress more clearly, it can be straightforwardly shown that Ψ˜ alsominimises the following functional:JY (ψ˜) =∫Ωra∫ |∇aΨ∗+∇aψ˜|0χ(s)dsdΩ+∫ΩraτYH|∇aψ˜|dΩ−∫Ωψ˜.∇b dΩ. (4.22)A similar procedure can be repeated to turn this minimisation problem into a saddlepoint problem. The Uzawa algorithm above will solve this problem if we replacethe second step with:q = θM|M| −∇aΨ∗ if |M|> raτYH(0,0) if |M| ≤ raτYHThe condition |M|> raτYH is the yielding criteria.102Slice modelLet’s assume the gradients in axial direction are much smaller than those in az-imuthal direction; i.e.∂∂ξ ∂∂φ(4.23)then (4.1a) simplifies to:∂∂φ(raHτwsgn(∂Ψ∂φ)+bφ)= 0 (4.24)where sgn(x) is the sign function. Integrating with respect to φ and rearranging theequation, we obtain:τw =Hra|G−bφ | (4.25)where G is the modified pressure gradient which is constant in φ ; i.e. G = G(ξ ).The wall shear stress can be uniquely calculated with G, given the local rheologicaland geometrical parameters. The procedure is as follows: On the slice ξ = ξi,1. Guess some G and find τw.2. Compute Rep = Rep(n,He,Hw). To compute Hw, we need to dimensionl-ize τw with our global scaling and then non-dimensionlize it with the localscaling of our hydraulic module. More specifically,He =τYcτHw =τwcτ=Hracτ|G−bφ |Rep = Rep(n,He,Hw)where the rescaling factor cτ iscτ =κ22−npτ∗(ρˆn(2Hˆ)2n) 12−nIn the above expression τˆ∗ is the global scale for stress, and other parameters(ρˆ, Hˆ,n, κˆp) are evaluated locally. Notice that Rep is explicitly given as a103function of its parameters. Therefore, this step is fast and straightforward.3. Construct mean velocity and then integrate to find stream function. Morespecifically,Rep =61−nρˆWˆ 2−n0 (2Hˆ)nκˆp=61−nρˆWˆ ∗2−n(2Hˆ)nκˆpH2−nr2−na|∂Ψ∂φ|2−n = B|∂Ψ∂φ|2−n⇒|∂Ψ∂φ|=(RepB) 12−n⇒∂Ψ∂φ=(RepB) 12−nsgn(G−bφ ) (4.26)4. Enforce the flow rate by requiring the following residual function to vanish:R =Ψ(1,ξi)−Ψ(0,ξi)−2Q half annulusR =Ψ(2,ξi)−Ψ(0,ξi)−4Q full annulus5. If |R| > ε , change G accordingly. Otherwise, the algorithm is converged. εis the tolerance of iterations.4.3 Solving for ckEquation (4.1c) is an advection diffusion equation. Let us first rewrite (4.1c) in aconservative form. For the sake of simplicity, we drop the subscripts and bars andwrite:∂c∂ t+(v,w) ·∇ac = δ0pi(es ·∇a[DT es ·∇ac]+ 12H∇a · [2HD¯∇ac])+δ0pi(es ·∇aH)(es ·∇ac)DT −D∗TH(4.27)104We multiply the left hand side of (4.27) by an Hra and write:Hra(∂c∂ t+(v,w) ·∇ac)=∂ (raHc)∂ t+∇a · [raHc(v,w)]− c∇a · [raH(v,w)]=∂ (raHc)∂ t+∇a · [raHc(v,w)](4.28)Note that we invoked the continuity equation to simplify the above relation. Simi-larly on the right hand side, we can write:Hra (es ·∇a[DT es ·∇ac]) = 1s∇a · (DT∇a · [raHc(v,w)]es) (4.29)wheres =√v2+w2Unfortunately, the last term on the right hand side cannot be written in a conserva-tive form, so we have:∂ (raHc)∂ t+∇a · [raHc(v,w)] = δ0pi1s∇a · (DT∇a · [raHc(v,w)]es)+δ0pira∇a · [D¯H∇ac]+δ0pira(es ·∇aH)(es ·∇ac)(DT −D∗T )(4.30)There are a number of challenges in solving this equation:• The velocity field in the left hand side of (4.30) implicitly depends on c. Thismeans that simplistically speaking, the left hand side of (4.30) is a quasi-linear conservative law, which is known that can develop shocks. Therefore,any method employed need a shock-capturing feature.• Numerical diffusion is inevitably present here. This is particularly prob-lematic, because numerical diffusive will contaminate the natural diffusionpresent here due to turbulent diffusion and Taylor dispersion.105• With the choice of discretization explained earlier, implicit time discretiza-tion of (4.1c) leads to non-sparse matrices. On the other hand, explicit im-plementation requires inhibitively small time steps for stability.We analyze these challenges separately:4.3.1 AdvectionIn order to study the advection separately, for now we ignore the diffusive anddispersive terms on the right hand side of (4.30) and write:∂U∂ t+∇a.(F,G) = 0 (4.31)whereU = Hrac, F = Hracv and G = Hracw.We test three common numerical algorithms for simulating conservation laws.These three algorithms are:1. Flux Corrected Transport (FCT) algorithm of Zalesak [251]: FCT schemerelies on calculating the advective fluxes using two methods: I) method onewhich is highly accurate, but may produce unphysical values in the case ofa non-smooth concentration. II) method two which is a lower order method,but is guaranteed to produce physical values at all times. When the solutionis not differentiable, the algorithm chooses method (II), otherwise method (I)is employed. This switch is conducted via calculating anti-diffusive fluxes.The algorithm proceeds as follows:(a) Compute low order fluxes: F lowi+1/2, j and Glowi, j+1/2(b) Compute high order fluxes: Fhighi+1/2, j and Ghighi, j+1/2(c) Compute anti-diffusive fluxes:Ai+1/2, j = Fhighi+1/2, j−F lowi+1/2, j, Ai, j+1/2 = Ghighi, j+1/2−Glowi, j+1/2(d) Limit anti-diffusive fluxes such that the solution does not get unphysi-cal values.ACi+1/2, j =Ci+1/2, jAi+1/2, j, ACi, j+1/2 =Ci, j+1/2Bi, j+1/2106(e) Compute the time advanced low order solutionqloi, j = qni, j−1∆φ∆ξ(Fi+1/2, j−Fi−1/2, j +Gi, j+1/2−Gi, j−1/2)(f) Apply the limited antidiffusive fluxes:qn+1i, j = qloi, j−1∆φ∆ξ(ACi+1/2, j−ACi−1/2, j +ACi, j+1/2−ACi, j−1/2)(4.32)The subtlety of the algorithm lies in finding suitable choice of limiting func-tions 0≤Ci+1/2, j,Ci, j+1/2 ≤ 1. As C varies from 0 to 1, the algorithm movesfrom a purely low order scheme to a purely high order scheme. Severalchoices of limiting functions are recommended in the literature; see Zalesak[251, 252].In this work, we use a 1st order upwind difference and a 2nd order centraldifference for the low and high order schemes, respectively. As for the lim-iting functions, we first compute the upper and lower bounds:P+i = max(Ai−1/2, j,0)−min(Ai+1/2, j,0) (4.33a)Q+i =(qmaxi −qloi)∆φ (4.33b)R+i = min(1,Q+i /P+i)if P+i > 0, otherwise R+i = 0 (4.33c)P−i = max(Ai+1/2, j,0)−min(Ai−1/2, j,0) (4.34a)Q−i =(qloi −qmini)∆φ (4.34b)R−i = min(1,Q−i /P−i)if P−i > 0, otherwise R−i = 0 (4.34c)and thenCi+1/2, j = min(R+i+1,R−i ) when Ai+1/2, j > 0min(R+i ,R−i+1) when Ai+1/2, j ≤ 0107Ci, j+1/2 can be calculates similarly. In the above procedure, two physicalvalues are employed: qmaxi and qmini . These are “physically-motivated upperand lower bounds on the solution in the next timestep” [252].2. Algorithm of Nessyahu and Tadmor [168] (NT): This is an improvementof the well-known Lax-Friedrichs (LxF) method. LxF method descritizes(4.31) as following: (for simplicity, we drop ra here)Un+1i+ 12 , j+12=14(Uni, j +Uni+1, j +Uni, j+1+Uni+1, j+1)− ∆t2∆ξ[F(Uni+1, j)−F(Uni, j)+F(Uni+1, j+1)+F(Uni, j+1)]− ∆t2∆φ[G(Uni, j+1)−G(Uni, j)+G(Uni+1, j+1)+G(Uni+1, j)] (4.35)LxF is simple, but only first order accurate. Improvement of the accuracy,while keeping the algorithm still simple, has been made by Nessyahu andTadmor [168]. The details of multi-dimensional implementation are fullyexplained in Jiang et al. [123]. It basically modifies (4.35) as follows:Un+1i+ 12 , j+12=14(Uni, j +Uni+1, j +Uni, j+1+Uni+1, j+1)− ∆t2∆ξ[F(Un+12i+1, j)−F(Un+ 12i, j )+F(Un+ 12i+1, j+1)+F(Un+ 12i, j+1)]− ∆t2∆φ[G(Un+12i, j+1)−G(Un+ 12i, j )+G(Un+ 12i+1, j+1)+G(Un+ 12i+1, j)]+116(U ′i, j−U ′i+1, j +U ′i, j+1−U ′i+1, j+1)+116(U 8i, j−U 8i, j+1+U 8i+1, j−U 8i+1, j+1)(4.36)whereUn+12i, j =Ui, j−∆t2∆ξf ′i, j−∆t2∆φg8i, j.In this notation ′ and 8 denote discrete slopes in ξ and φ directions. In orderto avoid oscillatory behavior and to ensure the scheme is total variationdiminishing (TVD), the slopes are found with a slope limiter. Here we use108min-mod limiter defined belowv′j = MM(θ(v j+1− v j), 12 (v j+1− v j−1) ,θ(v j− v j−1))(4.37)whereMM(x1,x2, ...) :=min j x j if x j > 0 for all jmax j x j if x j < 0 for all j0 otherwise(4.38)θ is a parameter that is introduced to enhance stability of the scheme. Nor-mally, 1≤ θ ≤ 2. Note that this is a staggered scheme. In order to turn thisinto a nonstaggered scheme a reconstruction of solution may be necessary.3. Semi-discrete method of Kurganov and Tadmor [138]: This is the next fam-ily of central difference scheme for simulation of hyperbolic equations. Inthis method, the numerical fluxes are calculated to form nonlinear system ofODE’s for the discrete unknowns Ui, j(t). Then, higher-order stable time dis-cretizations, such as 2-step or 3-step Runge-Kutta algorithms are employedto integrate these ODE’s in time. More specifically,ddtUi, j(t) =−Hi+1/2, j(t)−Hi−1/2, j(t)∆φ− Ki, j+1/2(t)−Ki, j−1/2(t)∆ξ(4.39)where the numerical fluxes are given byHi+1/2, j(t)=F(U+i+1/2, j(t))+F(U−i+1/2, j(t))2− ai+1/2, j(t)2(U+i+1/2, j(t)−U−i+1/, j2(t))Ki, j+1/2(t)=G(U+i, j+1/2(t))+G(U−i, j+1/2(t))2− ai, j+1/2(t)2(U+i, j+1/2(t)−U−i, j+1/2(t)),(4.40)109in which the intermediate terms areU+i+1/2, j(t) =Ui+1, j(t)−∆φ2U ′i+1, j (4.41a)U−i+1/2, j(t) =Ui, j(t)+∆φ2U ′i, j (4.41b)andU+i, j+1/2(t) =Ui, j+1(t)−∆ξ2U 8i, j+1 (4.42a)U−i, j+1/2(t) =Ui, j(t)+∆ξ2U 8i, j. (4.42b)Here, ai+1/2, j(t) and ai, j+1/2(t) denote the local maximum speed, i.e. max(U ′)or max (U 8), respectively and the derivative terms are computed using theminmod limiter (4.37) to avoid oscillatory behavior. This system of ODE’sis then integrated in time using a 2-step or 3-step Runge-Kutta algorithm.Stability Condition Stability of the formulations given above requires that onesatisfies the so called CFL-condition:CFL <CFLcritical. (4.43)For the FCT algorithm CFLcritical = 1, and for SD, CFLcritical = 1/8. We define ourCFL number, quite conservatively, by:CFL =(max(v)∆φ+max(w)∆ξ)∆t. (4.44)In the following, we test these three methods using two benchmark problems(see the Problem 1 and 2 defined below and Tables 4.1 and 4.2). Generally we cansay all these method are better than a first order scheme, but they are not secondorder. Quite surprisingly, FCT was found to be noticeably faster and equally accu-rate. Contours of c for the Problem 2 are plotted in Figure 4.3 for different choiceof mesh size and scheme. Interestingly, the FCT appears to be slightly better at lowmesh sizes, compared to the other two schemes. We therefore chose this schemefor the rest of our computational examples.110Problem 1: We take (ξ ,φ) = [0,1]× [0,1], (v,w) = (0.1,0.1) and ra = H = 1with periodic boundary. The initial condition isU = c =1 φ ,ξ ≤ 0.250 otherwise(4.45)i.e. a square blob that moves diagonally.Table 4.1: L2-norm of error for the three methods in problem 1 (computedsolution - analytical solution) calculated at t = 1.Mesh Size\Method FCT NT SD20×20 0.0056 0.008 0.007540×40 0.0023 0.0027 0.002580×80 7.6×10−4 9.4×10−4 9.3×10−4160×160 2.7×10−4 3.3×10−4 3.4×10−4320×320 1.0×10−4 1.2×10−4 1.2×10−4Problem 2: We take (φ ,ξ ) = [0,1]× [0,1], (v,w) = (0.0,0.1) and ra = H = 1with inflow boundary conditions at the bottom boundary:c = 1 at ξ = 0 ,∂c∂ξ= 0 at ξ = 1 ,∂c∂φ= 0 at φ = 0,1 (4.46)The initial condition is c = 0 everywhere.4.3.2 DiffusionNow we consider the diffusion and dispersion terms on the right hand side of (4.1c).We define intermediate variablesJ = DT∇a · [raHc(v,w)]es (4.47a)K = D¯H∇ac (4.47b)111Table 4.2: L2-norm of error for the three methods in problem 2 (computedsolution - analytical solution) calculated at t = 1.Mesh size\Method FCT NT SD20×20 0.0037 0.0068 0.006640×40 0.0013 0.0028 0.002680×80 4.77×10−4 0.001 0.0010160×160 1.69×10−4 3.79×10−4 4.0×10−4320×320 5.0×10−5 1.63×10−4 1.58×10−4which we discretize as:Ji, j = DT,i, j(vi, j,wi, j)[ 1∆φ(Hi+1/2, jci+1/2, jvi+1/2, j−Hi−1/2, jci−1/2, jvi−1/2, j)+1∆ξ(Hi, j+1/2ci, j+1/2vi, j+1/2ra, j+1/2−Hi, j−1/2ci, j−1/2vi, j−1/2ra, j−1/2)](4.48)Ki, j = D¯i, jHi, j[1ra, j1∆φ(ci+1/2, j− ci−1/2, j)+1∆ξ(ci, j+1/2− ci, j−1/2)](4.49)Notice that as our descritization is staggered (see Figure 4.1 ), several variableabove shall be reconstructed with interpolation. For example, azimuthal velocityvector is computed at positions (i, j + 1/2) (i.e. diamonds in Figure 4.1 ). Tocalculate v at positions (i, j), as needed above, we may simply write:vi, j =12(vi, j−1/2+ vi, j+1/2).Similarly, we can find wi, j, and so on.112            FCT method 𝜉 𝜙 NT method SD method Figure 4.3: Contour of c in benchmark Problem 2 at t = 9. Columns fromleft to right are FCT, NT and SD schemes. Rows from top to bottom aremesh sizes: 20×20, 40×40, 80×80, 160×160 and 320×320113Now the right hand side of (4.30) discretizes like:RHSi+1/2, j+1/2 =δ0pi1si+1/2, j+1/2[1ra, j+1/21∆φ(Ji+1, j+1/2− Ji, j+1/2)+1∆ξ(Ji+1/2, j+1− Ji+1/2, j)]+δ0pi[1∆φ(Ki+1, j+1/2−Ki, j+1/2)+ ra, j+1/21∆ξ(Ki+1/2, j+1−Ki+1/2, j)]+(DT,i+1/2, j+1/2−D∗T,i+1/2, j+1/2)ra, j+1/2×[1ra, j+1/21∆φvi+1/2, j+1/2(Hi+1, j+1/2−Hi, j+1/2)+wi+1/2, j+1/21∆ξ(Hi+1/2, j+1−Hi+1/2, j)]×[1ra, j+1/21∆φvi+1/2, j+1/2(ci+1, j+1/2− ci, j+1/2)+wi+1/2, j+1/21∆ξ(ci+1/2, j+1− ci+1/2, j)]×(4.50)To include the diffusion and dispersion terms in the FCT scheme, we need tomodify (4.51) toqn+1i, j = qloi, j−1∆φ∆ξ(ACi+1/2, j−ACi−1/2, j +ACi, j+1/2−ACi, j−1/2)+RHSnni+1/2, j+1/2∆t.(4.51)wherenn =n for explict schemen+1 for implicit scheme.(4.52)Approach 1: If we take nn = n, the algorithm is explicit in time, meaning that(4.51) can be solved at each time step to update the solution. However, in additionto the CFL condition (4.43), stability of diffusive terms requires:α =max(DT ,D∗T , D¯)[min(∆φ ,∆ξ )]2∆t < αcritical. (4.53)114The stability condition (4.53) is rather conservative, because the dominant diffusionterm is Taylor dispersion, which is active in the direction of streamlines, which isin turn mostly axial (in vertical wells). Therefore, the relevant length scale is ∆ξ(and not min(∆φ ,∆ξ )). Nonetheless, we proceed with the above definition.Theoretically, αcritical = 0.5, but we take values between 0.2 to 0.4 for robust-ness. Recall that the variables in (4.53) are all dimensionless. In a dimensionlesssetting, typically we get D¯ DT ∼ D∗T ∼ 0.001− 0.01, ∆φ ' 0.01 and ∆ξ ' 1.This means that ∆t . 0.001. Roughly, in a dimensional setting, the stability condi-tion requires ∆tˆ . 0.01 s. This stringent stability condition makes the simulationsprohibitively slow.Approach 2: Alternatively, if we take nn = n+1, the algorithm is implicit intime, and a nonlinear system of equations is needed to be solved. This approachhas the benefit that ∆t is governed only by the CFL condition (4.43). However, thesystem of equations is astronomically large. In addition, as we need to interpolate(v,w,c) at spatial position where they are not defined, the mass matrix is no longertri-diagonal.Approach 3: To alleviate this problem, we chose a third approach:• We separately compute the largest ∆t that satisfy (4.43) and (4.53). We callthese two values ∆tCFL and ∆tα , respectively.• If ∆tCFL < ∆tα , we set ∆t = ∆tCFL and use the explicit formulation. This nor-mally happens when the diffusive terms are very small, and ∆tCFL is suitablylarge.• If ∆tα < ∆tCFL, we take ∆t = ∆tα , but we will only update the velocity fieldevery m time steps, where m = div(∆tCFL,∆α), where the function div(x,y)computes the quotient of x divided by y. In other words, the concentrationequation is updated by ∆tα , but the streamfunction equation is updated by∆tCFL. Notice that this only introduces a second order splitting error in time.One might decide to choose a cap for m. This approach allows us to circum-vent the stringent condition (4.53) without the hassle of working with largematrices.1154.3.3 Displacement exampleTo explore the accuracy of our computational approach in solving the conservationequation (4.1c), we showcase a displacement example, and consider two diagnostictests, as discussed below.We take annular dimensions that are more typical of a laboratory flow loopsetting, with rˆi = 6.5 cm, rˆo = 9 cm, Zˆ = 20 m. The annulus is assumed to bevertical β = 0 and uniformly eccentric with e = 0.5. We restrict the simulation toonly one half of the annulus: assuming symmetry at the wide and narrow sides.The displaced and displacing fluids both have identical properties:ρˆ1 = 1100 kg/m3, κˆ1 = 0.002, Pa.s0.9, n1 = 0.9, τˆY,1 = 0 Pa.The flow rate is constant and equal to Q = 0.01 m3/s, equivalent to a meanvelocity of ˆ¯W = 0.82 m/s. The flow rate is large enough that the displacement isfully turbulent.At each timestep the stream function is found by solving (4.1a) using theUzawa algorithm presented §4.2.4. The concentration enters the stream functionequation through the buoyancy field b and through the local fluid properties. Hav-ing found the stream function we construct the velocity field from (4.1b) and ad-vance the fluid concentrations in time by solving (4.1c). We use 30×200 meshcellsin the azimuthal and axial directions, respectively. The concentration equation, inparticular, is solved using FCT scheme under the following conditions:i) We ignore the diffusion and dispersion terms on the right hand side of (4.1a).The timesteps are chosen dynamically, to satisify the CFL condition (4.43).ii) We include the diffusion and dispersion terms on the right hand side of (4.1a)and use Approach 1 of §4.3.2 to compute their flux. The timesteps arechosen dynamically, to satisify both stability conditions (4.43) and (4.53).We take CFLcritical = αcritical = 0.4.iii) Same as ii), except we use Approach 3 of §4.3.2.Our goal is to test the accuracy of our advection and diffusion schemes, aswell as to demonstrate the effect of diffusion and dispersion on the displacement116W N W N W NFigure 4.4: Turbulent displacement in an eccentric annulus: e = 0.5. Colormaps of the concentration are shown in dimensionless coordinate sys-tem (φ ,ξ ) at different dimensionless times as the front progresses alongthe annulus. White lines are streamlines with spacing ∆Ψ= 0.25. a) Nodispersion or diffusivity (D¯ = DT = D∗T = 0); b) Diffusion and disper-sion are present, and computed using the Approach 1 of §4.3.2; c) Dif-fusion and dispersion are present, and computed using the Approach 3of §4.3.2.interface.Figure 4.4 presents a colourmap of the fluid concentrations as they advancealong the annulus: fluid 1 (red) is displaced by fluid 2 (blue). The wide and narrowsides are denoted with W and N. As will be explained later in Chapter 5, the eccen-tricity of the annulus has led to faster flows on the wider side (W) than the narrowside (N). The interface constantly elongates as a consequence. In Figure 4.4a wehave set to zero all terms on the right hand side of (4.1c); i.e. no diffusion or dis-persion. In Figures 4.4b and 4.4c, the diffusive and dispersive fluxes are computedusing Approach 1 and Approach 3 of §4.3.2, respectively.Figure 4.4a essentially shows the performance of the FCT scheme in advectingthe front. Numerical smearing (diffusion/dispersion) of the front is present but iskept to a few cells in width. For this flow in particular (1D) there are no secondaryflows that can advectively mix intermediate concentrations. The front is sharpenedby mesh refinement of course, at the expense of longer computational times.117Figures 4.4b-c show the significant effect of DT , in comparison to Figure 4.4a.Notice that as commented and illustrated earlier in Chapter 2, turbulent diffusivityis one or two orders of magnitude smaller than Taylor dispersion, and in primarilyresponsible for the smearing of interface. Indeed, if we switch the turbulent diffu-sion on or off, no discernible difference can be identified. Furtheremore, althoughthe term D∗T is in fact larger than DT (see Figure 3.4), since the streamlines areparallel and es ·∇ack,0 is only in the ξ direction, the last term of (4.1c) does nothave any contribution in this example.In order to measure the accuracy of our FCT advection scheme, we computethe average concentration of displacing fluid:c¯2(t) =∫ 10 ra(ξ )H(φ ,ξ )c2(φ ,ξ , t) dφ∫ 10 ra(ξ )H(φ ,ξ ) dφ. (4.54)Notice that the denominator is equal to unity, by construction. (4.54) is in factthe efficiency of the displacement (which is used later in Chapter 6) and is definedbased on the conserved quantity: raHc2. Alternatively, we can also calculate theaveraged concentration, by computing the inlet and outlet flux:c¯2,q(t) =∫ t0qin(t˜)−qout(t˜)dt˜, (4.55)where here qin and qout are the azimuthally-averaged volumetric flux of displacingfluid entering and leaving the annulus from the bottom and top, respectively. Morespecifically,qin =∫ 10Glow(φ ,0)+AC(φ ,0)dφ ,qout =∫ 10Glow(φ ,ξbh)+AC(φ ,ξbh)dφ ;see §4.3.1.Figure 4.5 plots c¯2 and c¯2,q as a function of time. The green line marks the the-oretical breakthrough time with piston-like displacement, hence the curved line inFigure 4.5a is indicating some dispersion ahead of the breakthrough. More impor-tantly, we observe that the two averaged concentrations are almost identical, whichproves the FCT scheme is properly implemented (i.e. we are not losing mass).1180 20 40 60 8000. 20 40 6010 -810 -710 -610 -510 -410 -310 -210 -1Figure 4.5: a) Profile of c¯2 (black circles) and c¯2,q (red dashed line) vs timefor the displacement example shown in Figure 4.4a. The green linemarks the arrival time. b) Red line is the L2 norm of difference in thetwo solutions shown in Figures 4.4a and 4.4b. Blue line is the L2 normof difference in the two solutions shown in Figures 4.4b and 4.4cThe second test we conduct is to see if the approximation of Approach 3 is ac-curate, compared to Approach 1. In Figure4.5b, the blue line shows the L2 norm ofthe difference between the two solutions of c2 shown in Figure 4.4b (Approach 1)and Figure 4.4c (Approach 3). As this figure demonstrates, the error is.O(10−7).To gain some insight about this value of error, we have also computed a similar er-ror between Figures 4.4a and 4.4b, which is shown by the red line in Figure 4.5b.comparison confirms that error associated with the approximation of Approach 3is suitably negligible.In order to evaluate the accuracy of our scheme in a more complicated displace-ment regime, we consider an example where the displacement regime is mixed.Here the properties of displaced fluid is given byρˆ1 = 1100 kg/m3, κˆ1 = 0.005, Pa.s, n1 = 1, τˆY,1 = 5 Pa, (4.56)and the propties of displacing fluid is given byρˆ2 = 1200 kg/m3, κˆ2 = 0.005, Pa.s, n2 = 1, τˆY,2 = 0 Pa. (4.57)1190 20 40 60 80 10000. 4.6: Same as Figure 4.5a, except the displaced and displacing fluidproperties are changed to (4.56) and (4.57), respectively.Notice that the displaced fluid now has a yield stress and displacing fluid is nowheavier. The density difference here assists the displacing fluid to yield the dis-placed fluid around the annulus. In addition, both fluids are slightly more viscous,so that the displacement is in mixed regime.Figure 4.6 plots c¯2 and c¯2,q as a fuction of time. Again the green line marks thetheoretical breakthrough. We observe that the difference between the two averagesis slightly more amplified, which is due to the complexity of displacement example(e.g. the velocity field).4.4 ConclusionsIn this chapter, we analyzed the 2D displacement model that we derived earlier inChapter 3 from a computational point of view. The following remarks can be madehere:• We derive a weak solution for the stream function equation (4.1a). In addi-tion, we proved that the weak problem has a unique solution.• We construct an algorithm based on the variational form of the stream func-tion equation (4.1a). The algorithm is guaranteed to converge, and is partic-120ularly suitable for yield stress fluids.• A less computationally expensive algorithm for solving the stream functionequation (4.1a) is also derived which rely on neglecting the axial variationsin the annulus.• As for the concentration equation (4.1c), we have reviewed some of the com-mon approaches in solving conservation equation, and show that they arecomparable in terms of accuracy and speed.121Chapter 5Fully turbulent displacementflowsIn the previous chapters we derived our 2D displacement model, and studied themodel from a computational point of view. In particular, we designed robust al-gorithms to solve the model accurately. Starting from this chapter, we begin toexplore various primary cementing scenarios, with the focus on annular displace-ment in turbulent and mixed regimes. The cementing parameters chosen here areall in ranges commonly found in industrial applications, which we review moreexplicitly in Chapter 6.This chapter specifically discusses displacement examples that are fully tur-bulent. Although fully turbulent displacement flows are relatively uncommon inpractice, the examples studied here elucidate the key differences between laminarand turbulent displacements and paves the way to understand more complicatedmixed regime displacement that will be studied later in Chapters 6 and 7.A version of this chapter is published in Maleki and Frigaard [150].5.1 Simulation parametersAll simulations presented in this chapter are restricted to only one half of the an-nulus; i.e. assuming symmetry at the wide and narrow sides. This choice is notjustified if we have unstable displacement, as it will favor one azimuthal location122against others in terms of the onset and growth of instabilities. We will studyunstable displacement flows later in Chapter 7. We are particularly interested incementing scenarios pertinent to surface casing where the flow is more likely tobe turbulent. As explained earlier, surface casings normally have larger radii, andthere is often a wider pore-frac envelope, so more scope for high flow rates. There-fore, we take an annulus with inner and outer radii of rˆi = 16.5 cm (Dˆi = 13”) andrˆo = 19.0 cm (Dˆo = 15”). A cemented section is typically a few hundred meters,but the flow develops in a much shorter distance along the well, which suggests asfar as the displacement flows and efficiency are concerned, a much shorter simu-lated cementing section suffices. Therefore, we take the cemented section lengthto be ξˆbh = 150 m. We have used nφ = 30 mesh cells in the azimuthal direction andnξ = 300 cells along the annulus. This will give an azimuthal and axial mesh sizesof ∆θ = 6◦ and ∆ξˆ = 0.5 m. Similar spatial resolution will be used in the followingchapter too, where we study mixed regime displacement flows. Such resolution iscommonly found in the literature; e.g. see Tardy [225], Kragset et al. [135].In the examples presented below, we show the contours of concentration offluids present in the displacement. These can be thought as snapshots of the dis-placement. In all figures, the annulus is unwrapped into a channel with varyingwidth and the narrow and wide sides are marked with N and W on the horizontalaxis. Only half of the annulus is shown. Unless specified otherwise, time is re-ported in dimensionless units; see the derivations in Chapter 3. The displaced anddisplacing fluids are indexed with 1 and 2, and colored red and blue, respectively.Streamlines are depicted with white lines. Several figures are accompanied by amap of displacement regime. These maps highlight laminar, transitional and tur-bulent regions in dark gray, light gray and white, respectively. The regions withimmobilized (unyielded) mud are highlighted in black.5.2 Rheology is not very relevantOur first example establishes a general rule that we will use throughout the thesis.The rule says as long as the displacement is fully turbulent, the rheology of thefluids does not play any significant role in the displacement dynamics; i.e. thedisplacement front is not affected much by varying the rheology. To show this, we123consider the displacement example shown in Figure 5.1 . This example describesa displacement job in a vertical well with eccentricity of e = 0.5. The rheologicalparameters of the displaced fluid are kept constant, while those of displacing fluidare varied. Notice that as we change the power-law index of the displacing fluid,the consistency is also modified such that the nominal effective viscosity remainsconstant. The nominal effective viscosity is computed using the mean velocity andmean gap width as the nominal shear rate:ˆ˙γ∗ =ˆ¯Wrˆo− rˆi , µˆe f f =κˆ ˆ˙γ∗n+ τˆYˆ˙γ∗. (5.1)In our first example, the properties of displaced fluids are:ρˆ1 = 1200 kg/m3,n1 = 1, κˆ1 = 0.001 Pa.s and τˆY,1 = 0 Pa.. (5.2)while, those of displacing fluids areFluid A: ρˆ2 = 1250 kg/m3,n2 = 1.0, κˆ2 = 0.0010 Pa.s and τˆY,2 = 0 Pa.Fluid B: ρˆ2 = 1250 kg/m3,n2 = 0.7, κˆ2 = 0.0024 Pa.s0.7 and τˆY,2 = 0 Pa.Fluid C: ρˆ2 = 1250 kg/m3,n2 = 0.4, κˆ2 = 0.0059 Pa.s0.4 and τˆY,2 = 0 Pa.(5.3)Note that there is a positive density difference between the displacing and displacedfluids (ρˆ2 > ρˆ1), but we kept it small enough so that it does not mask any effectcaused by the variation of rheological parameters. The flow rate in all three casesis Qˆ = 0.0142 m3/s (= 5.46 bbl/min) or equivalently the mean velocity is ˆ¯W = 0.5m/s. The flow rate is large enough that both displaced and displacing fluids are fullyturbulent around the annulus. The displacement snapshots are shown in Figure 5.1.We observe despite of the variations in n and κˆ , no noticeable changes happen inhow the interface of displacement advances.If we reduce the flow rate such that the flow regime falls into laminar, then therheology of either fluid starts to become more relevant. For example, in Figure 5.2,we repeat the previous simulation in Figure 5.1a, except we have reduced the flowrates to Qˆ= 0.00071 m3/s (= 0.273 bbl/min), or equivalently ˆ¯W = 0.025 m/s. Hereboth fluids are fully laminar. We observe the displacement front is slightly more124a)W Nb)W Nc)W NFigure 5.1: Effect of rheological parameter in fully turbulent displacementflows. Panels in each subfigure show displacement snapshots at threedifferent times, with white lines denoting the streamlines ∆Ψ = 0.25.The displacements are fully turbulent. e = 0.5 and Qˆ = 0.0142 m3/s (=5.46 bbl/min), ˆ¯W = 0.5 m/s. Displaced and displacing fluids propertiesare given by (5.2) and (5.3), respectively. a) Fluid A; b) Fluid B; c)Fluid C.flat, and is much more dispersive. This presumably means stronger secondary flowsdevelop near the interface in the laminar case (see the large deviation of streamlinesnear the interface) compared to the turbulent case, which are responsible for thelarge dispersive mixing at the interface.To analyze this more carefully, in Figure 5.3 we have plotted contours of ax-ial velocity and wall shear stress at t = 100 for all three fluids in Figure 5.1. InFigure 5.3a, it appears that the axial velocities are identical for all three fluids. Infact, further inspection confirms that L2 norm of the differences of axial velocitiesamong the three simulations are of the order of iteration tolerance (≈ O(10−5)).Surprising here is that behind the displacement front, where the fluids have dif-ferent shear-thinning indices n2, the azimuthal distribution of turbulent velocitiesis very insensitive to n2, although clearly varying with φ . On the other hand, the125W NFigure 5.2: Same as Figure 5.1 a, except the flow rate is reduced to Qˆ =0.00071 m3/s (= 0.273 bbl/min), ˆ¯W = 0.025 m/swall shear stress contours (Figure 5.3b) show that behind the interface the valuesof wall shear stress for the three fluids are quite different; i.e. larger values for theFluid A and smaller values for the Fluid C. This is not surprising, because differentrheological parameters results in different values of Reynolds number (Rep), whichin turn leads to having variations of wall shear stress among the three simulations.However, what is surprising is that the interface front in the three simulation hasidentical position, despite the difference in the wall shear stress. To answer this, inFigure 5.4 we plot the profiles of wall shear stress behind the interface (solid lines)and at the interface (dashed lines). Quite remarkably, we see the difference in thewall shear stresses almost disappears at the interface, which could be the reasonthe displacement progresses similarly in all three examples.We remark here that the scaling we used to non-dimensionalize the wall shearstress fields in Figure 5.3b is a laminar stress scale. More specifically, we have:τw =τˆwτˆ∗lam, τˆ∗lam = κˆ1 ˆ˙γ∗n1 + τˆY,1. (5.4)126a) b)Figure 5.3: a) Contour of axial velocity (w), corresponding to the displace-ment example shown in Figure 5.1 at t = 100. From left to right, panelsare Fluid A, Fluid B and Fluid C. b) Same as a), except the contoursshow wall shear stress (τw). Wall shear stress is non-dimensionalizedwith a laminar velocity scale give by (5.4).Recall that the nominal shear rate ( ˆ˙γ∗) is defined based on the averaged gap widthand mean velocity (5.1). The laminar stress scale is based on the displaced fluid,and does not change. That is why the stress field remains unchanged for the dis-placed fluid in Figure 5.3. We may choose instead to scale the wall shear stresswith a turbulent stress scale. :τˆ∗turb =12ρˆ2 ˆ¯W 2 f f , (5.5)where heref f = f f(n2, κˆ2, τˆY,2, ˆ¯W)is the turbulent friction factor calculated for the displacing fluid based on a repre-1270 0.2 0.4 0.6 0.8 1010203040Figure 5.4: Profile of wall shear stress behind the interface (solid lines) and atthe interface (dashed lines), corresponding to the displacement exampleof Figure 5.1. Black, red and blue colors denote Fluid A, Fluid B andFluid C, respectively.Figure 5.5: Same as Figure 5.3, except the wall shear stress is scaled with aturbulent stress scale.128sentative mean velocity. Here the stress scale is based on the displacing fluid, andvaries for each set of displacement examples shown. The contours of wall shearstress, scaled with the turbulent stress scale above are plotted in Figure 5.5. Theturbulent stress scale defined in (5.5) encapsulates any variation of stress, and a re-sult the dimensionless wall stress field are identical in the displacing fluid. On thehand, we see that the dimensionless stress fields in the displaced fluid vary quitesignificantly.In our second example, we consider a slightly more complicated displacementexample. Here we viscosify the displaced fluid and add some moderate yield stress:ρˆ1 = 1200 kg/m3,n1 = 1, κˆ1 = 0.005 Pa.s and τˆY,1 = 8 Pa., (5.6)while the properties of the displacing fluid are:Fluid D: ρˆ2 = 1300 kg/m3,n2 = 1.0, κˆ2 = 0.0010 Pa.s and τˆY,2 = 0 Pa.Fluid E: ρˆ2 = 1300 kg/m3,n2 = 0.7, κˆ2 = 0.0034 Pa.s0.7 and τˆY,2 = 0 Pa.Fluid F: ρˆ2 = 1300 kg/m3,n2 = 0.4, κˆ2 = 0.0115 Pa.s0.4 and τˆY,2 = 0 Pa.(5.7)Notice that larger density difference is employed here, because the mud is moreviscous and has a moderate yield stress; i.e. more challenging. In addition, we runthe simulations at a higher flow rate: Qˆ = 0.0426 m3/s (= 16.38 bbl/min), whichis equivalent to the mean velocity of ˆ¯W = 1.5 m/s. The consistency of displacingfluid is chosen according to its value of power-law index, such that the effectiveviscosity based on mean shear rate (5.1) is equal among the three simulations.The snapshots of displacements together with the contours of flow regime areshown in Figure 5.6. Notice here that since the displaced fluid is more viscous,and has some yield stress, it undergoes a regime change around the annulus, as aresult of variation of gap thickness (eccentricity of the annulus). On the wide sidewith larger velocity, the displacement is in the transitional regime, while on thenarrow side, the displacement is laminar (or even static). One interesting feature inFigure 5.6a-b is that although the mud ahead of interface remains unyielded, at theinterface it yields, and therefore, the interface progresses.129a) b)c)Figure 5.6: Effect of rheological parameter in fully turbulent displacementflows. Left panels in each subfigure show displacement snapshots atthree different times, with white lines denoting the streamlines ∆Ψ =0.25. The right panels at each subfigure show the contour of regime:white, dark gay and light gray represent regions turbulent, transitionaland laminar flow regimes. The black regions denote unyielded fluid.e = 0.5 and Qˆ = 0.0426 m3/s (= 16.38 bbl/min), or equivalently ˆ¯W =1.5 m/s. Displaced and displacing fluids properties are given by (5.6)and (5.7), respectively. a) Fluid D; b) Fluid E; c) Fluid F.130Quite interestingly, although the displaced fluid regime changes around theannulus, the displacement front remains identical in the case of Fluids D and E.Further changing the rheology of displacing fluid to those of Fluid F results in hav-ing the displacing fluid then being unable to mobilize the mud on the narrow side.This is an interesting observation, because the displacing fluid is still turbulent, butis not able to move the mud on the narrow side. This could also be inferred fromFigure 5.4. Here Fluid C (which has properties similar to Fluid F) generates thesmallest wall shear stress. The generated wall shear stress is not sufficient to yieldthe mud on the narrow side.The conclusion is that as long as the two displacing and displaced fluids arefully turbulent, varying their rheological parameters has little effect on the dis-placement front. However, if either fluids’ flow regime becomes laminar, then therheological parameters of either of fluid may become relevant. In the exampleof Figure 5.6, it is the displaced fluid that is flowing partially laminar, but thenchanging the rheology of the displaced fluid influences how the displacement frontmoves. This suggests that the flows of the two (or more) fluids are strongly cou-pled.Of course, the notion that rheology is not important in fully turbulent flows isto some extent inherent in the scaling of the stresses. The notion that rheologybecomes relatively irrelevant in the fully turbulent flows is reported for the singleflows of yield stress fluids [102]. In the above we have simply verified that this isalso the case for displacement flows. The interesting aspect for cementing is thatfrequently we do deal with mixed regime flows (see Chapter 6), simply because thefluids are fairly viscous and often we have restrictions on the frictional pressuresin the wellbore.5.3 Turbulence vs buoyancyIn an eccentric well, the flow tends to be faster on the wide side, because in a narrowannulus, the wall shear stress scales linearly with the gap width and a larger wallshear stress induces a larger flow rate. For the very same reason, the displacementis slower on the narrow side. It is perhaps astonishing that when this same featureis present both far downstream and far upstream of an “interface”, the interface131may itself propagate at uniform (mean) speed along the annulus. However, thesesteady displacements have been shown to arise for laminar flows, both theoretically[179, 181] and numerically [180, 37]. In the laminar flow these arise via positivegradients of density and frictional pressure. In turbulent flows, we have just seenthat rheology is not important.A positive density difference (ρˆ2− ρˆ1 > 0) competes against the tendency forthe interface to elongate along the wide side, promoting a steady displacement.Here we provide a simple explanation that elucidates the role of buoyancy stresses,and the type of motions they induce near the interface. Recall that the buoyancyvector was defined asb =ra(ρ−1)Fr2(cosβ ,sinpiφ sinβ ) .In a vertical well with uniform geometry, ra = 1 and β = 0, thereforeb = (bφ ,bξ ) =(ρ−1Fr2,0), (5.8)from which it follows that∇.b =1ra∂bφ∂φ+∂bξ∂ξ=1Fr2∂ρ∂φ.If the displacing fluid is heavier than the displaced fluid, elongation of the interfaceon the wide side creates a negative azimuthal gradient of density, thus ∇.b < 0.Let’s now consider the following simple toy model∇2Ψ+ f = 0 (5.9a)0≤ φ ≤ 1, 0≤ ξ ≤ L (5.9b)Ψ(0,ξ ) = 0, Ψ(1,ξ ) = 2 (5.9c)∂Ψ∂ξ= 0 at ξ = 0,L (5.9d)Problem (5.9a) is a simpler version of (3.60), where f = f (φ ,ξ ) plays the role of∇.b and all other nonlinearities are thrown out. However, (5.9a) is elliptic just as is(3.60) and we expect qualitatively similar behavior, i.e. we expect that the solution132of (5.9a) will respond in the same mechanistic way according to the sign of thesource term f . Using f to mimic the buoyancy gradient should illustrate how thedensity stable configuration promotes stabilizing fluid motions. For the sake ofsimplicity, we take:f (φ ,ξ ) = −1 if L−ε2 ≤ ξ ≤ L+ε2 ,0 if otherwise ,(5.10)i.e. the interface lies on a band with thickness ε and its center at height L/2 (seeFigure 5.7a). The problem (5.9) with the source term given by (5.10) can be readilysolved numerically. We used a simple second order finite difference scheme tosolve this PDE. The solution Ψ is plotted in Figure 5.7b. Notice here that the effectof the source term here is obscured by the mean axial flow. To see the buoyancydriven motions more clearly, in Figure 5.7c, we subtract the mean velocity andplot arrows that represent magnitude and direction of the secondary flow near theinterface. Of course, as (5.9a) is linear we can also explicitly deconstruct and solvefor the secondary flow, but this is only illustrative. We see that counter-clockwiseflows develop near the interface that move the displacing fluid to the narrow sideand displaced fluid to the wide side. This mechanism clearly competes against thebias that eccentricity creates in the flow and is the source of stabilization.Notice that the sign of the source term f was crucial here. If we flip the sign(i.e. the lighter fluid displaces the heavier fluid), then the secondary flow movesin the opposite direction, which means these buoyancy-induced flows intensify thebiasing effect of eccentricity. This is the primary reason that Effective LaminarFlow method recommends to have the displacing fluid at least 10% heavier thanthe displaced fluid [48].We will now explore this in a more realistic displacement case. In laminardisplacement, the wall shear stress is determined by not only the gap thickness(i.e. eccentricity), but also by the rheology of the two fluids, leaving the displace-ment dynamics more complicated. In turbulent regime however, the competitionis rather simpler, because as shown earlier, the rheology is only weakly relevant133a)W N0Lb) c)W NFigure 5.7: a) The source term f (5.10); b) Solution Ψ to the problem intro-duced in (5.9). c) Fluid motions near the interface, after the mean axialvelocity is subtracted.and the turbulent wall shear stress at fixed flow rate is predominantly influencedby the eccentricity. Figure 5.8 investigates the competition between the eccentric-ity (through turbulent wall shear stress) and buoyancy stresses. We use the dis-placement scenario of Figure 5.1a, except here we vary the density of displacingfluid to illustrate different possible scenarios. Starting from no density difference(ρˆ1 = ρˆ2 = 1200 kg/m3) in Figure 5.8a, we gradually increase the density of dis-placing fluid to ρˆ2 = 1215 and finally 1230 kg/m3.In Figure 5.8a the two fluids have identical densities. As a result, we observethe interface elongates. As we increase the density difference in Figure 5.8b-c,the interface velocity become more and more azimuthally uniform (the differencein wide and narrow side velocities diminishes at the interface). Elongation of theinterface is suppressed and the flow is more steady.1 To see this more clearly,1Since Turbulent displacement flows fall into Taylor dispersion regime, the interface continuously134a) b) c)W NFigure 5.8: Competition between turbulent stresses and Buoyancy stresses.Panels in each subfigure show displacement snapshots at three differenttimes, with white lines denoting the streamlines ∆Ψ = 0.25. The dis-placements are fully turbulent: e = 0.5 and Qˆ = 0.0142 m3/s (= 5.46bbl/min), ˆ¯W = 0.5 m/s. Displaced and displacing fluids propertiesare given by (5.2) and (5.3), except the displacing fluid density is a)ρ2 = 1200 kg/m3; b) ρ2 = 1215 kg/m3 and c) ρ2 = 1230 kg/m3.we look at the interface velocity on the wide and narrow sides. Figure 5.9 plotsthe difference in wide and narrow side axial velocities (wW −wN) along the wellfor the three simulations shown in Figure 5.8. The colors black, red and bluedenote ρˆ2 = 1200,1215 and 1230 kg/m3, respectively. The solid lines are plottedat t = 50 and dashed lines are at t = 100. For the black line, the two fluids have nodensity difference. Thus, the wide side velocity is always larger than the narrowside velocity. However, for the two other cases, we can see the differential velocityshrinks, as we cross the interface. More interestingly, although not clear fromFigure 5.8b, the displacement example with ρˆ2 = 1215 is still not fully steady,because as Figure 5.9 shows, the narrow side velocity is slightly smaller than thediffuses. Therefore, strictly speaking, the interface will always be unsteady. Nonetheless, turbulentflows can still be steady in the sense that the interface moves uniformly around the annulus. Here,we have adopted the latter notion of a steady displacement.1350 0.5Figure 5.9: Profile of differential velocity (wW −wN) along the annulus, cor-responding to the simulations in Figure 5.8 at t = 50 (solid lines)and t = 100 (dashed lines). The colors black, red and blue denoteρˆ2 = 1200,1215 and 1230 kg/m3, respectively.wide side velocity at the interface. Perhaps this would have been more apparent,if the simulation was conducted in a longer annulus, and the displacement hadmore time to elongate. Further increasing the density ρˆ2 = 1230 kg/m3 completelyeliminates any differential velocity at the interface, indicating that a fully steadydisplacement is achieved.Figure 5.10 plots the contours of wall shear stress, scaled with the turbulentstress scale introduced in (5.5), and the φ -component of the buoyancy vector (bφ ),as well as the divergence of buoyancy vector ∂bφ/∂φ , all corresponding to thesimulations shown in Figure 5.8. In Figure 5.10a, once there is a density differencebetween the two fluids, we can see the wall shear stress contours change rapidlyat the interface, apparently becoming close to uniform along the interface. As thefluids are miscible the interface is a line of intermediate fluid properties. The dom-inant turbulent stresses scale with the square of the velocity and the friction factor.136a) b)W N-7-6-5-4-3-2-10c)W N-14-12-10-8-6-4-20Figure 5.10: a) Contour of wall shear stress (τw), scaled using a turbulentstress scale (5.5), corresponding to the displacement example shownin Figure 5.8 at t = 150. From left to right, panels are ρ2 = 1200,1215and 1230 kg/m3; b) Same as (a), except the contours show bφ ; c) Sameas (a), except the contours show ∂bφ/∂φ .If the interface moves steadily the only variation in wall shear stress along the inter-face comes from the fully turbulent friction factor, which is slowly changing withH for fixed rheology and velocity. This explains the apparent uniformity.The jump in bφ is also apparent across the interface. The mechanism that makesthe interface steady is, simplistically speaking, governed by ∂bφ/∂φ . The contoursof ∂bφ/∂φ are reminiscent of the f function (5.10) which is plotted in Figure 5.7a.We see an increasingly negative buoyancy gradient focused around the narrow sideinterface, which we have seen has the effect of inducing a stabilizing secondaryflow.Our second example probes into the competition between buoyancy and turbu-lent stresses in a more complicated displacement setting. Here the displaced fluidhas the following properties:ρˆ1 = 1200 kg/m3,n1 = 1, κˆ1 = 0.005 Pa.s and τˆY,1 = 3 Pa.. (5.11)137We consider three displacing fluids with three different values of density:ρˆ2 = 1200,1250 and 1300 kg/m3,n2 = 1, κˆ2 = 0.001 Pa.s and τˆY,2 = 0 Pa..(5.12)Snapshots of the displacement are shown in Figure 5.11. Here, since the displacedfluid is more viscous, and has a yield stress, it experiences a change in regimearound the interface. In Figure 5.11a, there is no density difference, therefore thedisplacing fluid is unsteady, but nevertheless we see that the turbulent stresses influid 2 are sufficient to remove the laminar fluid 1 on the narrow side. The displacedfluid on the narrow side moves far slower than on the wide side to the extent thata secondary displacement front develops on the narrow side. Adding some densitydifference in Figure 5.11b, the interface moves more uniformly. Eventually, inFigure 5.11c, the density difference is sufficient to make the displacement steady.This dynamic is more clearly depicted in Figure 5.12. The differential velocityis plotted at t = 50 (solid lines) and t = 100 (dashed lines) and the colors black,red and blue denote ρˆ2 = 1200,1250 and 1300 kg/m3, respectively. It is apparentthat two displacement fronts are developing in the case of ρˆ2 = 1200 kg/m3. Thedifferential velocity shrinks when ρˆ2 = 1250 kg/m3 and then finally vanishes whenρˆ2 = 1300 kg/m3.5.4 Is turbulence necessarily good?In the cementing community it is widely believed that the more turbulent the flow,the more efficient the displacement (e.g. see Nelson and Guillot [167], Lavrov andTorsæter [140]). However intuitively, as the flows becomes more turbulent (e.g. byincreasing the flow rate), the turbulent wall shear stress increases and eventuallydominates the flow, rendering both rheology and density differences unimportant inthe displacement. As a result, no mechanism remains present in the displacement tomake the displacement front steady. Therefore, one would expect the displacementto become unsteady. We will investigate this hypothesis here. We consider thedisplacement example in Figure 5.1a and change the flow rate. In Figure 5.13athe flow rate is reduced to Qˆ = 0.00071 m3/s (= 0.273 bbl/min), or equivalentlyˆ¯W = 0.025 m/s. As a result, the effect of buoyancy is stronger and the displacement138a) b)c)Figure 5.11: Competition of turbulent stresses vs Buyancy stresses. Rightpanels in each subfigure show displacement snapshots at three differenttimes, with white lines denoting the streamlines ∆Ψ= 0.25. Left pan-els in each window show the contours of displacement regime: white,light gray and dark gray represent turbulent, transitional and laminarflow regimes. e = 0.5 and Qˆ = 0.0426 m3/s (= 16.38 bbl/min), orequivalently ˆ¯W = 1.5 m/s. Displaced and displacing fluids propertiesare given by (5.11) and (5.12), respectively. a) ρ2 = 1200 kg/m3; b)ρ2 = 1250 kg/m3 and c) ρ2 = 1300 kg/m3 kg/m3.1390 0.5 1 1.5Figure 5.12: Profile of differential velocity (wW−wN) along the annulus, cor-responding to the simulations in Figure 5.11 at t = 50 (solid lines)and t = 100 (dashed lines). The colors black, red and blue denoteρˆ2 = 1200,1250 and 1300 kg/m3, perfectly steady. Notice that the displacement regime is also fully laminar here.In 5.13b, we increase the flow rate by a factor of 20. The displacement regimebecomes fully turbulent, and the displacement is still perfectly steady. Finally,in Figure 5.13c, we increase the flow rate to Qˆ = 0.142 m3/s (= 54.6 bbl/min),or equivalently ˆ¯W = 5.0 m/s. Interestingly, we observe the displacement is nowqualitatively similar to the case where there was no density difference, illustratedin Figure 5.8a, suggesting that the density difference is no longer effective.To explain this more comprehensively, in Figure 5.14 we have plotted the con-tours of τw, bφ and ∂bφ/∂φ for the displacement examples shown in Figure 5.13b-c. The sharp change in the wall shear stress (Figure 5.14a) has disappeared in thecase of the displacement with largest flow rate, which has left the wall shear stresscontours very similar to those in Figure 5.10a, for the case of no density differ-ence. Additionally, we can see the jump in bφ has also disappeared, which leads140a)W Nb)W Nc)Figure 5.13: Same as Figure 5.1a, except we have changed the flow rates: a)Qˆ = 0.00071 m3/s (= 0.273 bbl/min), or equivalently ˆ¯W = 0.025 m/s;b) Qˆ = 0.0142 m3/s (= 5.46 bbl/min), or equivalently ˆ¯W = 0.5 m/s; c)Qˆ = 0.142 m3/s (= 54.6 bbl/min), or equivalently ˆ¯W = 5.0 m/ ∂bφ/∂φ being insignificant. Recall that ∂bφ/∂φ is responsible for making theinterface steady, and in its absence, the interface becomes unsteady again.It is worth mentioning here that a mean velocity as large as ˆ¯W = 5 m/s, as inFigure 5.13c, is uncommon in industrial practice. However, the flow can locallyachieve velocities as high as this. For example, this can easily happen, when aheavy mud is to be displaced with a low viscous lightweight wash, as will be shownlater in Chapter 7.The question is now when is the displacement too turbulent? In other words,when is velocity too large that the buoyancy vector and its azimuthal gradient van-ish. The buoyancy vector in the vertical wells has only a φ -component. If we usea turbulent stress scale like (5.5) to calculate the Froude number, we can furthersimplify bφ :bφ =ρ−1Fr2= (ρ−1) gˆ(rˆo− rˆi)12ˆ¯W 2 f f= Ri c2, (5.13)141a) b)W N-10-9-8-7-6-5-4-3-2-10c)W N-12-10-8-6-4-20Figure 5.14: a) Contour of wall shear stress (τw), scaled using a turbulentstress scale (5.5), corresponding to the displacement example shownin Figure 5.13 at t = 100. The left and right panels in each subfigurebelong to the displacement examples shown in Figure 5.13b ( ˆ¯W = 0.5m/s) and Figure 5.13c ( ˆ¯W = 5 m/s), respectively. b) Same as (a), ex-cept the contours show bφ ; c) Same as (a), except the contours show∂bφ/∂φ .where Ri is the Richardson number defined by:Ri =(ρˆ2− ρˆ1) gˆ(rˆo− rˆi)12 ρˆ1ˆ¯W 2 f f. (5.14)Ri is the ratio of buoyancy stresses over turbulent stresses. Evidently, we expectdisplacement flows become too turbulent, when Ri is sufficiently small. To seethe role of Ri more clearly, we conduct a parametric study in which we vary theeccentricity and flow rate of the displacement example in Figure 5.13. We areinterested to identify different displacement flows, e.g. steady or unsteady laminar,steady or unsteady mixed regime and steady or unsteady turbulent (too turbulent).Figure 5.15 shows the result of our parametric study. Here the displacementparameters are exactly those in Figure 5.13, except we play with eccentricity and142flow rate. The density difference is kept constant, and as we increase the flowrate (or equivalently mean velocity), Richardson number decreases. A number ofremarks can be inferred here:• It appears that the criteria to avoid too turbulent flows is simply given by:Ri≥ 1. (5.15)In other words, if the turbulent stress scales are larger than representativebuoyancy scale, then the flow become too turbulent, and therefore unsteady.• To see why (5.15) is approximately correct, we can see from 5.13 than anystable interface will produce a buoyancy gradient of size Ri, since then c2changes slowly from wide to narrow. The wall shear stresses scale with thegap width, meaning the gradient is proportional to e. Hence the balance ofthese terms leads to (5.15).• The condition (5.15) is only a guideline and a necessary condition, and notsufficient, for steady flows. When the eccentricity is too large, even at largevalues of Ri, the flow becomes unstable.• Although (5.15) is a useful prediction, we need to be a little cautious aboutits generality and that of Figure 5.13. We constructed this example by in-creasing Ri through lowering the flow rates - leading eventually to laminarflows. An alternate would be to vary the density difference at fixed flow rate,and probably there are other possibilities.Finally, before we close this section, we aim to analytically derive the axial ve-locity profile for the case of highly turbulent flows. Notice that the streamlines inFigure 5.13c are almost parallel even at the interface, as opposed to those in Figure5.13 b, which have a tweak near the interface. This suggests that the flow is soturbulent that the density difference is not appreciated. For such highly turbulentflows, the displacement dynamics is solely determined by the gap size (eccentric-ity). In addition, since the streamlines are nearly parallel everywhere, the azimuthal1430.1 0.2 0.3 0.4 0.5 0.610 -210 -110 010 110 2Figure 5.15: Steady (green symbols) vs unsteady (red symbols) displacementfor different values of eccentricity e and Richardson number Ri. Cir-cles denote displacement flows where both displaced and displacingfluids are turbulent. Squares denote displacement flows where the dis-placing fluid is turbulent, and displaced fluid is laminar. Diamondsdenote displacement flows where either displaced (with no star) or dis-placing (with a star) fluid has a mixed regime.velocity vanishes and the flow is unidirectional everywhere. Following the closuremodel presented in §2.2, we see as τw→ ∞,Rep ∼ H1−n/2w logHw ∝ τ1−n/2w logτw, (5.16)where Hw is another dimensionless form of the given by (2.21). Neglecting the logterm,Rep ∝(∇ΨH)2−n∼ τ1−n/2w ⇒ ∇Ψ ∝ H√τw (5.17)Now assuming no density difference and fully developed flow, the wall shear stress144linearly scales with the gap size, therefore∇Ψ ∝ H3/2.Upon satisfying the imposed flow rate, we findW =∇Ψ2H=√H∫ 10 H3/2 dφ. (5.18)Notice that the velocity profile above can only be obtained if we neglect thelog term in (5.16). The alternative approach for deriving the velocity profile semi-analytically is to use the slice model approach described in §4.2.4. In fact, thedifference between the above derivation and the slice model is that the slice modeldoes include the log term in (5.16). To test the accuracy of this analysis, we com-pute and plot the velocity field using our 2D simulations for a highly turbulentdisplacement. Then, we compare this with the velocity profile of the slice modelas well as the velocity profile given by (5.18). For the two displacement examplesin Figure 5.13b-c, we compute the velocity profile far from the interface and nearthe interface using both full 2D simulation and slice model. The results are plottedin Figure 5.16. We see that the velocity profile given by (5.18) is somewhat cor-rect, at least for the case where the flow is too turbulent (i.e. Figure 5.13c). Thediscrepancy between the actual and the analytical velocity profile can be attributedto neglect of log term in the estimation of Rep. In fact, inclusion of these terms isequivalent to the slice model, which happens to be very accurate, compared withthe solution obtained by the 2D simulations.5.5 ConclusionsIn this chapter, we primarily focused on fully turbulent displacement flows. Al-though fully turbulent flows are relatively uncommon in primary cementing, wherelaminar or mixed regimes flows are more prevalent, our analysis in this chapterwill help us to understand mixed regimes flows, which we address in Chapter 6,following.Our numerical experiments in this chapter evolved around three main themes:145a)0 0.2 0.4 0.6 0.8φwb)0 0.2 0.4 0.6 0.8φwFigure 5.16: Axial velocity profile, corresponding to the simulations in Fig-ure 5.13b-c, far from interface (blue) and on the interface (red). Solidlines show result of 2D simulations and markers show result of slicemodel simulations. Solid black line is the velocity profile of (5.18). a)Qˆ = 0.0142 m3/s; b) Qˆ = 0.142 m3/s.1. Role of rheology: We show that if the flow is fully turbulent, varying therheology of either fluids does not significantly influence the displacementoutcome. This statement holds as long as both fluids remain fully turbulent.More interestingly, we show that if either fluid falls into laminar or mixedregimes, then the rheological parameters begin to become relevant. Thismeans that controlling turbulent displacement flows and ensuring steady dis-placement is more difficult than laminar displacement flows, because in lam-inar flows one can play with both the viscous and buoyancy forces to ensurea steady displacement. In turbulent flows however, the only tool left is buoy-ancy forces. An implication of this is that in highly inclined and horizontalwells, turbulent flows are bound to be unsuccessful at removing the mudfrom eccentric annuli.2. Competing effects of buoyancy and eccentricity: We explained how ec-centricity of the annulus creates an azimuthal bias in the flow. In addition,we constructed a simple problem analogous to our 2D model, to illustratehow density differences can aid or compete against the biasing effect of ec-centricity. More specifically, given a positive density difference (heavierdisplacing fluid), a counterclockwise velocity field developed about the in-146terface that move displacing fluid from the wide side to the narrow side, anddisplaced fluid from narrow side to the wide side. This mechanism allows theinterface to stabilize and progress steadily. More complicated examples arealso studied, where we gradually increase the density difference and observehow this changes the dynamic of displacement. Simplistically speaking, therole of buoyancy driven secondary flows is more or less similar in laminarand turbulent displacement flows. The results of our study debunk the earlierpresumption that “gravitational forces are not important when displacing inturbulent flow” [166].3. Too much turbulence: We demonstrate that if the flow becomes too turbu-lent, the density difference loses its ability to compete against the biasingeffect of eccentricity and as a result, the interface becomes unstable. In ad-dition, we identified a dimensionless parameter that characterizes the notionof too turbulent. Finally, in the case of a too turbulent flow, we derive avelocity profile analytically, and compared it with those computed using our2D model.A key conclusion therefore of this chapter is that the long-standing industrypractice of preferring a turbulent displacement flow over laminar is over-simplistic and questionable. If the flow rate is increased too much whenturbulent, steady displacements do not occur.147Chapter 6Mixed regimes displacementflowsIn the previous chapters we focused on fully turbulent displacement flows. Morelikely however, is that primary cementing displacement flows fall onto laminaror mixed regimes. Displacement flows with mixed regime can happen because thetwo fluids have different rheological parameters, resulting in one being laminar andthe other being turbulent. In this case, the flow regime changes across the displace-ment front. More interestingly, the change in flow regime can also happen in theazimuthal direction, governed by the annulus eccentricity. As pointed out earlier in§5.3, displacement flows tend to be faster on the wide side of annulus and sloweron the narrow side. Given enough variation in gap thickness (i.e. large enougheccentricity), the flow can be turbulent on the wide side, transitional or laminar onthe narrow side [167]. Examples of such flow configuration were presented in theprevious chapter. More dramatically, in some cases with large eccentricity values(e.g. e& 0.6), the flow can be turbulent on the wide side, and stuck on the narrowside. Such azimuthal variations are a key aspect of 2D simulators compared to therule based system, which are generally based on 1D hydraulic calculations.It is widely agreed upon that one of the most critical parameters in displace-ment flows during primary cementing is the eccentricity (standoff) of the annulus[167]. In Chapter 5, Figure 5.15 presented examples where a successful displace-ment with a steady front is turned into an unsteady unsuccessful displacement,148when the eccentricity is increased. Moreover, even after primary cement is fin-ished and the cement is set, stress distribution inside the cement sheath depends onthe eccentricity of the annulus [100]. This means a more eccentric annulus is morevulnerable to thermal or hydraulic stresses, and as a result more likely to developcracks that compromise the integrity of the cement.Eccentricity is controlled via the use of centralizers, which are fitted to theouter wall of the casing, designed to exert normal forces when in contact with theborehole wall to align the casing with the borehole. A range of centralizers existand there is no standard geometry/mechanical design. Centralizers spacing variesquite substantially, typically from as close as 9 m apart to as far as 40 m apart,depending on the operator’s design choices. In addition, operational realities of-ten override design choices, e.g. in long cemented sections the risk of the casinggetting stuck as it is lowered into the well is significant and centralizers representmechanical obstructions. The effectiveness of centralization can be inferred fromlogging measurements taken after the cement job. Positioning of centralizers isdesigned using a range of models; see Juvkam-Wold and Wu [124], Blanco et al.[25], Guillot et al. [100]. Figure 6.1 shows a typical profile of eccentricity mea-sured by ultrasonic logs [100]. The centralizers position are indicated by the bluediamonds. It may seem surprising that even in a vertical section of wellbore theannulus is not fully concentric. In fact, eccentricity is relatively large and variesconsiderably along the well. The maximum API standard recommendation for ec-centricity is 33% [29].Our model derived in Chapter 3 is capable of simulating wells with varyingeccentricity (see Appendix D, as an example). However, for the sake of simplicityof analysis, we choose to fix the value of eccentricity. It is important to realizethat eccentricity can dominate any other effects. Eccentricity values are rarely re-ported in primary cementing jobs in the literature, nor is there any routinely appliedpost-placement test that measures eccentricity. Also, depending on jurisdiction themechanical design of the centralization program might not be documented andstored external to the operator or service company, (e.g. with a regulator). Thismakes it hard to realistically assess the actual eccentricities in wells. To accountfor these, we will use two ranges of eccentricity in our simulations of this chapter:e = 0.3− 0.4 (mildly eccentric annulus, standoff = 70-60%) and e = 0.6 (highly149Figure 6.1: A typical profile of standoff along an annulus (standoff is 1− e).The blue points show the position of centralizers. The picture is takenfrom [100].eccentric annulus, standoff = 40%). Of course, in a horizontal well this could bemuch larger.In addition to the eccentricity, well geometry (inner and outer radii) variesalong the well. Annulus inner diameters can start at anything up to 20" (51 cm)and can end as small as 4" (10 cm) in the producing zone. In this chapter we willconsider two sizes of casing; described below.Another important family of parameters relevant in the primary cementing isfluids density and rheology, as well as the pumping rates. We have surveyed theliterature to collect a range of realistic parameters that are reported by several op-erators in the technical literature, as case studies, and also from private commu-nications. The results are collected in Table 6.1. A number of remarks are worthmentioning here:• The table has unfortunately many blank cells, indicating the lack of data in150the literature. As an example, many operators register and report the totalvolume of the pumped fluids, whereas from a fluid mechanics point of view,the total volume is largely irrelevant (it is the flow rate that matters).• As pointed out above, eccentricity measurements (if there are any) are almostnever reported. Ironically, eccentricity is perhaps the most critical parameterneeded for the simulations, and also affecting the actual cement job.• The rheological measurements reported here are typically collected usingFANN viscometers and industry standard techniques. In such techniques, theconsistency and yield stress are fitted using relatively few shear rate points,there are in-house variations in fitting methods, systematic extrapolation er-rors, etc. Therefore, some generosity of interpretation is needed, on top ofnatural variations in the fitted parameters due to variations in the fluids.Our focus in this chapter is to identify how flow regime can influence effi-ciency of displacement. It is widely believed in the industry that turbulent flowsare more effective in terms of mud removal and cement placement. However, thescientific evidence supporting this appears to be scant. In Chapter 1 we reviewedthe literature on this and pointed out the weakness in the scientific evidence. In thefollowing we primarily limit our analysis to displacement of mud with a spacer.This is because cement slurries are typically relatively dense and viscous, and aremostly pumped in laminar regime. Thus, fluid design possibilities are more fo-cused at having the “right” spacer relative to the mud. At the end of the chapter,we will briefly explore spacer-cement displacement too.In this chapter, we fix the properties of the mud toρˆ1 = 1200 kg/m3,n1 = 1, κˆ1 = 0.01 Pa.s and τˆY,1 = 10 Pa. (6.1)and play with the properties of the spacer and the flow rate. Notice that the mudhas somewhat typical properties (i.e. it is not rheologically too extreme or tooheavy). The key question is that, given a mud with a moderate viscosity and yieldstress, which spacer displaces the mud more efficiently? A heavy highly viscousspacer that flows in laminar regime or a low viscous lightweight spacer that flowsin turbulent regime or something in between?151Table 6.1: Range of density and rheological parameters as well as pumpingrates for the mud, preflush and cement slurry. Red readings are in SI.Ref Geometry Mud Preflush CementDˆi(in)Dˆo(in)β(◦)e(%)ρˆ (ppg)(kg/m3)κˆ(cP)τˆY (lb/ft2)(Pa)ρˆ (ppg)(kg/m3)κˆ (cP) τˆY (lb/ft2)(Pa)Qˆ (bbl/min)ˆ¯W0 (m/s)ρˆ (ppg)(kg/m3)κˆ (cP) τˆY (lb/ft2)(Pa)Qˆ (bbl/min)ˆ¯W0 (m/s)[10] 9.625 12.25 0 9.2, 1100 55.5 46.5, 22.2 10.2,122210.5, 1258 54 39, 18.6[10] 10.75 12.25 0 9.2, 1100 55.5 46.5, 22.2 10.2,122213.5, 1617 186 41, 19.6[157]5.5 8.75 0 10-10.5,1198-1258[157] 7 8.75 0 10-10.5,1198-1258[66] 10.75 13.6250 14.9,178515.8,189320-100 10-15,4.8-9.65, 0.37 17.8, 2132 70-190 3-20, 1.4-9.65, 0.37[66] 10.75 13.6250 14.9,178515.8,189320-100 10-15,4.8-9.65, 0.37 17.8, 2132 70-190 3-20, 1.4-9.65, 0.37[29] 9.62510.75 90 38-459.1, 1090 8.4, 1010 1-2 0, 0 16.5, 3.7 15.9, 1900 5.1-5.3,1.14-1.29[29] 9.62510.75 90 38-459.1, 1090 8.4, 1200 35 6.7, 3.2 5.3 1.29 15.9, 1900 5.1-5.3,1.14-1.29[29] 9.62510.75 90 38-459.1, 1090 8.4, 1450 70.5 10.9, 5.2 5.3, 1.29 15.9, 1900 5.1-5.3,1.14-1.29[69] 7 9.875 0 15, 1800 19.1, 2300 4.4, 0.47[184]13.37514.75 0 10, 1198 300.8(n=0.8)9.57, 4.6 11, 1318 350.7(n=0.7)4.79, 2.3 2-6, 0.27-0.8112, 1437 500.8(n=0.8)2.39, 1.1 2-6, 0.27-0.81[187]11.87 16 0 12, 1437 14.2,170110, 0.45 16.5, 1977 8, 0.36[95] 7.75 9.875 60 8.7, 1042 8.34, 0.58 15.3, 1833 24-64 4.4-7.4,2.1-3.55, 0.29[95] 5 6.5 90 8.7, 1042 8.34, 1.26 15.3, 1833 24-64 4.4-7.4,2.1-3.55, 0.43[33] 10.75 13.62 0 9.9, 1186 300 12.3, 5.9 3, 0.22 16, 1917 8, 0.59[186]6.6 8.875 45 noisy 12.5,15001.31 15, 1800 0.9[186] 10 11.62 45 8.3, 1000 12.5,15001.23 15, 1800 0.75[243]3.5 4.5 2.5-3.5, 1.6-2.27-11, 4.5-7.1[1] 11.7-19.1,1400-23000.08-1 0.62-17,0.4-11152As we change the physical properties and flow rate to test different flow regimes,it is important to notice that spacer design is typically constrained by the formationfracture pressure and the pore pressure (the pore-frac envelope). This means that ifthe frictional pressure drop is too large (i.e. pumping a highly viscous spacer at alarge flow rate), the spacer can fracture the formation or alternately allow an influx.Which of these is more likely is very well dependent, but in either case there is africtional pressure constraint. In order to perform a more realistic analysis in com-paring fluid designs, we keep the pumping capacity constant. More specifically, weimpose that the total frictional pressure drop generated by the displacing fluid, overthe length of well, down the pipe and up in the annulus, should be less than 150 psi(= 1034 kPa). The value 150 psi is representative of a typical safety margin, butis nominal in that different well plans would have different total frictional pressurelosses.To proceed, we will consider two casing sizes, representing a surface casingand a production casing.6.1 Surface casingWe consider an annulus with the following geometrical parameters:Dˆi = 13”(rˆi = 16.5 cm), Dˆo = 15”(rˆo = 19 cm), ξˆbh = 500 m (6.2)Notice that displacement flows typically develop within a few diameters from theentrance, which is much shorter than the cementing section length. Therefore, forthe sake of a better spatial resolution, we will only simulate the bottom 150 metersof the well.Seven fluids with different properties are listed in Table 6.2 as the displacingfluid. These candidates represent a wide range of parameters, covering from lam-inar low Reynolds displacement to highly turbulent high Reynolds displacements.For each candidate, the flow rate is maximum flow rate possible without violatingthe pressure constraint. The flow rate is computed using the 1D hydraulic proce-dure, as laid down in Chapter 2.Fluids A1, A2 and A3 are all significantly heavier than the mud and they all have153Table 6.2: Candidate preflush fluids for displacement in the surface casing.caseρˆ2 (ppg)(kg/m3)n2 κˆ2(τˆY,2(lb/100ft2)(Pa)Qˆ (bbl/min)(m3/s)µˆe f f(Pa.s)features turbulentwhene = 0.3?turbulentwhene = 0.4?turbulentwhene = 0.6?A1 11.3,13501 0.04 0, 0 0.039, 1.38 0.04 highly viscous, noyield stressno transitional transitionalA2 11.3,13501 0.01 4.2, 2 0.043, 1.50 0.043 moderately viscous,small yield stresspartiallyturbulentpartiallyturbulentpartiallyturbulentA3 11.3,13500.5 0.30 0, 0 0.049, 1.72 0.036 shear thinning, noyield stresspartiallyturbulentpartiallyturbulentpartiallyturbulentAp 10.0,12001 0.04 0, 0 0.039, 1.38 0.04 no density differ-ence, highly viscous,no yield stresstransitional transitional partiallyturbulentB 11.3,13501 0.001 0, 0 0.056, 1.99 0.001 low viscous, noyield stressfully tur-bulentfully tur-bulentfully tur-bulentBp 10.0,12001 0.001 0, 0 0.060, 2.13 0.001 no density differ-ence, low viscous,no yield stresspartiallyturbulenthighlyturbulenthighlyturbulentC 11.3,13501 0.04 10.6, 5 0.016, 0.55 0.27 highly viscous, highyield stressno no noapproximately similar effective viscosity based on their flow rates. The effectiveviscosity is computed using the mean velocity and mean gap width as the nominalshear rate, as illustrated by (5.1).Figure 6.2 shows the snapshots of displacement together with the contours offlow regime at three different times, when the annulus is highly eccentric (e= 0.6).The details on how to interpret the simulation figures, together with the numericalparameter used in the simulation were explained in §5.1. We observe that the flowregime is in transition to turbulence for Fluid A1 and partially turbulent for FluidsA2 and A3. In all cases however, the mud remains either in laminar or transitionalregime, due to its larger yield stress. The change in the flow regime, both axiallyalong the well and azimuthally around well is clearly depicted here. Despite thechange in the flow regime from laminar and transitional in the case of Fluid A1to turbulent in the case of Fluids A2 and A3, the displacement outcome does notappear to have improved significantly.The displacement scenarios discussed above are all unsteady, meaning that theinterface is faster on the wide side and slower on the narrow side. This leads tocontinuous elongation of the interface and accumulation of mud that is left behindon the narrow side. Ideally, we would like to avoid this. Two different directionsmay be pursued to improve the displacement efficiency: i) reduce the viscosity154of the spacer and enhance turbulence (Fluid B) and ii) increase the yield stress ofthe spacer and rely on viscoplastic stresses (Fluid C). The displacement snapshotsfor these two choices are shown in Figure 6.3 . In case of Fluid B (Figure 6.3a) the turbulent regime expands and is found all around the annulus within FluidB. The interface is still progressing unsteadily, however the wide and narrow sidevelocity difference has shrunk slightly (differential velocity criteria has improved),as can be seen by the large volume of mud that is displaced on the narrow side.The displacement is of course improved, which appears to be due to the turbulentregime. On the other hand, for Fluid C, the displacement has deteriorated, as themud on the narrow side barely moves.Before analyzing the displacement efficiency more closely, we also considertwo preflushes that are not any heavier than the mud (fluids Ap and Bp). Thedisplacement snapshots for these two fluids are plotted in Figure 6.4 . Comparedto their counterpart examples with density difference, we observe that these twofluids displace the mud very poorly, leaving a large layer of mud unyielded on thenarrow side. This may seem intuitive, but bear in mind that one strategy to enhancedisplacement quality that is often cited in literature [258] is to use a lightweightpreflush that can be pumped in turbulent regime. Figure 6.4 disproves this ideaentirely. In Chapter 7, we will further investigate the use of lightweight preflushes.Also notice that the pressure limit is less of a concern for these two fluids, meaningthat we might be able to pump them at higher flow rates, as we have decreasedthe static pressure component. However, this will not improve the displacement,because the displacement here enters the too turbulent regime (see the study in§5.4), where increasing the flow rate has no effect on the displacement.To compare the preflush candidates in Table 6.2 more precisely, it is customaryin the literature to quantify the displacement using a volumetric efficiency η(t),which is the percentage of mud that is displaced. Here we compute the efficiencyin the bottom 100 meters of the well. Mathematically, this is equivalent to:η(t) =∫ ξη0∫ 10 raHc2(φ ,ξ , t)dφdξ∫ ξη0∫ 10 raHdφdξ, ξη = 100/(pi rˆa) (6.3)155a) b)c)Figure 6.2: Effect of flow regime in a largely eccentric surface casing. Foreach subfigure, left panels show displacement snapshots at three dif-ferent times, with white lines denoting the streamlines ∆Ψ = 0.25 andright panels shows the corresponding flow regime map. In the regimemaps, dark gray, light gray and white regions are laminar, transitionaland turbulent, respectively and black regions are unyielded fluid. Wellgeometry is given by (6.2) with e = 0.6, displaced fluid properties aregiven by (6.1) and displacing fluid properties are given in Table 6.2 : a)case A1; b) case A2 and c) case A3.156a) b)Figure 6.3: Same as Figure 6.2 except a) case B; b) case C.a) b)Figure 6.4: Same as Figure 6.2 except a) case Ap; b) case Bp.Recall that ξˆ = ξ × (pi rˆa). For a uniform well, (6.3) is simplified toη(t) =1ξη∫ ξη0∫ 10Hc2(φ ,ξ , t)dφdξ . (6.4)Notice that the above definition of efficiency might be somewhat deceptive. Thisis because the volume of annulus on the narrow side is smaller than the wide side,therefore when the mud on the wide side is displaced successfully, the value of157volumetric efficiency grows rapidly. This might happen in spite of having the mudleft behind on the narrow side, but that will not be noticed, because the volumeof narrow side is smaller, and does not influence volumetric efficiency as much.Nevertheless, from the perspective of well leakage, a residual mud channel is asevere problem. As an example, for an annulus with e = 0.6, the widest quartileof annulus has a volume 3.25 times larger than that of the narrowest quartile. Thisnumber grows to 6.15, if the eccentricity is e = 0.8. This is particular problematic,because in annuli with high eccentricity, the value of volumetric efficiencies canreach as high as 80-90%, even if the displacement is poor on the narrow side.In fact, this is the case for the displacement example shown above. Figure 6.5aplots the volumetric efficiency η as a function of time (t) for all the seven preflushcandidates in Table 6.2. Although none of displacement examples can be calledsuccessful, as clearly illustrated in Figures 6.2-6.4, efficiency values are as high as90%.To account for this factor, we define a more stringent measure of efficiency,which is solely based on the displacement on the narrow side. More specifically,we only look at the displacement in the narrowest quartile of the annulus:ηN(t) =∫ ξη0∫ 134Hc2(φ ,ξ , t)dφdξ∫ ξη0∫ 134Hdφdξ=4piξη(pi−2√2e) ∫ ξη0∫ 134Hc2(φ ,ξ , t)dφdξ(6.5)Figure 6.5b plots the narrow side displacement efficiency ηN vs time for all theseven preflush candidates in Table 6.2 . As expected, the narrow side efficiencyreflects a better picture of the displacement quality. We observe roughly two-thirdof the mud in the narrowest quartile of the annulus is left behind. In fact, the bestscore is for Fluid B, and then Fluids C and A2, all at around 30-35%. This is in-teresting, because the laminar displacement (Fluid C) performed almost equallygood as the partially turbulent displacements (Fluids A1 and A2), and fully turbu-lent displacement (Fluid B). More critically, the Fluids Ap and Bp which are bothflowing in fully turbulent regime did not move the mud on the narrow side at all,and their efficiency score remains zero. These observations suggest that the notionthat “turbulent flow cementing yields improved results and reduces the amount ofremedial work required”[32] needs some adjustment.158a)0 50 100 150 20000. 50 100 150 20000. 6.5: Displacement efficiency as a function of time. Well geometryis given by (6.2) with e = 0.6, mud properties are given by (6.1) andpreflush properties are given in Table 6.2. The green line indicates the(dimensionless) arrival time, based on the mean velocity. a) volumetricefficiency η ; b) narrow side efficiency ηN .Upon closer inspection, it appears that the single parameter that has made thedisplacement examples above unsuccessful is the eccentricity of the annulus. Toelucidate the critical role of eccentricity, we have repeated the above simulationsfor a slightly less eccentric annulus (e = 0.4). Figure 6.6 shows the snapshots ofdisplacement examples for five fluid candidates in Table 6.2, when e = 0.4. TheFluids Ap and Bp are not shown here, as their displacement performance remainspoor. Because the annulus is less eccentric, the velocity profiles are slightly moreuniform, but still similar flow regimes are found for the different fluid candidates.For example, the flow regime varies from fully turbulent for Fluid B to partiallyturbulent for Fluid A2 to transitional for Fluid A1, and finally to fully laminarfor the Fluid C. Although the displacement regimes remain relatively unchanged,the displacement efficiency is improved significantly, as shown in Figure 6.7 a.Here four candidate fluids have reached a narrow side efficiency of 90% or higher.Namely, Fluids A2 and B efficiency reaches 99%, which could be attributed tothe turbulent regime. Fluid C also did almost as well. More critically, Fluid A3and Ap and Bp did not perform well, despite the fact that they were in turbulent or159a) b)c) d)e)W NFigure 6.6: Same as Figure 6.2, except e = 0.4 and a) case A1; b) case A2; c)case A3; d) case B and e) case C.160a)0 50 100 150 200 25000. 50 100 150 20000. 6.7: Narrow side displacement efficiency (ηN) vs time (t). Well geom-etry is given by 6.6, mud properties are given by 6.1 and spacer proper-ties are given in Table 6.3. The green lines indicate the (dimensionless)arrival time, based on the mean velocity. a) e = 0.4 and b) e = 0.3.transitional regimes. In Figure 6.7 b, we further decrease the eccentricity to e= 0.3and Fluids A1, A2, B and C reach to 0.95% narrow side efficiency or higher.6.2 Production casingFor the second part of our analysis, we take a smaller annulus with the followinggeometrical parameters:Dˆi = 4”(rˆi = 5.1cm), Dˆo = 6”(rˆo = 7.6cm), ξˆbh = 1500m (6.6)This represents a production casing. Notice that the annulus length is now muchlonger, which will result in having smaller flow rates, as the total pressure drop isstill limited to 150 psi. Similar to the previous section, we only simulate the bottom150 m of the well to study displacement (although the entire flowpath is consideredin calculating the maximal flow rates).Five fluids with different properties are listed in Table 6.3 as the displacingfluid. Similar to the previous section, the maximum allowed flow rates are com-puted using the 1D hydraulics model of Chapter 2. Although the physical pa-161rameters vary quite significantly, the flow rates remain relatively small due to thepressure drop limit, and as a result, in most cases only laminar flows are present inthe annulus.Table 6.3: Candidate preflushes for displacement in the production casing.caseρˆ2 (ppg)(kg/m3)n2 κˆ2(τˆY,2(lb/ft2)(Pa)Qˆ (bbl/min)(m3/s)µˆe f f(Pa.s)features turbulentwhene = 0.3?turbulentwhene = 0.6?A1 1350 1 0.04 0 0.0039,0.380.04 highly viscous, no yield stress no noA2 1350 1 0.01 1 0.0063,0.480.051 moderately viscous, small yieldstressno noA3 1350 0.5 0.25 0 0.0057,0.560.05 shear thinning, no yield stress no noB 1350 1 0.001 0 0.0085,0.840.001 low viscous, no yield stress fully turbu-lentpartiallyturbulentC 1350 1 0.01 2.5 0.0005,0.051.3 highly viscous, high yield stress no noFigure 6.8 shows the narrow side displacement efficiency ηN as a function oftime for two values of eccentricity e = 0.6 and e = 0.3. For the more eccentricannulus, the displacement is relatively poor on the narrow side for all preflush can-didates. Notice that here only Fluid B is in turbulent regime, and the rest are inlaminar regime. Nonetheless, even Fluid B does not reach any satisfactory effi-ciency. More interestingly, the fluid that outperforms the other candidates is FluidC, which has the largest rheological parameters and smallest Reynolds number.Fluid C even performs better than Fluid B, which is in turbulent. When the eccen-tricity is reduced to e = 0.3, the displacement outcome is substantially improved.Here, Fluid B reaches perfect displacement (ηN = 1), and then Fluids A1, A2 andC with narrow side efficiency of 95%.Similar to the previous section, it appears that the displacement regime onlymarginally influences the displacement outcome. Importantly, it is not the turbulentregime that outperforms other displacement regime. Indeed, in the more eccentricannulus, the highly viscous low Reynolds displacement displaced the mud betterthan other candidates.162a)0 100 200 300 400 500 600 70000. 100 200 300 400 500 600 70000. 6.8: Narrow side displacement efficiency (ηN) vs time (t). Well geom-etry is given by 6.6, mud properties are given by 6.1 and spacer proper-ties are given in Table 6.3. The green lines indicate the (dimensionless)arrival time, based on the mean velocity. a) e = 0.6; b)e = Removing the preflushIn the two previous sections, we were investigating how to design an ideal preflushbased on its ability to remove mud from the annulus and achieve a high displace-ment efficiency. Another aspect of the design is to see how these preflush candi-dates are removed by the cement slurry. In particular, is there one that is removedeasier or harder compared to the other ones? We choose a cement slurry with thefollowing properties:ρˆ2 = 1550 kg/m3,n2 = 1, κˆ2 = 0.05 Pa.s and τˆY,2 = 5 Pa. (6.7)The cement is highly viscous and has a moderate yield stress. Therefore, we wouldexpect that it flows only in laminar regime. However, the preflush candidates canstill flow in laminar or turbulent regimes, depending on the displacement flow rate.For simplicity, we opt to work with same geometry as in §6.1 . The displaced fluids(the preflush that is to be removed by the cement) are those listed in Table 6.2 . Wekeep the flow rate as indicated in Table 6.2 . We aim to see whether the flow regimeof the displaced fluid influences the displacement or not.163a)W Nb)c) d)e)W NFigure 6.9: Same as Figure 6.2, except the displacing fluid properties aregiven by (6.7) and the displaced fluid properties are given in Table 6.2:a) case A1; b) case A2; c) case A3; d) case B and e) case C.164a)0 50 100 150 200 250 30000. 50 100 150 200 250 30000. 6.10: Displacement efficiency as a function of time. Well geometryis given by (6.2) with e = 0.6, displacing fluid properties are given by6.7 and displaced fluid properties are given in Table 6.2. The green lineindicates the (dimensionless) arrival time, based on the mean velocity.a) volumetric efficiency η ; b) narrow side efficiency ηN .Figure 6.9 plots the displacement snapshots together with contours of flowregime for all seven preflush candidates in Table 6.2. Although the annulus isrelatively largely eccentric, the displacement outcomes are satisfactory. More pre-cisely, Figure 6.10 shows the volumetric efficiency (η) as well as the narrow sideefficiency (ηN). Although η > 0.95 is achieved in all displacement cases, the dis-placement on the narrow side is poorer, as indicated in Figure 6.10 b. The bestpreflush candidates for removal are Fluids A3 and C. Notice that Fluid A3 per-formed very poorly in terms of mud removal. More interestingly, the fully turbulentcandidates, Fluids B and Bp are harder to remove than the more viscous laminarcandidate, Fluid C.Compared to the two previous sets of example in annuli with e = 0.6, the dis-placements show in Figure 6.9 have higher scores, both in terms of the overallefficiency and the narrow side efficiency. This is primarily due to the large den-sity difference between the cement slurry and the preflush. The other contributingfactor is the larger rheological parameters of the cement compared to those of thepreflushes. These two factors compete against the effect of eccentricity.1656.4 ConclusionThis chapter presented a number of interesting simulations, mostly in mixed flowregimes. We explored different displacement scenarios to identify if the displace-ment regime has any effect on the displacement. In particular, we tested the notionthat turbulent displacement is always preferred than laminar. Our analysis showsthat• Far more important than the flow regime is the effect of annulus eccentricity.Our simulations consistently confirmed that in a largely eccentric annulus(e.g. e & 0.6), displacement of a mud with moderate yield stress and 10%density difference is generally unsuccessful, regardless of flow displacementregimes. On the other hand, in a mildly eccentric annulus (e.g. e . 0.3)displacement is typically successful.• The other key parameter in achieving successful displacement is having suf-ficient density difference between the displacing and displaced fluids. Inparticular, reducing displacing fluid density to achieve turbulent displace-ment was shown to be extremely unsuccessful, no matter how turbulent theflow is.• There is no clear indication that turbulent displacement always outperformslaminar displacement. On the contrary, we showcased examples that thehighly viscous low Reynolds displacement flow achieved a better displace-ment efficiency compared to any other choice, including partially and fullydisplacement flows.We feel the value of this chapter is not specifically in the example considered,as arguably slightly different parameters might favour particular fluid design strate-gies. We see the contribution as threefold: i) First, we hope that this leads to otherresearchers correctly describing the fluid design problem in terms of operationalconstraints and then experimenting within those constraints to see which designperforms better. ii) Second, in selecting measures of success, volumetric bias inthe displacement efficiency needs to be countered. Our narrow side efficiency is166offered as one sensible measure that targets the typical problem area. iii) Lastly, al-though industrial practice likes simple statements/rules, selecting a turbulent flowregime as being “better” does not stand up to serious analysis and this is an areawhere engineers need to work hard on specific wells with simulation tools such asthese, before making a design decision.167Chapter 7Using washes for primarycementingIn the two previous chapters, we primarily focused on density stable displacementflows, meaning that a heavier fluid displaces a lighter fluid in the annulus. Inthis chapter, we consider the opposite scenario where a lighter fluid displaces aheavier fluid (i.e. density unstable). This is particularly relevant in understandingthe effect of low viscous lightweight preflushes in primary cementing. Intuitively,it is difficult to assume how such a fluid, which is in most cases water, can displacea heavy mud in the annulus. A version of this chapter has appeared in Maleki andFrigaard [148].Washes are a category of preflushes. Preflushes are often used as part of thesequence of fluids pumped in primary cementing. Usually two functions are servedby preflushes: I) to wash the drilling fluid ahead, by a combination of turbulenceand chemical reaction; II) to provide a chemically compatible spacer between thelead slurry and the drilling mud. Preflushes can be generally divided into two cate-gories: I) Light weight, low viscosity preflushes known as washes and II) weightedand viscosified preflushes known as spacers. In some cases a wash precedes aspacer, but often only a single preflush is used. In using a wash, protocols suggestthat a minimum contact time is met. The contact time here refers to the time takenfor the wash to pass a position in the annulus. Typically, this minimum contacttime is 10 minutes, although shorter contact times (e.g. 5 minutes) may be recom-168mended when the flow is fully turbulent [167].Washes may be water-based or oil-based. Rheologically, they are generallyNewtonian fluid solutions (e.g. water). They are designed to wash the walls of theannulus free from residual fluids (and any remaining solids), to leave the annuluswater-wet for the cement slurry. In addition, they should break any static gelationof the mud, mobilizing the mud in general. The low viscosity and density of thesefluids allows them to be pumped in turbulent flow regimes. Turbulent displacementis believed to be more effective in terms of cleaning the annulus walls [167, 140],although in operational practice, turbulent flow may not be achieved by all the fluidin the sequence, e.g. due to pump limitation or pore/frac restrictions.Spacers are heavier preflushes with viscosifying constituents. Because of theirlarger density and viscosity, spacers are more often pumped in laminar regime. Thetwo roles of a spacer are: (i) to separate cement slurry from the drilling mud; (ii)to aid displacement of the drilling mud by careful rheological design. Mud-slurryseparation is often necessary because the drilling mud and cement slurry can bechemically incompatible.In this chapter, we primarily focus on washes. Our intention here is to evaluatethe efficiency of displacements involving a wash. In particular, we challenge thenotion that light weight low viscous preflushes provide effective chemical cleaningof the annulus and address the following questions: I) Does the preflush mobilizethe mud around the annulus? II) In the context of using washes, is turbulent dis-placement more successful in displacing the drilling mud? III) Does the preflushprovide the pre-designed contact time needed for chemical cleaning?7.1 Displacement parametersTo focus our study we adopt a range of physical and geometrical parameters typicalof wells drilled in North Eastern British Columbia, when setting surface casing.We consider a vertical annulus of length 450 m, with inner and outer radii of rˆi =24.5 cm and rˆo = 31.0 cm, respectively. For simplicity we assume the annulus isuniformly eccentric, and 3 values of eccentricity are studied; e = 0.1,0.3 and 0.5.As we emphasize the role of the preflush here, we keep the rheology and den-sity of mud and cement constant throughout this chapter. It is assumed that all ce-169menting fluids can be characterized rheologically as Herschel-Bulkley fluids. Thefollowing physical parameters are assumed:Mud: ρˆ1 = 1200 kg/m3, n1 = 1, κˆ1 = 0.02 Pa.s, τˆY,1 = 5 Pa,Cement: ρˆ3 = 1700 kg/m3, n3 = 0.6, κˆ3 = 0.4 Pa.s0.6, τˆY,3 = 7 Pa.The subscripts “1” and “3” denote mud and cement, respectively. We havesaved the superscript “2” for the preflush.The cement slurry properties are within the typical range for oilfield cementsand the mud is not particularly extreme, rheologically speaking. Indeed, in theabsence of the preflush the cement slurry is able to effectively displace the mud,even at e = 0.5. We now vary the density and viscosity of the wash to exploredifferent flow scenarios.Preflush: ρˆ2 = 1050,1100 and 1200 kg/m3, n2 = 1, κˆ2 = 0.001,0.02 Pa.s andτˆY,2 = 0 Pa.For all simulations we assume the annulus is initially filled with the drilling mud.The cementing commences with pumping a preflush volume calculated to give 10minutes of contact time. To study the displacement in both laminar and turbulentregimes, we consider two flow rates: Qˆp f = 0.015 and 0.075 m3/s. The associatedmean velocities are 0.13 and 0.66 m/s. The cement slurry follows the preflush atfixed flow rate Qˆcs = 0.015 m3/s. Note that this reduction in flow rate (for theturbulent case) is to ensure that only the dynamics and behaviour of the preflushdisplacement are varied. We make comparisons between different cases at the timewhen the cement slurry has progressed half way along the annulus. We start belowby using a relatively viscous preflush (κˆ2 = 0.02 Pa.s). The final examples use amore realistic value of viscosity (κˆ2 = 0.001 Pa.s, i.e. water) and show the dynamicof displacement is unchanged.7.2 Laminar washIn this section we consider several displacement scenarios in which the preflush isin laminar flow regime. Our first three examples start with a rather heavy and vis-cous preflush (green fluid in all figures). The density and viscosity of the preflushare ρˆ2 = 1200 kg/m3 and κˆ2 = 0.02 Pa.s and the preflush flow rate is Qˆp f = 0.015m3/s. Figure 7.1 shows snapshots of the displacement (top panels) and flow regime170W NFigure 7.1: Laminar pre-flush displacement in a nearly concentric annulus;e= 0.1. The wide and narrow sides of annulus are marked with a W andN, and only half of the annulus is shown. Preflush density and viscosityare ρˆ2 = 1200 kg/m3 and κˆ2 = 0.02 Pa.s and the pre-flush flow rate isQˆp f = 0.015 m3/s. Top panels: Red, green and blue fluids are mud,preflush and cement slurry respectively. Bottom panels: static regionsare shaded black and moving laminar regions are highlighted in gray. Infuture figures, we show turbulent regions in white.(bottom panels) when the well is nearly concentric (e = 0.1). In the top panels, thered, green and blue fluids represent mud, preflush and cement slurry, respectively.In the bottom panels, the black regions are stationary (where yield stress is notsurpassed and the fluid is, therefore, stuck). The gray regions are moving laminarregions (and in following figures, we show turbulent regions in white).The preflush high shear viscosity and density is identical to those of mud. How-ever, the mud has a yield stress (of 5 Pa.) and the annulus is slightly eccentric.171W NFigure 7.2: Same as Figure 7.1, except e = 0.3These combine to favour flow of the preflush along the wide side. We observethat where the preflush flows on the wide side, the wall shear stress generated bythe preflush is insufficient to mobilize the mud on the narrow side: a static mudchannel is formed. Interestingly, the mud flowing alone (before/after) the wash isin fact mobile on the narrow side. When the cement slurry enters the annulus, bothfluids are fully displaced. This interesting effect is due to buoyancy, i.e. the pres-sure gradient is approximately equal around the annulus. The lighter fluid requirespressure gradient to move upwards and hence the pressure gradient is reduced inthe mud also.What this example illustrates is that even if the well is only slightly eccentric,the preflush displacement is likely to be restricted to the wide side of the annulus.As might be expected, if the annulus is more eccentric, the preflush will be con-tained within a tighter region of the annulus. Figures 7.2 and 7.3 confirm this. In172W NFigure 7.3: Same as Figure 7.1, except e = 0.5Figure 7.2 we observe as the annulus becomes more eccentric (here e = 0.3), thepreflush moves more rapidly in a narrower channel on the wide side of annulus. Ef-fectively, this leaves a larger mud layer stuck on the narrow side as the mud passes.However now at these larger eccentricities, the mud is also static on the narrowside after the wash passes. Interestingly, the cement slurry itself, is stationary onthe narrow side. When we further increase the eccentricity, the preflush movesfaster in a narrower channel and becomes partially turbulent as shown in Figure7.3 (notice the turbulent regions are highlighted in white in the lower panels).In the previous three examples the preflush is effectively contained within thewide side of annulus and appears to have no impact on the mud on the narrow side.In the next few example, we will see that once we reduce the density of preflush(more representative of preflushes used in the industry), the preflush displacementis even less successful. Figure 7.4 and 7.5 show displacement where the preflush173W NFigure 7.4: Same as Figure 7.1, except e = 0.3 and ρˆ2 = 1100 kg/m3density is reduced to 1100 and 1050 kg/m3 respectively. The value of eccentricity isnow fixed at e= 0.3 and all other parameters are kept constant. We observe in bothcases, the preflush displacement becomes unstable which leads to fingering type ofinstabilities. The instability originates with the lower density of the preflush. Thefingering instabilities spread the preflush randomly around the annulus. Although alarger volume of mud is mobilized as a result of the spreading compared to previousexamples in Figure 7.2, the notion that the preflush might provide a buffer layer,separating the mud and cement slurry, has proved to be flawed. Notice here thatalthough our laminar model does not have any mixing mechanism (because Pe 1;see Bittleston et al. [24]), there is still significant dispersion due to secondary flowsdriven by the fingering instabilities. This dispersive mechanism is clearly strongerin Figure 7.5 where the density of the preflush is lowest.174W NFigure 7.5: Same as Figure 7.1, except e = 0.3 and ρˆ2 = 1050 kg/m37.3 Turbulent washIn this section we study the displacement of mud using a preflush in turbulent flowregime. In particular, we are interested to understand if the turbulent displacementcan make the preflush displacement successful, i.e. mobilize the mud around theannulus and separate the mud and the cement slurry. Similar to previous section,we start with a preflush with density and rheology identical to those of mud. Toensure the displacement is turbulent, the flow rate is increased to 0.075 m3/s duringpre-flushing. All other physical parameters are kept constant for now.The first example starts with the heaviest preflush (ρˆ2 = 1200 kg/m3) in a nearlyconcentric annulus (e = 0.1). Figure 7.6 shows the snapshots of the displacementas well as the flow regime. In contrast to the laminar case, here we observe thatthe preflush mobilizes the mud on both wide and narrow sides of annulus. The175W NFigure 7.6: Turbulent pre-flush displacement in a nearly concentric annulus;e = 0.1. Properties are same as Figure 7.1, except e = 0.1 and Qˆp f =0.075 m3/s. Regions shaded light gray are in transitional regime.narrow side, however, moves at a lower speed which leads to the elongation ofthe interface. Comparing Figures 7.1 and 7.6 clearly shows that the turbulent dis-placement is more effective in mobilizing the mud and providing cleaning on thewalls. Once the eccentricity is increased however, the static mud channel againforms on the narrow side. Figure 7.7 shows the turbulent displacement of preflushwhen e = 0.3. The preflush on the wide side rapidly travels the annulus in turbu-lent regime, and leaves the mud on the narrow side stationary. Note that in thesetwo examples, as the cement slurry enters the annulus the flow rate is reduced toQˆcs = 0.015 m3/s, which is why the preflush becomes laminar. Without this reduc-tion the preflush advances more rapidly along the wide side by-passing the mud, asin the first few panels of Figures 7.6 and 7.7.176W NFigure 7.7: Same as Figure 7.1, except e = 0.3 and Qˆp f = 0.075 m3/s.The previous example considered a dense preflush, which resulted in having arelatively sharp interface between the fluids. The next examples investigates tur-bulent displacement when the preflush is lighter than the mud (ρˆ2 = 1100 kg/m3).Figure 7.8 shows snapshots of the displacement and flow regime. As expected theinterface between the preflush and mud becomes unstable and fingering instabili-ties mix the preflush around the annulus leaving a mixture of preflush and mud. Wemay expect that this mixture has a smaller yield stress stress than the mud aloneand therefore is more easily mobilized.7.4 Actual contact timeAs discussed earlier, industrial protocols suggest a minimum contact time that thewash must be pumped in the annulus such that cleaning objectives are met. A typ-177W NFigure 7.8: Same as Figure 7.1, except ρˆ2 = 1100 kg/m3 and Qˆp f = 0.075m3/s.ical designed minimum contact time is 10 minutes [167]. To analyse the notion ofcontact time, we measured the actual contact time (tˆACT ) everywhere inside the an-nulus, during each displacement. As the fluids mix, contact with the preflush needsdefining. At any position within the annulus and at any time during the displace-ment, if the concentration of preflush is larger than 20% of the total concentration,we regard that the mud is in contact with the preflush.Figure 7.9 shows the contours of contact time associated with laminar preflush-ing, as studied. From left to right, the density of preflush is reduced, which resultedin fingering instabilities (illustrated previously in Figures 7.4 and 7.5). We observethat the localised nature of the preflush confines contact time to a narrow region onthe wide side (left panel). The buoyancy-induced instabilities do mix the preflushmore effectively around the annulus and as a result, the 10-minute minimum con-tact time criterion is met in a larger area, although coverage is still marginal (centreand right panel).While actual contact time gives a better picture of the displacement providedby the preflush, it is critically important to realize that the contact time itself maynot guarantee cleaning objectives are met. The idea of the wash is based on havingturbulent flow at the same time as contact. Therefore a second and more tellingmeasure of contact time is the actual turbulent contact time (tˆATCT ), which consid-ers the contact time at which preflush is flowing in turbulent regime.Figure 7.10 compares the actual contact timeand the actual turbulent contact178W N0 200 400 600W N0 200 400 600W N0 200 400 600Figure 7.9: Actual contact time (in seconds) for laminar preflush. e= 0.3 andκˆ2 = 0.02 Pa.s. From left to right, ρˆ2 = 1200, 1100 and 1050 kg/m3.Top row: Qˆp f = 0.015 m3/s (laminar). Bottom row Qˆp f = 0.075 m3/s(turbulent)time for a turbulent displacement with two different preflush densities. Lookingat the top row, we see once the preflush density is reduced, the minimum (10-minute) contact time is satisfied in a wider area. As explained earlier, this is due thebulk dispersion driven by the viscous fingering and secondary flows. However, thebottom row shows that when we consider actual turbulent contact time, apart froma thin layer on the wide side, nowhere else in the annulus satisfies the minimum 10-minute contact time. Reducing the density of preflush does not improve the actualturbulent contact time, perhaps even exacerbates it. A similar story can be foundin Figure 7.11 where the actual contact time and actual turbulent contact time arecompared in a turbulent displacement in which vary the annulus eccentricity. Hereagain we observe the actual turbulent contact time is barely met only in the wideside of annulus, even if the well is only slightly eccentric.7.5 Low viscous washThe examples illustrated above are all for a preflush with a relatively high viscosity(κˆ2 = 0.02 Pa.s), while in many situations washes have viscosities similar to the179W N0 200 400 600W N0 200 400 600W N0 200 400 600W N0 200 400 600Figure 7.10: Actual contact time (top row) vs actual turbulent contact time(bottom row) in seconds. κˆ2 = 0.02 Pa.s. e = 0.3. Left column: ρˆ2 =1200 kg/m3; Right column: ρˆ2 = 1100 kg/m3viscosity of water (∼ 0.001 Pa.s). The final few examples we study investigate awash with such low viscosity and low density. Figure 7.12 and 7.13 show snapshotsof the displacements two different flow rates.Figure 7.12 is a turbulent displacement with a low viscous lightweight pre-flush, the closest example to common current industrial practice. Here the inter-face between the preflush and mud is doubly unstable, because of both the lowerdensity and lower viscosity of the preflush compared to those of the mud (at highshear). Consequently, the fingering instabilities and bulk dispersion effects are180W N0 200 400 600W N0 200 400 600W N0 200 400 600W N0 200 400 600W N0 200 400 600W N0 200 400 600Figure 7.11: Contours of the actual contact time (top row) vs actual turbulentcontact time (bottom row) in seconds. κˆ2 = 0.02 Pa.s. Qˆp f = 0.075m3/s. From left to right, e = 0.1,0.3 and 0.5.much stronger, rapidly mixing the preflush around the annulus. The resulting mix-ture has a lower yield stress (see Figure 7.12, bottom row). Note that here thepreflush is turbulent at the lower imposed flow rates, due to the lower viscosity.Interestingly, when we increase the flow rate, we see the mixing mechanismlargely disappears, suggesting that the interface is now not as unstable as the previ-ous example. This is because when the flow rate is increased, the turbulent stressesdominate the buoyancy stress. Effectively, this means the density difference be-tween the preflush and mud is not felt. Thus the interface remains fairly stable181W NFigure 7.12: Same as Figure 7.1, except ρˆ2 = 1050 kg/m3 and κˆ2 = 0.001Pa.s. The bottom row indicates the yield stress of the mixture at thedifferent times.(Figure 7.13).The above observation can also explain the actual contact time and the actualturbulent contact time shown in Figure 7.14. When the flow is more turbulent here,mixing is significantly reduced and therefore the contact time is only high in thewide side.7.6 ConclusionIn this chapter, we have demonstrated that when using low viscosity and den-sity washes, the wash progressively advances ahead of the lead slurry, channelingrapidly up the wide side of the annulus. Even when fully turbulent, it is ineffective182W NFigure 7.13: Same as Figure 7.1, except ρˆ2 = 1050 kg/m3, κˆ2 = 0.001 Pa.sand Qˆp f = 0.075 m3/s.W N400 600W N0 200 400 600W N0 200 400 600W N0 200 400 600Figure 7.14: Actual contact time (top row) vs actual turbulent contact time(bottom row) in seconds. ρˆ2 = 1050 kg/m3 and κˆ2 = 0.001 Pa.s. leftcolumn Qˆp f = 0.015 m3/s. right column Qˆp f = 0.075 m3/s.183at displacing mud from around the annulus. Furthermore, the advance along thewide side of the annulus drains the volume of fluid which separates the cementfrom the drilling mud. Thus, the idea that the preflush provides a barrier betweenslurry and and mud, shielding incompatibility is largely invalid.We have also computed the actual contact time and actual turbulent contacttime from our simulations. Our threshold of > 20% wash concentration is quitegenerous, but still we are unable to ensure a uniform distribution of contact times.One should note too that due to the localization observed, increasing wash volumesdoes not necessarily help with contact time. This largely invalidates the motivationof measuring chemical cleaning efficiency through a bulk contact time. If thisconcept is to be rescued it needs local computations, as here, and more attentionneed to be paid to the fluid mechanics of the actual displacement flow.Also, still missing in such designs is a mechanical measure of local wall shearstresses. Studies such as Allouche et al. [9], Zare et al. [254] show that staticresidual mud layers can persist at the wall wherever the yield stress is not overcomeby the wall shear stress of the displacing fluid. Thus, it is not only advisable tocompute a contact time but also a measure of effectiveness of mechanical removal,based on wall shear stress and yield stress. As illustrated in Guo et al. [101], suchmeasures often agree well with post-job evaluation techniques.Unlike in previous similar studies such as Guillot et al. [99] where relativelycomplex scenarios were studied, here the setting is about as simple as can be: avertical surface casing in a mildly eccentric well with a modest drilling fluid tobe displaced. This reinforces the suggestions of Guillot et al. [99] that washes areoften ineffective. The parameters chosen for this study were representative of localwells. In the context of typical wells currently drilled in British Columbia, a sec-ond issue with lightweight washes comes from cementing the production casing.Commonly the production casing runs to surface, but only a 200m overlap of ce-ment with the surface casing is required. Therefore, if the production casing is notcemented to surface the interval above top of cement will contain a mix of washand drilling mud, the former of which offers no protection against casing corrosionand is typically chemically loaded, i.e. ideally washes should be circulated fromthe well.184Chapter 8Tracking displacement interfaceusing suspending particlesThe outcome of a primary cementing operation may have many defective featuresthat are not detected during the cementing operation. For example, mud chan-nels and wet micro-annuli may form, annular volumes are only approximatelyknown, fluid losses may occur, eccentricity may result in a top of cement that isnon-uniform (higher on the wide side) and the fluids may move significantly post-placement. Such imperfections call for a robust and sensitive assessment technol-ogy. Currently, the quality of a cement job quality is evaluated using a variety oflogging methods (CBL, VDL, USIT, etc). The most common form is CBL (cementbond log), which measures acoustic amplitude, travel time and attenuation. A va-riety of tools are available providing different spatial resolutions, depths, loggingspeeds etc. Interpretation of the signals infers information about how well the ce-ment is bonded to the inside of the casing. CBL readings generally indicate the topof cement fairly clearly and other large-scale features/defects, see e.g. Tardy et al.[227]. However, the CBL is only run when the cement has fully set, which mayinvolve a costly time delay. Thus, CBL is not run on the majority of cement jobs,directly afterwards, although many wells will have CBL run later in their lifetime,before decommissioning. In addition, since logging is only performed after the ce-ment is set, no corrective measures can be pursued if the cement job went wrong,except performing remedial cementing, which are expensive and can jeopardize185the well integrity.In this chapter we explore the feasibility of a new methodology for tracking theinterface between two fluid stages pumped consecutively during primary cement-ing (spacer–mud or slurry–spacer). In Chapters 6-7, we saw that a density differ-ence between the successive fluids pumped in the annulus greatly facilitates thedisplacement. The main idea here is to exploit this significant density difference,in order to design a tracer particle of intermediate density to sit at the interface. Thelocation of the particles then allows us to evaluate the cement job more accuratelythrough identifying the location of the fluid interfaces and of cementing defects.Tracking of the particles can happen using radioactive or electromagnetic means(e.g. see Elshahawi et al. [69]), but is not addressed here. Instead we focus entirelyon the dynamics of the particles and the mud displacement process.In the following, we initially present a simply toy model and explore how abuoyant particle would behave in an simplistic displacement flow system. Thisleads to a design rule for the proposed process, that we test in both concentric andeccentric configurations. We then extend the concept using our 2D annular model(Chapter 3) and present results for Newtonian-Newtonian fluid displacements, andthen shear-thinning and yield stress fluids. Finally, we discuss applicability andlimitations of this method. A version of this chapter has been submitted for publi-cation and is under revision [149].8.1 A simplified view of annular displacements and inter-face tracking8.1.1 Annular cementing fluid mechanicsAs mentioned earlier, most cemented annuli are eccentric, even when wells arevertical; see e.g. Guillot et al. [100]. Nevertheless, a moderately eccentric well canstill be cemented successfully. Pelipenko and Frigaard [181] showed that success-ful laminar primary cementing is synonymous with a steady state traveling wavedisplacement front. In this chapter, we simply accept that, for some combinationof density and frictional pressure hierarchies between displacing and displaced flu-ids, there is a steady state. The interface between the two fluids advances at steady186speed wˆi upwards along the well. Since the fluids are incompressible, wˆi = ˆ¯w,the mean velocity pumped. As discussed in depth earlier, both ahead of the in-terface and behind the interface, the annular eccentricity results in velocities thatvary around the annulus, i.e. a wider gap means larger wall shear stresses whichthen lead to larger shear rates and a larger flow rate (vice versa on the narrow side).Therefore, in the far-field the fluid velocity along the annulus wˆ varies from wˆW onthe wide side to wˆN on the narrow side of the annulus, where:wˆW ≥ wˆi = ˆ¯w≥ wˆN .As we approach the steadily advancing interface, this implies a redistribution offluid from the far-field flows to match with the steady interface motion. This isachieved via secondary flows near the interface, which flow from the wide sidetowards the narrow side in the displacing fluid and from narrow to wide side in thedisplaced fluid, downstream of the interface. Note that the precise wˆW and wˆN willbe different in displaced and displacing fluids, due to rheology differences betweenthe fluids.8.1.2 Particle migrationFor now we neglect the secondary flows near the interface and address the questionof whether a particle, released into the fluid upstream of the interface can catch theinterface? We therefore suppose that the flow domain consists of two fluids withdensities ρˆ1 and ρˆ2 where fluid 2 is below fluid 1 and the system is density stable(i.e. ρˆ1 < ρˆ2). The interface moves with steady speed wˆi, having position zˆi = wˆitˆ.We assume that the fluid advances at speed wˆ, which may be different from theinterface velocity; see Figure 8.1.The particle diameter is dˆp, its position is denoted zˆp(tˆ) and its vertical motionis determined by the following momentum balance:mˆpd2zˆpdtˆ2= FˆD+ FˆB, (8.1)where mˆp denotes the particle mass, FˆD and FˆB are drag and buoyancy forces,respectively. The particle’s initial position and initial velocity are zˆp,0 and wˆp,0,187Figure 8.1: Schematic of our 1D simplified displacement model.respectively. The particle volume is Vˆp = pi dˆ3p/6 and the particle density is ρˆp. Forsimplicity, we model FˆD using Stokes drag:FˆD =−3pi dˆpµˆ(dzˆpdtˆ− wˆ),with µˆ the fluid viscosity. Equation (8.1) becomes:ρˆpVˆpd2zˆpdtˆ2=−3pi dˆpµˆ(dzˆpdtˆ− wˆ)+∆ρˆ gˆVˆp, (8.2)where ∆ρˆ = ρˆ − ρˆp, in which ρˆ is the fluid density. Considering Figure 8.1, thedensity of the fluid can be written as:ρˆ = ρˆ2+H (zˆp− wˆitˆ)(ρˆ1− ρˆ2) , (8.3)where H (x) is the Heaviside function, (i.e. H (x) = 1 for x > 0, and zero other-wise). Let us assume that the particle density is taken intermediate between that ofthe two fluids: ρˆp = 0.5(ρˆ1+ ρˆ2). The density difference in (8.2) simplifies to∆ρˆ =ρˆ2− ρˆ12[1−2H (zˆp− wˆitˆ)] . (8.4)We see that there are two competing effects in (8.2). The drag term drives theparticle to move at the speed of the fluid wˆ, and in the absence of the buoyancy188force we observe that the particle speed converges exponentially to wˆ. On the otherhand the term ∆ρˆ above switches sign as the interface is crossed, which keeps theparticle accelerating towards the interface position.We define dimensionless variables as follows:zp =zˆpdˆp, t =tˆdˆp/wˆi, w =wˆwˆiThe dimensionless form of (8.2) is:Rep18d2zpdt2+dzpdt−1 =−∆w+ 118Bu(1−2H (zp− t)) (8.5)whereRep =ρˆwˆidˆpµ, Bu =(ρˆ2− ρˆ1)gˆdˆ2pwˆiµˆ, ∆w = 1− wˆwˆi= 1−w.The particle Reynolds number (Rep) is a measure of the ratio of inertial to viscousforces over the scale of the particle. Since dp ≈ 1 mm, Rep ≈ 1−50. The seconddimensionless parameter is the Buoyancy number (Bu) which is a ratio of buoyancyto viscous forces. The last dimensionless parameter is the velocity deficit (∆w)which measures the difference between the interface velocity and far field velocity:∆w < 0 (interface slower than far field velocity) happens in the wide side of theannulus and ∆w > 0 happens in the narrow side.We can further simplify to study convergence to the interface position, by defin-ing:X =18(zp(t)− t)Repand τ =18Rept.The momentum balance becomesd2Xdτ2+dXdτ=−∆w+ Bu18(1−2H (X)) , (8.6)which we can write (8.6) in the form of two first order ODEs:X ′ =V, (8.7a)V ′ =−V −∆w+ Bu18(1−2H (X)) (8.7b)189Evidently X(τ) measures distance from the interface and V (τ) is the relative veloc-ity of particle and interface. When (X(τ),V (τ))→ (0,0) the particles are movingat the interface position and velocity. We consider two cases below.8.1.3 Concentric annulus (wˆ = wˆi)First of all, in a concentric annulus we must expect that wˆ = wˆi, which means that∆w= 0. We find that the system (8.7) always has an equilibrium point (XEP,VEP)=(0,0), which is globally stable. To show this we can use as a Lyapunov functionΓ(X ,V ) =12V 2+Bu18|X |. (8.8)The function Γ is positive in the neighbourhood of the equilibrium point (0,0) andconsists of an increasing sequence of nested curves. We take the time derivative∂Γ∂τ=∂Γ∂X∂X∂τ+∂Γ∂V∂V∂τ= −Bu18[1−2H (X)]V +V[−V + Bu18(1−2H (X))]= −V 2 ≤ 0. (8.9)This proves that the equilibrium point (0,0) is Lyapunov stable, i.e. the phase pathsof the system (8.7) cross the level sets of Γ inwards only.To illustrate this, Figure 8.2 shows the phase plane associated with the system(8.7) when ∆w = 0. The (blue) arrows represent the direction field, always tangentto the solutions of (8.7). The red dots are different examples of initial position andvelocity (X0,V0). Particles starting at different initial condition follow differentphase paths to reach to the equilibrium point, but all converge asymptotically.Changing the value of Bu will affect the particle phase paths followed, butnot the equilibrium point, nor the asymptotic behaviour of the system. IncreasingBu increases the amplitude of the velocity V and consequent oscillations aboutthe equilibrium point. The particle Reynolds number plays a role primarily in the190a)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVb)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVc)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVd)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVFigure 8.2: Phase plane for the system of (8.7) when ∆w = 0, showing thedirection field (blue arrows) and sample phase paths emanating fromselected initial position and velocity (indicated by red dots): a) Bu = 1;b) Bu = 10; c) Bu = 20; d) Bu = 100.length- and timescales. In dimensional terms:zˆp− wˆitˆ = X18wˆidˆ2pνˆ, tˆ =τ18dˆ2pνˆ,where νˆ = µˆ/ρˆ ≈ 10−6−10−4 m2/s. For dˆp∼ 1mm, this viscous-diffusive timescalevaries: dˆ2p/νˆ ≈ 10−2−1s, and the length-scale is advective: the distance moved bythe interface over this timescale.1918.1.4 Eccentric annulus (wˆ 6= wˆi)We now consider wˆ 6= wˆi, for which ∆w is non-zero. As discussed, this correspondsto an eccentric annular flow with interface moving at steady state speed wˆi andparticle within fluid moving at a different speed. If we again consider the system(8.7), we see that X ′ = 0 when V = 0. Along V = 0 we see that V ′ changes signfrom positive to negative, provided that:Bu18> |∆w|. (8.10)Assuming (8.10), if we were to regularize the Heaviside function, we could com-pute a position (XEP,0) of an equilibrium point and it is evident that XEP→ 0 as thebuoyancy term becomes discontinuous. Note too that a smoothed buoyancy termwould anyway be physically reasonable for miscible fluids.We would normally expect |∆w| < 1, although in cases of extreme eccentric-ity the wide side velocity could violate this. Therefore, an approximate stabilitycriterion is:Bu > 18. (8.11)Note that for an extremely eccentric annulus we likely would not have a steadydisplacement anyway.We can explore this behaviour by plotting the phase plane associated with (8.7)for various Bu and ∆w; see Figure 8.3. In Figures 8.3a & b, we have Bu = 1and ∆w = ±0.1, so that (8.10) is not satisfied. Although some of the phase pathsappear initially to approach (0,0), they move past and eventually asymptoticallyX(τ)→∓∞ with small constant V (τ). For ∆w > 0 this represents the narrow sideof the annulus and ∆w < 0 represents the wide side.For larger Bu = 10 with ∆w = 0.1, (8.10) is satisfied and (0,0) is again a glob-ally stable equilibrium point that particles with different initial conditions eventu-ally approach (Figure 8.3c). The behaviour is similar to that for ∆w= 0. For larger∆w = 0.7, e.g. the narrow side on a significantly eccentric well, Figure 8.3d showsthat the phase paths again have X(τ)→−∞ as the particle is left behind the inter-face. Finally, Bu = 20 and ∆w = 1 restores the globally stable equilibrium point(Figure 8.3e).192a)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVb)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVc)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVd)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVe)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5XVFigure 8.3: Phase plane and sample solutions for the system (8.7). The bluearrows show the direction field and the red curves show solutions start-ing from different initial position and velocity (indicated by red dots):a) Bu = 1,∆w = 0.1; b) Bu = 1,