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Fluid mechanics causes of gas migration : displacement of a yield stress fluid in a channel and onset… Zare Bezgabadi, Marjan 2018

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Fluid mechanics causes of gas migration: Displacement ofa yield stress fluid in a channel and onset of fluid invasioninto a visco-plastic fluidbyMarjan Zare BezgabadiB.Sc Mechanical Engineering, Isfahan University of Technology, 2007M.Sc Mechanical Engineering, Isfahan University of Technology, 2012A 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)August 2018c©Marjan Zare Bezgabadi, 2018The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:Fluid mechanic causes of gas migration: Displacement of a yield stress fluid ina channel and onset of fluid invasion into a visco-plastic fluid.submitted by Marjan Zare Bezgabadi in partial fulfillment of the requirementsfor the degree of Doctor of Philosophyin Mechanical EngineeringExamining Committee:Ian A. Frigaard, Mechanical Engineering & MathematicsSupervisorJohn Stockie, Mathematics, SFUSupervisory Committee MemberAnthony Wachs, Chemical and Biological Engineering & MathematicsSupervisory Committee MemberBoris Stoeber, Mechanical Engineering & Electrical and Computer EngineeringUniversity ExaminerRoger Beckie, Earth, Ocean and Atmospheric SciencesUniversity ExaminerJohn Tsamopoulos, Chemical Engineering, University of PatrasExternal ExaminerAdditional Supervisory Committee Members:Mark Martinez, Chemical and Biological EngineeringSupervisory Committee MemberiiAbstractThis thesis studies the buoyant miscible displacement flow of a Bingham fluid bya Newtonian fluid and the invasion of miscible and immiscible fluids into a yieldstress fluid. The objective of the former study is to characterize the residual layerthickness and identify the flow regimes within the range of governing flow param-eters. In the latter, the aim is to capture the invasion pressure of the invading fluidsinto a yield stress fluid, understand the actual invasion process and quantify theeffect of yield stress and other influencing physical parameters.We start the first part of the thesis with density stable displacements. We showthe different parametric effects on the residual layer thickness and present a noveland computationally efficient method for predicting the long-term behaviour of theresidual wall layers. We then extend this study to density unstable displacementand show that static residual wall layers can exist for yield stresses below the min-imum for density stable regimes. These layers are partially static and may also bethicker than the fully static layers encountered in density stable flows. We also finda range of hydrodynamic instabilities, which we map out parametrically, givingapproximate onset criteria. The predictive method for density stable flows is ex-tended to density unstable configurations and appears able to predict the occurrenceof stable displacements.In the second part, we study invasion flows into a vertical column of yieldstress fluid through a small hole. We first examined the invasion of water, usingboth experimental and computational methods. We find that the invasion pressuredepends on the yield stress of the fluid and the height of the yield stress column.However, the invasion process is initially localised close to the hole. Similar resultswere found with glycerin solutions. Interfacial stress effects were then tested withiiia density-matched silicon oil and air, which resulted a non-local invasion. In sum-mary, we find that miscible fluids penetrate locally at significantly lower invasionpressures than immiscible fluids.Finally, for both parts of the thesis, there are a number of useful consequenceshelping to understand the mechanisms leading to gas migration.ivLay SummaryLeakage of oil and gas wells and consequent emission of greenhouse gases hasbeen a focus of the industry for the past 2-3 decades and periodically arouses publicconcern and regulatory scrutiny. Significant attention has consequently been paidto identifying imperfections in the design and execution of those processes that aremeant to seal a well during construction and later production. Principal amongstthese is the complex process of primary cementing. This thesis is built around thefluid mechanics of well leakage as a consequence of primary cementing. We studyin detail two flows that are at the heart of this problem. These flows bring insightinto the three major root causes of gas migration: existence of entrance spacesand paths for gas/other formation fluids, invasion and migration, and a down-holepressure imbalance between formation fluids and those inside the casing.vPrefaceThe four chapters describing the research in this thesis are the results of the researchof the author, Marjan Zare, during the course of her PhD studies at UBC, under thesupervision of professor Ian Frigaard. The following papers have been publishedand/or are submitted for journal publication:• M. Zare, A. Roustaei, and I.A. Frigaard. “Buoyancy effects on micro-annulus formation: Density stable displacement of NewtonianBingham flu-ids.” J. Non-Newtonian Fluid Mech., 247:22-40, 2017.The author of this thesis was the principal contributor. A. Roustaei assistedin performing the computations and contributed to manuscript edits. I.A.Frigaard supervised the research and was involved in the concept formation,analysis and editing of the paper.• M. Zare and I.A. Frigaard. “Buoyancy effects on micro-annulus formation:Density unstable displacement of Newtonian-Bingham fluid displacementsin vertical channels.” J. Non-Newtonian Fluid Mech., 260:145-162, 2018.The author of this thesis was the principal contributor. I.A. Frigaard super-vised the research and was involved in the concept formation, analysis andediting of the paper.• M. Zare, A. Roustaei, K. Alba, and I.A. Frigaard. “Invasion of fluids intoa gelled fluid column: Yield stress effects.” J. Non-Newtonian Fluid Mech.,238:212-223, 2016.The author of this thesis was the principal contributor. The author supervisedco-op students, Q. L. Roberts and M. Ward, in running the experiments. A.viRoustaei performed the computations and K. Alba was involved in earlystages of experimental planning and contributed to manuscript edits. I.A.Frigaard supervised the research and was involved in the concept formation,analysis and editing of the paper.• M. Zare, and I.A. Frigaard. “Miscible and immiscible fluids’ invasion intoa viscoplasic fluid.” Phys. Fluids., 30:063101, 2018.The author of this thesis was the principal contributor. The author supervisedvisitor and co-op students, L. Gassamann and A. Dworschak, in running theexperiments who are acknowledged in the paper. I.A. Frigaard supervisedthe research and was involved in the concept formation, analysis and editingof the paper.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Industrial Motivation . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Primary cementing . . . . . . . . . . . . . . . . . . . . . 31.1.2 Gas migration causes . . . . . . . . . . . . . . . . . . . . 51.1.3 Why is primary cementing difficult? . . . . . . . . . . . . 71.1.4 Problems investigated . . . . . . . . . . . . . . . . . . . 101.2 Yield stress fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Displacement flows . . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Hydrodynamic instabilities . . . . . . . . . . . . . . . . . . . . . 201.4.1 Fingering instability . . . . . . . . . . . . . . . . . . . . 20viii1.4.2 Rayleigh-Taylor instability . . . . . . . . . . . . . . . . . 221.4.3 Kelvin-Helmholtz instability . . . . . . . . . . . . . . . . 221.4.4 Other types of multi-layer flow instabilities . . . . . . . . 241.5 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 252 Buoyancy effects on micro-annulus formation: density stable Newtonian-Bingham fluid displacements in vertical channels . . . . . . . . . . 282.1 A 2D model for buoyant displacement in vertical channel . . . . . 332.1.1 Scope of study . . . . . . . . . . . . . . . . . . . . . . . 362.1.2 Computational method . . . . . . . . . . . . . . . . . . . 372.1.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Variations in layer thickness . . . . . . . . . . . . . . . . . . . . 382.2.1 Computation of residual layers . . . . . . . . . . . . . . . 412.2.2 Effect of varying m and BN . . . . . . . . . . . . . . . . . 422.2.3 Effect of varying buoyancy, χ∗ . . . . . . . . . . . . . . . 462.2.4 Effect of varying B = BN/m . . . . . . . . . . . . . . . . 482.2.5 Summary: moving and static layers . . . . . . . . . . . . 492.3 Results: displacement front classifications . . . . . . . . . . . . . 522.3.1 Classifying the flow types . . . . . . . . . . . . . . . . . 552.3.2 Lubrication displacement model . . . . . . . . . . . . . . 602.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . 632.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 663 Buoyancy effects on micro-annulus formation: density unstable Newtonian-Bingham fluid displacements in vertical channels . . . . . . . . . . 673.1 2D model for density unstable displacement in vertical channel . . 703.1.1 Dimensionless groups and scope of the study . . . . . . . 723.1.2 Computational method . . . . . . . . . . . . . . . . . . . 733.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2.1 Flow regime panoramas . . . . . . . . . . . . . . . . . . 883.2.2 Onset of instability . . . . . . . . . . . . . . . . . . . . . 883.2.3 Static wall layers . . . . . . . . . . . . . . . . . . . . . . 903.2.4 Predicting regimes . . . . . . . . . . . . . . . . . . . . . 101ix3.3 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 1063.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 1094 Invasion of fluids into a gelled fluid column: yield stress effects . . . 1104.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.2 Fluid invasion simplified . . . . . . . . . . . . . . . . . . . . . . 1134.3 Experimental description . . . . . . . . . . . . . . . . . . . . . . 1154.3.1 Fluid characterization . . . . . . . . . . . . . . . . . . . . 1164.3.2 Experimental evolution, calibration and repeatability . . . 1174.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.4.1 A typical invasion experiment . . . . . . . . . . . . . . . 1194.4.2 Invasion and transition stages . . . . . . . . . . . . . . . 1204.4.3 Post-invasion propagation . . . . . . . . . . . . . . . . . 1254.5 Computational predictions . . . . . . . . . . . . . . . . . . . . . 1264.5.1 The model problem and numerical method . . . . . . . . 1264.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 1344.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 1375 Onset of Miscible and Immiscible Fluids invasion into a ViscoplasticFluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1 Dimensional analysis of invasion . . . . . . . . . . . . . . . . . . 1405.2 Experimental description . . . . . . . . . . . . . . . . . . . . . . 1435.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.2.2 Fluids used . . . . . . . . . . . . . . . . . . . . . . . . . 1445.3 Invasion of miscible liquids . . . . . . . . . . . . . . . . . . . . . 1455.3.1 Mixing and invasion stages . . . . . . . . . . . . . . . . . 1455.3.2 From transition to rupture . . . . . . . . . . . . . . . . . 1485.3.3 Fracture and arrest . . . . . . . . . . . . . . . . . . . . . 1515.4 Invasion of Immiscible Fluids . . . . . . . . . . . . . . . . . . . 1535.4.1 Invasion and propagation of Rhodorsil oil . . . . . . . . . 1545.4.2 Air invasion . . . . . . . . . . . . . . . . . . . . . . . . . 1575.5 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 159x5.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 1636 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . 1656.1 Buoyancy effects on micro-annulus formation: Newtonian-Binghamfluid displacements in vertical channels . . . . . . . . . . . . . . 1656.1.1 Scientific contributions . . . . . . . . . . . . . . . . . . . 1656.1.2 Industrial implications . . . . . . . . . . . . . . . . . . . 1716.2 Invasion of fluids into a gelled fluid column . . . . . . . . . . . . 1736.2.1 Scientific contributions . . . . . . . . . . . . . . . . . . . 1736.2.2 Industrial implications . . . . . . . . . . . . . . . . . . . 1786.3 Thesis limitations . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . 182Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185A A lubrication displacement model . . . . . . . . . . . . . . . . . . . 204A.0.1 The functions Ik,p & Jk,p . . . . . . . . . . . . . . . . . . 207B Flow regime panoramas . . . . . . . . . . . . . . . . . . . . . . . . . 209C Development of the experimental setup . . . . . . . . . . . . . . . . 214D Pressure reduction in hydrating cement slurries . . . . . . . . . . . 218D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219D.2 An introduction to the chemistry of the Cement . . . . . . . . . . 222D.3 Chemical kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 224D.3.1 Continuum approach . . . . . . . . . . . . . . . . . . . . 228D.3.2 Governing Equation . . . . . . . . . . . . . . . . . . . . 231D.4 Herschel-Bulkley extended suspension balance model (SBM) . . . 232D.4.1 Suspension transport . . . . . . . . . . . . . . . . . . . . 234D.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240xiList of TablesTable 4.1 Rheological measurements of Carbopol solutions used in theexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Table 5.1 Physical properties of the invading liquids. . . . . . . . . . . . 144Table 5.2 The volume of domes before fracture/splashing and the maxi-mum strain rate of Carbopol during expansion of the dome. . . 148Table D.1 The index of the species and the rate of consumption/productionof each in liquid phase. . . . . . . . . . . . . . . . . . . . . . 228Table D.2 The index of the species and the rate of consumption/productionof each in solid phase. . . . . . . . . . . . . . . . . . . . . . . 228xiiList of FiguresFigure 1.1 Schematic of the process of primary cementing in which ce-ment slurry is pumped down the casing to displace drillingmud from the annulus. The original picture is taken from [6]. . 4Figure 1.2 Major contributing factors to failure of well integrity. Image istaken from [7] . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.3 Root causes of gas migration. Image is taken from [5]. . . . . 7Figure 1.4 Flow curves of ideal visco-plastic fluids: Newtonian, Binghamand Herschel-Bulkley models (n = 0.65). . . . . . . . . . . . 15Figure 1.5 Yield stress fluid elastically deform before yielding and exhibita shear thinning behaviour at high shear rates. Image is takenfrom [28] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 1.6 Schematic of displacement flow of two fluids with differentdensity and viscosity. . . . . . . . . . . . . . . . . . . . . . 21Figure 1.7 Two examples of fingering instability are shown here. Left: ASaffman-Taylor finger obtained with air injected in a viscousfluid, [66], Right: Red fluorescent fingering into Carbopol so-lution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 1.8 Mechanism of Kelvin-Helmholtz instability observed in a frameof reference moving at the average speed of the fluids, ∆Uˆ =(Uˆ1−Uˆ2)/2. . . . . . . . . . . . . . . . . . . . . . . . . . . 23xiiiFigure 1.9 Examples of a)Bamboo waves (BW) instabilities observed incore-annular flow experiments done by Joseph et al. [76]; b)the mushroom type instability which is observed in [90]; c)Inverse bamboo-type instability which is found in the experi-ments of [42] . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.1 Primary cementing of an oil well. a) Fluid-fluid displacementin a narrow eccentric annulus modelled as a longitudinal chan-nel. b) Channelling of mud due to eccentricity, illustrated ina cross-section e.g. [8]. c) Formation of a fluid-filled micro-annulus (modelled here). . . . . . . . . . . . . . . . . . . . . 29Figure 2.2 Comparison of layer thickness h for varying µ2 with µ1 =0.01, τ1,Y = 0.2, τ2,Y = 0.5: (+) data from Fig. 12(d) in [54],(◦) Fig. 3 in [55] and (•) this study. . . . . . . . . . . . . . . 38Figure 2.3 Example of a two-dimensional displacement flow for (Re,χ∗,BN ,m)=(100,200,50,10). Left: concentration profile at successivetimes. Right: Strain rate colourmap at the same times. . . . . 40Figure 2.4 Variations in hmax for χ∗ = 0, 20, 50, 100, 200, 400, 2000. . 41Figure 2.5 Left: χ∗= 20,BN = 50,m= 0.1; Right: χ∗= 400,BN = 50,m=10. (a) & (d) Concentration field c at t = 25. The areas of dis-placed fluid where second invariant of stress has not exceededthe Bingham number are marked with a (+) symbol. (b) &(c) plot a channel cross-section showing (+) Concentration,(◦) axial velocity, and (4) shear rate profiles. These cross-sections are at x = 25 and relate to the displacements of (a) &(d), respectively. . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 2.6 Thickness of residual layer h plotted against viscosity ratio m:a) χ∗ = 20; b) χ∗ = 200; c) χ∗ = 400; d) χ∗ = 1000. . . . . . 44Figure 2.7 Thickness of residual layer h plotted against an effective vis-cosity ratio m(1+ B): a) χ∗ = 20, Re = 0.1; b) χ∗ = 20,Re = 20; c) χ∗ = 1000, Re = 5. . . . . . . . . . . . . . . . . 45xivFigure 2.8 Two-dimensional displacements: displaced fluid is red, dis-placing one is blue. (a) Rheological parameters are (Re,χ∗,BN ,m)=(2,400,5,0.1); (b) (Re,χ∗,BN ,m)= (2,400,5,10); times (rightto left): t = 2, 7, 12, 17, 22 . . . . . . . . . . . . . . . . . . . 46Figure 2.9 Residual layer thickness variation plotted against χ∗ at eachBN is plotted: a) BN = 1; b) BN = 5; c) BN = 10; d) BN = 50.The broken lines in c & d denote hmax. . . . . . . . . . . . . . 47Figure 2.10 Variation of residual layer thickness with B = BN/m: a) χ∗ =20; b) χ∗ = 400. . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 2.11 Two-dimensional displacements: displaced fluid 2 is red, dis-placing fluid 1 is blue: a) (Re,χ∗,BN ,m) = (0.1,20,0,0.1); b)(Re,χ∗,BN ,m) = (0.1,20,50,0.1). Approximate times of sim-ulation (left to right): t = 2, 7, 12, 17, 22. . . . . . . . . . . 50Figure 2.12 Variation of residual layer thickness with: a) χ∗/m and B andb) χ∗/m and 1+B. . . . . . . . . . . . . . . . . . . . . . . . 51Figure 2.13 Spatiotemporal plots of c¯(x, t) for: a) dispersive regime, χ∗ =20,BN = 0,m= 3; b) shock regime (plug type), χ∗ = 20,BN =50,m = 0.1; c) spike regime χ∗ = 1000,BN = 10,m = 3. . . . 53Figure 2.14 Front velocity variation against viscosity ratio, m, at each χ∗ isplotted: (a) χ∗ = 20, (b) χ∗ = 200, (c) χ∗ = 1000, (d) χ∗ = 2000 54Figure 2.15 Plot of h against h f : green symbols hmax = 0; black symbolsh≥ hmax > 0; red symbols h≤ hmax and hmax > 0. . . . . . . . 55Figure 2.16 Interface propagation for χ∗ = 400,BN = 50,m= 10, obtainedfrom: (a) 2D computational results, (identified as frontal shock);(b) two-layer model. . . . . . . . . . . . . . . . . . . . . . . 57Figure 2.17 Interface propagation for χ∗ = 200,BN = 1,m = 3, obtainedfrom: (a) 2D computational results, (identified as a spike); (b)two-layer model. . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 2.18 Dispersive χ∗= 20,BN = 1,m= 3, obtained from: (a) 2D com-putational results, (identified as dispersive); (b) two-layer model. 57xvFigure 2.19 Classification of computed h for hmax > 0: a) frontal shock (redsymbols), h vs h f ; spike (black symbols), h vs hs; b) h≤ hmax(red symbols), h vs h f ; h≥ hmax (black symbols), h vs hs. Thesymbol shapes denote values of m as in Fig. 2.15. . . . . . . . 59Figure 2.20 Comparison of the flow classification of our 2D simulationsand of the lubrication model (both with BN = 0) with the flowregime map of [41] for density stable displacement of twoNewtonian fluids. Small symbols from our lubrication model:blue - dispersive; green - frontal shock; red - spike. Largesymbols from our flow classification: spike 4; frontal shock+; dispersive ×. . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 2.21 Displacement flow classifications: a) 2D computations; b) lu-brication model computations. Symbols: spike - 4; frontalshock - +; dispersive - ×. . . . . . . . . . . . . . . . . . . . 63Figure 3.1 Primary cementing of an oil well. Fluid-fluid displacement ina narrow eccentric annulus modelled as a longitudinal channel. 68Figure 3.2 Examples of observed flows. Left: Stable- (Re,χ,BN ,m) =(15,10,5,10), Right: RT- (Re,χ,BN ,m) = (25,500,5,3). Topfigures show a red-blue colourmap of the concentration. Belowthe figures show a spatiotemporal colourmap (yellow-blue) in-dicating the y-averaged concentration c¯y(x, t), and the insetsshow the same variable plotted against x/t late in the simulation. 74Figure 3.3 Examples of observed flows. KH: Left-(Re,χ,BN ,m)= (150,100,1,0.33),Right-(Re,χ,BN ,m) = (50,100,1,0.1). See Fig. 3.2 for de-scription. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 3.4 Examples of observed flows. KH: Left-(Re,χ,BN ,m)= (15,100,1,0.33),Right-(Re,χ,BN ,m) = (25,50,0,0.33). See Fig. 3.2 for de-scription. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 3.5 Examples of observed flows. I.B.M: Left-(Re,χ,BN ,m)= (270,180,1,10),Right-(Re,χ,BN ,m) = (500,100,5,10). See Fig. 3.2 for de-scription. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78xviFigure 3.6 Examples of observed flows. Footprinting: (Re,χ,BN ,m) =(180,180,5,10). Left: the concentration colourmap. Right:contours of τ(u)−BN , the grey shaded contour indicating whereτ(u)< BN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 3.7 Panorama of flow types observed for m = 0.1 and BN = 1.Markers indicate data position in (Re,χ) plane and flow clas-sification:  - stable,4 - KH,© - IBM; filled symbols - RT. 82Figure 3.8 Panorama of flow types observed for m = 0.1 and BN = 5.ForBN = 10 the flows are similar to BN = 5. Markers as in Fig. 3.7. 83Figure 3.9 Panorama of flow types observed for m= 1 and BN = 1. Mark-ers as in Fig. 3.7. . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 3.10 Panorama of flow types observed for m = 1 and BN = 5. ForBN = 10 the flows are similar to BN = 5. Markers as in Fig. 3.7. 85Figure 3.11 Panorama of flow types observed for m= 10 and BN = 1 .Mark-ers as in Fig. 3.7. . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 3.12 Panorama of flow types observed for m = 10 and BN = 5. ForBN = 10 the flows are similar to BN = 5. Markers as in Fig. 3.7. 87Figure 3.13 Stable and unstable regions for m=0.1: a) BN = 0; b) BN = 1;c) BN = 5; d) BN = 10; - stable, 4 - KH, © - IBM; filledsymbols - RT. . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 3.14 Stable and unstable regions for m=0.33: a) BN = 0; b) BN = 1;c) BN = 5; d) BN = 10. Markers as in Figs. 3.13 . . . . . . . . 90Figure 3.15 Stable and unstable regions for m=1: a) BN = 0; b) BN = 1; c)BN = 5; d) BN = 10. Markers as in Figs. 3.13 . . . . . . . . . 91Figure 3.16 Stable and unstable regions for m=3: a) BN = 0; b) BN = 1; c)BN = 5; d) BN = 10. Markers as in Figs. 3.13 . . . . . . . . . 92Figure 3.17 Stable and unstable regions for m=10: a) BN = 0; b) BN = 1;c) BN = 5; d) BN = 10. Markers as in Figs. 3.13 . . . . . . . . 93Figure 3.18 a) Plot of hmax using the definitions in [103]: extending to χ <12. b) Location of regimes 1-4 in the (yi,χ) plane. c) Variationof the maximal shear stress in fluid 2 max(|τ2,xy|), for χ =10, 100, 1000. d) Plot of hmax,a for χ = 0, 10, 50, 120, 190. . 95xviiFigure 3.19 Values of yi = 0.5− h computed from our 2D simulations: a)m≤ 1; b) m > 1. Symbols (M,◦,♦) represent fully static, par-tially static and moving layers, respectively; BN = 1, 5, 10 areblue, green and red, respectively. The filled symbols are stableand empty symbols unstable. The lines represent transitionsbetween regimes 1-4, according to the fully static wall layeranalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 3.20 Variation of static wall layer position with interface positionfor a) (BN ,χ) = (1,113.78); b) (BN ,χ) = (5,70); c) (BN ,χ) =(5,220); d) (BN ,χ) = (10,90). Each symbol represents a spe-cific m, such that m = (0.1,0.33,1,3,10) are shown respec-tively with (×,♦,+,∗,◦). Inset figures show τxy and U(y) atdifferent m values for each case: a) m = 1, b) m = 0.33, c)m = 1 and d) m = 0.1. . . . . . . . . . . . . . . . . . . . . . 99Figure 3.21 The value of yY at each yi for a)(BN ,m) = (5,0.1), b)(BN ,m) =(5,10), c)(BN ,m) = (10,0.1), and d)(BN ,m) = (10,10). . . . 101Figure 3.22 Variation of maximum static wall layer thickness hmax,p withχ at: a) BN = 5; b) BN = 10. Each symbol represents a spe-cific m: m = (0.1,0.33,1,3,10) are shown respectively with(×,♦,+,∗,◦). The value of hmax,a at similar χ is shown withthe broken line pattern in b. . . . . . . . . . . . . . . . . . . 102Figure 3.23 Plot of h against hmax,p; (M,◦,♦) mark fully static, partiallystatic, and moving layer regimes respectively. BN = 1, 5, 10are blue, green and red, respectively. The filled symbols arestable and empty symbols unstable. . . . . . . . . . . . . . . 103Figure 3.24 Thickness of residual layer against χ , for BN = 10. Sym-bols (M,◦,♦) denote fully static, partially static, and movingregimes, respectively. Each color represents a specific m: ma-genta, blue, black, red and green colors respectively show m=0.1, 0.33, 1, 3, 10. The lines plot hmax,p for m = 0.1, 0.33with aforementioned colors. . . . . . . . . . . . . . . . . . . 105xviiiFigure 3.25 a) Flow regimes from the 1D thin film model of Chapter 2(see detail in Appendix A): ()- Dispersive, or there is onlya frontal shock, (4)- There are two shocks: front and trailing,(◦) - The only possible solution is that the channel plugs upand displaces with a speed VS = 1. b) Flow regimes classifiedfrom the 2D model; () - stable, (◦) - KH. In both figures,BN = 1, 5, 10 are blue, green and red, respectively. Markersize in both figures is chosen to decrease with BN . . . . . . . . 106Figure 4.1 Schematic setup: a) conceptual problem; b) experimental ap-paratus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 4.2 Example flow curves for 4 tests using C = 0.15% (wt/wt) Car-bopol solution. The instantaneous viscosity technique is usedto estimate the yield stress of the solution. . . . . . . . . . . . 117Figure 4.3 Observed stages in the invasion/penetration process . . . . . . 121Figure 4.4 Examples of dome surface shapes after “transition” showingsmooth (a & b) to granular (c-e). White broken lines are aguide to the eye for the surface of the tank and machined hole. 121Figure 4.5 Dimensionless invasion pressures Pi (red circles) and transitionpressures Ptr (blue squares). Error bars indicate the variabilityof measured pressures over repeated experiments. . . . . . . . 122Figure 4.6 Illustrations of transition dome shapes for a range of differ-ent Carbopol concentrations, C, and heights, H. Snapshots fora given C and H set correspond to different experiments, re-peated to reduce variability of the data. . . . . . . . . . . . . 123Figure 4.7 Variations in transition dome size for different Carbopol con-centrations, marked by circles (C=0.15%), squares (C=0.16%)and diamonds (C=0.17%), and dimensionless Carbopol heights,H. Each data point is obtained from more than 3 different ex-periments to ensure reliability. . . . . . . . . . . . . . . . . . 124Figure 4.8 Images of Carbopol fractures after the transition stage for a)C= 0.17%, H = 5.175; b) C= 0.15%, H = 5; c) C= 0.15%, H =5.025; d) C = 0.16%, H = 5.1; e) C = 0.16%, H = 5.075. . . 125xixFigure 4.9 Model geometry computed: a) axisymmetric column with cen-tral flat hole at the bottom; b) axisymmetric column with hemi-spherical incursion of invading fluid. . . . . . . . . . . . . . . 127Figure 4.10 Computed invasion pressures for different column height, H,and hole radii, rh. The dashed line denotes the Poiseuille flowyield limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Figure 4.11 Example stress fields for H = 2. Top row rh = 0.05: τrz, τθθ ,τrr, |τ| (left to right). Bottom row rh = 0.1: τrz, τθθ , τrr, |τ|(left to right). . . . . . . . . . . . . . . . . . . . . . . . . . . 132Figure 4.12 Evolution of the stress fields with H = 8, 12, 15, 16, 16.25for rh = 0.05: left panel τrz; right panel |τ|. . . . . . . . . . . 133Figure 4.13 Invasion pressures, for a hemispherical invading dome: a) smallrd ; b) full range of rd . . . . . . . . . . . . . . . . . . . . . . . 134Figure 5.1 Schematic of the experimental setup. . . . . . . . . . . . . . . 144Figure 5.2 Invasion pressures of miscible fluids against height of the Car-bopol column:  G58; • G45; N water from Chapter 4. . . 147Figure 5.3 Recirculatory vortices formed inside a glycerin dome duringtransition stage. Here Carbopol concentration is 0.15% wt andH = 15.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Figure 5.4 Illustrations of transition dome shapes observed during G45invasion for a range of different Carbopol concentrations, C,and heights, H. Snapshots for a given C and H set correspondto different experiments, repeated to reduce variability of thedata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Figure 5.5 Illustrations of transition dome shapes observed during G58invasion for a range of different Carbopol concentrations, C,and heights, H. Snapshots for a given C and H set correspondto different experiments, repeated to reduce variability of thedata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Figure 5.6 Examples of fracture and arrest stage in: a) H2O, b) G58 c)G45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152xxFigure 5.7 A stepped ramp pressure profile for increasing the applied pres-sure during immiscible fluids injection. . . . . . . . . . . . . 154Figure 5.8 Example invasion of R550 into a density matched Carbopolsolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Figure 5.9 An example of flow developement after injecting R550 intothe Carbopol. Here concentration of Carbopol is 0.15% wtand H = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Figure 5.10 Invasion pressures of immiscible fluids against height of theCarbopol column:  R550, • Air. . . . . . . . . . . . . . . 156Figure 5.11 A typical sequences of bubble formation in Carbopol. In thisexample, concentration of Carbopol is 0.16% wt and H = 10. 157Figure 5.12 The invasion pressure (Pi) of the air versus the number of re-peat experiments made into the same Carbopol. . . . . . . . 159Figure 5.13 The volume of the bubbles versus the number of repeat exper-iments made into the same Carbopol. . . . . . . . . . . . . . 160Figure 5.14 The velocity of the bubbles versus the number of repeat exper-iments made into the same Carbopol. . . . . . . . . . . . . . 160Figure 5.15 Rising velocity of the bubbles against their volume. . . . . . 161Figure A.1 Schematic of displacement geometry. Fluids 1 and 2 are sepa-rated by an interface y = yi(x, t). . . . . . . . . . . . . . . . . 205Figure B.1 Panorama of flow types observed for m = 0.33 and BN = 1.Markers indicate data position in (Re,χ) plane and flow clas-sification:  - stable,4 - KH,© - IBM; filled symbols - RT. 210Figure B.2 Panorama of flow types observed for m = 0.33 and BN = 5.Markers as in Fig. B.1. . . . . . . . . . . . . . . . . . . . . . 211Figure B.3 Panorama of flow types observed for m= 3 and BN = 1. Mark-ers as in Fig. B.1. . . . . . . . . . . . . . . . . . . . . . . . 212Figure B.4 Panorama of flow types observed for m= 3 and BN = 5. Mark-ers as in Fig. B.1. . . . . . . . . . . . . . . . . . . . . . . . 213xxiFigure C.1 Development of experimental setup: a) Initially, a pressureregulator was used to inject air into Carbopol. b) The pressureregulator was replaced by a manometer setup. Pressure wasapplied by adding extra water to its container to make differ-ence between the fluid levels. c) A scissor jack used to adjustthe water column height. d) The apparatus is equipped with anautomated scissor jack. . . . . . . . . . . . . . . . . . . . . . 216Figure C.2 The preliminary results obtained by injecting water (dyed witha black ink) into Carbopol. . . . . . . . . . . . . . . . . . . . 217Figure D.1 Schematic of cement undergoing hydration process including:I) Pre-induction, II) Induction, III) Acceleration, IV) Deceler-ation stages. . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Figure D.2 Cement compartment before and during the hydration process[175]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229xxiiAcknowledgmentsFirst and foremost, I wish to thank my supervisor, Professor Ian Frigaard, for hisimmense patience, constant support and valuable expertise. It was an amazing ex-perience to work with someone so knowledgeable and affable at the same time.Attending conferences and meeting with people from the complex fluids commu-nity and the oil and gas industry, enabled me to develop my own professional skills- Thank you Ian, for all the opportunities you gave me. I feel very privileged tohave worked with you.I would also like to thank Professor John Stockie for his advice and guidanceon the study of cement hydration modeling. Our lengthy discussions on reviewingthe related literature were all to my advantage, thanks!I gratefully acknowledge Dr. Ali Roustaei for his assistance during my com-putational studies.During the past few years, several undergraduate and visiting interns workedwith me on the fluid invasion project in Chapter 4 & Chapter 5. My sincere thanksto all of them for their crucial contribution in tackling that challenging and complexproblem.A very special thanks to my beloved parents, who taught me to have hope andfollow my dreams. Every member of my family, I love you all and thanks forsupporting me through all stages of my education.Last but certainly not least, I want to thank my beloved companion who stoodby me every step of this endeavour. I don’t think I would have been able to see itthrough without you; you were not just the only family I had here, you were alsomy closest friend with your solid support, and love...thank you so much!The following organizations are gratefully acknowledged for their financialxxiiisupport: NSERC and Schlumberger through CRD project 444985-12, BCOGRISthrough project EI-2016-09 and PTAC through project 16-WARI-02, as well asUBC through the 4YF programme. This research was also enabled in part by in-frastructure provided from Compute Canada/Calcul Canada (www.computecanada.ca).xxivDedicationTo my familyxxvChapter 1IntroductionLeakage of oil and gas wells and consequent emission of greenhouse gases havebeen a focus of the industry for the past 2-3 decades, periodically arousing publicconcern and regulatory scrutiny. Any preliminary study of the problem confirmsthat it is a complex phenomenon with causes that combine mechanical, chemical,thermo-mechanical and geo-mechanical processes. Due to this complexity, to un-derstand and mitigate gas migration we need to divide the processes of constructingand sealing the well into smaller pieces that can be studied in detail. In this the-sis, we focus specifically on fluid mechanic causes of gas migration related to theprimary cementing operation.Two key problems addressed here are: (i) formation of a wet micro-annulusduring the mud removal stage of primary cementing; (ii) invasion of fluids intothe cement slurry after placement, driven by a pressure imbalance between annularand formation fluids. The wet micro-annulus refers to a layer of drilling mud thatremains stuck to the walls of the annulus throughout the process of displacement ofthe drilling mud, i.e. as a sequence of fluids are pumped into the well. Fluid inva-sion driven by pressure difference, is inherent in many well operations (production,gas kicks, etc), but here the focus is on the detail of the process at the pore-leveland how this might be influenced/controlled.While the above gives the important industrial motivation for the thesis, it isalso fundamentally a thesis in fluid mechanics and specifically in the mechanics ofyield stress fluids. Drilling muds, cement slurries and many spacer fluids are fine1colloidal suspensions characterized rheologically as yield stress fluids. Washes(fluids having a density and viscosity close to that of water or oil), gas and oil aretypically Newtonian. Thus, the fluid mechanical advances in this thesis are madethrough study of two idealized flows that capture the essence of the two industrialprocesses of interest: yield stress fluid displacement along a channel and fluidinvasion into a column of yield stress fluid.This chapter serves as an introduction to the industrial background, the fluidmechanical background and to the displacement flows and instabilities encoun-tered. The individual chapter introductions also contain additional detailed reviewof relevant literature for each problem. An outline of this chapter is as follows.• In Section 1.1 we introduce the industrial problem of gas migration, whichprovides the motivation for the two parts of this thesis and connects them.• The complexity of the fluid mechanic analysis of the problem stems fromthe fact that we are dealing with yield stress fluids. These fluids have bothnon-linear flow curve behavior and a singularity in their effective viscosity.These fluids are introduced in Section 1.2.• Section 1.3 reviews the background literature underlying fluid displacementflows.• During our study of yield stress fluid displacements we come across a rangeof hydrodynamic instabilities, which we classify in broad terms. In Sec-tion 1.4 as a preamble to this study, we briefly discuss the better known fluiddynamical instabilities that are relevant to our flows.• Finally, we close this chapter with an outline of the thesis structure and ob-jectives, in Section 1.5.1.1 Industrial MotivationThe motivation for our study comes from the chronic problem of so-called annulargas migration, or its observable consequence: surface casing vent flow (SCVF).Hydrocarbons and other gases are present in the earth’s geological strata and gen-erally remain trapped beneath caprock for millennia. In penetrating impermeable2strata with an oil or gas well, we immediately give a pathway for these gases toleak to the surface either directly or via some other permeable strata. Therefore,all instances of gas leakage around a wellbore are evidence of the failure of thehydraulic seal around the well. This seal is created by cement during the processof primary cementing.Well leakage occurs worldwide. In Canada there have been over 500,000 wellsdrilled. The incidence of leakage is hard to quantify. Three recent studies of rele-vance to Western Canada are [1–3], from which we can conclude that somewherebetween 5-20% of wells are leaking, incidence grows with age of the wells, thereare strong local variations (probably geological), and there is a wide distribution interms of the severity of leakage. These features are not unique to Western Canada.Similar studies can be found for most jurisdictions around the world that allow asimilar level of public access to data. Equally, the techniques used to cement oiland gas wells in Canada are globally applied in the industry.Consequences of gas migration are varied. Emissions of methane, a potentgreenhouse gas, are a serious environmental issue. Hydrogen sulfide emissionsare potentially deadly. Slow leakage around the well damages subsurface ecology.Partial leakage can contaminate aquifers and water sources. If the gas is migratingfrom an intended production zone, the reduced reservoir pressure can compromiseproduction. Finally, there are instances where the gas invasion and migration occurduring cement hydration (a focus of this thesis), which can lead to loss of wellcontrol. Although most well control incidents occur during drilling operations(kicks), those that occur during cementing are harder to deal with and are oftenmore severe. A recent example is the Deepwater Horizon oil spill disaster in theGulf of Mexico [4].1.1.1 Primary cementingAs commented above, well leakage occurs only when the primary cemented sealfails. Primary cementing is an operation performed on every well and has beencommonly used since the 1930s. During this process (see [5] for details), a steelcasing is inserted into the newly drilled section of the well which is full withdrilling mud. The flowpath in the well is similar to when drilling fluids flow down3the inside of the casing to the bottom-hole (through a check valve) and then returnupwards in the annulus formed by the outer wall of the casing and the inner wallof the borehole (or a previous casing).In the first step, the in-situ drilling mud is circulated in order to condition thedrilling mud and increase its mobility. A sequence of fluids is then pumped downthe inside of the casing, returning upwards in the annulus. Typically, washes andspacers are pumped as pre-flushes ahead of the cement slurry. Usually the cementslurry (or slurries) follow, separated from the other fluids before and after by 2rubberised plugs that fit inside the casing, preventing mixing. The spacers/washare both to aid mud removal and to provide a buffer between potentially chemicallyincompatible liquids; see Fig. 1.1.CasingCement SlurryWell formationSpacer/Drilling mudFigure 1.1: Schematic of the process of primary cementing in which cementslurry is pumped down the casing to displace drilling mud from theannulus. The original picture is taken from [6].The main objective of this process is to provide zonal isolation, i.e. to pre-vent water, gases and oil in one geological zone from hydraulically connecting toanother zone. Once set and if not contaminated or disturbed, cement is not verypermeable. Thus, failure of the seal (loss of well integrity) is generally due to therebeing some residual fluid left behind (i.e. drilling mud is not fully removed), or for4some other reason that the cement does not bond well with the casing or formation.This means there is a clear motivation to focus on the displacement process of mudremoval.1.1.2 Gas migration causesGas migration problems may occur at different stages during the life of the well.The first stage occurs during the cementing operation until the cement has beenplaced. The second stage is post-placement, occurring between the end of theprimary cementing operation and the setting or hydration of the cement. The thirdstage, post setting, happens after the cement has set. Some authors describe thedifferent stages as immediate, short term, medium and long term gas migration. Itis the first two stages in which fluid flows are critical. Note that although it hasbecome common to categorize gas migration in this way, there is some ambiguityin terminology, e.g. the actual gas invasion/migration may not occur in one of thesestages, but the cause of later migration might occur.Most well integrity problems that lead to gas migration thus develop from faultsin the cementing. As examples, poor bonding at the cement-casing and/or cement-formation interfaces can develop from cement deterioration with time, such asshrinkage or formation of cracks. Mud channels and wet micro-annuli result fromineffective displacement of the drilling mud, but are felt later once the cement hashydrated and the in situ mud has dried. Using the wrong density fluids, or design-ing the cement slurry with an early gelation, or allowing contamination, etc. areall examples of factors that can contribute to gas migration. These are illustratedschematically in Fig. 1.2.All the imperfections listed above and illustrated in Fig. 1.2 can lead to one ormore of the following features. At a basic level, it is believed that all 3 root causesmust be present for gas invasion/migration to occur; see Fig. 1.3.• An imbalance in the pressure, i.e. lower pressure inside the annulus than thepore pressure of the gas in the surrounding rock formation.• A pore or other entrance for formation fluids at the cement-formation inter-face.5fluid densities are too high. Also, considera-tion must be given to the free-fall or U-tub-ing phenomenon that occurs during cementjobs.3 Therefore, cement jobs should bedesigned using a placement computer simu-lator program to assure that the pressure atcritical zones remains between the pore andfracture pressures during and immediatelyafter the cement job.Any density errors made while mixing aslurry on surface may induce large changesin critical slurry properties, such as rheologyand setting time. Inconsistent mixing alsoresults in placement of a nonuniform col-umn of cement in the annulus that may leadto solids settling, free-water development orpremature bridging in some parts of theannulus. This is why modern, process-con-trolled mixing systems that offer accurate37Spring 1996and knowing what can be done to minimizeor counteract their effects.In the past, various techniques have beendeveloped to tackle individual factors thatcontribute to gas migration. However, gasmigration is caused by numerous relatedfactors. Only by addressing each factor sys-tematically can a reasonable degree of suc-cess be expected. There is no single “magicbullet” for gas migration.This article summarizes the current stateof knowledge about gas migration, drawingon field expertise from Dowell, and onexperimental work carried out predomi-nantly at Schlumberger Cambridge Research(SCR) in England. Much of this experimentalwork is unpublished.Setting the SceneSuccessfully cementing a well that haspotential for gas migration involves a widerange of parameters: fluid density, mudremoval strategy, cement slurry design(including fluid-loss control and slurry freewater), cement hydration processes,cement-casing-formation bonding and setcement mechanical properties (above).Although gas may enter the annulus by anumber of distinct mechanisms, the prereq-uisites for gas entry are similar. There mustbe a driving force to initiate the flow of gas,and space within the cemented annulus forthe gas to occupy. The driving force comeswhen pressure in the annulus adjacent to agas zone falls below the formation gas pres-sure. Space for the gas to occupy may bewithin the cement medium or adjacent to it. To understand how, and under what cir-cumstances, gas entry occurs, a review ofthe main mechanisms, including cementhydration and resultant pressure decline,follows. First, however, no cementing articleis complete without emphasizing that goodcementing practices are vital.2 To effectivelycement gas-bearing formations the centralpillars of good practice—density control,mud removal and slurry design—are criti-cal, and here is why.Density: Controlling the driving force—Gas can invade and migrate within thecement sheath only if formation pressure ishigher than hydrostatic pressure at the bore-hole wall. Therefore, as a primary require-ment, slurry density must be correctlydesigned to prevent gas flow during cementplacement. However, there is a danger oflosing circulation or fracturing an interval if1. Bol G, Grant H, Keller S, Marcassa F and de RozieresJ: “Putting a Stop to Gas Channeling,” Oilfield Review3, no. 2 (April 1991): 35-43.2. Bittleston S and Guillot D: “Mud Removal: ResearchImproves Traditional Cementing Guidelines,” OilfieldReview 3, no. 2 (April 1991): 44-54.3. Cement free-fall or U-tubing occurs when the weightof the slurry causes it to fall faster than it is beingpumped. This must be considered when designingdisplacement rates and pumping schedules. Wrong density Poor mud/filter-cake removal Premature gelation Excessive fluid lossHighly permeable slurry High shrinkage Cement failure under stress Poor interfacial bonding■Major contributing parameters during the cementing process, in the order that they typically occur. Incorrectcement densities can result in hydrostatic imbalance. Poor mud and filter-cake removal leaves a route for gasto flow up the annulus. Premature gelation leads to loss of hydrostatic pressure control. Excessive fluid loss con-tributes to available space in the cement slurry column for gas to enter. Highly permeable slurries result inpoor zonal isolation and offer little resistance to gas flow. High cement shrinkage leads to increased porosityand stresses in the cement sheath that may cause a microannulus to form. Cement failure under stress helpsgas fracture cement sheaths. Poor bonding can cause failure at cement-casing or cement-formation interfaces.Figure 1.2: Major contributi g fact r to failure of well integrity. Image istaken from [7]• Pathways in the ce ented annulus through which the invading fluids canmigrate up to the well head.Although we focus on gas invasion/migration, the above 3 causes are equally ap-plicable for liquids.6Gas migrationPath to migrateSpace for entry Annular pressureFormation pressure≤Figure 1.3: Root causes of gas migration. Image is taken from [5].1.1.3 Why is primary cementing difficult?There are many reasons why primary cementing is a technically difficult opera-tion to both perform effectively in a field setting and to understand physically ormechanically. Below we try to group some of these factors into three main settings.Fluids: The fluid displacement problem is to remove the drilling fluid and replacewith a cement slurry. Both fluids are fine colloidal suspensions, rheologicallycharacterised as shear-thinning yield stress fluids. The drilling mud proper-ties are determined by the drilling operation. The density is needed to bal-ance formation pressures and the yield stress is important for cuttings trans-port. This can typically be in the range 5−20Pa, but could be larger for mudswhich dehydrate or are not properly conditioned. Cement slurries generallyhave density larger than the drilling mud, designed partly for primary wellcontrol and partly as a consequence of desired mechanical properties whenset. Typical density ranges might be 1700− 1900kg/m3 (although there aremany lightweight options too) and yield stresses are typically < 10Pa. Be-tween these two (largely constrained) fluids can be pumped various pre-flushfluids (spacers and washes). For these, the fluid properties are in principleopen to be designed, i.e. viscosity/rheology, miscibility/compatibility and7density (within the limits of primary well control).In practice, mud conditioning practices vary, as does the degree to which thesolids were cleaned before cementing, so the mud in the well may be differ-ent rheologically from that believed to be there. The preflushes and cementare mixed at the well site, where water quality/chemistry can be different tothat in lab tests where the fluids design was carried out. Pumping and mixingequipment may control and monitor density, i.e. particularly of the cementslurry, but rheology is not measured and adjusted on the fly. So there is someuncertainty about what is pumped into the wellbore to displace the drillingmud.Geometry: Wellbores are long and thin. A typical cemented section could rangefrom 200m to greater than 2km, with mean annular diameters decreasingfrom 30− 40cm to 10− 20cm, from surface casing to production casing.As a guide, the mean annular gap is 20− 30mm, so that curvature effectsare typically negligible at the annular gap scale, near the top of the annulus.The mean gap width however is not the best indicator of difficulty, as annuliare typically eccentric. Even when vertical, centralisers are needed to keepthe casing approximately centred, but different design methodologies andproducts mean that a uniformly concentric annulus is a rarity. The situation isworse with wells which are inclined and horizontal. Thus, due to eccentricitythe narrow side of the well may be much smaller than the mean.Once we consider annular gaps that may be in the range 5− 15mm on thenarrow side of the annulus, we see why mud removal is an issue. As anintuitive example, a 25Pa yield stress will keep a hydrogel static between2 plates separated by 5mm, i.e. suspending its own weight in air. Heredensity differences between fluids are also present, but lower, in the range100−300kg/m3. Thus for typical muds and eccentricities we see that we arecementing in parameter ranges where drilling muds should be expected tobecome stuck on the narrow side of the annulus, without the aid of pumping(frictional stresses) to mobilise them. Indeed, bridging of the narrow sideannular gap to form a mud channel has been long recognised as a hazard [8].Moving away from regular annuli, other sources of non-uniformity are also8there in the well. Centralizers, casing joints, washouts and residual cuttingsbeds are routine examples of constrictions and enlargements that effect thegeometry locally. Geomechanical stresses may also act to deform the bore-hole away from circular. Collectively, these irregularities can either affect thelocal displacement flow dynamics or can affect calculation of fluid volumesneeded for the well.Operational factors: As well as the physical parameters, operationally primarycementing is a contracted service brought to the well site. Drilling depart-ments/divisions in operating companies contract service companies to per-form cementing. From the drilling perspective, cementing is both a cost anda delay in drilling further. Additional services, such as e.g. running a caliperto determine hole size, or running logging tools after cementing, are alsoviewed as costs. Running additional centralisers in a tight hole is a risk ofthe casing getting stuck. Therefore, most additional well operations beyondthe basics of cement placement are viewed as costs and delays to drilling,and consequently not embraced voluntarily. The point is that these opera-tions that can be seen as reducing the uncertainty, improving the geometry,better conditioning the mud, or later testing the cement placement, do not getan immediate valued benefit. Defects in cement, e.g. that lead to gas migra-tion, may be caused during primary cementing but may not be felt until muchlater, i.e. during production of the well. Although these defects may be detri-mental over many years, the associated costs are not felt at the operationalstage of well construction, nor passed back to the drilling divisions.In the above we have concentrated on physical factors that are most relevant fromthe fluid mechanics perspective of primary cementing. Other considerations alsoexist, such as materials choices and geological influence (e.g. temperature and fluidloss). From a fluids engineering perspective, the above listed issues really amountto there being significant uncertainty and variability in the primary cementing op-eration. In response, this means that the physical understanding needed for theprocess and any model-based predictions made, have to be robust rather than pre-cise.91.1.4 Problems investigatedHaving understood the primary cementing environment, we now focus in on twoproblems that are treated extensively in this thesis. These problems are both rele-vant to the causes of gas migration and are interesting scientifically in dealing withthe mechanics of yield stress fluids in multi-fluid flows.In the first part of this thesis we investigate fluids displacement in the annulussurrounding the casing, which is often incomplete due to a residual layer of the in-situ fluid remaining on the walls. If the displaced fluid (drilling mud) was purelyviscous, residual wall layers would drain over time (or might mix). Consideringtypical velocities and gap widths, the advective timescale of a typical cementingoperation is very long, so situations that drain slowly are not a primary concern.The yield stress in the drilling mud allows it to resist the imposed stresses duringdisplacement and hence to remain static in the annulus, attached to the walls. Theexistence of thin residual mud layers has been termed a (fluid-filled or wet) micro-annulus. As the gap is thin, it is apparent that the layering does not need to beannular, i.e. the main dynamics should be determined by the 2D displacement flowalong the annular gap.This points the way for our study, which simplifies the annulus to a channel,i.e. a longitudinal section of the annulus. We simplify the wide range of differentrheologies by focusing on the key feature of the drilling mud that allows a layer toremain static, i.e. the yield stress, and consider only viscous effects in displacing.The simplest non-trivial and relevant combination of fluids here is a Newtonianfluid displacing a Bingham fluid. We have seen that density differences often oc-cur between cementing fluids and include these in our study, with the displacingfluid being either lighter or heavier than the displaced fluid. Finally, to limit thedirection of buoyancy, we only consider vertical channels (annuli). In this case weknow that many different effects will be observed at different inclinations, but thevertical orientation is most relevant to cementing of surface casing, which is the keybarrier for gas migration. The above simplifications lead to an idealised problemthat is addressed in detail in Chapter 2 and Chapter 3. Even with these simplifica-tions the flows studied are governed by 4 dimensionless parameters. Without thesesimplifications, the full annular flow with 2 shear-thinning yield stress fluids would10be governed by 12 dimensionless parameters - which is intractable to study in anycomplete way.The industrial objective of this part of the thesis is to understand the physicalphenomenon of how a static residual mud layer forms during the displacement flow.Since density differences are always present in the primary cementing, we’d liketo know how buoyancy affects the efficiency of the displacement process in termsof preventing the layer formation or reducing the thickness of this layer. Equally,with the pre-flushes that are pumped there is some freedom to vary the fluid rheol-ogy and we’d like to know the effect on wall layers of e.g. changing the viscosityratio between fluids, or pumping faster/slower (although we restrict our study tolaminar regimes). In vertical annuli there has long been an industrial recommen-dation to pump the fluids in a sequence of increasing density (mud, spacer, cementslurry) to have a positive density difference which aids in displacing the fluids, andincreasing frictional pressure (viscosity); see e.g. chapter 5 in [5]. Thus, we hopeto understand the effect of recommended practices on the wet micro-annulus for-mation. On the other hand, displacing from below with a less dense fluid is alsonot uncommon, e.g. using a chemical wash or lightweight spacer. One question inusing such fluids might be whether they can destabilize the flow and reduce staticwall layer thickness.As well as fully static wall layers, there is a temporal aspect to the displacementtoo. It is possible to observe a layer next to the wall but it doesn’t mean thatthis layer is necessarily static; this layer could be moving or partially static. Welook at these possibilities and see that really we want to be able to estimate howthick the layers next to the wall can be and make a long term prediction abouttheir behavior. The long time behaviour of the wall layers is a crucial factor indetermining the efficiency of the primary cementing later, in blocking a crucialconduit for gas migration. On the other hand, a more critical look at a cementingdisplacement reveals that not all parts of the well are the same from the temporalperspective. Near the bottom of the cemented well, the fluid-fluid displacement(and associated time for drainage and cleaning) is much longer than that near thetop of the cemented interval.A more long term objective in studying these flows is to develop understand-ing that can be used in field settings as part of the cementing design. One tool for11doing this is a two-dimensional annular simulator that has been developed specif-ically to study these displacement flows. This model relies on the narrow gap inthe annulus to simplify the Navier-Stokes equation: a Hele-Shaw gap-averagingapproach. A number of experimental and numerical displacement studies withHele-Shaw model approach have been carried out [9–18] and are implemented inSchlumberger proprietary software, the WELLCLEAN II simulator, which is cur-rently in use (for more detail see [19]). This type of simulator is also becomingmore popular generally in the industry. Such simulators rely on averaging the vari-ables across the gap, which loses the resolution that is needed for determining walllayers and micro-annuli. However, these simulators do give some local predictionof fluid concentrations, wall shear stresses and average velocities, throughout thecementing operation. Therefore, ideally we would like to model the layer forma-tion in a way that is complementary and which can eventually be coupled to thesimulators to predict the occurrence of wall layers.The second part of this thesis relates to the invasion phase of gas migration,after the displacement flow has stopped and the slurry is in the annulus as hydrationbegins. Conventionally, it is perceived by industry that any pressure imbalancewill lead to invasion of formation fluids into the annulus (see Fig. 1.3). In essence,this suggests that the interfacial tension and the rheology of the cement slurry donot have any effect on the invasion pressure. From a mechanics perspective, thisassumption that any pressure imbalance leads to gas invasion is valid only if the in-situ fluid is a purely viscous fluid and if capillary effects are negligible. However,cement slurries behave as yield stress fluids and so we might expect that they willrequire a minimum pressure imbalance to initiate invasion.The chief question raised here is what the effect of the yield stress is on theinvasion process. As hydration commences cement slurries thicken - so does in-creasing the gel strength increase the resistance against the invasion of the fluids?Is the invasion pressure different for formation gases than for formation brines oroil? What are the other key parameters, rather than the yield stress, that determinethe sufficient pressure imbalance for invasion to occur? These are the topics westudy in this second part of my thesis. The industrial motivation is clear: not onlythe invasion process is not well understood/studied, but also each insight gainedmay give ideas for how to mitigate this chronic problem. More fundamentally, this12study allows us to re-examine Fig. 1.3 and reconsider whether all 3 root causes arenecessary.As with the first problem, we study the invasion problem in a simplified setting,that we describe fully in Chapter 4 and Chapter 5. We consider a tall column ofstatic yield stress fluid with a single small “hole” in the bottom wall of the containerthrough which fluids may invade. We test a range of different fluids as invadingfluids, in each case increasing the pressure at the hole continually until invasionoccurs. We use transparent fluids and various visualization techniques in order toobserve the actual invasion event, which we describe later in detail. The singleinvasion hole is to represent an isolated pore in the rock. In hydrocarbon bearingrocks, permeabilities vary widely. Representative pore sizes can be in the µm rangefor conventional reservoirs, down to the nm range for many tight/unconventionalreservoirs, an example of the latter being the Montney shale found in Northeast-ern British Columbia [20]. For porosity ranges less than (or equal to) 4%, poresabutting the newly drilled borehole are essentially isolated holes on a surface.1.2 Yield stress fluidsA significant number of fluids used in our daily life like toothpaste, hair gel andpeanut butter, as well as working fluids in geophysical and industrial settings, arecategorized as viscoplastic fluids. As discussed in Section 1.1, the working flu-ids in cementing operations (e.g drilling mud, cement slurry some preflushes), arealso viscoplastic fluids. The key feature of viscoplastic fluids is their characteristicyield stress. More precisely if such fluids are subjected to a stress more than theiryield stress they deform and flow, but otherwise they exhibit a solid-like behav-ior. Hence, yield stress fluids exhibit a potential twofold response to the appliedstresses: deformation or plastic flow [21]. Bingham [22] presented the first andsimplest model for explaining the behavior of these fluids. In tensorial form theconstitutive equations for a Bingham fluid are:τˆi j = (µˆp+τˆYˆ˙γ) ˆ˙γi j, ⇔ τˆ > τˆY (1.1a)ˆ˙γi j = 0, ⇔ τˆ ≤ τˆY . (1.1b)13Here µˆp is plastic viscosity, τˆ is deviatoric stress tensor and ˆ˙γ is strain rate tensor.The second invariant of the strain rate and shear stress are defined as:τˆ =√12τˆi jτˆi j, (1.2)ˆ˙γ =√12ˆ˙γi j ˆ˙γi j. (1.3)In this thesis, we adopt the notational convention of showing all dimensional quan-tities with the ·ˆ accent and all dimensionless quantities have no accent.According to the Bingham model, yielding occurs once the second invariant ofthe stress tensor at each point surpasses the yield stress of the fluid. This assump-tion relies on the von Mises yield criterion. There are two popular generalizationsfor the Bingham equation that take the shear thinning behaviour of the fluid intoaccount, the Herschel-Bulkley [23] and Casson models:τˆ = τˆy+ κˆ ˆ˙γn, ⇔ τˆ > τˆY , (1.4)τˆ1/2 = τˆ1/2y + µˆp ˆ˙γn, ⇔ τˆ > τˆY , (1.5)where κˆ and n are the consistency and power law index, respectively. The Bing-ham model is a specific case of the Herschel-Bulkley equation, when n = 1. TheHerschel-Bulkley (HB) model also includes a power-law shear-thinning behavior(n < 1) after yielding, but note that all three models are shear-thinning as the effec-tive viscosity decreases with ˆ˙γ . For many practical viscoplastic fluids the power-law index is found to vary in a range of 0.2-0.8; see e.g. [24]. The variable power-law index provides flexibility for fitting experimental data and hence the Herschel-Bulkley model is generally more applicable than the Bingham model and is widelyused. On the other hand, the Bingham model retains its popularity as being math-ematically simpler and having fewer parameters. Example flow curves are plottedin Fig. 1.4.In practice, real yield stress fluids exhibit time dependent behavior in terms ofviscoelastic and thixotropic responses [25], that cannot be described by the afore-mentioned models. In other words, for real viscoplastic fluids, the effective vis-cosity is not only a function of yield stress and degree of shear thinning of the140 10 20 30 40 50 60 70 80 90 1000102030405060708090100Newtonian FluidIdeal viscoplastic fluid (Bingham model)Shear thining viscoplastic fluid (HB model)Figure 1.4: Flow curves of ideal visco-plastic fluids: Newtonian, Binghamand Herschel-Bulkley models (n = 0.65).fluid, but also depends on the shear history of the fluid: ηˆ( ˆ˙γ, tˆ). The shear his-tory dependency of the fluid can be neglected and the fluid can be considered as asimple yield stress fluid, only if the time scale of the problem is much larger thanthe rheological time scale of the fluid (which could be a relaxation timescale or athixotropic/structural timescale).In addition, real viscoplastic fluids are not completely rigid when the imposedstress is less than the yield stress. Instead, the fluids undergo some elastic defor-mation; see [26–28]. Hence the stress in a simple yield stress fluid, below the yieldpoint, can be assumed to be a linear function of its elastic modulus (Gˆ) as follows,τˆ = Gˆγˆ ⇔ τˆ ≤ τˆY . (1.6)The non-ideal behavior of the yield stress fluids becomes more significant in thetransition between the solid and fluid regimes. The presence of viscoelasticity,shear banding and hysteresis are reported in this regime for a wide range of vis-coplastic fluids; see [29–32].As mentioned in the previous section, in this thesis we focus on the effect ofthe yield stress of the fluid on two flows that are very relevant to gas migration pro-cesses in an oil and gas well. Therefore, in order to simplify these problems and150 2 4 6 8 10 12 14 16 18 20020406080100120140160180  Shear stress (Pa)Shear rate (s-1) Yield stress:YW0,0 0,5 1,0 1,5 2,0020406080100120Shear stress (Pa)Strain( ) YW J W|YJSummary: yield stress fluid behavior   Y GW W W Jd Ÿ     nY Y KW W W W Jt Ÿ  Elastoplastic Viscoplastic57Figure 1.5: Yield stress fluid elastically deform before yielding and exhibit ashear thinning behaviour at high shear rates. Image is taken from [28]capture the leading order effect of having a yield stress, the Bingham model is usedto model the rheology of the fluids in our numerical computations. Furthermore,in our experimental study we use Carbopol solutions. These are widely consideredas model viscoplastic fluids to work with, due to both stability and transparency.Carbopol does not show significant thixotropic behavior and the Herschel-Bulkleymodel can be considered as a good approximation to describe its flow curve behav-ior. Other rheological effects do manifest at stresses close to the yield stress, but inpractice there are no ideal fluids better to work with.1.3 Displacement flowsFlow of displacement fluids and removal of visco-plastic fluids from interior ge-ometries have been the subject of many studies in a variety of contexts, e.g. bio-medical, cleaning of food processing equipment, cementing of oil and gas wells.Here we are concerned with the initial formation of micro-annuli during displace-ment of the drilling mud from the annulus surrounding the steel casing. Cementedannuli are a few centimeters wide and 100’s of meters long. The fluids involvedare typically miscible, but due to the geometry and flow rates, Pe´clet numbers (Pe)are very large. Mathematically, we study the limit Pe→ ∞ of (laminar) miscibledisplacement flows. An equivalent limit is that of immiscible displacement flows16with infinite Capillary number Ca. Thus, we draw insight from both miscible andimmiscible displacement flows and review some of the most relevant studies.The study of immiscible displacement of Newtonian fluids in capillary tubeswas initiated more than 50 years ago by Taylor [33], and then by Cox [34, 35]. Itwas shown that for two viscous fluids the residual layer thickness asymptotes to aconstant value as Ca→ ∞. At a fixed Ca, the residual layer thickness interestinglydecreases as the viscosity ratio of displaced to displacing fluid increases (denoted min our study) see [36], which is counter-intuitive, i.e. the thinnest residual layers arefound for gas-liquid displacement. In the miscible fluid context, Chen and Meiburg[37], Petitjeans and Maxworthy [38] studied displacements of a viscous fluid witha less viscous fluid in capillary tubes computationally and experimentally, respec-tively. It was observed that at large Pe (& 105), the residual layer thickness alsoasymptotes to a constant value, but the thickness of this layer increases with vis-cosity ratio m. Some comparison is made with the large Ca limit, although it ispointed out that the original results of [33] do not extend to this limit. This appar-ent discrepancy has been discussed by [39]. They attributed the discrepancy to thefact that the residual layers are not stationary for fluids with comparable viscos-ity, which affects the calculation of residual layer thickness in [38] via the methodreported by Taylor [33]. This is shown more clearly in the recent experiments of[40].Buoyancy effects were included in [38], but these were not particularly signif-icant at large Pe. Buoyancy was studied in more detail by Lajeunesse et al. [41],who studied density stable displacements of miscible fluids in a Hele-Shaw cell,both experimentally and theoretically. Flows were studied over a range of viscos-ity ratios and buoyancy numbers, and not strictly confined to low Re. We applythe analysis used in [41] to analyzing the flow types observed in our numericalsimulations; see later in Chapter 2 and Chapter 3.Whereas Newtonian fluids continuously deform when a shear stress is applied,yield stress fluids do not have to deform. Therefore, in the case of yield stressfluid, static residual layers may develop on the walls. Displacement of the yieldstress fluids in pipes have been studied in both miscible [42–46] and/or immiscible[47–51] scenarios. Others have studied similar flows in Hele-Shaw geometries[52, 53]. All the aforementioned studies confirm the presence of static residual17layers or zones/channels, due to the yield stress of the displaced fluid.Among the studies of miscible displacement of yield stress fluids, Alloucheet al. [54] studied the miscible displacement of two visco-plastic fluids in a planechannel numerically and determined sufficient conditions for the non-existence ofa static wall layer using a simple 1D model. Gabard and Hulin [43] experimentallystudied miscible displacements of non-Newtonian fluids with zero and non-zeroyield stresses by less viscous and mostly Newtonian fluids of the same densityin a vertical tube. These were generally at low-moderate Re and for yield stressdisplaced fluids. The experiments showed a steadily moving front leaving behinda static layer of uniform thickness, qualitatively analogous to the flows in [54].In [54] predictions were made of the static layer thickness that represented thethickness of the computed layers reasonably well and depended primarily on thedownstream fluid flow. However, deeper examination in [55] showed that the pre-dictions of [54] could not account for observed variations in layer thickness as Reis increased, and prediction of the layer thickness without computation remains anunsolved problem.Among the studies of immiscible displacement of viscoplastic fluids, Soaresand Thompson [51] have extended the numerical methodology and analysis of [36]to immiscible inertia-less flows in pipes and channels, encompassing: Newtonian-Newtonian channel flows [56]; power law displacing Newtonian pipe flows [57];visco-plastic displacing Newtonian channel flows [58]; visco-plastic displacingvisco-plastic channel flows [59]. In each case, the methodology is very similar.Two-dimensional simulations are carried out over broad ranges of Ca at differ-ent rheological parameters and there is a theoretical analysis of the upstream anddownstream 1D flows. The latter can be used to predict the thickness of the resid-ual layer, provided one knows the pressure gradient and total flow rate, plus thefluid properties. This analysis essentially samples the parameter space of multi-layer flows, for different non-Newtonian fluid pairs. A similar analysis is presentedin [60], focusing on the relationship between the interface position and frictionalpressure gradient, and the 2 individual flow rates of the fluid layers. Soares andThompson instead argue that the interface position cannot be determined from thetotal flow rate and the frictional pressure gradient, for certain critical values of vis-cosity ratio (see [36]), and they extend this approach in their later work. For viscos-18ity ratios below the critical values only recirculatory streamline patterns are found,whereas above the critical viscosity ratio, both bypass and recirculatory streamlinepatterns are found (as well as transitional parameter ranges). Within this param-eter space, necessary conditions for the bypass patterns are found by comparingthe downstream centreline velocity with the front velocity, which is analogous tothe method of [54]. The computational method of Soares and Thompson is cer-tainly superior to that used here (and in [54]) for looking at steady state streamlinepatterns. The only negative aspect is that some of the steady state streamlines ofSoares and Thompson might not be found in a transient flow due to hydrodynamicinstabilities and other transients.Gas displacement of visco-plastic fluids has also been studied by various au-thors, typically in capillary tubes e.g. [48–51], and along ducts e.g. [61, 62]. Withrespect to our work, these studies lack the effects of buoyancy and of viscositydifferences. Poslinski et al. [47] studied gas penetration into a tube filled withviscoplastic and power-law fluids, both theoretically and experimentally. Theyshowed that the thickness of the residual layer is always smaller than that of New-tonian fluids, irrespective of the capillary number, provided that the power lawindex is smaller than unity. In such displacement flows static layers also routinelydevelop, but the displacement dynamics at the front are different as the displacingfluid is inviscid.Taghavi et al. [63] studied heavy-light and light-heavy displacement of vis-coplastic fluids in a channel oriented close to horizontal, using a thin-film/lubricationmodelling approach. They showed that a yield stress in the displacing fluid en-hances the displacement flow process, while a yield stress in the displaced fluidleads to completely static residual wall layers. A more detailed study of buoyanthorizontal channel displacements using a 2D transient numerical simulation wasrecently made by Eslami et al. [64]. Aside from these two studies, the role ofbuoyancy in the liquid-liquid displacement of yield stress fluids and consequentthickness of the residual layer are missing in the literature and remain open ques-tions. In this thesis, we study the displacement of a yield stress fluid along a ver-tical channel, considering the effects of buoyancy forces in the same or oppositedirection as the flow; see Chapter 2 and Chapter 3.191.4 Hydrodynamic instabilitiesOne of the main concerns in primary cementing is achieving an efficient displace-ment of the drilling mud by the preceding washes and/or spacers and a completeplacement of cement in the annulus. In laminar flows as discussed earlier (seeSection 1.1.4), it is recommended to keep a 10% density hierarchy between flu-ids pumped; see e.g. [5]. However, this is not universally adopted and many jobdesigns include a chemical wash (basically water), which is generally much lessdense than the fluid which is displacing. This mechanically unstable configurationis one in which fluids mixing and hydrodynamic instability may develop. Otherdisplacement flows result in regions of multi-layer flow, between fluids of differentrheology and density. This type of flow is also often destabilized. Although inter-face instability might mostly be regarded as a threat to a successful displacement,one can also imagine that instabilities might reduce the thickness of residual walllayers or might mobilize static layer. Thus, below we review some of the under-lying types of hydrodynamic instability. Later we will indeed find many of thesephenomena in the computational study of Chapter 3, although it has not been ourintention to study them directly.The basic concepts of hydrodynamic instability, both experimentally and theo-retically, were introduced by Kelvin, Reynolds, Taylor, Rayleigh and others abouta century ago. The ideas and concepts were initially developed for Newtonianand inviscid fluids in idealised setups. These studies were extended to many non-Newtonian fluids in the same flow geometries over the past 30-40 years as extendedclassical theories. At the same time, industrial and practical applications shifted thefocus to the stability of two-phase and multi-fluid flows. Extensive studies havebeen performed to determine the effect of different flow parameters, including vis-cosity and density differences, and the effect of the yield stress on stabilizing themulti-phase flows.1.4.1 Fingering instabilityThe interface between two fluids, which are moving in a direction perpendicular tothe interface, can remain stable or unstable depending on the density and viscos-ity gradient. Taylor [65] made a direct analogy between displacement flows in a20porous media and those in a Hele-Shaw cell. He derived a general condition, thatthe interface is stable if the combination of viscous stresses and buoyancy terms ispositive, and otherwise it is unstable, see Figure 1.6.xˆDˆyˆgˆFluid 2: 𝜌"#, ?̂?#Fluid 1: 𝜌"1 , 𝜇"1𝑉)i f (µˆ1kˆ1− µˆ2kˆ2)Vˆ +(ρˆ1− ρˆ2)gˆ > 0, ⇒ Stablei f (µˆ1kˆ1− µˆ2kˆ2)Vˆ +(ρˆ1− ρˆ2)gˆ < 0, ⇒UntableFigure 1.6: Schematic of displacement flow of two fluids with different den-sity and viscosity.The indices 1 & 2 in the above mentioned relations refer to the displacing anddisplaced fluids, respectively, Vˆ and k are the mean velocity of the interface andthe permeability. The permeability for fluid flow between two parallel plates iskˆ = Dˆ2/12. More complex flows can continue to finger at smaller scales and thehydrodynamic instability that appears is generically known as a fingering insta-bility. Perhaps the most common scenario is when a less viscous fluid displacesa more viscous fluid, which is called viscous fingering. As illustrated in Fig. 1.7(left), the displacing fluid channels through the displaced fluid. In the case wherethe fluids are immiscible this is known as the Saffman-Taylor instability [66].The viscous fingering instability that develops in the displacement of a yieldstress fluid was studied in [67]. They conducted an experimental study and ob-served a branched pattern at low velocity and a single finger at high velocity. Anexample of the fingering instability that occurs due to viscosity differences betweenin Newtonian and viscoplastic fluids is shown in Fig. 1.7 (right).21Figure 1.7: Two examples of fingering instability are shown here. Left: ASaffman-Taylor finger obtained with air injected in a viscous fluid, [66],Right: Red fluorescent fingering into Carbopol solution.1.4.2 Rayleigh-Taylor instabilityThe instability induced by a less dense fluid displacing one with higher density iscalled Rayleigh-Taylor (RT) instability after the pioneering work done by Rayleigh[68] and Taylor [65]. The RT-instability is inviscid in origin and the driving forcecan be understood using the vorticity equation. By considering a 2D fluid flow inwhich the fluids are inviscid, the vorticity equation reduces to,DωDtˆ=1ρˆ2∇ρˆ×∇pˆ (1.7)We see that the vorticity created on the right-hand side will be in a direction thatincreases the perturbation of the horizontal interface separating heavy and lightfluids. The RT instability is often characterized by the emergence of a mushroomshaped interface.1.4.3 Kelvin-Helmholtz instabilityOne of the best-known instabilities is that at the interface between two parallel fluidlayers moving at different speeds, which is known as the Kelvin-Helmholtz (KH)instability. This instability was discovered first by von Helmoheltz [69], Kelvin[70] and is explained in detail in [71, 72]. Although strictly applicable to flows in22pˆ < 0pˆ > 0UˆUˆFigure 1.8: Mechanism of Kelvin-Helmholtz instability observed in a frameof reference moving at the average speed of the fluids, ∆Uˆ = (Uˆ1 −Uˆ2)/2.infinite domains, similar flows manifest in stratified duct flows. The basic mecha-nism of this instability is explained by Charru [73] in terms of a “Bernoulli effect”as follows. Consider a perturbed interface between two fluid layers, moving withspeeds Uˆ1 and Uˆ2, as shown in Fig. 1.8. Above the interface, the fluid is accel-erated because of a reduction in the area perpendicular to the flow. This velocityincrease above the crests of the interface leads to a pressure reduction, according tothe Bernoulli equation. This in turn amplifies the perturbation, due to the pressuredifferential across the interface. The perturbation will be swept along the inter-face by the flow and develops a roll-wave structure at the interface. The roll-wavestructure at the interface is called the KH Instability. This instability is essentiallygoverned by inertia and not by viscosity.We show later in Chapter 3 that counter-current flow may create in displace-ment process when buoyancy is in the opposite direction of the imposed flow andthe stresses generated by the buoyancy are larger than wall stress. This provides asimilar setting for the formation of KH type instability which is described above.231.4.4 Other types of multi-layer flow instabilitiesStability of multi-layer flows for immiscible and miscible fluids have been exten-sively studied with well-established methods in the cases of gravity-driven flows[74], pressure-driven flows (e.g. core-annular flow) [75–79], plane Poiseuille flows[80], and shear-driven flows (e.g. plane Couette flow) [81].Basically, interfacial instabilities are caused by a discontinuity/jump of the ve-locity profile of the base flow at the interface due, to viscosity/density difference.The magnitude of inertial stresses and the length scale of the vorticity disturbancesdeveloped at the interface are the origin of the instability [82]. In the theoreticalapproach, many have taken a linear stability analysis and studied the instabilitiesdeveloped by short-wave [83] or long-wave [82, 84, 85] disturbances. Bamboowave (BW) instabilities are observed in core-annular flow experiments performedby Joseph et al. [76] in a vertical tube, see Fig. 1.9a. In their experiment, the corefluid (oil) had higher viscosity than the outer fluid (water). The fluids were immis-cible and capillary instabilities were damped by shear-stabilization, which leads tothe regime of wavy flow. The stability of miscible fluids in plane Poiseuille flow,plane Couette flow and core-annular flow has been studied by [86–88] and by [89],respectively. It was shown that small amount of diffusion, present when the flow isin a high Pe´clet number regime, can in fact be destabilizing. Hence, miscible fluidsare potentially more unstable than immiscible fluids; see [89].Another type of instability that may occur in multi-layer flow systems is themushroom type instability which was first observed for Newtonian-Newtonianflows by [90], see Fig. 1.9b. The same type of instabilities have been predictedcomputationally by [91–93]. These studies confirm that mushroom instabilitiesappear in the core-annular flow only when the more viscous fluid is at the wall.A number of authors have studied instabilities in multi-layer flow with non-Newtonian fluids, e.g. [94–96], using methods that are well-established, althoughcomplex to implement for visco-plastic fluids. The difficulty arises when the baseflow is not completely yielded. With the assumption that the viscoplastic fluid layerat the interface is completely yielded, a linear stability analysis can be applied with-out mathematical difficulty. Sahu and Matar [97] studied instability of two-layerflow, with Newtonian and Herschel-Bulkley fluid layers, by applying the aforemen-24tioned assumption. Those cases which have fully unyielded base are uncondition-ally linearly stable, since a small disturbance must be able to raise the equilibriumstress above the yield value. Frigaard et al. [98] solved a Poiseuille flow of a Bing-ham fluid for partially yielded base flows. The linear stability analysis has beenapplied over the yielded flow domain, with boundary conditions imposed at theperturbed yield surfaces. In Poiseuille flow, the fluid moving between the centralplug and wall has a velocity profile that is equivalent to a mixed Poiseuille-Couetteflow in a channel of reduced width. The Couette component stabilizes the viscousmodes [99, 100]. It was shown in [101, 102] that if the fluid next to the free-surfaceor interface in multi-layer flow is unyielded, the interface will remain stable. Ingeneral, it can be concluded that yield stress of the fluid plays a stabilizing role inmost multi-layer flow systems.Gabard [42] studied iso-density miscible displacements in which a more vis-cous shear thinning fluid with zero and non-zero yield stresses is displaced by aless viscous fluid. The shear thinning fluid displacements were characterized byslowly evolving interfacial instabilities of inverse-bamboo type, which further re-duced the initially symmetric residual wall layer, see Fig. 1.9.c. However, theynoticed that interfacial instability is not generally significant while displacing ayield stress fluid.1.5 Scope of the thesisIn this thesis, we mainly study two problems related to the root causes of gas mi-gration from oil and gas wells.• The focus of the first phase of this thesis is on wet micro-annulus formation.Here we are concerned with the formation of micro-annuli during displace-ment of the drilling mud in the primary cementing process. This layer can beformed while drilling mud (a yield stress fluid) is being displaced in the an-nulus surrounding the casing. The layer of yield stress fluid left on the wallcan potentially remain static. The fluids involved are typically miscible, butdue to the geometry and flow rates, Pe´clet numbers (Pe) are very large. Westudy miscible displacement of a yield stress fluid by a Newtonian fluid inthe vertical direction, using computational methods. In particular, we are in-25a) c)b)Figure 1.9: Examples of a)Bamboo waves (BW) instabilities observed incore-annular flow experiments done by Joseph et al. [76]; b) the mush-room type instability which is observed in [90]; c) Inverse bamboo-typeinstability which is found in the experiments of [42]terested in evaluating the effects of a positive and negative density differenceon the efficiency of displacement and developing a long-term prediction ofthe dynamics of this layer. We simplify by considering only a longitudinalsection of the annulus (a channel). These studies are explained in Chapter 2and Chapter 3. Further detail on the industrial motivation and objectives hasbeen given in Section 1.1.• As discussed in Section 1.1, one of the root causes of gas migration is apressure imbalance between formation fluids and cement slurry in the annu-lus surrounding the casing. The cement slurry is categorized as a yield stressfluid and we expect that the yield stress will resist against fluid invasion. In26the second part of the thesis we test this hypothesis and also examine otherparameters which might influence the critical pressure at which formationfluids can invade into a static yield stress fluid. We study the invasion of aseries of miscible and immiscible fluids into a vertical column of yield stressfluid through a small hole, using both experimental and computational meth-ods. This serves as a simplified model for understanding the invasion of gasinto the cemented wellbores. This study is presented in detail in Chapter 4and Chapter 5.Apart from the motivational connection between the above two areas, stemmingfrom gas migration, we recognise a second common theme: the invasion/displace-ment of a yield stress fluid by another fluid. The aim is to make both fundamentaland practical industrial contributions in this thesis.We also outline our work on a third problem related to gas migration. A com-mon perception is that gas invasion is instigated by a drop in the pore pressure asthe solids structure becomes self-supporting within the cement slurry. This expla-nation seems like a plausible mechanism but has not been studied at a fundamentallevel with the aim of giving a quantitative and usable description in model form. Wedevelop a continuum level model for this later in the thesis, but have not progressedfar in its analysis. The model must bridge between models for a non-Newtoniansuspension flow and porous media flow, capturing the mechanics of the transitionbetween these two, but must also model the essentials of the cement hydration. Themodel outline and our preliminary analysis are given in Appendix D.27Chapter 2Buoyancy effects onmicro-annulus formation: densitystable Newtonian-Bingham fluiddisplacements in verticalchannels 1This chapter studies miscible displacement flows in long vertical channels, in whichthe displaced fluid has a yield stress and the displacing fluid does not and the con-figuration is density stable. As is common in such flows, the initial displacementis incomplete (not fully efficient) and a residual layer of the in-situ fluid can beleft behind on the walls. In the case that the displaced fluid is purely viscous theresidual layers drain over time or mix. However, if the residual fluid has a yieldstress, residual layers may remain indefinitely; see [54]. The aim of this chapteris to predict the long-time behaviour of residual layers (as t → ∞), in terms of thedimensionless groups of the flow. In particular we are interested in the effects ofthe stable buoyancy gradient on the flow, which differentiates this study from the1A version of this chapter has been published in Journal of non-Newtonian Fluids Mechanics[103].28FormationCasingxˆDˆyˆ0UˆgˆFluid 2:lightyield stressFluid 1:heavyNewtoniana)CasingChannelling from static mud filling the narrow sideb)xˆyˆgˆFluid 2:drilling mudFluid 1:spacer/ slurryWet micro-annulus from static residual mud layersFormationCasingc)Figure 2.1: Primary cementing of an oil well. a) Fluid-fluid displacementin a narrow eccentric annulus modelled as a longitudinal channel. b)Channelling of mud due to eccentricity, illustrated in a cross-sectione.g. [8]. c) Formation of a fluid-filled micro-annulus (modelled here).iso-density flows studied in [43, 54, 55].The chief motivation behind our study is the operation of primary cementing,which occurs when oil and gas wells are constructed; see [5]. The main objectiveof primary cementing is to provide complete and permanent isolation of differentfluid-bearing rock formations by sealing the drilled hole hydraulically with cement.During this process, a steel casing is inserted into the newly drilled section of thewell. The in situ drilling mud must be fully replaced with cement slurry betweenthe casing and the formation. To this end, a number of fluids (e.g. wash, thenspacer, then cement slurry) are pumped down the inside of the casing and return29upwards through the narrow annulus between the outside wall of the steel casingand the inside wall of the surrounding rock formation, see Fig. 2.1a. Drilling mudstypically have a yield stress, which is important during drilling for cuttings trans-port. The displacing fluids (wash, spacer and eventually the slurry) have variedrheology.The yield stress in the drilling mud allows it to resist the imposed stressesduring displacement and hence to remain static in the annulus, attached to the walls.The existence of thin residual mud layers has been termed a (fluid-filled or wet)micro-annulus. Cemented annuli are rarely concentric and residual layers tend tobe thicker on the narrow side of the annulus where the shear stresses are reduced.In extreme cases, drilling mud layers can bridge between the annulus walls to forma mud channel, usually at the narrowest point [8], although this process is notfully understood [104]. Wet micro-annuli dehydrate during setting of the cement,(dry micro-annuli may also form due to shrinkage), and these layers provide thepossibility for gas (or liquids) from one geological zone to hydraulically connectto another zone, i.e. it provides both entry points for fluid invasion and paths alongwhich fluid can migrate. See Fig. 2.1b & c for an illustration of these fluid flow-related defects.Here we are concerned with the initial formation of micro-annuli during dis-placement. Cemented annuli are a few centimeters wide and many 100’s of me-ters long. The fluids involved are typically miscible, but due to the geometry andflow rates, Pe´clet numbers (Pe) are very large. Mathematically, we study the limitPe→ ∞ of (laminar) miscible displacement flows. An equivalent limit is that ofimmiscible displacement flows with infinite Capillary number Ca. Thus, we drawinsight from both miscible and immiscible displacement flows.The study of immiscible displacements in capillary tubes was initiated morethan 50 years ago by Taylor [33], and many studies have followed. For two viscousliquids the residual layer thickness asymptotes to a constant value as Ca→ ∞. Atfixed Ca the residual layer thickness interestingly decreases as the viscosity ratio ofdisplaced to displacing fluid increases (denoted m in our study) see [36], which iscounter-intuitive, i.e. the thinnest residual layers are found for gas-liquid displace-ment. In the miscible fluid setting [37, 38] studied displacements of a viscous fluidwith a less viscous fluid in capillary tubes computationally and experimentally, re-30spectively. It was observed that at large Pe (& 105) the residual layer thickness alsoasymptotes to a constant value but that this layer thickness increases with viscosityratio m. Some comparison is made with the large Ca limit, although it is pointedout that the original results of [33] do not extend to this limit. This apparent dis-crepancy has been discussed by [39], who attribute the discrepancy to the fact thatthe residual layers are not stationary for comparable viscosity fluids, which affectsthe calculation of residual layer thickness in [38] via the method of [33]. This isshown more clearly in the recent experiments of [40].Buoyancy effects were included in [38], but these were not particularly signif-icant at large Pe. Buoyancy was studied in more detail by Lajeunesse et al. [41],who studied density stable displacements of miscible fluids in a Hele-Shaw cell,both experimentally and theoretically. Flows were studied over a range of viscos-ity ratios and buoyancy numbers, and not strictly confined to low Re. We refer to[41] in analyzing the flow types observed in our numerical simulations.Whereas Newtonian fluids continuously deform when a shear stress is applied,yield stress fluids do not have to deform. Therefore, residual layers of yield stressfluid can remain forever. Allouche, et al. [54] studied the miscible displacementof two visco-plastic fluids in a plane channel numerically and determined suffi-cient conditions for the non-existence of a static wall layer using a simple 1Dmodel. Gabard and Hulin [43] experimentally studied miscible displacements ofnon-Newtonian fluids with zero and non-zero yield stresses by less viscous andmostly Newtonian fluids of the same density in a vertical tube. These were gen-erally at low-moderate Re and for yield stress displaced fluids; the experimentsshowed a steadily moving front leaving behind a uniform thickness static layer,qualitatively analogous to the flows in [54]. In [54] predictions were made ofthe static layer thickness that represented the computed layers thickness reason-ably well and depended primarily on the downstream fluid flow. However, deeperexamination in [55] showed that the predictions of [54] could not account for ob-served variations in layer thickness as Re was increased, and predicting the layerthickness remains unsolved.Here we revisit this problem and also consider the effects of a positive densitydifference to aid in displacing the fluids. This is indeed usual in primary cementingfor laminar flows as the displacing fluids are usually pumped in a sequence of in-31creasing density (mud, spacer, cement slurry); see e.g. [5]. Due to the large numberof dimensionless parameters involved in displacement of two shear-thinning yieldstress fluids with yield stress in a 3D annulus, we simplify both geometrically andrheologically in considering a vertical plane channel, with a heavier Newtonianfluid displacing a lighter Bingham fluid in the upwards direction. This is the sim-plest configuration and rheology to allow static residual layers, as well as beingable to explore viscosity ratio and buoyancy effects.Static residual layers are found in gas-liquid displacements of visco-plastic flu-ids along ducts, e.g. [48–50], but with respect to our work, these studies lack theeffects of buoyancy and viscosity differences. A number of others have studiednon-Newtonian liquid-liquid displacement flows. Soares and Thompson have ex-tended the numerical methodology and analysis of [36] to immiscible inertia-lessflows in pipes and channels, encompassing: Newtonian-Newtonian channel flows[56]; power law displacing Newtonian pipe flows [57]; visco-plastic displacingNewtonian channel flows [58]; visco-plastic displacing visco-plastic channel flows[59]. In each case the methodology is very similar. Two-dimensional simulationsare carried out over broad ranges of Ca at different rheological parameters andthere is a theoretical analysis of the upstream and downstream 1D flows.The latter can be used to predict the residual layer thickness, provided oneknows the pressure gradient and total flow rate, plus the fluid properties. Thisanalysis essentially samples the parameter space of multi-layer flows, for differentnon-Newtonian fluid pairs. A similar analysis is presented in [60], focusing on therelationship between the interface position and frictional pressure gradient, and thetwo individual flow rates of the fluid layers. Soares & Thompson instead arguethat the interface position can not be determined from the total flow rate and thefrictional pressure gradient, for certain critical values of viscosity ratio (see [36]),and they extend this approach in their later work. For viscosity ratios below thecritical values only recirculatory streamline patterns are found whereas above thecritical viscosity ratio, both bypass and recirculatory streamline patterns are found(as well as transitional parameter ranges). Within this parameter space, necessaryconditions for the bypass patterns are found by comparing the downstream cen-treline velocity with the front velocity, which is analogous to the method of [54].However, the computational method of Soares & Thompson is certainly superior32to that here (and in [54]) for looking at steady state streamline.The structure of this chapter is as follows. In Section 2.1 we explain theflow studied, introducing model equations and benchmarking the computationalmethod. The results come in two sections. Firstly, in Section 2.2 we explore vari-ations in layer thickness with the key dimensionless parameters, showing that theresidual layers depend primarily on the Bingham number B and a buoyancy num-ber χ∗/m (explained below). We see also that the layers may be classified as eitherfully mobile (for a sufficiently small ratio of yield stress to viscous stress of the dis-placing fluid), or potentially static. The latter are either fully static or will becomestatic at long times as the mobile part of the film drains and the wall layers thin.In Section 2.3 we look at the relationship between the types of residual layer andthe behaviour of the displacement front, finding that the long-time behaviour ofthe three wall layer types appears to correspond to three types of frontal behaviour.We then develop a simplified lubrication-style displacement model that exhibits thesame three frontal behaviours and use this as a predictive tool for the residual layertype. The chapter ends in Section 2.4 with a discussion and conclusions.2.1 A 2D model for buoyant displacement in verticalchannelAs illustrated schematically in Fig. 2.1a (left), we consider a plane channel flowas simplified model for a narrow vertical annulus. The vertical channel of widthDˆ is initially filled with fluid 2 which is displaced upwards by fluid 1, against thedirection of gravity. It is assumed that the displacing fluid is Newtonian and thedisplaced fluid is a Bingham fluid of different density: ρˆ2 < ρˆ1, see Fig. 2.1a (right).This model relates for example to the situation where a viscous spacer displaces adrilling fluid/mud. In general, the spacer and the mud would have a more complexshear rheology but this simplification allows us to study the effects of buoyancy,yield stress and viscosity ratio combined in their most simple setting.In the case considered the two fluids are miscible and an appropriate modelframework consists of the concentration-diffusion equation coupled to the Navier-Stokes equations. Defining fluid 1 as having concentration c = 1 (and fluid 2 is33c = 0), the following dimensionless equations govern the flow of the mixture.Re[1+φ(c)At︸ ︷︷ ︸BA][ut +u ·∇u] =−∇p+∇ · τ− ReFr2φ(c)eg (2.1)∇ ·u = 0 (2.2)ct +u ·∇c = 1Pe∇2c︸ ︷︷ ︸LPA(2.3)where u, p, τ denote the velocity, the pressure and the deviatoric stress, respec-tively. The function φ(c) = 1−2c and eg = (−1,0)The equations have been made dimensionless via the following scaling:x =xˆDˆu =uˆUˆ0t =tˆDˆ/Uˆ0pˆ = ρˆ gˆxˆ+µˆ1Uˆ0Dˆp τ =τˆµˆ1Uˆ0/Dˆ(2.4)where µˆ1 is the viscosity of the displacing Newtonian fluid, the mean density is:ρˆ =(ρˆ1+ ρˆ1)/2, gˆ is the gravitational acceleration and Uˆ0 is the mean displacementvelocity.There are 6 dimensionless groups that influence the flow. However, the termsmarked LPA (large Pe´clet approximation) and BA (Boussinesq approximation) areneglected for the remainder of this chapter, with the following rationale. First,as we consider laminar displacements, the Pe´clet number (Pe) is based on themolecular diffusivity and Pe 1 for all practical cementing displacements. Thus,(2.3) approximately models the concentration as a scalar field that is advected withthe fluid. Note however that the concentration influences the buoyancy throughφ(c), and also the rheology. Secondly, the Atwood number is defined as: At =(ρˆ2− ρˆ1)/(ρˆ2 + ρˆ1) ≤ 0, representing a dimensionless density difference. Herewe shall make the Boussinesq approximation: |At|  1, which still allows densitydifferences to generate strong buoyancy forces (comparable to those in cement-ing flows) but neglects the effects of the different densities on the acceleration ofthe individual fluids. This is partly convenience and partly to reduce parametriccomplexity.The remaining 4 dimensionless groups include the Reynolds number (Re) and34densimetric Froude number (Fr):Re =ρˆUˆ0Dˆµˆ1, Fr =Uˆ0√|At|gˆDˆ. (2.5)The other two dimensionless groups are related to rheological effects. In the puredisplacing fluid, the constitutive law is:τ1(u) = γ˙(u) (2.6)where γ˙(u) is the strain rate tensor:γ˙(u) = ∇u+(∇u)T (2.7)The constitutive law for the displaced Bingham fluid is:τ2(u) =[m+BNγ˙(u)]γ˙(u) ⇐⇒ τ2(u)> BN , (2.8)γ˙(u) = 0 ⇐⇒ τ2(u)≤ BN (2.9)Where γ˙(u) and τ2(u) are defined by:γ˙(u) =[122∑i, j=1[γ˙i j(u)]2]1/2, (2.10)τ2(u) =[122∑i, j=1[τ2,i j(u)]2]1/2, (2.11)The viscosity ratio (m) and Newtonian Bingham number (BN) are defined as:m =µˆ2µˆ1, BN =τˆY Dˆµˆ1Uˆ0(2.12)where τˆY denotes the yield stress of the Bingham fluid and µˆ2 denotes the plasticviscosity. We remark that the Newtonian Bingham number defined in this waymeasures the ratio of the yield stress of the displaced fluid to the viscous stress ofthe displacing fluid, which is often the salient balance. The combination B= BN/m35is the more usual Bingham number, which defines the plasticity of the downstreamflow.Precise ranges for the industrial application are difficult to define. First, wehave simplified both rheologies considerably, and second, we have restricted ourstudy here to laminar displacements. Generally however, Re > 10 (and up to tran-sitional values), although studying low Re displacements is also of general interest.We might have Fr ∈ [0.01,∞), although here we study only a stabilizing densitydifference (so the upper limit is restricted). Certainly, buoyancy can dominate bothinertial and viscous effects for common density differences, i.e. Re/Fr2 can be verylarge as well as Fr very small. In terms of viscosity ratio m, note that a displacingspacer fluid may be designed to be more or less viscous than the displaced mud,hence a range of m are of interest. Finally, the Bingham number ranges from 0 to(potentially) many hundreds depends on operational factors.Boundary conditions for the simulation are no-slip on the walls of the channel,the inflow (at x = 0) is a Newtonian Poiseuille flow. The initial velocity field is asteady flow of fluid 2 along the channel (c = 0). For t > 0, fluid 1 enters at x = 0:c(0,y, t) = 1, t > 0. Other boundary conditions are: cy = 0 at the channel walls andcx = 0 at the exit.2.1.1 Scope of studyThe main method that we use for the results presented in this thesis is computationof two-dimensional (2D) transient simulations of the displacement flow along achannel of unit width and length L = 30. For the flows of principal interest, whenthe wall layers are not removed, fluid 2 is relatively viscous and this computationallength is adequate to allow the identification of the steady characteristics of thedisplacement front.Our study is based on approximately 500 simulations, which cover the follow-ing parameters systematically: m = 0.1, 0.3, 1, 3, 10; BN = 0, 1, 5, 10, 50, andvarious combinations of Re and Fr. The latter are chosen such that the parameter:χ∗ =2ReFr2(=|ρˆ2− ρˆ1|gˆDˆ2µˆ1Uˆ0), (2.13)36adopts values χ∗ = 20, 200, 400, 1000, 2000, with the modified Froude numbertaking values Fr2 = 0.01, 0.1, 1, 2. We cover this range of χ∗ while restricting to(formally) laminar Reynolds numbers: Re = 0.1−2000. The parameter χ∗ repre-sents the balance of buoyancy to viscous stresses. The wide range of Re exploredensures that we encounter the same values of χ∗ for flows that are both viscous andinertia dominated. Similarly the range of Fr means that the imposed flow velocityis both above and below that at which buoyancy and inertia are balanced. Our mainaim is to study the effect of (m,BN ,χ∗) on both the thickness of residual wall layersand the type of interface propagation observed in the displacements.2.1.2 Computational methodThe numerical discretization used is a mixed finite element/finite volume scheme,analogous to that in [55], but with inclusion of buoyancy terms as in [44, 105].We use a structured rectangular mesh, refined towards the walls, with 56 elementsfor the width of channel and 250 along the length. To solve the scalar transportequation, a finite volume method with a MUSCL scheme is used for advectiveterm to preserve reasonably sharp interfaces between the two fluids and to keepc ∈ [0,1]. To include the exact Bingham constitutive model, we use the augmentedLagrangian method [106], which helps to resolve unyielded parts of the fluid cor-rectly, as is important in identifying the static layers of fluid close to the walls.The algorithm is implemented using PELICANS, a C++ object oriented platformdeveloped at IRSN, France, for the solution of PDEs and distributed under the Ce-CILL license agreement [107, 108]. Readers may consult [44, 55, 105] for furtherdetails on computational aspects.2.1.3 ValidationWe have validated the code by benchmarking our results with those obtained byWielage-Burchard & Frigaard [55] and Allouche et al. [54], who considered thedisplacement flows of two iso-dense fluids in a plane channel. Dimensionlessnumbers from [54] (r,µ1,µ2,τ1,Y ,τ2,Y ) are translated to those in the study herevia: BN = 2mτ2,Y/µ2, m = µ2, Re = (r+1)/µ1, Fr = ∞.We have evaluated the thickness of the (static) residual layers for µ1 = 0.01, τ1,Y =37720 0.02 0.04 0.06 0.08 0.1h0.040.060.080.10.120.140.160.180.20.22Figure 2.2: Comparison of layer thickness h for varying µ2 with µ1 =0.01, τ1,Y = 0.2, τ2,Y = 0.5: (+) data from Fig. 12(d) in [54], (◦) Fig. 3in [55] and (•) this study.0.2, τ2,Y = 0.5 and for µ2 ∈ [0.001−0.1]. The results are plotted in Fig. 2.2 againstthose obtained by [55] (who used the PELICANS code) and those obtained by [54](who used a commercial CFD software FIDAP). As expected, our results coincidewith those of [55] and show an offset with the results of [54]. The latter computa-tions used a rather coarse discretization and also a viscosity regularization method,instead of the augmented Lagrangian method. Both of these tend to reduce thelayer thickness.2.2 Variations in layer thicknessIn iso-density displacements at constant imposed flow rate, where the yield stressof the displaced fluid is sufficiently large, the flows are often observed to evolveinto a steadily moving stable finger-like displacement: see examples in [54, 55].The flow then consists of 3 distinct regions.38(a) A plane Poiseuille flow, downstream from the front, which has a central rigidplug of width yy, which can be found from the cubic equation:4y3y−3yy(1+4mBN)+1 = 0 : yy ∈ [0,1/2]. (2.14)This is found numerically. The speed of the plug region is:Vplug =BNm(1/2− yy)22yy,and outside of the plug, the velocity decreases parabolically to zero at thewalls. Note that yy and the solution here is parameterized only by B= BN/m.(b) A 2D region of flow around the finger tip, which requires numerical solutionof (2.1)-(2.3) to resolve properly. This may be done either as a transientproblem, e.g. [54, 55], or in a steady frame of reference, e.g. [58, 109].(c) A 1D multi-layer flow upstream of the front. In this region fluid 1 movessteadily between two static layers of fluid 2, through a channel of reducedwidth 2yi.We expect that the above type of flow also exists in the presence of a density differ-ence and this is indeed the case, as shown in Fig. 2.3. Here, with modest buoyancy(χ∗ = 200,BN = 50,m = 10 ) the front rapidly attains a steady shape moving atconstant velocity through fluid 2 as shown, leaving behind a near-uniform layerof fluid 2 at the walls. The right panel of Fig. 2.3 shows the strain rate at thesame times as the colourmaps of concentration. We observe zero strain rate in thedownstream plug region and in the newly formed residual wall layers behind thefront.Static layers of fluid 2 are a key feature of the flow and may be understoodvia a simple analysis of the 1D multi-layer region. The flow in (−yi,yi) is simply aNewtonian Poiseuille flow, driven by a modified pressure gradient: −[px+χ∗/2] =3/(2y3i ), such that the following flow rate constraint is satisfied:∫ yi−yiU(y) dy = 1. (2.15)39  00.10.20.30.40.50.60.70.80.9  2468101214Figure 2.3: Example of a two-dimensional displacement flow for(Re,χ∗,BN ,m) = (100,200,50,10). Left: concentration profile atsuccessive times. Right: Strain rate colourmap at the same times.We may consider the flow symmetric about y = 0 and consider only y ∈ [0,0.5].For the density stable flows considered, the wall shear stress isτw = 3/(4y3i )+χ∗(1/2− yi).The first term in this expression is the wall shear stress found without buoyancy andthe second term represents the increase in wall shear stress due to buoyancy. Thethickness of the static layer of fluid 2, (1/2− yi), is not determined, although thissimple analysis does give a value for the maximal possible static layer thickness.Any interface location yi ≥ 0 admits a static wall layer, as long as:τw =34y3i+χ∗(1/2− yi)≤ BN . (2.16)40BN0 6 10 20 30 40 50 60 70 80 90 100h00.050.10.150.20.250.3χ∗= 2000χ∗= 0χ∗= 100χ∗= 400Figure 2.4: Variations in hmax for χ∗ = 0, 20, 50, 100, 200, 400, 2000.Observe that τw decreases monotonically with yi and τw→ ∞ as yi→ 0+. There-fore, a minimal yi,min can be found, for which the yield stress is attained at the wall,i.e. τw = BN :4χ∗y4i,min− (2χ∗−4BN)y3i,min−3 = 0. (2.17)Alternatively, we may define hmax = 1/2− yi,min as the maximal static layer thick-ness. Fig. 2.4 shows variations in hmax. As is intuitive, the maximal layer thicknessdecreases with increasing the buoyancy parameter χ∗ and increases with BN . ForBN ≤ 6 there can be no static layer.2.2.1 Computation of residual layersBelow we shall explore variations in residual layer thickness. In order to compute aresidual layer thickness, we wait until the front has exited the channel and compute41the average thickness of residual layer over a fixed length of 10, near the end of thechannel: 15 < x < 25. As well as computing the layer thickness, we distinguishnumerically whether or not the residual layer is static. This is done by comparingτ2 and BN over the entire thickness of the residual layer.Examples of both a static layer and a moving layer are shown below in Figs. 2.5a& d. The regions of displaced fluid where the second invariant of the stress has notexceeded BN are marked with a (+/green) symbol. The central panel (Figs. 2.5b& c) shows the variation in the concentration, velocity and strain rate across thechannel, at x = 25, for the displacements in Figs. 2.5a & d, respectively. In accor-dance with the stress distribution, the strain rate profile in the static layer is zerothroughout the residual layer while in the moving layer we see a thin yielded layerclose to the wall. The velocity profiles in Fig. 2.5 adopt a parabolic Newtonianprofile within the displacing fluid.2.2.2 Effect of varying m and BNThe thickness of residual layers as m increases is illustrated in Fig. 2.6 for 4 dif-ferent χ∗ and for different BN . The first observation is that the layer thicknessgenerally increases with m, which is intuitive as the displacing fluid has a tendencyto finger more at increasing m. Note that for χ∗= 20 the buoyancy gradient is com-parable to the modified pressure gradient of fluid 1 flowing alone in the channel.As χ∗ is increased further (Figs. 2.6b-d) the stabilizing buoyancy gradient becomesdominant and the increase in layer thickness with m at each χ∗ becomes modest.Secondly, it is interesting to observe that the layer thickness often appears todecrease with BN . This occurs only for modest buoyancy effects and for larger min the range studied; see e.g. Figs. 2.6a & b. This same effect was observed in[54] for iso-density displacements, but is quite counter-intuitive. As the buoyancyis sufficiently increased (Figs. 2.6c & d) we see instead the expected increase in hwith BN , although the differences are marginal at large χ∗.We note that the increase in h with m is contrary to that reported with viscosityratio in [36, 56–59]. Combined with the complex behaviour with respect to BN ,it might be thought that a different interpretation of the viscosity ratio is required,than that captured in m. For example in [110], the authors argue that the yield stress42  510152025−0.4−0.200.20.40.2 0.4 0.6 0.8−0.5 0 0.5−0.500.511.52C,Ux0246810γ˙  a510152025−0.4−0.200.20.40 0.2 0.4 0.6 0.8−0.5 0 0.5−0.500.511.522.5C,Ux024681012γ˙a) c) d)b)Figure 2.5: Left: χ∗ = 20,BN = 50,m = 0.1; Right: χ∗ = 400,BN = 50,m =10. (a) & (d) Concentration field c at t = 25. The areas of displaced fluidwhere second invariant of stress has not exceeded the Bingham numberare marked with a (+) symbol. (b) & (c) plot a channel cross-sectionshowing (+) Concentration, (◦) axial velocity, and (4) shear rate pro-files. These cross-sections are at x = 25 and relate to the displacementsof (a) & (d), respectively.43a)m10-1 100 101h00.050.10.150.20.25BN =0BN =1BN =5BN =10BN =50b)m10-1 100 101h00.050.10.150.20.25BN =0BN =1BN =5BN =10BN =50c)m10-1 100 101h00.050.10.150.20.25BN =0BN =1BN =5BN =10BN =50d)m10-1 100 101h00.050.10.150.20.25BN =0BN =1BN =5BN =10BN =50Figure 2.6: Thickness of residual layer h plotted against viscosity ratio m: a)χ∗ = 20; b) χ∗ = 200; c) χ∗ = 400; d) χ∗ = 1000.must be included in the definition of the viscosity scale used, as in [59]. Here, wehave relatively strong buoyancy effects present in our computations and generallysignificant inertia. Taking the smallest buoyancy (χ∗ = 20) and plotting h againstm(1+B) (= m+BN), is equivalent to the effective viscosity scaling used in [59].Figure 2.7a & b shows this variation for Re = 0.1 and Re = 20. Interestingly, forthe small m(1+B) the layer thickness h increases, whereas at larger m(1+B) wesee a steady decrease in h. At larger χ∗, Fig. 2.7c, there is essentially no variationin h with m(1+B). To conclude, using an effective viscosity scale here results inrelative insensitivity of the data, which is the reverse trend from that expected. The44a) m(1 +B)0 20 40 60h00.050.10.150.20.25m =0.1m =0.33m =1m =3m =10b) m(1 +B)0 20 40 60h00.050.10.150.20.25m =0.1m =0.33m =1m =3m =10c) m(1 +B)0 20 40 60h00.050.10.150.20.25m =0.1m =0.33m =1m =3m =10Figure 2.7: Thickness of residual layer h plotted against an effective vis-cosity ratio m(1+B): a) χ∗ = 20, Re = 0.1; b) χ∗ = 20, Re = 20; c)χ∗ = 1000, Re = 5.arguments made in [110], while reasonable for some flows, do not appear to haveled to any advantage in [59], nor do they here.The final physical phenomena observed by increasing m involve changes tothe shape of the frontal region. Depending on χ∗ and m, the dispersive flows atthe front may either be eliminated, i.e. steady finger propagation, or can form intospike that advances ahead of the main front, e.g. Fig. 2.8. More detail about thisinterface transition is mapped out and discussed in Section 2.3.45Figure 2.8: Two-dimensional displacements: displaced fluid is red, displac-ing one is blue. (a) Rheological parameters are (Re,χ∗,BN ,m) =(2,400,5,0.1); (b) (Re,χ∗,BN ,m) = (2,400,5,10); times (right to left):t = 2, 7, 12, 17, 222.2.3 Effect of varying buoyancy, χ∗The reduction of h when increasing χ∗ is shown more clearly in Fig. 2.9, at fixedBN . Regardless of BN and m we see a monotone decrease in h, which suggests thatefficient mud removal can be achieved by sufficiently increasing χ∗. The maximallayer thickness obtained from (2.17) is also marked in Figs. 2.9c & d (dashed line).Note that according to (2.16) there is no static layer for any BN ≤ 6 and hence noline marked in Figs. 2.9a & b.46a)χ∗0 500 1000 1500 2000h00.050.10.150.20.25m =0.1m =0.33m =1m =3m =10b)χ∗0 500 1000 1500 2000h00.050.10.150.20.25m =0.1m =0.33m =1m =3m =10c)χ∗0 500 1000 1500 2000h00.050.10.150.20.25m =0.1m =0.33m =1m =3m =10d)χ∗0 500 1000 1500 2000h00.050.10.150.20.25m =0.1m =0.33m =1m =3m =10Figure 2.9: Residual layer thickness variation plotted against χ∗ at each BNis plotted: a) BN = 1; b) BN = 5; c) BN = 10; d) BN = 50. The brokenlines in c & d denote hmax.The simulations have shown that locally we have a 2-stage displacement pro-cess. In the first stage the two-dimensional frontal region passes. This front gen-erates stresses that displace a portion of the fluid only, but determine an initialresidual layer thickness for the second stage. In the second stage the flow ispseudo-one-dimensional: a film draining problem. It seems that the film drain-ing stage shows one of 3 behaviours. First, we may have residual layers buthmax = 0. In this case the residual layers are slowly moving and will eventuallybe removed, e.g. Figs. 2.9a & b. Secondly, we may have h≥ hmax > 0. In this case(e.g. Fig. 2.9c) the layers are still moving and consequently be reduced over time.47a) B0 100 200 300 400 500h0.050.10.150.20.25b) B0 100 200 300 400 500h00.020.040.060.080.10.120.14Figure 2.10: Variation of residual layer thickness with B=BN/m: a) χ∗= 20;b) χ∗ = 400.We expect h→ hmax at sufficiently large times. Note that the speed of this removalprocess will depend strongly on m, with the larger m retarding the removal, as isobserved in Fig. 2.9. Thirdly, we have cases for which the frontal stage leavesh < hmax but also hmax > 0. Here there is no further draining and the layer is static,e.g. below the broken line in Fig. 2.9d.Finally, note that the same χ∗ values have been achieved at different Re and Fr(hence the multiple points marked in Fig. 2.9). The spread of data is not extremewhich suggests that χ∗ is of primary importance rather than (Re,Fr) individuallyin these flows.2.2.4 Effect of varying B = BN/mThe ratio B=BN/m describes the plasticity of the downstream flow: the plug widthwidens as B→ ∞, approaching 1/2 asymptotically, as B−1/2 → 0. Similarly, theplug speed approaches 1 as B→ ∞. Figure 2.10 shows the decrease in h with Bat two fixed values of χ∗, and earlier we have noted a decrease with BN . Fromthe perspective that larger BN increases hmax and that increasing BN/m gives anincrease in the effective viscosity of fluid 2, this decrease in h is counter-intuitive.The only compatible explanation for this trend is that the downstream plugvelocity limits the velocity of the displacement front in some way, and the dis-48placement front velocity is related to the residual layer thickness in any steadypropagation via mass-conservation. This type of explanation was explored in [54],leading to a prediction h ≈ hcirc(B) = 0.5− yY/[B(0.5− yY )2]. However, the pa-rameter range explored in [54] was relatively small. In [55] it was shown that thisapproximation could not account for observed variations in h at different Re, andearlier here we have observed variations with m and χ∗ that also refute the utilityof hcirc(B) from [54].Nevertheless, the global effect of increasing B is evident and needs explanation.It appears that what we are observing is related to the displacement front. Toillustrate, we show two simulations in Fig. 2.11, at the same parameters (χ∗,m) =(20,0.1), but with increasing BN (hence also B). The Newtonian displacementBN = 0 (Fig. 2.11a) is dispersive with a spike forming at the centre. However,as B increases this is replaced by a steady uniform (shock-like) front at BN = 50(Fig. 2.11b).2.2.5 Summary: moving and static layersTo summarize the variations in computed residual layers Fig. 2.12 plots h in boththe (χ∗/m,B)-plane and the (χ∗/m,1+ B)-plane. In Fig. 2.12a our Newtoniancomputations are missing from the log-log plot. The size of the symbols (andcolour) reflects h. It is evident that overlapping points in this plane have verysimilar h and this data collapse h= h(χ∗/m,B) is a key result, i.e. the computationscovered a range of (BN ,m,Re,Fr). The distinction between B and (1+B) is one ofinterpretation: B controls the downstream velocity profile, but (1+B) characterizesthe effective viscosity.Our interpretation of this is that B and (potentially) BN/χ∗ appear to controlthe first stage of frontal displacement, whereas the film draining process is largelygoverned by χ∗/m. Note that χ∗/m measures the ratio of buoyancy stress to theviscous stress of the Bingham fluid (i.e. due only to the plastic viscosity), which isthe main balance in any wall layer that is sheared. Also it seems that once we haveχ∗/m > 104, regardless of B = BN/m, the displacement is largely efficient.As we have observed, and discussed above, the other main transition of rele-vance is between moving and stationary layers. In Fig. 2.12 we see that this change49Figure 2.11: Two-dimensional displacements: displaced fluid 2 is red, dis-placing fluid 1 is blue: a) (Re,χ∗,BN ,m) = (0.1,20,0,0.1); b)(Re,χ∗,BN ,m) = (0.1,20,50,0.1). Approximate times of simulation(left to right): t = 2, 7, 12, 17, 22.from moving to static is not mirrored by any sharp change in h. Note however thatour computations measure h at a finite time late in the simulations, but not as t→∞.In the well cementing context the axial length of the well is very large com-pared to the gap width, so that the local displacement timescale over which a layercontinues to thin may be significant. A moving layer may appear to be betterthan a static layer operationally. However, if eventually the moving layer becomesstatic with h = hmax, this scenario may be worse than an initially static layer with50ha)10−2 10−1 100 101 102 103100101102103104105Bχ∗/m  00.050.10.150.20.25hb)100 101 102 1031001011021031041051+Bχ∗/m  00.050.10.150.20.25Figure 2.12: Variation of residual layer thickness with: a) χ∗/m and B and b)χ∗/m and 1+B.51h < hmax. On the other hand, many drilling muds are thixotropic and a longer pe-riod of (even low) shear may degrade the mud allowing for better displacement bysubsequent displacing fluids. In either case, a practical goal is to be able to predictif the layers are moving or not. This long time behaviour is predictable throughknowledge of hmax, and then comparing with h, but the latter requires expensivecomputation.2.3 Results: displacement front classificationsAs observed in the previous section, displacement flows are characterised not onlyby the residual layer thickness after the displacement front has passed, but alsoby the behaviour of the displacement front and mobility of the residual layer. Inconsidering our results, three characteristic regimes have been observed at the dis-placement front: dispersive front, shock and spike. The shock regime consists ofa large fraction of the centre of the channel displacing at constant speed and frontshape, (e.g. Fig. 2.3 & Fig. 2.11b). The spike regime has a similar interval ofconstant front speed and shape (shock-like), but close to the channel centre thefront speed is faster than the shock speed and moves ahead forming a spike shape(e.g. Fig. 2.8(right) & Fig. 2.11a). The dispersive regime consists of a monotonedecrease in front speeds from the channel centre to edge.Regardless of flow regime, each displacement appears to converge to a constantfront speed (at the channel centre) within a few time units (= channel widths) of thestart. This is best observed from a spatiotemporal plot of c¯y(x, t), which denotes theaverage of c(x,y, t) with respect to y. An example is shown in Fig. 2.13 illustratingthe 3 different regimes. In each of these cases we can see that a front speed Vf iseasily defined. Figure 2.14 shows the variations in Vf with m for different χ∗. Thefront velocity generally increases with m, decreases with BN and with χ∗. Thesetrends are largely inverse to those of h, as reported in the previous section.The relationship between h and Vf becomes clear if the displacement is as-sumed to proceed in steady state, meaning that a fixed frontal shape travels at speedVf . In this case, from mass conservation we may predict a residual layer thickness,say h f , on each wall:h f =12[1− 1Vf]. (2.18)52xt(a) 0 5 10 15 20 25 3005101520xt(b) 0 5 10 15 20 25 3005101520xt(c) 0 5 10 15 20 25 3005101520Figure 2.13: Spatiotemporal plots of c¯(x, t) for: a) dispersive regime, χ∗ =20,BN = 0,m= 3; b) shock regime (plug type), χ∗ = 20,BN = 50,m=0.1; c) spike regime χ∗ = 1000,BN = 10,m = 3.53a)m10-1 100 101Vf11.11.21.31.41.51.6BN =0BN =1BN =5BN =10BN =50b)m10-1 100 101Vf11.11.21.31.41.51.6BN =0BN =1BN =5BN =10BN =50c)m10-1 100 101Vf11.11.21.31.41.51.6BN =0BN =1BN =5BN =10BN =50d)m10-1 100 101Vf11.11.21.31.41.51.6BN =0BN =1BN =5BN =10BN =50Figure 2.14: Front velocity variation against viscosity ratio, m, at each χ∗ isplotted: (a) χ∗ = 20, (b) χ∗ = 200, (c) χ∗ = 1000, (d) χ∗ = 2000The layer thickness h f corresponds to the residual layer thickness only in the casethat the residual layer is stationary. This in turn occurs when the yield stress isnot exceeded (h < hmax), or if the displacing fluid slips, e.g. in gas-liquid displace-ments. This distinction is illustrated very well e.g. in the recent study [40]. Fig-ure 2.15 plots the computed h against h f . Those cases for which hmax = 0 arecoloured green and eventually will be displaced. If hmax > 0, flows with h ≥ hmaxare coloured black and these residual layers are moving. For h < hmax the symbolsare red.We observe that the green and black symbols are both found over a broad range54hf = (1! 1=Vf )=20 0.05 0.1 0.15 0.2h00.050.10.150.20.25m =0.1m =0.33m =1m =3m =10Figure 2.15: Plot of h against h f : green symbols hmax = 0; black symbolsh≥ hmax > 0; red symbols h≤ hmax and hmax > 0.of h, but generally for h f > h. Note that for h f > h the measured front velocityis faster than that implied by a steady traveling wave. This type of behaviour iscertainly found in displacements that are either fully dispersive or which developa spike ahead of the main front. In contrast the red symbols are found only veryclose to the line h f = h. When h f = h we have a steadily travelling front shape andsince h≤ hmax it follows that all red symbols represent static layers.The question now is whether by analyzing the front velocity and identifyingthe type of flow, plus knowing whether hmax > 0, we can predict situations wherewe find static residual layers (red symbols), as opposed to those in which the layeris still moving (black symbols in Fig. 2.15)?2.3.1 Classifying the flow typesIn order to relate the displacement front behaviour to the type of residual layer it isnecessary to classify the displacement fronts. Although spatiotemporal diagrams55such as Fig.2.13 are suggestive of which frontal behaviour occurs, a more deductiveprocedure is needed. First, we replot c¯y(x, t) against the similarity variable x/t, atthe later times in each simulation. As in each regime we approach a steady Vf atthe centreline, this suggest advective behaviour more generally. Indeed, in eachof the 3 regimes we see a collapse of the averaged concentration at later timesonto single curve monotonically decreasing curve: c¯y(x/t). For such curves therange of observed x/t indicates the range of observable interfacial velocities, sayVi =Vi(c¯y), where Vi(1) = 0 and Vi(0) =Vf .Examples of this similarity rescaling for computed c(x,y, t) are shown in Figs. 2.16a,2.17a & 2.18a. The data collapse onto a single curve c¯y(x/t) is clearly evidentnear the end of the computed displacement times. However, there remains somesmoothing of c¯y(x/t) at the Vi where we appear to be converging to a jump (orshock). Reasons for this smoothing are two-fold. Firstly as noted, we are evalu-ating c¯y(x/t) only at large finite times, to predict similarity behaviour as t → ∞.Secondly, the numerical solutions we have seen earlier also exhibit some smooth-ing of c(x,y, t) close to the interface. This smoothing results from a combinationof numerical diffusion and (largely physical) dispersion of any intermediate con-centrations by secondary flows (meaning velocities relative to the front speed).Despite these imperfections, we may infer that in the case of an exact solution ofthe concentration equation (2.3) with Pe =∞, and in an infinitely long channel, wewould see convergence of c¯y(x/t) to profiles with shock discontinuities for frontaland spike regimes.Therefore, to classify the flow from the actual c¯y(x/t) we proceed as follows.1. We calculate Vi(c¯y), to precision tol1 = 0.01.2. We calculate repeated values of Vi(c¯y) (within precision tol1) to establishpotential shock speeds. The most common repeated value is taken as theshock speed Vs, provided also that |Vi(c¯y)−Vs| < tol2 = 0.04 over at least40% of the channel width.3. If unable to identify a Vs from the above criterion, the displacement is clas-sified as dispersive. If Vs has been identified, we classify as a frontal shockif Vf −Vs ≤ tol3 = 0.06, and as a spike if Vf −Vs > tol3.560 0.2 0.4 0.6 0.8 1 1.2 1.400.20.40.60.81Vs =1.33xtCy  a)t =11.2t =12.2t =13.2t =14.2t =15.2t =16.2t =17.2t =18.2t =19.2t =20.2t =21.2t =22.20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.10.20.30.40.5yiVi Vs =1.543b)Figure 2.16: Interface propagation for χ∗ = 400,BN = 50,m = 10, obtainedfrom: (a) 2D computational results, (identified as frontal shock); (b)two-layer model.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.600.20.40.60.81Vs =1.16xtCy  a)t =10.2t =11.2t =12.2t =13.2t =14.2t =15.2t =16.2t =17.2t =18.2t =19.2t =20.20.2 0.4 0.6 0.8 1 1.2 1.4 1.600.10.20.30.40.5yiVi Vs =1.183b)Figure 2.17: Interface propagation for χ∗ = 200,BN = 1,m = 3, obtainedfrom: (a) 2D computational results, (identified as a spike); (b) two-layer model.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.600.20.40.60.81xtCy  a)t =10.2t =11.2t =12.2t =13.2t =14.2t =15.2t =16.2t =17.2t =18.2t =19.20.2 0.4 0.6 0.8 1 1.2 1.400.10.20.30.40.5yiVi Vs =1.483b)Figure 2.18: Dispersive χ∗ = 20,BN = 1,m = 3, obtained from: (a) 2D com-putational results, (identified as dispersive); (b) two-layer model.57Figs. 2.16a, 2.17a & 2.18a show the result of this type of classification. In Figs. 2.16a& 2.17a the broken vertical line indicates the identified Vs in each case, and oncomparing with Vf , Fig. 2.16a was classified as a frontal shock and Fig. 2.17a as aspike. No shock was identified in Fig. 2.18a, hence classified as dispersive. Evi-dently no such classification is perfect. The precision tol1 is numerically motivated.The precision tol2, in combination with the % of the channel width, is essentiallya threshold on the steepness of the (smoothed) shock and whether the jump is sig-nificant. The use of a modal value to identify Vs is because the shock need not beat the channel centre, e.g. for a spike. The third tol3 should exceed tol2 and is thena threshold above which we identify a distinct spike, i.e. due to the monotonicityof c¯y and smoothing of c(x,y, t) at the front, Vf > Vs in all cases. The values ofthese parameters have been fixed following sensitivity calculations, to conform toour perceived interpretation of observed flow regimes.With the above classification we now return to the representation of Fig. 2.15.We consider all the computed data for which hmax > 0. The mass conservationprinciple that defines h f via (2.18) relies on 2 assumptions: (i) that the front movesat steady speed; (ii) that the residual layer is static. In the case that the residual layermoves with positive velocity, we would expect h > h f . Instead we have observedthat h < h f , which means the front does not move at steady speed, which happense.g. if the flow is characterized as a spike. Therefore, if the flow is characterized asa frontal shock, we plot h against h f , as before, but for spikes, we assume that themain displacement is via the contact shock. Consequently, h ≈ hs, on each wall,wherehs =12[1− 1Vs]. (2.19)We plot the computed h against hs. This is shown in Fig. 2.19a. Note that therewere no dispersive regime flows for hmax > 0.In Fig. 2.19b we replot the data from Fig. 2.15, restricting again only to param-eters for which hmax > 0 (i.e. we delete the green points). For the data with h< hmaxwe again plot h against h f . The data with h≥ hmax is plotted against hs. ComparingFigs. 2.19a & b it becomes clear that nearly all spike regime flows have h ≥ hmaxand nearly all frontal shock regime flows have h < hmax. In other words, classify-ing the type of wall layer appears to be equivalent to classifying the displacement58a) hf ; hs0 0.05 0.1 0.15 0.2h00.050.10.150.20.25b) hf ; hs0 0.05 0.1 0.15 0.2h00.050.10.150.20.25Powered by TCPDF (www.tcpdf.org)Powered by TCPDF (www.tcpdf.org)Figure 2.19: Classification of computed h for hmax > 0: a) frontal shock (redsymbols), h vs h f ; spike (black symbols), h vs hs; b) h ≤ hmax (redsymbols), h vs h f ; h ≥ hmax (black symbols), h vs hs. The symbolshapes denote values of m as in Fig. 2.15.59front regime. Furthermore, from the spike regime results it seems that h ≈ hs is areasonable approximation to the layer thickness as the front passes. However, atlong times since these also correspond to h > hmax we expect an evolution towardsh = hmax as t→ ∞.Compared to Fig. 2.15, the proximity of the data with h≥ hmaxto the line h = hs is notable: although mobile, the wall layers are slowly movingand approaching hmax.2.3.2 Lubrication displacement modelThe observations of the previous section lead to the question of whether we canclassify the displacement front regime in a simple way, and thereby classify thetype of residual layer. One potential method to predict the frontal behaviour is viaa simplified lubrication (thin film) model of displacement, as developed in e.g. [54].Our belief that this method will be successful stems from the work of Lajeunesse etal. [41], who successfully used a lubrication model to identify/predict these same 3different displacement flow regimes, but for a Newtonian-Newtonian displacement.Derivation of the lubrication model is described in Appendix A, and leads to thefollowing evolution equation for the evolution of the (symmetric) interface positionyi(x, t):∂yi∂ t+∂∂xq(yi;χ∗,m,BN) = 0. (2.20)Here q = q1 is the areal flux of the displacing fluid. This relation governs theinterface motion and can be analysed using kinematic wave theory. In particular,the speed of the interface is given by:Vi(yi) =∂q∂yi(yi), (2.21)at any value yi for which the interface height is continuous. However, (2.20) ishyperbolic and nonlinear. Thus, shock development is common. When a shockdevelops the height and speed of the shock are determined by mass conservationconsiderations. It follows that the prediction of shock heights and the propagationspeed of the interface at different yi is wholly determined from q(yi) which maytherefore be used to give characterization of the displacement equivalent to that ofthe 2D flow.60Benchmarking the lubrication modelTo demonstrate the effectiveness of this approach, we follow [41]. For Newto-nian fluids (BN = 0) and with appropriate re-scaling of the lengths, our model isidentical with that of Lajeunesse et al. [41], who used it to successfully predictfrontal regimes with experimental displacements in a vertical Hele-Shaw cell. Theexperiments also showed 3 characteristic behaviours termed no shock (dispersive),contact shock (spike) or frontal shock (shock). These 3 different flow regimes wereplotted in the (M,U)−plane, (in terms of our parameters M =m and U = 12m/χ∗).They then analyzed their experiments, also categorizing into the 3 regimes, andfound a good match.Here we analyze our numerical experiments for BN = 0, classifying them againinto the 3 flow regimes as described in the previous section, and we plot the resultsagainst the regimes of [41] in Fig. 2.20. Nearly all our Newtonian displacementflows are spike type displacements. We see that the regime prediction from thelubrication model is a reliable predictor of the behaviour of the 2D simulations.We also verify the lubrication model predictions, by analyzing q(yi) and classifyingdirectly the pseudo-1D approach.Regimes for BN > 0Analyzing q(yi) to predict displacement flow types for BN > 0 displacements isin principle the same as for Newtonian fluid displacements. Figs. 2.16b, 2.17b &2.18b show the results of analyzing Vi(yi) from (2.21) to classify the lubricationmodel displacements at the same parameters as in each of Figs. 2.16a, 2.17a &2.18a. The classification agrees between the two displacement flow models (2Dand lubrication) for these cases.For wider parametric verification the parameterization (M,U) does not allow usto delineate easily the regimes found for BN > 0. Instead, following Fig. 2.12, wesee that our computed results appear to be described well in the (B,χ∗/m)-plane.We therefore plot in this plane the results of classifying our 2D flows (Fig. 2.21a)and the lubrication model flows (Fig. 2.21b). There is clearly a very reasonableagreement on the classification. The points at which the classification disagreesare confined to the transition region between spikes and frontal shocks, the precise6110−1 100 101 10210−310−210−1100101102103MU  0.51.51.5-1.50.5-0.5Figure 2.20: Comparison of the flow classification of our 2D simulations andof the lubrication model (both with BN = 0) with the flow regime mapof [41] for density stable displacement of two Newtonian fluids. Smallsymbols from our lubrication model: blue - dispersive; green - frontalshock; red - spike. Large symbols from our flow classification: spike4; frontal shock +; dispersive ×.location of which is sensitive to the numerical tolerances in our classification al-gorithm. Although promising in terms of flow classification, a cautionary remarkis that the layer thickness predicted by the lubrication model is unlikely to reflectthe actual layer thickness from 2D computations. As already noted in [54] thelubrication model underestimates the stresses in the flow and consequently oftenover-predicts the layer thickness.62a)10−1 100 101 102 103100101102103104105Bχ∗mb)10−1 100 101 102 103100101102103104105Bχ∗mFigure 2.21: Displacement flow classifications: a) 2D computations; b) lu-brication model computations. Symbols: spike -4; frontal shock - +;dispersive - ×.2.4 Summary and DiscussionThis study has explored density stable displacements of a Bingham fluid by a New-tonian fluid along a channel, over a broad range of viscosity ratios, Bingham,Froude and Reynolds numbers, with the latter 2 parameters selected to providevarying χ∗ = 2Re/Fr2. In total approximately 500 2D displacement simulationshave been performed and analyzed. The main objective has been to understand andpredict residual wall layer thicknesses that result and the effects of the different pa-rameters.Many of the parametric effects are intuitive: increasing m increases h and in-creasing χ∗ decreases h. Thus as expected, increasing buoyancy or the displacingfluid viscosity improves the displacement. Counter-intuitive however, is the de-crease in h for increasing B, meaning the plasticity of the displaced fluid. Thiseffect is most evident at small to moderate buoyancy: for large buoyancy the resid-ual layers are anyway diminished. A summary of our results is given in Fig. 2.12,which shows that h is described very well as a function only of (B,χ∗/m).The residual wall layer thickness h is measured near the end of our 2D channelsimulations. As well as the size, a critical feature is whether or not the wall layer63is moving. This depends on h and hmax, the maximal static wall layer thickness,which is easily computed and hmax = hmax(χ∗,BN); see Fig. 2.4. If hmax = 0 (whichoccurs for BN ≤ 6) the wall layer is moving and will eventually be removed. Ifinstead hmax > 0 we have 2 possibilities: either h < hmax and the wall layers arestatic, or h ≥ hmax and the wall layers are moving. If the wall layers are movingand h ≥ hmax, it follows that h will decrease and approach hmax at which point itmay become static.The second part of our results has focused at prediction of the residual layer be-haviour by classifying the displacement front behaviours. Our displacement flowshave shown 3 characteristic behaviours: frontal shock, spike and dispersive. Thefirst interesting result from examining our data is that when hmax > 0, we have onlyobserved frontal shocks and spikes. Secondly, we have shown that frontal shocksappear to correspond to h < hmax, and spikes correspond to h(t)≥ hmax.This led to the interesting possibility of qualitatively predicting the residuallayer behaviour from a simplified model (as opposed to carrying out a full 2D dis-placement). To this end we developed a lubrication/thin-film approach and havedemonstrated that flow classification from the lubrication model is able to distin-guish/predict the 2D displacement type, in terms of the frontal behaviour, and con-sequently the residual layer behaviour.The observed characteristic behaviours are common to many other displace-ment flows. In Newtonian-Newtonian displacements, the wall layers are not strictlyuniform as they thin during drainage, but are often approximated as constant thick-ness. Spike-type miscible displacements in Newtonian-Newtonian flows were firstobserved in [38, 41]. The change from frontal shock to spike type flow is related toa transition in the streamline pattern, measured in a frame of reference at the shockspeed; see [38]. Here we have a diffuse interface and imprecise calculation of Vs,so that working in a moving frame is not a good diagnostic method for analyzingour 2D simulations. This type of computation has however been performed ef-fectively by Freitas et al. [58, 59] using an interface-fitted mesh, considering bothimmiscible Newtonian and visco-plastic fluids.Industrially we may adopt the following rubric. (i) Compute hmax(χ∗,BN) ifBN > 6 (otherwise the displacement is effective: h(t)→ 0 as t→ ∞). (ii) Analyzeq(yi) to construct Vi(yi) and classify as either spike or frontal shock. (iii) If a64frontal shock, expect a uniform static wall layer to form during the displacementand to remain, with thickness h < hmax. (iv) If a spike, locally expect an initiallymoving wall layer, following passage of the displacement front, and eventuallyh→ hmax as t → ∞. Thus, only 3 long time behaviours occur: h(∞) = 0, hmax orh(∞) < hmax (frontal shock). Although lacking the precise layer thickness in thefrontal shock case, h(∞) is bounded and the above steps require only very quickcalculations, i.e. solution of algebraic equations. In considering how one mightimprove this (i.e. in the case of frontal shocks, with h(∞) < hmax), the first optionis of course 2D computation, either as here or using a front-fixing method suchas [36, 58] (at large Ca). Although the computations of [36, 58] are attractiveand probably quicker than the transient computations here, they are restricted toRe = 0 and proceed by assuming a steady state configuration. In particular, someof the recirculatory patterns predicted are not likely to be found in practice as theinterface evolves away from the steady state, e.g. the formation of spikes occursas explained in [38]. In an idealistic laboratory setup, one could do reasonablywell without computation. Following the type of analysis in [36], if one monitoredthe total pressure drop and flow rate as the displacement progressed, it would bepossible to predict the pressure gradient from the multi-layer flow region and hencecompute the interface position. This however requires differentiating the pressuredrop with respect to time, which would be problematic in noisier industrial settingswith limited data, such as primary cementing.In the primary cementing context, although the rubric presented is a significantadvance in our ability to predict micro-annulus formation, it leaves the question ofwhat is best? Dispersive or spike type displacements result in moving residual lay-ers. The timescale for these layers to be removed will increase with m (potentiallyalso with B) and these mobile layers should be regarded as potential sources of con-tamination. We have seen no evidence of instability in our simulations, but theseare of limited duration. The velocity profiles in these draining layers have pseudo-plugs at the interface, i.e. yielded at the wall only but with a plug velocity thatslowly varies along the channel. Visco-plastic lubrication studies would suggeststability (e.g. [111]) but viscous-viscous theories (e.g. [112]) often show instabilitywhen the more viscous fluid abuts the wall. On the other hand, the frontal shocksproduce a static layer h(∞)< hmax as soon as the front passes. These layers will be65hydrodynamically stable and remain static. There is no contamination risk from in-terfacial mixing, but such layers are likely to dry during cement hydration, forminga porous conduit. In this context, the spike displacements at large m appear to bethe worst case: not only do we suffer the contamination risk of mobile layers overa timescale ∝ m, but also the layers at best thin towards h(∞) ∼ hmax > 0, whichremain on the walls.2.5 Concluding remarksAll in all, in this chapter we thoroughly studied the effect of the governing fluiddynamic parameters (Re,BN ,m,Fr) on the thickness of the residual layer devel-oping in a density stable displacement of a Bingham fluid with a Newtonian one.In addition, we presented a method for predicting the long-term dynamics of thislayer (i.e static, removed, or gradually moving) from the displacement front typewhich was predicted from our 1D-model.Density unstable displacement in the annulus is the other possible scenarioencountered in the primary cementing process. We would like to investigate if itis possible to make similar predictions for the long-time behavior of the residuallayer developing in a density unstable displacement using the 1D model, as well.In addition, we aim to determine if flow instabilities improve the efficiency of thedisplacement or not. These are the questions that we address in Chapter 3.66Chapter 3Buoyancy effects onmicro-annulus formation: densityunstable Newtonian-Binghamfluid displacements in verticalchannels 1In Chapter 2 we studied miscible density stable displacement of a Bingham fluidwith a Newtonian fluid. In this chapter, we study miscible density unstable dis-placement of the same fluids in a long vertical channel. In such flows, the initialdisplacement is often incomplete (not fully efficient) and a residual layer of thein-situ fluid can be left behind on the walls. In the case that the displaced fluid ispurely viscous the residual layers drain over time, or may mix. However, if theresidual fluid has a yield stress, residual layers may be static and remain indefi-nitely, as identified by Allouche et al. [54]. The aim of the chapter is to understandthe flow regimes that are present and to explore the phenomenon of a static resid-ual wall layer in terms of the dimensionless groups of the flow. In particular we1A version of this chapter has been published in Journal of non-Newtonian Fluids Mechanics[93]67FormationCasingxˆDˆyˆ0UˆgˆFluid 2:heavyyield stressFluid 1:lightNewtonianFigure 3.1: Primary cementing of an oil well. Fluid-fluid displacement in anarrow eccentric annulus modelled as a longitudinal channel.are interested in the effects of an unstable buoyancy gradient on the flow, mean-ing that the flow is in the upwards direction and a lighter fluid displaces a heavierfluid from below. This differentiates this study from the iso-density flows studiedin [43, 54, 55], and the density stable flows in Chapter 2.As in Chapter 2, a key motivation for our study is the operation of primarycementing, which occurs when oil and gas wells are constructed; see [5]. The mainobjective of primary cementing is to provide complete and permanent isolation ofdifferent fluid-bearing rock formations by sealing the drilled hole hydraulicallywith cement. During this process, a steel casing is inserted into the newly drilledsection of the well. The in-situ drilling mud must be fully replaced with cementslurry between the casing and the formation. To this end, a number of fluids (e.g.wash, then spacer, then cement slurry) are pumped down the inside of the casingand return upwards through the narrow annulus between the outside wall of thesteel casing and the inside wall of the surrounding rock formation, see Fig. 3.1.Drilling muds typically have a yield stress, which is important during drilling forcuttings transport. The displacing fluids (wash, spacer and eventually the cementslurry) have varied rheology.The yield stress in the drilling mud allows it to resist the imposed stressesduring displacement and hence to remain static in the annulus, attached to the walls.The existence of thin residual mud layers has been termed a (fluid-filled or wet)micro-annulus. Cemented annuli are rarely concentric and residual layers tend to68be thicker on the narrow side of the annulus where the shear stresses are reduced.In extreme cases, drilling mud layers can bridge between the annulus walls to forma mud channel, usually at the narrowest point [8].Density differences are always present in primary cementing. A conventionalcement slurry has density ρˆ ≈ 1800kg/m3. Conventional drilling muds have den-sities in the range 1000− 1600kg/m3. Although more dense than the mud, thecement slurry is preceded by one or more preflushes, which are (partly) designedto keep a buffer between cement and mud. The preflushes can range from chem-ically charged water (a wash) through to viscous spacer fluids of varying density.Thus, displacing from below with a less dense fluid is not uncommon.Here we study the potential initial formation of micro-annuli during the abovedisplacement, i.e. due to ineffective removal of drilling mud from the walls. Thechannel geometry adopted represents a longitudinal section of the annulus (seeFig. 3.1). Cemented annuli are a few centimeters wide and 100’s of meters long. Tocapture the essential competition of yield stress with viscous, inertial and buoyancystresses, as well as the effects of viscosity ratio, we take the simplest possiblenon-trivial scenario of a Newtonian fluid displacing a Bingham fluid. For brevitywe move directly to the problem at hand. Further relevant literature is reviewedextensively in Chapter 2 which we would only repeat.In Chapter 2 we studied miscible density stable displacement of a yield stressfluid with a Newtonian one and discussed the main results of such setting on theresidual layer thickness and it’s long time behaviour. We complete this study bydelving into density unstable displacement in the same idealised setting. We studymiscible displacement flows in long vertical channels, in which the displaced fluidhas a yield stress and the displacing fluid does not.The structure of the chapter is as follows. In Section 3.1 we explain the flowstudied, introducing the model equations and dimensionless groups. The results(Section 3.2) are in 3 sections. First we characterize the range of flow regimes interms of the instabilities observed: Section 3.2.1 & Section 3.2.2. Next we look atthe phenomenon of static residual wall layers Section 3.2.3. Finally, we close byassessing whether we can predict flow regimes in our 2D simulations from simplermodels, Section 3.2.4. The chapter ends in Section 3.3 with a summary.693.1 2D model for density unstable displacement invertical channelThe displacement model is identical with that in part 1, [103], as we outline briefly;see also Fig. 3.1. A vertical channel of width Dˆ is initially filled with fluid 2 whichis displaced upwards by fluid 1, against the direction of gravity. It is assumedthat the displacing fluid is Newtonian and the displaced fluid is a Bingham fluid ofdifferent density: ρˆ2 > ρˆ1. This model relates for example to the situation wherea chemical wash or lightweight viscous spacer displaces a drilling fluid/mud inlaminar regime. A wash is Newtonian. A spacer fluid and the drilling mud wouldgenerally have a more complex shear rheology, i.e. shear-thinning and with a yieldstress (mud). This simplification mimics that in Chapter 2.The model consists of the concentration-diffusion equation coupled to the Navier-Stokes equations. Defining fluid 1 as having concentration c = 1 (and fluid 2 isc = 0), the following dimensionless equations govern the flow of the mixture.Re[1+φ(c)At︸ ︷︷ ︸BA][ut +u ·∇u] =−∇p+∇ · τ− ReFr2φ(c)eg (3.1)∇ ·u = 0 (3.2)ct +u ·∇c = 1Pe∇2c︸ ︷︷ ︸LPA(3.3)where u, p, τ denote the velocity, the pressure and the deviatoric stress, respec-tively. The function φ(c) = 1−2c and eg = (−1,0)The equations have been made dimensionless via the following scaling:x =xˆDˆu =uˆUˆ0t =tˆDˆ/Uˆ0pˆ = ρˆ gˆxˆ+µˆ1Uˆ0Dˆp τ =τˆµˆ1Uˆ0/Dˆ(3.4)where µˆ1 is the viscosity of the displacing Newtonian fluid, the mean density is:ρˆ =(ρˆ1+ ρˆ1)/2, gˆ is the gravitational acceleration and Uˆ0 is the mean displacementvelocity.There are 6 dimensionless groups that influence the flow. However, the termsmarked LPA (large Pe´clet approximation) and BA (Boussinesq approximation) are70neglected, as explained in Chapter 2 (i.e. see [103]] also). The remaining 4 dimen-sionless groups include the Reynolds number (Re) and densimetric Froude number(Fr) (similar to those defined in eqs. 2.5 with the difference that Atwood is positivenumber here.):Re =ρˆUˆ0Dˆµˆ1, Fr =Uˆ0√AtgˆDˆ, (3.5)here At = (ρˆ2− ρˆ1)/(ρˆ2 + ρˆ1) > 0 is the Atwood number, which does not appearindependent of Fr. The other 2 dimensionless groups are related to rheologicaleffects. In the pure displacing fluid 1, the constitutive law is:τ1(u) = γ˙(u) (3.6)where γ˙(u) is the strain rate tensor:γ˙(u) = ∇u+(∇u)T (3.7)The constitutive law for the displaced Bingham fluid 2 is:τ2(u) =[m+BNγ˙(u)]γ˙(u) ⇐⇒ τ2(u)> BN , (3.8)γ˙(u) = 0 ⇐⇒ τ2(u)≤ BN (3.9)Where γ˙(u) and τ2(u) are defined by:γ˙(u) =[122∑i, j=1[γ˙i j(u)]2]1/2, (3.10)τ2(u) =[122∑i, j=1[τ2,i j(u)]2]1/2, (3.11)The viscosity ratio (m) and Newtonian Bingham number (BN) are defined as:m =µˆ2µˆ1, BN =τˆY Dˆµˆ1Uˆ0(3.12)where τˆY denotes the yield stress of the Bingham fluid and µˆ2 denotes the plastic71viscosity.Boundary conditions for the simulation are no-slip on the walls of the channel,the inflow (at x = 0) is a Newtonian Poiseuille flow, and outflow conditions areimposed at the channel exit (x=L). The initial velocity field is a steady flow of fluid2 along the channel (c = 0). For t > 0, fluid 1 enters at x = 0: c(0,y, t) = 1, t > 0.Other boundary conditions are: cy = 0 at the channel walls and cx = 0 at the exit.3.1.1 Dimensionless groups and scope of the studyThe Reynolds and Froude numbers have their usual physical meaning. Note thatwith the Boussinesq approximation, we may consider that Re is the Reynolds num-ber of the displacing Newtonian fluid. Often lightweight spacers and washes areturbulent, but here we consider laminar flows which might correspond to eitherlower flow rates or the flow of these fluids in the narrower part of the annuluswhere the mud may be difficult to remove and local velocities are small. Thus, awide range of laminar Re could be encountered.The (Newtonian) Bingham number BN describes the ratio of the yield stress ofthe displaced fluid to the viscous stress of the displacing fluid, which is relevant inthe deposition of residual wall layers of mud (wet micro-annuli), where buoyancyis not significant. The combination B = BN/m is the Bingham number, whichdefines the plasticity of the downstream flow.Buoyancy is an important driving mechanism in our flows and is measuredin different ways. First, the densimetric Froude number Fr measures the relativestrength of inertial stress, from the imposed flow, to the buoyancy stress. Secondly,we can see in (3.1) that the difference in (vertical) body force between the two purefluids isχ = 2ReFr2, (3.13)which measures the ratio of buoyancy stress to the viscous stress in fluid 1. This isalso important in determining residual wall layers.A third parameter related to buoyancy effects is the ratio Re/Fr, which wenote is independent of the imposed flow rate. The parameter 2(Re/Fr)2 is the72Archimedes number (Ar):Ar =ρˆ(ρˆ2− ρˆ1)gˆDˆ3µˆ21.This is usually interpreted physically as measuring the destabilizing influence ofbuoyancy stresses to the ability of the viscous stresses to stabilize the flow. Thisinterpretation needs refinement as clearly the viscosity of both fluids should play apart here. In natural convection studies the Gra¨tz number plays a similar role to Ar,albeit with the density difference arising from thermal expansion. Later we plotour flow stability classifications in the (Fr,Re/Fr) plane (i.e. (Fr,√Ar).Here we focus on the phenomenon of residual wall layers (wet micro-annuli),formed in laminar flows. Our study is based on around 800 simulations, whichcover the following parameters: m = 0.1, 0.33, 1, 3, 10; BN = 0, 1, 5, 10, andvarious combinations of Re and Fr in the range of Fr2 ∈ (0.1 : 50) and Re∈ (0.25 :500). These values are chosen so as to give values of:χ = 0.1, 0.5, 1, 5, 10, 50, 70, 90, 100, 120, 180, 500.3.1.2 Computational methodThe numerical discretization used is a mixed finite element/finite volume scheme,analogous to that in [55], but with inclusion of buoyancy terms as in [44, 105].We use a structured rectangular mesh, refined towards the walls, with 56 elementsfor the width of channel and 250 along the length (L = 30 for all computations).To solve the scalar transport equation, a finite volume method with a MUSCLscheme is used for advective term to preserve reasonably sharp interfaces betweenthe two fluids and to keep c ∈ [0,1]. To include the exact Bingham constitutivemodel, we use the augmented Lagrangian method [106], which helps to resolveunyielded parts of the fluid correctly, as is important in identifying the static layersof fluid close to the walls. Further detail and validation of the algorithm is given inSection 2.1.3 of the study in Chapter 2.73xt0 5 10 15 20 25 30051015200 100.51 Vs =1.58xtCyxt0 5 10 15 20 25 30051015200 1 200.51 Vs =2.04xtCyFigure 3.2: Examples of observed flows. Left: Stable- (Re,χ,BN ,m) =(15,10,5,10), Right: RT- (Re,χ,BN ,m) = (25,500,5,3). Top fig-ures show a red-blue colourmap of the concentration. Below the fig-ures show a spatiotemporal colourmap (yellow-blue) indicating the y-averaged concentration c¯y(x, t), and the insets show the same variableplotted against x/t late in the simulation.743.2 ResultsA wide range of different flows were observed in our results, as we outline below.In the first place, particularly for values of χ . 10, the flows were mostly stableand symmetric. The stable flows looked similar to those for small χ∗ in Chapter 2,i.e. without significant buoyancy. Note that χ = 12 corresponds dimensionlesslyto the axial pressure gradient required to push fluid 1 at the mean velocity.At combinations of large Re and large χ we found a variety of interesting in-stabilities, which we illustrate below. Firstly, we observed frontal instabilities ofRayleigh-Taylor type (RT), with a characteristic mushroom shaped (displacement)front. These frontal instabilities could develop with a stable layered structure be-hind the flow (see Fig. 3.2), or with instabilities developing also in the layers be-hind. These are not strictly Rayleigh-Taylor instabilities as we also have an im-posed displacement flow. In the top panels of Fig. 3.2 we show the evolving flow,in terms of the concentration colourmap at successive times. Below the colourmapsis shown a spatiotemporal plot of the width-averaged concentration, c¯y(x, t):c¯y(x, t) =∫ 1/2−1/2c(x,y, t) dy, (3.14)ranging from 0 to 1. We observe the sharp yellow-blue interface signifying theadvancing front. The tails of the front stretch backwards along the sides of the pen-etrating fluid stream, moving backwards relative to the first front and terminating ina diffuse region. We see the more diffuse transition of this tail end, moving down-stream in the spatiotemporal plot, significantly slower than the advancing front.Finally, the strong buoyancy force (χ = 500 here) appears to pin the interface tothe wall at the inflow.The evident linear advance of the fronts in the spatiotemporal figures suggeststhat the flow, despite the instability at the front, is essentially advective. To quantifythe speeds of the displacement interface we replot c¯y(x, t) against the similarityvariable x/t. At large times the averaged concentrations c¯y(x/t) appear to collapseonto a single curve, shown in the insets of the spatiotemporal plots in Fig. 3.2.The horizontal axis indicates the front speeds observed in the flow. The centrallypositioned Rayleigh-Taylor front is advancing essentially as a kinematic shock,75xt0 5 10 15 20 25 30051015200 0.5 100.51 Vs =1.36xtCyxt0 5 10 15 20 25 30051015200 0.5 100.51 Vs =1.38xtCyFigure 3.3: Examples of observed flows. KH: Left-(Re,χ,BN ,m) =(150,100,1,0.33), Right-(Re,χ,BN ,m) = (50,100,1,0.1). See Fig. 3.2for description.76xt0 5 10 15 20 25 30051015200 0.5 1 1.500.51 Vs =1.66Vs =0.12xtCyxt0 5 10 15 20 25 30051015200 0.5 1 1.500.51 Vs =1.61Vs =0.32xtCyFigure 3.4: Examples of observed flows. KH: Left-(Re,χ,BN ,m) =(15,100,1,0.33), Right-(Re,χ,BN ,m) = (25,50,0,0.33). See Fig. 3.2for description.77xt0 5 10 15 20 25 3005101520250 0.5 1 1.500.51 Vs =1.61xtCyxt0 5 10 15 20 25 30051015200 0.5 1 1.500.51 Vs =1.53xtCyFigure 3.5: Examples of observed flows. I.B.M: Left-(Re,χ,BN ,m) =(270,180,1,10), Right-(Re,χ,BN ,m) = (500,100,5,10). See Fig. 3.2for description.78with speed Vs = 2.04.The second set of instabilities that we have observed appears to be related torelative motion along the sides of the advancing front. These occur at significant χand Re, typically with m< 1. The initial displacement appears stable, with a finger-like front of fluid 1 penetrating steadily through fluid 2. The layered structurealong the sides of the penetrating finger is essentially 1D and the combinationof significant χ and low m allows for some negative velocities within the walllayer. This counter-current layered flow appears to destabilize upstream of thefront, resulting in interfacial waves that advance downstream at a speed slowerthan the front. Thus, both the stable and unstable lengths of the interface grow intime. We label this type of instability as Kelvin-Helmholtz (KH), as they appearto be associated with the parallel flow in which the fluids move at different meanspeeds (eventually counter-current at large enough χ).Examples are shown in Figs. 3.3 & 3.4. In both cases we see a stable front,with no sign of RT instabilities at the front. The spatiotemporal plots and insetfigures reveal a sharp shock-like front, moving at speed significantly less than theRT example of Fig. 3.2. The instabilities show up as interfacial waves on the spa-tiotemporal plots, with positive wavespeeds slower than the leading displacementfront. We see from the inset figures that the instabilities have variable wavespeeds,but appear to be confined within a narrow range of amplitudes (measured in c¯y).It is apparent in these figures that initial instabilities grow at different speedsand frequencies, e.g. the left column of Figs. 3.3 & 3.4 both have only a singleinterfacial wave by the time that the front exits the channel, whereas multiple wavesexist for the other two examples. Also the morphology of the waves is somewhatdifferent, e.g. Fig. 3.4 (right) seems to initiate as a bamboo wave, whereas Fig. 3.3(right) has more of a roll wave structure. Bamboo waves were observed in theexperimental study of [113] where viscous oil in the core and water in the annulus.This type of instability was studied further by [114–116] theoretically.It has not been our intention to study the onset of instabilities in these sys-tems. A number of authors have studied multi-layer instabilities with these fluids,e.g. [96, 109, 117–119], and the methods are well established although complexto implement. It is worth remarking that for the KH examples shown here wehave BN = 0, 1, the fluid at the interface is yielded and the stability problem is79Figure 3.6: Examples of observed flows. Footprinting: (Re,χ,BN ,m) =(180,180,5,10). Left: the concentration colourmap. Right: contoursof τ(u)−BN , the grey shaded contour indicating where τ(u)< BN .essentially that of 2 shear-thinning fluids in parallel flow, as opposed to the visco-plastic lubrication case [109, 111, 120] where instabilities are suppressed. Thus,we feel that study of the hydrodynamic stability problem could be interesting forlater study.Here we are more interested in displacement front behaviour and residual lay-ers. For the low values of BN here these downstream residual layers, althoughbeautifully uniform, are mobile. In contrast to the RT flows, the KH flows all havea trailing front that moves slowly along the wall from the entrance to the channel.The profiles of c¯y(x/t) collapse onto a single curve for c¯y close to 1 and 0, i.e. forthe leading and trailing fronts. At intermediate values there appears to be a rangeof c¯y over which no collapse is achieved, i.e. due to the instabilities observed.For m > 1 but similar ranges of (Re,χ) we find qualitatively different instabil-ities, as illustrated in Fig. 3.5. Here the displaced fluid is more viscous than thedisplacing fluid, BN is small enough not to allow a fully static residual layer and80instead we observe a slow draining of the residual film. Instabilities are both ofinverse bamboo type (left panel) and mushroom type (right panel). The formerhave been observed in displacement flows of xanthan along circular pipes [42, 43],and the mushroom instabilities were observed and characterised for Newtonian-Newtonian flows by [90, 121] as well as being predicted computationally; see[120, 122]. These both occur in multi-layer systems with the more viscous fluid atthe wall. The initially stable interface observed in Fig. 3.5 (left) is also found inthe experiments of [42, 43]; presumably the timescale for growth of instabilities inthese situations is controlled by the more viscous outer layer. The mushroom insta-bilities appears to initiate closer to the displacement front than the inverse bamboo,but the front itself generally remains stable (no RT instability). These instabilitiesare closely related and we classify them as inverse bamboo/mushroom (IBM).With regard to the spatiotemporal plots and wavespeeds, we see that similar tothe KH instabilities the waves travel at a range of different speeds, all slower thanthe leading front. The interface speed approaches zero as c¯y → 1 and at interme-diate values we again see no fixed interface speed over the range of c¯y for whichinstabilities are observed.Finally we show an interesting and novel phenomenon that appears to be re-lated to the inverse bamboo regime. Figure 3.6 (left) shows the colourmap for whatappears to be an IB instability, emerging late in the displacement upstream of theadvancing front. In Fig. 3.6 (right) is plotted τ(u)−BN , with the grey shaded con-tour indicating where τ(u) < BN . In the displaced fluid these regions correspondto unyielded flow regions (plugs). We see that static wall layers are found in theregion where the uniform stream of displacing fluid advances. The static layers areonly found adjacent to the wall and not throughout the residual layer. Note thatthis contrasts with the density stable flows in [103] where the maximal stresses inthe residual layer are experienced at the wall. Further upstream in these flows wesee that the onset of instabilities is matched by a patterning of the static residuallayer. This type of footprinting of the static layer was first observed in iso-densedisplacements studied in [54]. They were also studied in the context of pulsatingdisplacement flows by [55]. Here it appears that they result from the instability.81100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure 3.7: Panorama of flow types observed for m = 0.1 and BN = 1. Markers indicate data position in (Re,χ) plane and flow classification: - stable,4 - KH,© - IBM; filled symbols - RT.82100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure 3.8: Panorama of flow types observed for m = 0.1 and BN = 5.For BN = 10 the flows are similar to BN = 5. Markers as in Fig. 3.7.83100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure 3.9: Panorama of flow types observed for m = 1 and BN = 1. Markers as in Fig. 3.7.84100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure 3.10: Panorama of flow types observed for m = 1 and BN = 5. For BN = 10 the flows are similar to BN = 5. Markers as in Fig. 3.7.85100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure 3.11: Panorama of flow types observed for m = 10 and BN = 1 .Markers as in Fig. 3.7.86100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure 3.12: Panorama of flow types observed for m = 10 and BN = 5. For BN = 10 the flows are similar to BN = 5. Markers as in Fig. 3.7.873.2.1 Flow regime panoramasWe now explore pictorially the different displacement flow regimes in the (Re,χ)plane for m = 0.1, 1, 10 in Figs. 3.7-3.8, 3.9-3.10,and 3.11-3.12, respectively.Each figure plots the displacement sequences and classifications within the rangeRe ∈ [1,100]. The flow regimes for m = 0.33, 3 are presented in Appendix B. Ascommented above, for lower χ the flows are largely stable, but even here we seetwo distinct front types: dispersive and frontal shocks (we return to these later inthe chapter). For m≤ 1 the instabilities are mainly of KH type, sometimes with RTfrontal instabilities (larger χ and significant Re).For m > 1 the instabilities are largely of IBM type, with some RT at the front.The unstable flows occur with increasing Re at the larger values of BN . For BN = 10there is very little instability, except at larger Re. Increasing BN tends to stabilizeall the flows. In terms of stability classification we see little qualitative differencebetween BN = 5, 10 (hence we show only the results for BN = 5).3.2.2 Onset of instabilityA more quantitative image of the flow regimes and limits of stability is presentedin Figs. 3.13-3.17. These figures present the full range of computations, at succes-sively increasing values of m. The different flow classification symbology followsthat used earlier in Fig. 3.7 and the symbols are coloured according to χ . First, forthe smaller values of m (Figs. 3.13 & 3.14), we see that the main transition is fromstable to KH type instabilities. This transition appears to occur across a criticalvalue χ = χc(m,BN). The values of χc in these figures are intended simply as aguide to the eye and have not been fitted. Nevertheless we observe that χc appearsto increase with BN and mildly with m.Although we characterize these as KH instabilities, we note that an analysis ofthe onset would include the influence of viscosity, which is not explicitly presentin the KH analysis. The transition with χ corresponds to a buoyancy-viscous stressbalance. This is suggestive that the driving mechanism for the transition is themulti-layered base flow, i.e. that along the sides of the displacing finger, which isdefined by this balance. Increased buoyancy leads eventually to a counter-currentflow, which often instigates KH type instabilities. Counter-current flows require8810−1 100100101FrRe/Fr  χc=40a) 0.10.51510507090100120180500χ10−1 100100101FrRe/Fr  χc=40b) 0.10.51510507090100120180500χ10−1 100100101102FrRe/Fr  χ c= 220c) 0.10.51510507090100120180500χ10−1 100100101102FrRe/Fr  χ c=2200.10.51510507090100120180500χFigure 3.13: Stable and unstable regions for m=0.1: a) BN = 0; b) BN = 1; c)BN = 5; d) BN = 10; - stable,4 - KH,© - IBM; filled symbols - RT.larger χ if either m or BN is increased. This dependence on the base flow (throughχ) explains the viscous (rheological) influence, i.e. rheology contributes to insta-bility implicitly through governing the multi-layer flow structure, even though theonset mechanism at the interface appears of KH type.For the smaller BN , as we increase χ , we first see KH instabilities alone andthen later with RT. For the two larger BN values the KH instabilities are accompa-nied by RT instabilities at the front. Presumably, counter-current flows are harderto drive at larger BN and for some of these flows the displaced fluid will be staticat the wall, which suppresses instability.As we increase to m= 1 (Fig. 3.15) the low BN transition with χ (i.e. Re∼ Fr2)is replaced with a weaker dependency: Re ∼ Fr8/5. An interpretation is that thebuoyancy-viscosity balance, influencing the base layered flow, is still the drivingforce, but that at higher flow rates inertial effects become important in driving thetransition. The larger BN flows are largely unaffected.8910−1 100 101100101102FrRe/Fr  χc=40a) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  χc=60b) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  χ c= 220c) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  χ c= 220d) 0.10.51510507090100120180500χFigure 3.14: Stable and unstable regions for m=0.33: a) BN = 0; b) BN = 1;c) BN = 5; d) BN = 10. Markers as in Figs. 3.13For m > 1, two changes are observed. First, the instabilities are now all IBMtype and are frequently associated with RT frontal instabilities. Secondly, the tran-sition now shows a decrease in Re/Fr with Fr: Re ∼ Fr4/5 at m = 3, decreasingto Re∼ Fr2/5 at m = 10. At larger BN we observe some simulations still classifiedas stable above these transitions. However, our simulations are in a finite channel(L = 30) and the IBM type of instability grow slowly, governed by the viscosityof the more viscous displaced fluid. Therefore, it may be that these flows willdestabilize at a later time, given a sufficiently long channel.3.2.3 Static wall layersWe now turn directly to the issue of wet micro-annuli, which are modelled here asthe emergence of a static residual layer of displaced fluid remaining at the walls.In Chapter 2 for density stable displacements (χ < 0) a maximal layer thicknesshmax was calculated and shown to depend only on (χ∗,BN), where χ∗ = −χ . The9010−1 100 101100101102FrRe/Fr  Re= 108/5 F r3/5a) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  Re= 108/5 F r3/5b) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  χ c= 220c) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  χ c= 220d) 0.10.51510507090100120180500χFigure 3.15: Stable and unstable regions for m=1: a) BN = 0; b) BN = 1; c)BN = 5; d) BN = 10. Markers as in Figs. 3.13definition of hmax(χ∗,BN) may be extended algebraically into the density unstableregime here. However, for large χ this produces erroneous results.Fully static wall layersTo explore the occurrence of static wall layers we proceed with an axial two-layermodel of a uniform section of channel. Assuming symmetry, we may consider onlyhalf of the channel, and the simplified momentum balances are:ddyτ1,xy = − f , y ∈ [0,yi), (3.15)ddyτ2,xy = χ− f , y ∈ (yi,0.5] (3.16)9110−1 100 101100101102FrRe/Fr  Re = 108/5Fr−1/5a) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  Re = 108/5Fr−1/5b) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  Re = 108/5Fr−1/5c) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  Re = 108/5Fr−1/5d) 0.10.51510507090100120180500χFigure 3.16: Stable and unstable regions for m=3: a) BN = 0; b) BN = 1; c)BN = 5; d) BN = 10. Markers as in Figs. 3.13where yi denotes the interface position and f is the modified pressure gradient,f ≡−∂ p∂x− ReFr2. (3.17)Boundary and interface conditions are:τ1,xy(0) = 0, τ1,xy(yi) = τ2,xy(yi), (3.18)U(y+i ) = U(y−i ), U(1/2) = 0. (3.19)Finally, the following flow constraint is satisfied:∫ 1/20Udy = 1/2. (3.20)The shear stresses are thus linear in each layer, with interfacial and wall shear9210−1 100 101100101102FrRe/Fr  Re = 10 3/2Fr −3/5a) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  Re = 10 3/2Fr −3/5b) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  Re = 10 3/2Fr −3/5c) 0.10.51510507090100120180500χ10−1 100 101100101102FrRe/Fr  Re = 10 3/2Fr −3/5d) 0.10.51510507090100120180500χFigure 3.17: Stable and unstable regions for m=10: a) BN = 0; b) BN = 1; c)BN = 5; d) BN = 10. Markers as in Figs. 3.13stresses:τi =− f yi, τw =− f2 +χ(12− yi) = τi+(χ− f )(12 − yi). (3.21)To extend the static layer concept of Chapter 2 (see detail in Appendix A ) , wefirst focus on the situation in which the flow is static throughout the layer of fluid2. In this case the velocity for fluid 1 (Newtonian) is easily solved and integratedto define the flow rate. Using (3.20) then determines f = 3/(2y3i ). Substitutingback into (3.21) we observe that τi < 0 always, but that the sign of τw depends onthe size of χ > 0. Herein lies the essential difference with Chapter 2, as differentregimes are possible.First, if χ < 0 then the stress remains negative and decreases throughout fluid2, meaning that the largest magnitude stresses are always found at the wall fordensity stable displacements. For density unstable displacements, provided that93χ ≤ f = 3/(2y3i ), then τ2,xy decreases across fluid 2, the largest stress is found atthe wall of the channel and if the fluid is static there, it will be static throughoutthe fluid 2 layer (regime 1). This scenario is analogous to that for the densitystable flow of Chapter 2 (see [103] also), leading to simple calculation of hmax.In particular, if χ < 12 then the maximum stress is at the wall for all yi. Usingthe same method as in Chapter 2 the maximal layer thickness is simply calculated.Figure 3.18a shows hmax(χ∗,BN) extended to χ∗ = −10 (χ = 10) as well as fora range of positive χ∗. We observe that in this regime, hmax = 0 for BN ≤ 6 andthat the density unstable χ values result in thicker maximal layers than for densitystable.Suppose now that χ > 3/(2y3i ). We see that τ2,xy now increases across fluid 2.Provided that:32y3i< χ <32y3i1(1−2yi) ,the largest magnitude shear stresses will be found at the interface and the shearstresses remain negative throughout the fluid 2 layer (regime 2). On increasing χfurther the stresses become positive close to the wall. Provided that:32y3i1(1−2yi) < χ <32y3i(1+2yi)(1−2yi) ,the largest magnitude shear stresses are still found at the interface and the shearstresses switch from negative near the interface to positive at the wall (regime 3).For either of regimes 2 or 3 to remain fully static requires BN > |τi|. Finally, notethat for32y3i(1+2yi)(1−2yi) ≤ χ,the largest shear stress in fluid 2 is again at the wall, but now is positive (regime 4).These 4 regimes are plotted in Fig. 3.18b.We see from the above discussion that for any fixed χ the maximal absoluteshear stress in fluid 2, may depend critically upon the regime and may vary with yi.In Fig. 3.18c we plot max(|τ2,xy|) for 3 different χ . For χ = 10, 100 we see a mono-tone decrease in max(|τ2,xy|) with yi, which happens in regimes 1-3. For χ = 1000we see that max(|τ2,xy|) is no longer monotone with yi. As we pass through regime94Figure 3.18: a) Plot of hmax using the definitions in [103]: extending to χ <12. b) Location of regimes 1-4 in the (yi,χ) plane. c) Variation of themaximal shear stress in fluid 2 max(|τ2,xy|), for χ = 10, 100, 1000. d)Plot of hmax,a for χ = 0, 10, 50, 120, 190.4 max(|τ2,xy|) increases and then decreases. This obviously becomes problematicfor extending the way in which we have defined and used hmax in Chapter 2.First let us suppose that χ < 12. For any BN > 6 we may compute yi,min,a:max(|τ2,xy|)(yi,min,a) = BN , which has a unique solution. Defining hmax,a = 0.5−yi,min,a leads to the maximal possible layer thickness for which all the fluid 2 layeris static. If now 12 ≤ χ < 197.49..., regime 4 is never entered and again (seeFig. 3.18c), max(|τ2,xy|) is monotone. Again hmax,a is uniquely defined. These95curves are plotted in Fig. 3.18d for a range of χ . For the smaller BN we have aregime 2 or 3 flow at hmax,a, whereas for larger BN as hmax,a increases we find againregime 1.If we consider χ > 197.49... we may enter regime 4 and have multiple solutionswhere max(|τ2,xy|)(yi,min,a) = BN . Although we can select the smallest of the (upto 3) solutions, the multi-valued solutions are hard to interpret. At the minimum itseems that we can have a jump in hmax,a as χ is increased, i.e. from a relatively thinlayer to a relatively thick layer. We have not analyzed this regime further.Partially static wall layersBefore moving on, we should explore the form of 1D solutions that are actuallyfound in regimes 2-4. The analysis of the previous section is purely based on thestress distribution, but the full 1D solution is anyway uniquely determined (mean-ing the modified pressure gradient f and the velocity U(y); see Chapter 2). Thereis therefore no guarantee that the fluid 2 layer will remain fully unyielded in theactual flow solutions.First in regimes 2 & 3, if the fluid 2 layer yields it yields first at the interface.Thus, the fluid 2 layer consists of a yielded layer y ∈ (yi,yY ) adjacent to the in-terface and a static layer y ∈ [yY ,0.5], attached to the wall. The position of theyield surface is yY = (χyi−BN)/(χ − f ), but the modified pressure gradient f isno longer given by f = 3/(2y3i ), as the yielded layer of fluid 2 also contributes tothe net flow rate. As fluid 2 yields, m also affects the flow. The interfacial velocityis found as: U(yi) = (yY − yi)2(χ− f )/(2m) and f is found from:2 f y3i3+(yY +2yi)(yY − yi)2(χ− f )3m= 1,which is simply (3.20). Thus, for small m we have the possibility that f is reducedfrom the fully static layer value and consequently relatively low stresses might befound close to the wall. Note too that the broken line in Fig. 3.18b denotes wherethe wall shear stress is zero, for fully static layers, suggesting that a small yieldstress might be sufficient to keep a static wall layer.Due to the flow constraint, f > 0 and hence τi < 0 always. This has the conse-quence that the yielded fluid 2 interfacial layer is pulled in the positive x-direction96by fluid 1, in all regimes. As we move from regime 2 to 3, the stresses on fluid2 begin to push fluid 2 downwards adjacent to the wall. However, the yield stressresists and is able to keep the layer static until entering regime 4. Now if the fluid2 layer yields, it does so at the wall: there will be no static wall layer and the fluid2 layer will move downwards against the flow.Two questions arise. Firstly, in parts of the flow where the streamlines areparallel to the x-axis, what are the flow types found in our 2D simulations, e.g. dowe find mobile, fully static or partially static mud layers? Secondly, in regime 2 &3 when the fluid 2 yields near the interface, can the thickness of a partially staticlayer be larger than the fully static maximum hmax,a?To answer the first question we have plotted in Fig. 3.19 the values of yi =0.5− h, measured near the exit of the channel for different χ . To calculate h wefirst select a time at which the front is at least 80% of the distance along the channeland define a sampling length of 25-35% of the channel length, behind the front.The layer thickness is taken from the average of c over this window (see Chapter 2for a description). To classify the flow type we need to estimate the wall andinterfacial stresses. We do this by defining narrow windows in y, close to thewall and straddling the interface, and averaging the shear stress τxy. This is thencompared with BN to define the regime.Figures 3.19a & b plot the data for m≤ 1 and m > 1, respectively. In each, thecolours of the symbols represent BN ; the filled symbols are stable and the emptysymbols are unstable; the shape classifies the flow regime within the parallel partof the flow: (M,◦,♦) represent fully static, partially static and moving layers, re-spectively. We see that there are indeed static wall layer flows in regimes 2 & 3 forall values of m. Moreover, we see that even for BN < 6 it appears that static walllayers may occur.Other interesting trends are observed in Fig. 3.19. First the layer thicknessappears to decrease with increasing BN ; see also [54]. Increasing χ generally leadsto instability (as we have seen above in Section 3.2.1). Mobile layers are foundat all χ whereas partially static wall layers appear to require significant χ as ispredicted by 1D model. Fully static layers have been found in regimes 1 & 2 & 3,for m ≤ 1, and only in regimes 2 & 3 for m > 1. For BN < 6 we found partiallystatic wall layers and only a few fully static for m ≤ 1 which are very close to the970.2 0.25 0.3 0.35 0.4 0.451001011021030.2 0.25 0.3 0.35100101102103Figure 3.19: Values of yi = 0.5− h computed from our 2D simulations: a)m ≤ 1; b) m > 1. Symbols (M,◦,♦) represent fully static, partiallystatic and moving layers, respectively; BN = 1, 5, 10 are blue, greenand red, respectively. The filled symbols are stable and empty symbolsunstable. The lines represent transitions between regimes 1-4, accord-ing to the fully static wall layer analysis.yielding threshold i.e. within numerical error.We now examine the second question, i.e. is hmax,a maximal? Whereas Fig. 3.18is constructed assuming a fully static wall layer, Fig. 3.19 has shown that thisis not the case in practice. To find the actual static layer thickness at any given(yi,χ,BN ,m) we solve the 1D model fully, which is done computationally as de-scribed in Chapter 2. We then evaluate whether or not layer 2 is static near the wall.The simplest way to represent this is to evaluate the position of the yield surface(y = yY ) at each value of yi. Since the stresses are linear in y there are unique posi-tions where ±BN are attained. From these we can classify whether the solution isin regime 1-4, i.e. according to the interfacial and wall shear stresses.Figure 3.20 shows variations in yY with yi for 5 different values of m for variouschoices of (BN ,χ). Note that if yY < yi we set yY = yi, indicating that the interfacialstress does not exceed the yield stress. Nominally we see a U-shaped variation ofyY (yi), with a single minimum, from which h= 0.5−yY defines the maximal staticwall layer. In increasing yi from 0 the fluid 2 layer thickness decreases and wetransition from flows of regime 1 or 2, through 3 or 4, then back through 3, 2, 1.The inset figures in Fig. 3.20 show the velocity and stress solutions for 4 selectedyi, at the values of m indicated in the captions. These show explicitly the different980 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.050.10.150.20.250.30.350.40.450.50 0.1 0.2 0.3 0.4 0.50120 0.1 0.2 0.3 0.4 0.5-15-10-5050 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.050.10.150.20.250.30.350.40.450.50 0.1 0.2 0.3 0.4 0.500.511.522.50 0.1 0.2 0.3 0.4 0.5-8-6-4-200 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.050.10.150.20.250.30.350.40.450.50 0.1 0.2 0.3 0.4 0.501230 0.1 0.2 0.3 0.4 0.5-20-100100 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.050.10.150.20.250.30.350.40.450.50 0.1 0.2 0.3 0.4 0.5-15-10-500 0.1 0.2 0.3 0.4 0.50123Figure 3.20: Variation of static wall layer position with interface position fora) (BN ,χ) = (1,113.78); b) (BN ,χ) = (5,70); c) (BN ,χ) = (5,220); d)(BN ,χ) = (10,90). Each symbol represents a specific m, such that m=(0.1,0.33,1,3,10) are shown respectively with (×,♦,+,∗,◦). Insetfigures show τxy and U(y) at different m values for each case: a) m= 1,b) m = 0.33, c) m = 1 and d) m = 0.1.regime solutions as we transition.We observe in Figs. 3.20a & c that for some interface values there is no value ofyY . At these intermediate values of yi we find only regime 4 solutions. In this casewe must examine which of the two branches that allow static wall layers yieldsthe minimum yY at the points of discontinuity of the curve. In general the smallervalues of m give rise to regime 4 solutions more readily, as does smaller BN . For99sufficiently large BN we do not find regime 4 solutions, e.g. Fig. 3.20d.In Figs. 3.20c & d we show a range of yi near to yi = 0.5, for which yY = yi,(for all m) with the interpretation that the entire layer is static. We note that BN = 5in Fig. 3.20c demonstrating that fully static layers exist below BN = 6 for densityunstable displacements. Figures 3.20a & b show that partially static wall layersexist over wide ranges of yi, even for BN = 1. The value of χ in Fig. 3.20a is theminimum value of χ on the broken line in Fig. 3.18b, i.e. where τw = 0 for a fullystatic wall layer (although here for BN = 1 the wall layer is partially static). Wehypothesize that we can find values of (χ,m) that give partially static wall layersfor arbitrarily small BN .To further understand the solution space in which partially (and fully) staticwall layers are found, Fig. 3.21 takes selected pairs (BN ,m) and plots admissible yYin the (yi,χ)-plane. By admissible, we mean those values that allow static layers.From this we see that the full U-shaped profile of yY vs yi is only obtained for anarrow band of χ: more common is that there are one or two branches of yi-valueswith admissible yY . Only for BN > 6 does this extend to χ = 0.From the solutions, as in Fig. 3.20, we can recompute maximal static walllayer, say hmax,p, that evaluates the static wall layer thickness using yY and yi, thenmaximizes the value for yi ∈ [0,0.5]. The computed hmax,p includes both fullyunyielded wall layers and partially unyielded wall layers. Figure 3.22 shows theresult of this calculation for BN = 5, 10. In Fig. 3.22a (BN = 5), below χ = 12there are no static wall layers: hmax,p = 0 for all m and the wall shear stress isnegative. As we increase χ above χ = 12, we see a transition to regime 2, withhmax,p > 0. The smaller values of m bifurcate at lower values of χ and we note thatthe thickness of the partially static layers is significant. Given the form of solutionsin Fig. 3.20, note that for each value of χ there are potentially 2 intervals of yi thatgive static h < hmax,p.In Fig. 3.22b (BN = 10), we may have a static wall layer for χ = 0. For thesesolutions the entire wall layer is static and as χ increases we see a change fromregime 1, to regime 2 (but fully static) and then regime 2 with partially static walllayer. The symbols overlap until the partially static layers are found; again smallerm bifurcate at lower χ . Also plotted in Fig. 3.22b is hmax,a (broken line), which isindependent of m. We see that hmax,p = hmax,a until the solutions bifurcate and that100Figure 3.21: The value of yY at each yi for a)(BN ,m) = (5,0.1), b)(BN ,m) =(5,10), c)(BN ,m) = (10,0.1), and d)(BN ,m) = (10,10).hmax,p > hmax,a thereafter. Surprisingly the maximal static layer thickness is largerwhen the layer is only partially yielded!3.2.4 Predicting regimesThe residual layer thickness h is taken from our 2D simulations and classified (asdescribed earlier) and plotted against hmax,p, calculated from the 1D model; seeFig. 3.23. Markers specify the flow regimes and the colors give the values for BN .The values of h, in the case of partially static layers, are calculated using only thethickness of the static part.Almost all of the 2D results found to have partially static layers lie below thehmax,p predicted by the 1D model. This is to be expected following the resultsfor iso-dense and density stable displacements [54, 55, 103], which show that theselection mechanism for layer thickness is generally determined at the displace-1010 0.05 0.1 0.15 0.2 0.25100101102hmax,pχa)0 0.05 0.1 0.15 0.2 0.25100101102hmax,pχb)Figure 3.22: Variation of maximum static wall layer thickness hmax,p withχ at: a) BN = 5; b) BN = 10. Each symbol represents a specificm: m = (0.1,0.33,1,3,10) are shown respectively with (×,♦,+,∗,◦).The value of hmax,a at similar χ is shown with the broken line patternin b.1020 0.05 0.1 0.15 0.2 0.25 0.300.050.10.150.20.250.3Figure 3.23: Plot of h against hmax,p; (M,◦,♦) mark fully static, partiallystatic, and moving layer regimes respectively. BN = 1, 5, 10 are blue,green and red, respectively. The filled symbols are stable and emptysymbols unstable.ment front. Also we have seen in the previous section that there are intervals ofadmissible partially static layers.In considering the cementing/micro-annulus context, the key question concernsthe long-term behaviour of these partially static layers. In stable flows we canexpect these to evolve according to a thin-film type of analysis/model as outlinedin Chapter 2. In general, as the yielded part of the layer is mobile, the displacedfluid is being washed away (drained) downstream and we see that yi(x, t) shouldincrease. Considering Fig. 3.21, if we are on the left branch of one of the shadedregions, increase in yi may result in a regime 4 solution (moving at the wall), andeventually back to a partially static layer on the right branch. However as discussed,partially static layers drain and yi(x, t) increases. If on the right branch, whetherthe layer persists at all depends on the behaviour as yi→ 0.5. For sufficiently largeBN the situation is as in Figs. 3.21c or d: eventually the layer evolves to a thin fully103static layer. On the other hand, if like Figs. 3.21a or b, there is no final fully staticlayer.Thus remarkably, we are back to a simple transition about BN = 6 to describethe long-time behaviour of the draining of wall layers. All these behaviours areasymptotic as t → ∞ and the speed of evolution of such thin-films depends on theareal flux within the displaced fluid. We can expect this to decrease with m and tobe proportional to (yY − yi)3. The latter suggests very slow draining of the yieldedpart of the wall layer, which may mean that partially static layers will persist overthe duration of primary cementing displacement.The moving layers in Fig. 3.23 generally have the largest thickness. At smallerBN these will be removed over long times (either directly or transitioning throughother regimes). Most of the unstable flows have moving layers. At larger BN(where thin fully static wall layers may be possible), it is possible that h evolves tothe maximal static wall layer hmax,a.Regarding the layers classified as fully static in Fig. 3.23, we observe that mostare below or just above hmax,p. The latter should not happen according to the 1D-model. This may be a consequence of averaging to determine h, or the interfacialand wall stresses (used to classify). We note that the thickness above hmax,p isgenerally 1-2 mesh cells. We regard these layers as having evolved from transientlayers towards hmax,p, and some may still be evolving. Note that classification ofour layers is based on averaged interfacial and wall stresses, not velocity. Thereare also a number of layers far below hmax,p, which will be fully static.The data for BN = 10 is plotted in Fig. 3.24, showing the classification for eachm. Again partially static layers have only the static part plotted. We observe astratification with m, i.e. thicker layers for larger m at each χ . With regard to thetransient layers, this is natural as what is plotted is simply a snapshot in time andlarge m have more viscous wall layers, so drain slower. As commented above, weagain see some h classified as fully static, lying within a few mesh cells of hmax,pFinally, we have used the same method as in part 1, to classify the displace-ment front behaviour of the associated 1D transient thin-film model (see Chapter 2and Appendix A where this is discussed in detail). The thin film model leads to ahyperbolic conservation law for the interface evolution yi(x, t). By analyzing theflux function directly it is possible to classify/predict the long time behaviour with-1040 20 40 60 80 100 1200.040.060.080.10.120.140.160.180.20.220.24Figure 3.24: Thickness of residual layer against χ , for BN = 10. Symbols(M,◦,♦) denote fully static, partially static, and moving regimes, re-spectively. Each color represents a specific m: magenta, blue, black,red and green colors respectively show m = 0.1, 0.33, 1, 3, 10. Thelines plot hmax,p for m = 0.1, 0.33 with aforementioned colors.out solving the thin-film equation numerically in time. Three frontal regimes havebeen identified for BN > 0 and with our density unstable configurations: 1 - frontdisperses or moves with a single shock ahead; 2- two shocks exist, the frontal ad-vances fastest and the trailing shock stretches the interface; 3- the shocks combineand the only mass-conserving solution is that the interface moves plug-like withspeed of 1. The 3 regimes are shown in Fig. 3.25a with symbols (,M,◦) respec-tively. Each BN is shown with a specific color: BN = 1, 5, 10 are in blue, greenand red, respectively and slightly smaller symbols have been used as BN increases,in order to be visible where the classifications overlap.The results of our 2D flows which are either stable or have instability of KHtype are shown in Fig. 3.25b, marked with (,◦) respectively. What is obvious bycomparing the figures, is that the stable flows generally coincide with configuration1 from the 1D model and that the KH type instability can be predicted from thetransition to plug-like or two-fronts. Note that the plug-like regime in the 1D modelis an intermediate regime where no solution with 2 shocks can be found. The 2shock solutions produce the characteristic differential velocity in 2 streams that arecharacteristic of KH type instabilities.10510−1 100 10110−1100101102103mχa)10−1 100 10110−1100101102103mχb)Figure 3.25: a) Flow regimes from the 1D thin film model of Chapter 2 (seedetail in Appendix A): ()- Dispersive, or there is only a frontal shock,(4)- There are two shocks: front and trailing, (◦) - The only possiblesolution is that the channel plugs up and displaces with a speed VS = 1.b) Flow regimes classified from the 2D model; () - stable, (◦) - KH.In both figures, BN = 1, 5, 10 are blue, green and red, respectively.Marker size in both figures is chosen to decrease with BN .This partial classification of flow type is useful and mimics that in Chapter 2for density stable flows, i.e. the 1D thin-film models predict dynamics if not actuallayer thickness. The IBM type instabilities were not predicted by this analysis of1D displacement front behaviour. This is not surprising as these instabilities appearto grow later, triggered by instability of the layers and not the front. On the otherhand the KH prediction appears to work here, but indirectly: the 1D model predictsa base flow that is characteristically unstable to KH type instabilities.3.3 Summary and discussionWe complete the study of displacement flows of a yield stress fluids by a Newtonianfluid along a vertical channel with the results presented in this chapter. Iso-densedisplacements were studied in [54, 55] and density stable displacement flows werestudied in Chapter 2. Of course, the results are only complete in considering aNewtonian fluid displacing a Bingham fluid. This is the simplest configuration toallow the key phenomena of a static residual wall layer and to study competingeffects of yield stress, buoyancy stress, viscous stress and viscosity ratio. More106rheological complexity can be introduced through consideration of two Herschel-Bulkley fluids.For density unstable displacement of a Bingham fluid with a Newtonian one,and χ ∈ (0.1,500), three types of instabilities have been identified: frontal instabil-ities of Rayleigh-Taylor (RT) type, interfacial roll-waves of Kelvin-Helmholz (KH)type and viscous-controlled inverse bamboo and mushroom (IBM) morphologies.Each classification is interpreted loosely as in practice other physical effects arepresent and the flows are not idealised for the study of individual instabilities.In general, increasing χ when the displaced Bingham fluid is less viscous thanthe displacing fluid leads towards a counter-current flow and KH instabilities areobserved in this scenario. This tends to be a combination of low m, weak BN andstrong χ . IBM instabilities are generated when m > 1 and a two-layer structure isinitially formed. Since m > 1, they have a slow growth rate controlled by the vis-cosity of the displaced fluid. Within the IBM classification we have also observedan interesting regular patterning of static wall layers (footprinting); see Fig. 3.6.The transition between stable and unstable regimes has been plotted in the(Fr,Re/Fr)-plane. It happens across a boundary of form: Re ∼ Frx, where x is2,8/5,4/5,2/5 depending on m and BN . The exponenent x decreases with increas-ing m values and slightly increases with BN . Our study has been phenomenologicalin this aspect, so these transitions are intended more as a guide rather than as anyserious attempt to curve-fit data. In terms of these parameters, although Re/Fr isuseful to plot against, this balance does not represent the viscosity of the displacingfluid at all: including m and BN would be a potential improvement.We have also explored static residual layers for density unstable flows. First,by assuming that fluid 2 is completely static, four flow regimes have been identifiedin our two layer model: 1- τxy decreases from zero in the center toward the wall,and the maximum absolute value will occur at the wall; 2- by increasing χ furtherthe slope of τxy is reduced in layer 2, but still the interfacial and wall shear stressare negative; 3- by increasing χ further the wall shear stress becomes positive, butis still smaller than the interfacial stress; 4- the wall shear stress is positive andbigger than the negative interfacial stress.The above classification was based purely on the stress distributions. The fluid2 layer next to the wall can be static or moving in all regimes, depending on107BN . However, the interesting fact is that the wall layer can be partially yieldedin regimes 2 & 3, such that a layer near the interface yields and is moved in theflow direction, but wall layer remains static. For regimes 1-3 we have extended theconcept of the maximal static wall layer to define hmax,a, where all of the fluid 2layer is static. However, it is apparent that when the wall layer yields in regimes2 or 3, there will be a partially static layer and potentially the partially static layercould be thicker than the fully static. This is because, by yielding and movingslightly the fluid 2 layer contributed to the flow rate, hence reducing the pressuregradient and shear stresses.The residual layer left on the wall in our 2D simulations has also been analysed.This layer can be: 1- fully static; 2- partially static; 3- fully moving. It is shownthat fully moving layers can happen at all χ,BN and m values whereas partiallystatic layers lie in regimes 2 & 3, as predicted by the 1D-model. For m ≤ 1, fullystatic layers for BN = 10 can develop in regimes 1, 2 and 3 but for m > 1 theyare distributed only in regime 2 & 3. For BN < 6, all the layers are moving orpartially static except a few fully static layers which are very close to the yieldingthreshold, i.e. within numerical error. Similar to the results obtained in Chapter 2the thickness of the residual layer increases with m and decreases with BN .As well as our stress-based analysis, we have solved the 2-layer model com-putationally, so that the stress profiles are now those actually found as part of thevelocity (and pressure gradient) solution, and not simply inferred from the momen-tum balances and constraint of a static wall layer. Using this, it is observed that atany given set of (BN ,χ,m), yY varies with yi over a U-shaped curve which eitherhas a single minimum or is discontinuous with the middle part broken. This leadsto the simple calculation of the minimal yY and hence a maximal static layer hmax,p.The maximal static layer hmax,p ≥ hmax,a and allows for partially static wall layers.The important results are that: (i) partially static wall layers can be formed forBN < 6, unlike for the density stable case; (ii) as discussed above, the partiallystatic wall layers can for certain parameters yield a layer thickness larger than fora fully static wall layer. In agreement with 1D-model results, the thickness of walllayers obtained from 2D-computations in fully static regime are mostly less thanhmax,p. A few cases have slightly thicker layers which may be due to the numericalprocedure.108Finally we have explored a thin-layer version of the 2-layer 1D model, as wasdone in Chapter 2 (see detail in Appendix A). Just as for the density stable case,the displacement front predictions from the 1D model are predictive of behaviouron the 2D simulations. We are able to predict stable flows and those susceptible toKH type instabilities.3.4 Concluding remarksIn a nutshell, in this chapter we analyzed the miscible displacement of a Binghamfluid with a less dense Newtonian and classified hydrodynamically stable and un-stable regimes with respect to the governing parameters of the flow (Re,Fr,BN ,m).The regimes of residual layer (static, moving, or partially static) are re-plotted ina map obtained from our stress-based analysis with respect to the governing pa-rameters of the problem. Given the flow parameters, this mapping facilitates theprediction of the layer regimes. In addition, our results confirm that our 1D-modelcan be used as a credible tool to predict stable flows and KH type instability in 2Dflows.109Chapter 4Invasion of fluids into a gelledfluid column: yield stress effects 1As a simplified model for understanding invasion of gas into cemented wellbores,we study the invasion of a Newtonian fluid into a vertical column of yield stressfluid through a small hole, using both experimental and computational methods.We find that the invasion pressure must exceed the static pressure by an amountthat depends linearly on the yield stress of the fluid and that (for sufficiently deepcolumns) is observed to increase with the height of the yield stress column. How-ever, invasion pressures far less than the Poiseuille-flow limit are able to yieldthe fluid, for sufficiently small hole sizes. Observed experimental behaviours inyielding/invasion show a complex sequence of stages, starting with a mixing stage,through invasion and transition, to fracture propagation and eventual stopping ofthe flow. Precise detection of invasion and transition pressures is difficult. Invasionproceeds initially via the formation of a dome of invaded fluid that grows in thetransition stage. The transition stage appears to represent a form of stress relax-ation, sometimes allowing for a stable dome to persist and at other times leadingdirectly to a fracturing of the gel. The passage from initial invasion through totransition dome is suggestive of elasto-plastic yielding, followed by a brittle frac-ture. Computed results give qualitative insight into the invasion process and also1A version of this chapter has been published in Journal of non-Newtonian Fluids Mechanics[123].110show clearly the evolution of the stress field as we change from local to non-localyielding.4.1 IntroductionGas migration has been a chronic problem in the completion of oil and gas wellsfor decades [124, 125]. This is a complex phenomenon with causes that com-bine chemical, geochemical and fluid mechanical processes. The consequences ofgas migration can range from gas emissions at the wellhead or into the surround-ing sub-surface ecosystem, to aquifer contamination and in extreme cases to wellblowouts. In addition, leakage lowers reservoir pressures impacting well produc-tivity. Thus, gas migration impacts health and safety, environment and economicaspects of hydrocarbon energy production.A key difficulty of understanding gas migration is that the causes manifest overdifferent timescales in the well construction process. Here we focus on early-midstage gas migration, which happens during the primary cementing of a well, and onrheology effects. In primary cementing [5] a steel casing is inserted into the newlydrilled borehole and cement slurry is placed into the annular space surroundingthe casing and bordering the reservoir. After the pumps are turned off the cementslurry is left to set within a narrow annular space (e.g. 2−3 cm mean annular gap,for mean diameters anywhere in the range 12−30 cm) that extends many 100’s ofmeters along the wellbore. The cement slurry which generally has modest yieldstress (∼ 5− 10 Pa) develops a significant gel strength during the early stages ofhydration, when it may still be regarded as a semi-solid yield stress suspension.During this stage the root cause of gas invasion into the well is pressure imbalance.Consider a tall column of static yield stress fluid with a single small “hole”in the wall of the container. We apply fluid under pressure through the hole andquestion whether or not the fluid invades and under what mechanism. In the caseof a purely viscous fluid within the container and if we neglect capillary effects,a pressure over-balance is sufficient to ensure that fluid can enter. How is thisinvasion question affected when the in situ fluid has a yield stress? Firstly, in anadmittedly idealised scenario, the fluid behaves locally as a rigid solid, blockingany invasion unless the over-pressure generates stresses sufficient to locally yield111the fluid around the hole. Secondly, if there is a net influx into the container it isnecessary for fluid to yield and displace all the way to surface, suggesting that theheight of fluid in the column is important. These are the topics we study in thischapter.Within the literature on viscoplastic fluids there are a number of works of rel-evance to fluid invasion and migration, although not directly to the problem westudy. Regarding the migration phase, much of the literature on droplet and bubblemotion is reviewed by Chhabra in the comprehensive text [126]. The question ofwhat yield stress is sufficient to prevent migration of a bubble was addressed byDubash & Frigaard [127, 128], but the bounds derived are fairly conservative asshown by Tsamopoulos et al. [129]. There has been much recent research on bub-ble motion, both experimental and computational, e.g. [130–133], and to a lesserdegree on droplets [134–136]. Others have studied displacement/injection flowsof yield stress fluids in pipes in both miscible [42–46] and/or immiscible [47–51]scenarios. Others have studied similar flows in Hele-Shaw geometries [52, 53].Here we study invasion through a small hole. In her thesis, Gabard [42] studiedeffects of moderate nozzle size variations on displacement flows and in [44–46]the authors have studied Newtonian-viscoplastic displacement flows in pipes, withinvasion hole the same size as the pipe, (i.e. initial separation of fluids via a gatevalve). Thus, these studies can be interpreted as a continuous variation of holesize. However, here our concern is the actual invasion/yielding stage as motion isinitiated within the yield stress fluid.The transient start-up flow of a (weakly compressible) viscoplastic fluid causedby imposing a sudden pressure drop has been studied in [137–140] in the contextof pipeline restart of waxy crude oils. In order to mobilize the gel in the pipe,a large enough pressure drop needs to be applied which is related to the pipelinelength and to the fluid yield stress and bulk compressibility [141]. Waxy crudeoil gel breaking mechanisms can range from adhesive (breakage at the pipe-gelinterface - partial slip) to cohesive (breakdown of the internal gel structure itself- yielding) [142]. There also exist a number of works in pharmaceutical contextsthat study high-speed injection of a liquid into a soft tissue mimicking needle-freejet injection of drug under the skin [143, 144]. Analysis of drug jet entry intoelastic polyacrylamide gels has revealed three distinct penetration stages namely112erosion, stagnation, and dispersion [143]. The jet removes the gel at the impactsite during the erosion phase leading to the formation of a distinct cylindrical hole.Thereafter, the jet comes to stagnation characterized by constant penetration depthand finally followed by dispersion of the liquid into the gel. During dispersionnearly symmetrical cracks of the injected fluid propagate within the gel. See also[144] for mathematical modeling of similar injection flows.An outline of this chapter is as follows. Section 4.2 introduces a setup in whichto study fluid invasion and performs a dimensional analysis of the simplest setup.The experimental method is outlined in Section 4.3. Our main experimental re-sults on invasion pressures and a qualitative description of fluid invasion stagesare given in Section 4.4. Section 4.5 presents results of a computational study ofinvasion pressures, using idealised yield stress fluids, which helps to understandthe transition between local yielding and a Poiseuille-type behaviour. The chapterends with a brief discussion.4.2 Fluid invasion simplifiedAs discussed above, the objective of our study is to understand the effects of theyield stress on the invasion (i.e. penetration) stage of gas migration. Consider there-fore the following simplified setup in which fluid invasion can occur. A column ofan ideal incompressible yield stress fluid of height Hˆ is contained in a tank of lateraldimension Rˆ, (e.g. uniform circular or rectangular cylinder). The fluid density andyield stress are denoted ρˆ and τˆY , respectively. A circular orifice (hole) of radiusRˆh in the base of the tank (Fig.4.1a) contains the invading fluid, which is assumedviscous and incompressible, with density ρˆi. The invading fluid is connected viatubing to a large reservoir (Fig.4.1b) that is maintained at a height Hˆi. The reservoirheight is controlled to ensure that the pressure Pˆh, within the invading fluid orificeat the entry to the tank, is Pˆh ≥ ρˆ gˆHˆ. The question we wish to address is whetheror not the invading fluid is able to penetrate into the static gelled column?For simplicity it is assumed that ρˆi = ρˆ , so that buoyancy forces do not play anyrole in the flow. Equally, we assume that the two fluids are completely miscible, soas to neglect capillary effects. In order to penetrate into the gelled column it is clearthat the visco-plastic fluid must yield. If Pˆh = ρˆ gˆHˆ, there is no driving pressure and113Yield stress fluid2𝑅#𝑃# = 𝑃#ℎ𝐻(2𝑅#ℎa)LaserLaser sheetGate valveMotorized scissor jackRed fluorescent water reservoirYield stress fluid25𝐻(b)𝐻(𝑖2. 002.500Figure 4.1: Schematic setup: a) conceptual problem; b) experimental appa-ratus.we may expect there to be no flow. Consider conceptually an experiment in whichHˆi is progressively and slowly increased, e.g. as in Fig.4.1b. We expect that at someheight we have sufficiently increased Pˆh > ρˆ gˆHˆ, such that fluid penetration occurs,causing the fluid in the tank to yield and flow. The invasion pressure Pˆi is definedas the value of Pˆh at which fluid penetration first occurs.Evidently, we expect that Pˆi depends on the static pressure in the yield stressfluid at the base of the tank, and thus on ρˆ , gˆ and Hˆ. As yielding is involved, τˆYalso naturally affects the flow. However, if we only consider the process of yieldingand ignore what may happen afterwards, other rheological parameters should notaffect Pˆi. We must also consider the two geometric parameters: Rˆ and Rˆh.Following a dimensional analysis and subtracting the static pressure field ev-erywhere, we find thatPi =Pˆi− ρˆ gˆHˆτˆY= f (H,rh), (4.1)where H = Hˆ/Rˆ and rh = Rˆh/Rˆ, i.e. the scaled invasion pressure should dependonly on the dimensionless height of the yield stress fluid column and on the holeradius, i.e. 2 dimensionless parameters define Pi. Note that in this analysis, by con-sidering only an ideal visco-plastic fluid (e.g. Bingham, Casson, Herschel-Bulkley,etc) we do not allow for other mechanical behaviours to influence the yielding pro-cess, e.g. elasto-plastic behaviour, thixotropy, etc. This simplification is purely toprovide a framework with which to design our experiments and understand ourresults.1144.3 Experimental descriptionTo explore the flow described in Section 4.2, we constructed a simple experimentas follows. Experiments were performed in a long Plexiglas cylinder of radiusRˆ = 3.175 cm (1.25′′) and height HˆC = 63.5 cm (25′′), with sealed base. ThePlexiglas cylinder was immersed in a rectangular tank filled with a glycerin so-lution, prepared to match the refractive index of the Plexiglas, hence minimizingoptical distortion. The inner surfaces of the fluid column are treated with a PEI(polyethylenimine) solution, following the method suggested by Metivier et al.[145], to avoid wall slip (evident in some early experiments after invasion occurs).The cylinder is filled with a Carbopol EZ-2 solution and water invades througha hole positioned in the centre of the base, of radius Rˆh = 0.3175mm. To balanceatmospheric pressure we control the injection pressure Pˆh with a manometer ar-rangement. The net injection pressure is then regulated by the difference in heightbetween the Carbopol column and a water reservoir; see Fig. 4.1b. Using thisdesign instead of a pressure regulator, leads to elimination of both atmosphericpressure and hydrostatic pressure. In order to enhance experimental repeatabilityat smaller heights of Carbopol, we added water on top of the Carbopol column sothat the total height of fluid column remained constant in each experiment, (seeSection 4.3.2).While injecting liquids, the planar Laser-Induced Fluorescence (planar-LIF)technique is used to identify the entry of the fluid once invasion occurs. Red flu-orescene powder, which has maximum excitation upon contact with 532nm greenlaser light, is used to dye the invading fluid.The experiments are recorded usingtwo cameras. A JAI AD-081 (Edmunds OPTICS) with 16mm compact Fixed Fo-cal Length Lens is used to record a small field of view of 2.5× 3cm2, around thehole with a spatial resolution of ≈ 30 pixels per mm. A second camera (NikonD800) records a larger propagation field of the invading fluid. The first camerarecords with 8 frames per seconds at 0.8 megapixels. The invading fluid propa-gation is recorded by at 720 pixel resolution with 60 frames per second, giving aspatial resolution of ≈ 10−15 pixels per mm.Note that we built our experimental setup and gradually developed it to removethe sources of experimental errors and uncertainties in invasion pressure measure-115ments. The earlier experimental setups and some of the preliminary results areillustrated in Appendix C.The experiment consists of gradually increasing the applied pressure withinthe invading fluid “pore” until the fluid enters the column. The precision of ourheight control system is 0.1µm, which means a static pressure of ≈ 0.001Pa canbe controlled by the height of the invading fluid through the automated scissor jacksystem.However, measurement of surface height is by a laser level projecting ontoa scale giving pressure to within 5 Pa.With fixed hole size and cylinder radius, the parameters expected to be rele-vant to the experiments are the aspect ratio of the column and the yield stress ofthe invaded fluid, see (4.1). Thus, to study the effects of these parameters on theinvasion pressure, we have performed experiments with a variety of concentrationsof Carbopol over many different heights of the Carbopol column.4.3.1 Fluid characterizationThe in-situ fluid is a solution of Carbopol EZ-2 (Lubrizol). Carbopol is a commonwater-based yield stress material used in many different applications. The structureof this material and its yield stress is mainly a function of the concentration, pH,and preparation procedure. For consistency, we used an identical procedure for ourCarbopol preparation and it was prepared daily for the experiments of the same dayto prevent time-dependency of the material. The desired concentration of Carbopolpowder was mixed with water for one hour, then it was neutralized by stirring inan appropriate amount of 0.1 g/l of sodium hydroxide solution (NaOH), and thenmixed for 23 hours. To avoid Carbopol evaporation, the container was coveredduring mixing time. For mixing the batch gently, we used a 3-blade stirrer withsmooth edges.Before starting each experiment, a series of controlled shear rate rheologicalmeasurements were conducted using a Gemini HR nano rheometer (Malvern In-struments), at 20◦C, representing the laboratory temperature. Shear rate sweepsover the range of 0.001−0.1s−1 were conducted, with data collected for 100 pointsat each shear test. To negate potential elastic effects, the time step between eachshear rate measurement was set to be 7 seconds, including delay and integration116=^ (Pa)10-1 100 101Instantaneousviscosity(Pas)101102103104Test 1Test 2Test 3Test 4Figure 4.2: Example flow curves for 4 tests using C = 0.15% (wt/wt) Car-bopol solution. The instantaneous viscosity technique is used to esti-mate the yield stress of the solution.time. Three different Carbopol concentrations were used, as outlined in Table 4.1.For each Carbopol the yield stress was estimated using the maximum viscositymethod [146]. This method is based on the theory that yield stress materials haveinfinite viscosity before yielding. A representative example of using this method isshown in Fig. 4.2 for C = 0.15% Carbopol. These tests were performed 4 times foreach sample, with a high degree of repeatability, but nevertheless there is a signif-icant range of maximum τˆ . The Carbopol concentrations were limited both aboveand below for practical reasons. Too low a concentration resulted in a very diffuseinterface and weak yield stress. The issue here is that yield stress measurement isrelatively low precision and we scale with the yield stress in our dimensional anal-ysis. Too high a concentration results in a large yield stress, which causes problemswith the filling protocol.4.3.2 Experimental evolution, calibration and repeatabilityOur initial experimental focus was on gas invasion. With the intention to investi-gate this in a small lab-scale experiment we identified the following stresses thatwould influence: (i) atmospheric pressure (105 Pa); (ii) static pressures (∼ 10cm117Fluid Shear rate range Yield Stress PHˆ˙γ τˆY(wt/wt) (s−1) (Pa)Carbopol 0.15% 0.001-0.1 4.3-5.9 6.5-7.05Carbopol 0.16% 0.001-0.1 6.2-7.3 6.5-7.05Carbopol 0.17% 0.001-0.1 7.06-8.53 6.5-7.05Table 4.1: Rheological measurements of Carbopol solutions used in the ex-periments.= O(103)Pa); (iii) yield stresses (∼ O(102)Pa); (iv) capillary stresses (dependingon hole size, ∼ O(10− 102)Pa); (v) buoyancy stresses. It became apparent thatin order to investigate yielding we would need to impose a pressure differential of∼ O(102)Pa, and hence control for stresses of comparable and larger magnitude.Elimination of atmospheric pressure effects and balancing static pressure ismost easily achieved via a manometer design, such as Fig. 4.1b. Some initial ex-periments using air as the invading fluid showed that the transition from yield-ing (i.e. invasion) to pinch-off and migration was strongly influenced by capillaryand buoyancy effects. Consequently, we decided to inject an isodense miscibleNewtonian fluid, water, into the Carbopol, thus eliminating capillary and buoyancystresses. In a typical experiment, we increase the injection pressure Pˆh by raisingthe reservoir continuously. Care was taken that the increase rate was slow enough(relative to the viscous timescale in the tube connecting to the reservoir) so thatthe invading fluid can be considered steady: hence Pˆh is determined purely fromhydrostatics, Pˆh = ρˆ gˆHˆi.Following these initial design considerations, further experimental protocolsand changes followed to improve repeatability. Our first tests were in square cross-section tanks. Late stages of these experiments showed asymmetric propagationof the invading fluid. To eliminate the possibility of tank geometry affecting flowsymmetry we then moved to the cylindrical tank. Next, we observed poor experi-mental repeatability at smaller heights of Carbopol. This problem was eliminatedby adding water on top of the Carbopol column so that the total height of fluidcolumn remained constant in each experiment, i.e. a height Hˆ of Carbopol with a118height HˆC− Hˆ of water on top. We postulate that there are some minor effects ofthe filling procedure that may allow for residual stresses in the gelled column. Theadditional static pressure of the water potentially minimizes these effects. Lastly itwas found that Carbopol may enter into the invasion hole and block it, prior to thestart of the experiment. To avoid this: (a) the hole was machined from below to beconical through the thickness of the base, i.e. Rˆh is the radius of hole at the entry tothe tank; (b) the tank filling procedure was carefully executed; (c) we limited theCarbopol concentration to 0.17% in this study. The latter of these kept the yieldstress reasonably low, so that in the event of Carbopol entering the hole duringfilling it could be easily removed.4.4 ResultsWe have conducted approximately 100 experiments at various τˆY and five differentHˆ. Experiments were repeated at the same column height and Carbopol concentra-tion, typically 3-5 times to reduce variability in the results. Before presenting ourgeneral results, we describe in detail the typical experimental observations.4.4.1 A typical invasion experimentThe invasion/penetration of water into Carbopol, namely flow yielding/initiation,is characterized by the following stages, illustrated in Fig. 4.3.1. Mixing stage: A very small mixed region of water-Carbopol develops di-rectly above the hole. The water is observed to mix into the Carbopol, butthere is no observable motion of the fluids i.e. no displacement. This stageis probably driven by either molecular diffusion or osmotic pressure. Theprecise extent of the region is hard to specify as it is diffuse, but a typicalthickness would be ∼ 0.1mm.2. Invasion stage: At the center of the mixed region, when the pressure is highenough, the water advances into the mixed region of the Carbopol column(invasion). Typically the invasion takes the form of a minuscule dome ap-pearing directly above the hole, (≈ 0.5mm radius). Within the dome theintensity of the LIF image is significantly brighter than in the diffuse mixing119stage. The invasion pressure Pˆi is recorded.3. Transition stage: If the pressure is not increased further, the invasion domebecomes progressively diffuse but does not grow. Therefore, the appliedpressure is increased beyond the invasion pressure until the small invasiondome is observed to oscillate and becomes significantly brighter. At thispoint the reservoir height is held constant: the pressure (Pˆtr) is not increasedfurther. These phenomena signify the onset of a second stage of the invasion,that we have called the transition stage.During transition, the small invasion dome expands. Although there is aflow into the dome from the reservoir, the change in static pressure balance isnegligible and the applied pressure at the hole Pˆtr can be regarded as constant.The expansion eventually slows as the dome expands, signalling the end ofthe transition stage. The interface of the dome ranges from smooth if theapplied stress is (relatively) large, to granular if the applied stress is small.This surface variation is illustrated in Figure 4.4.4. Fracture stage: At the end of the transition stage, a small finger or non-uniformity is observed to initiate a “fracture”. This either happens at the“transition” pressure, or after a slight further increase in applied pressure(needed only if the dome remains stable at the end of transition). During thefracture stage the water advances away from the dome in a dyke-like sheet,the edge of which can both finger and branch as it advances, see Fig. 4.8.5. Arrest stage: With no further increase in static pressure to drive the flow,eventually the invasion flow stops. Stopping is dependent on the Carbopolheight H(= Hˆ/Rˆ). Either the invading water penetrates fully to the surfaceof the Carbopol (for smaller H) or it may stop before reaching the surface(larger H).4.4.2 Invasion and transition stagesThe critical invasion pressures for different H, over all our experiments, are plottedin Fig. 4.5. After an initial stage at H ∼ 1, where the invasion pressure decreases120Applied PressureMixing      Invasion Transition starts Transition ends Fingering/ ArrestfracturingFigure 4.3: Observed stages in the invasion/penetration processa b c d eFigure 4.4: Examples of dome surface shapes after “transition” showingsmooth (a & b) to granular (c-e). White broken lines are a guide tothe eye for the surface of the tank and machined hole.slightly, we observe an approximately linear increase in invasion pressure withH. From a naı¨ve macroscopic perspective, we may expect this linear increase.Assuming that the material is incompressible, it is necessary for the invading fluidto displace fluid throughout the gelled column, i.e. volume is conserved. In thisscenario, a yield surface must extend to surface and hence the invasion pressureshould approximately scale with H.Although sometimes considered as such, Carbopol is not an “ideal” yield stressfluid. The yielding process is not abrupt, but instead we see yielding occurringmacroscopically over a stress plateau/range, across which we change from elastic121H0 5 10 15 20P i;P tr5678910111213Figure 4.5: Dimensionless invasion pressures Pi (red circles) and transitionpressures Ptr (blue squares). Error bars indicate the variability of mea-sured pressures over repeated experiments.strain to creep and plastic flow, eventually with nonlinear stress-strain rate charac-teristics; see e.g. [147]. As observed in [148], depending on the bonding strengthof the material (largely controlled by Carbopol concentration) either ductile-typeor brittle-type failure may happen at this stress plateau. Transition pressures arealso plotted in Fig. 4.5 and show an approximately linear increase with H. It isclear from our observations that the invasion process is not straightforward. Weinterpret the stage from invasion to transition as one in which the Carbopol is es-sentially elastic. The initial invasion dome is strongly localised. As the pressureis increased and the transition pressure is attained the dome oscillation is char-acteristic of elastic behaviour. On the other hand, the slow transitional yieldingas the dome grows under constant applied stress is characteristic of ductile-typeplastic yielding. Thus, we believe that the transition pressure is indicative of thiselastic-plastic threshold.Figure 4.6 displays a range of domes, for varying Carbopol concentrations andH, all recorded at the end of the transition stage. Note that some image variationhere is due to 2 different cameras being used. Another visual effect is that theilluminating laser sheet attenuates with depth, so that often one side of the dome122H=10, C=0.17%H=10, C=0.15%H=5, C=0.17%H=5, C=0.16%H=5, C=0.15%H=2, C=0.16%H=2, C=0.15%H=2, C=0.17%H=15, C=0.16%H=15, C=0.17%H=15, C=0.15%H=19, C=0.15%H=19, C=0.16%H=19, C=0.17%H=10, C=0.16%Figure 4.6: Illustrations of transition dome shapes for a range of differentCarbopol concentrations, C, and heights, H. Snapshots for a given C andH set correspond to different experiments, repeated to reduce variabilityof the data.appears brighter than the other. We observe that there is significant variability indome size at each Carbopol concentration and H. At least part of this is due tothe unstable nature of what is being measured. It is only the oscillation of theinvasion dome and a change in light intensity that signal transition: both measurescan be slightly subjective. If the pressure increase is not stopped at Ptr, the invasiondome tends to quickly expand as the pressure increases and moves directly intothe fracture mode. However, stopping the pressure increase at a suspected Ptr isdelicate, as evidenced by the fact that some transition domes readily fracture andothers need an additional pressure increase to do so. Figure 4.7 shows variationin the size of the domes for different yield stress and H. There is no consistenttrend apparent, with radii ranging between 1.2−2 mm. The initial size of invasiondome appears to be controlled by the hole radius (Rˆh = 0.3175mm) whereas finaltransition domes are 4-6 times the hole radius.One interpretation of the growth and stopping of the transition domes is as aform of relaxation process, i.e. the additional pressure imposed above the invasionpressure (Ptr−Pi) causes plastic yielding and growth. As the hydrostatic balance islargely unaffected during transition, the main effect is to impose the same pressure,123H0 5 10 15 20r d0.030.040.050.060.070.080.09Figure 4.7: Variations in transition dome size for different Carbopol concen-trations, marked by circles (C=0.15%), squares (C=0.16%) and dia-monds (C=0.17%), and dimensionless Carbopol heights, H. Each datapoint is obtained from more than 3 different experiments to ensure reli-ability.but over a larger dome surface, thus easing the stresses in the Carbopol. In thisinterpretation, the difference between transition and invasion pressures in Fig. 4.5represents the pressure that can be relaxed in this way.Our observation of both granular and smooth interfaces suggests a more com-plex and localized behaviour than that offered by macroscopic constitutive models.In the context of displacement flows, Gabard & Hulin [42, 43] have observed asimilar change in the rugosity of the interface in studying displacement flow of Car-bopol solutions from a narrow tube. With a glycerin solution displacing, smootherinterfaces result from faster displacement speeds. Equally, less viscous displacingfluids led to coarser interfaces. In both cases the larger applied stress (as here)leads to the smoother interface. Note that this effect is not believed to be related tomiscibility, as similar observations were made by de Souza Mendes et al. [49] ina gas-liquid displacement of Carbopol. Additionally, the diffusive timescales formiscibility effects are long relative to our experimental timescales.124a b c d eFigure 4.8: Images of Carbopol fractures after the transition stage for a) C =0.17%, H = 5.175; b) C = 0.15%, H = 5; c) C = 0.15%, H = 5.025;d) C = 0.16%, H = 5.1; e) C = 0.16%, H = 5.075.4.4.3 Post-invasion propagationThis study has focused on invasion/yielding rather than post-invasion propagation,which our ongoing work studies in more depth. Therefore, here we make a fewqualitative comments only. Some examples of fractures observed are illustrated inFigure 4.8. As we have seen above, the end of transition often results in coarsedome interface shapes. For such domes the initial localisation is evident. In otherdomes, presumably the growth/relaxation of the dome also results in some locali-sation of the Carbopol deformation and/or stress field. In either case, the fracturepropagation is characteristic of a more brittle-type failure. Observations from [148]indicate that brittle fractures are likely to happen in the presence of a localized de-formation in the fluid domain at very low shear rates.The fractures have the overall form of bladed dykes progressing upwards fromthe dome. The propagation itself often shows “viscous” fingering characteristics atthe penetrating fracture front/edge (i.e. with branching/splitting behaviour). Visu-ally the penetrating fronts can be diffuse and sometimes granular on small scales.The direction of propagation initially taken is not repeatable between experiments;see e.g. Figs. 4.8b & c, or d & e, both at very similar H and concentrations.1254.5 Computational predictionsAs an integral part of the experimental study described above, we have conducteda computational study, using idealized yield stress fluid models. Study of suchmodels provides only a baseline for understanding of the invasion process. Wehave seen that Carbopol exhibits elasto-plastic behaviours at low shear, and henceis not an ideal yield stress fluid. Therefore, in performing this type of computa-tional study, at the outset we acknowledge that at best qualitative agreement can beexpected.4.5.1 The model problem and numerical methodAs we are mainly interested in the invasion pressure, indicating onset of flow, ratherthan computing the actual velocity field and flow, we adopt the simplest model,i.e. a Bingham fluid. As most of the calculations involve stresses below the yieldstress, any other ideal yield stress fluid with a von Mises yield criterion would giveidentical results (e.g. Casson, Herschel-Bulkley, etc). The Bingham model isτˆi j = (µˆ+τˆY| ˆ˙γ|)ˆ˙γi j, ⇔ τˆY < |τˆ| (4.2a)ˆ˙γi j = 0, ⇔ τˆY ≥ |τˆ|. (4.2b)Here the plastic viscosity is µˆ , the deviatoric stress tensor is τˆ and strain rate tensoris ˆ˙γ . The norm of the strain rate and shear stress are defined as:|τˆ|=√12τˆi jτˆi j, (4.3)| ˆ˙γ|=√12ˆ˙γi j ˆ˙γi j. (4.4)On scaling stresses with the yield stress (as discussed earlier in Section 4.2), wecan write the model in non-dimensional form asτi j = (1+1|γ˙|)γ˙i j, ⇔ 1 < |τ| (4.5a)γ˙i j = 0, ⇔ 1≥ |τ|. (4.5b)126HAxis1rhσ.n = 0∂Ωwrz∂Ωp : σ.n = −Pina)HAxis1rdσ.n = 0∂Ωwrz∂Ωp : σ.n = −Pinb)Figure 4.9: Model geometry computed: a) axisymmetric column with cen-tral flat hole at the bottom; b) axisymmetric column with hemisphericalincursion of invading fluid.Lengths have been scaled with Rˆ and velocities with Uˆ = τˆY Rˆ/µˆ . The two di-mensionless parameters remaining are rh (the radius of the hole) and H (the heightof the column).We consider two similar but slightly different axisymmetric geometries, as il-lustrated schematically in Fig. 4.9. Firstly, we model directly the invasion stage ofthe experiment, by setting constant normal stress, σ .n = −Pin, at the flat hole, ofradius rh. Secondly, we impose constant normal stress, σ .n = −Pin, on the sur-face of the hemispherical dome of radius rd . We use this setup for understandingthe effects of the transition dome on the stress field. For both cases we solve theStokes equations in the axisymmetric geometry, using constitutive laws (4.5). Inboth cases, no-slip conditions are applied at the walls, symmetry conditions aresatisfied along r = 0, and zero stress is imposed at the top of the column.The main difficulty with models such as the Bingham model occurs at γ˙ = 0where the effective viscosity is singular. Robust numerical algorithms for vis-coplastic fluids were developed in 80’s by Glowinski and coworkers [106, 149,150], based on convex optimization methods. These algorithms produce unyielded127regions with true zero strain-rate and are well-known to be superior for studyingproblems involving flow onset/yielding; e.g. [151]. The implementation that weuse follows that in [152, 153], and we have used very similar numerical codesextensively in recent work where it was important to identify unyielded regions,e.g. [154, 155].As a brief overview, two extra fields are added to the velocity and pressure: arelaxed strain rate tensor γ˙r and Lagrange multiplier tensor T (that corresponds tothe deviatoric stress). An iterative procedure consisting of three steps is repeateduntil the desired convergence is achieved. The first step is a Stokes flow problemwith an additional source term: ∇ · [T− aγ˙r], where a > 0 is an augmentation co-efficient (typically 10≤ a≤ 50). The second and third steps are pointwise updatesof the relaxed strain rate and stress Lagrange multiplier. It is known that γ˙r andT converge to the exact Bingham flow strain rate and an admissible stress field[150], as does the velocity. The nice property of the relaxed strain rate γ˙r is thatit becomes exactly zero in unyielded parts of the flow, at each successive itera-tion. Thus, when γ˙r = 0 for the whole domain we may infer that invasion has nothappened. The steps of the algorithm tabulated in Algorithm 1.The Stokes subproblem at each iteration is solved using the finite elementmethod with Taylor-Hood element pair (P2−P1) for velocity and pressure (to ful-fill the inf-sup condition). The relaxed strain rate γ˙r and stress multiplier T uselinear discontinuous (P1d) elements to comply with the discrete compatibility con-dition between velocity field and these spaces; see [152]. The mesh starts relativelycoarse (40-60 points on ∂Ωp) and is refined slowly and adaptively after each con-verged solution. We ensure that the maximum edge size is < 0.04. Commonlyused meshes can have 50,000-200,000 points and for longer columns, may reachas high as 500,000 points in size. We have implemented this algorithm using Rhe-olef [156], a freely available C++ finite element library developed by P. Saramitoand co-workers.A typical procedure to find the invasion pressure is as follows. The objectivefunction is an evaluation of maxΩ ||γ˙|| and we compute the maximal Pi for whichmaxΩ ||γ˙|| = 0 within the bound. Twice the Poiseuille flow limiting pressure istypically a good upper bound for the invasion pressure. To find γ˙r at each step, wesolve the flow problem in the column of fluid for given invasion pressure Pi. We use128Algorithm 1 Augmented Lagrangian algorithm for invasionrepeatStep 1: Solve Stokes flow problem (minimization with respect to u):−a∇.γ˙(un) =−∇pn+∇.(T−aγ˙r)n−1σn.n =−Pin on Ωpσn.n = 0 on outletun = 0 on ΩwStep 2: Minimization with respect to γ˙r:γ˙rn ={0∥∥Tn−1+aγ˙(un)∥∥< 1{1− 1‖Tn−1+aγ˙(un)‖}Tn−1+aγ˙(un)1+a otherwiseStep 3: Maximization with respect to T :Tn = Tn−1+a(γ˙(un)− γ˙rn)until |γ˙rn− γ˙(un)| ≤ 10−6 or iteration count≥5000the augmented Lagrangian method at each step which itself consists of fixed pointiterations through 3 steps of an Uzawa algorithm. These 3 steps find the solution ofa saddle-point problem with respect to the three variables u, γ˙r and T as explainedin Algorithm 1. A flow is considered static (not invaded) provided that: (i) γ˙r = 0in the whole domain for 100 consecutive iterations, and (ii) the relaxed strain rateis close enough to the actual strain rate i.e. |γ˙rn− γ˙(un)|L2 ≤ 10−6. If not invadedwe increase Pi following the usual bisection strategy. We continue until ∆Pi < 0.01is satisfied as a tolerance.4.5.2 ResultsUsing the procedure described above, we have computed axisymmetric solutions,and from these have iteratively calculated the invasion pressures for a wide rangeof H and rh. The results are shown in Fig. 4.10 and are seen to depend significantlyon the hole radius rh. For H . 1 the invasion pressures Pi increase from zero. Thisrepresents the transition from invasion into a shallow layer, towards invasion into a129H0 5 10 15 20 25 30P i01020304050607080rh = 0:02rh = 0:05rh = 0:1rh = 0:2rh = 0:5Figure 4.10: Computed invasion pressures for different column height, H,and hole radii, rh. The dashed line denotes the Poiseuille flow yieldlimit.tall column.Considering first moderate rh & 0.1, on increasing H we see that Pi increaseslinearly with H, for H & 5. The linear increase is the same for each rh & 0.1,following the broken line marked in Fig. 4.10, which denotes the Poiseuille flowyield limit. More clearly, in a laminar Poiseuille flow along a length Hˆ driven bypressure drop ∆Pˆ, the wall shear stress is = Rˆ∆Pˆ/2Hˆ. This must exceed the yieldstress in order to flow. In the invasion context, with the scaling defined earlier, thisleads to: Pi ≥ 2H. For H & 5, at the point of yielding, these flows exhibit a 2Daxisymmetric region of yielded fluid fanning out from the hole and reaching thewalls of the tube at a development height Hd . Above Hd the velocity field becomesessentially 1D, following a Poiseuille profile just as the fluid begins to flow/yield.As H increases the Poiseuille section grows proportionately longer, accounting forthe linear increase.Secondly, we observe in Fig. 4.10 that the behaviour for small rh is quite dif-ferent from that described above (see rh = 0.02, 0.05). Evidently, as rh decreasesthe inflow pressure is increasingly localised and singular. For H > 1 we observea strong reduction in Pi below that of the invasion pressures for larger rh. It ap-130pears that the fluid adopts a more localised stress distribution in this range of H.Over this range of H the fluid yields significantly below the Poiseuille flow predic-tion. Eventually the pressure reduction terminates at some height and thereafter thePoiseuille gradient is followed. In fact Pi is larger for smaller rh in the Poiseuilledominated regime. For rh = 0.05 (as illustrated) the Poiseuille regime is attainedat H ≈ 16. For rh = 0.02 the Poiseuille regime is only attained at H ≈ 225, i.e. asrh is reduced, the reduction in Pi below the Poiseuille regime invasion pressure isfound over an increasingly wide range of H.To understand differences in the yielding patterns for both small and larger rh,examples of the stress distributions are presented in Fig. 4.11, for rh = 0.05, 0.1at H = 2, i.e. at the invasion pressure. For rh = 0.05 we observe that τrz and τrrchange sign within a small localised dome shape. Together with and aided by non-zero τθθ and τzz (of similar magnitude), we see that |τ| = 1 within a small domearound the hole. This dome is isolated from the walls of the cylinder, suggestingthat fluid within the dome would recirculate on yielding. This recirculation couldof course entrain invading fluid from the hole, but remains a local mobilizationrather than an invasion that extends to the top of the fluid column.On the other hand, for rh = 0.1, we see that although the qualitative distributionof τrz, τθθ and τrr is similar to that for rh = 0.05 close to the hole, the wedge ofnegative shear stress τrz, now extends out from the dome near to the hole, all theway to the wall. It combines with the stress components and is nearly able to yieldthe fluid to surface. As H increases moderately for rh = 0.1 (not shown) the shearstress increases and the Poiseuille-like distribution commences, extending alongthe column to surface.To see the transition from the strongly localized stress dome into the Poiseuilleflow regime at rh = 0.05, Fig. 4.12 shows τrz and |τ| for H = 8, 12, 15, 16, 16.25.As we see, the stress distribution local to the hole does not change qualitatively asH increases, but does penetrate increasingly far towards the wall. Before the entirecolumn begins to yield we can see that the shear stress adopts a 1D Poiseuilleprofile, e.g. for H = 15, 16, whereas the column only begins to displace at H =16.25 (below this motion is only local to the hole).Computed Pi for rh = 0.01 (experimental value) are approximately constantover the range of experimental H, being very similar to rh = 0.02 shown in Fig. 4.10.131Figure 4.11: Example stress fields for H = 2. Top row rh = 0.05: τrz, τθθ ,τrr, |τ| (left to right). Bottom row rh = 0.1: τrz, τθθ , τrr, |τ| (left toright).Comparing with the experimental results in Fig. 4.5 we observe 2 primary dif-ferences. First, the invasion pressures in the experiment are smaller by 10-30%.Secondly, although experimentally we see a short plateau (or decrease) at smallH and then increase in Pi with H, the increase is significantly smaller than thatof the Poiseuille flow gradient (Pi = 2H). The offset in invasion pressures is un-doubtedly due partly to the constitutive law: Carbopol is not a rigid solid below theyield value, but exhibits elasto-plastic behaviour that allows it to deform at lowerpressures than an idealised yield stress fluid. The slower increase in Pi with H(experimentally) suggests a localised stress distribution and reduction in Pi, e.g. asshown in Fig. 4.10 for rh = 0.05.In our experiments, following the initial very small invasion dome, the pressureis further increased until the transition pressure after which it grows at constantpressure. At the end of transition, domes are either stable or initiate fingering/frac-132 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1Figure 4.12: Evolution of the stress fields with H = 8, 12, 15, 16, 16.25 forrh = 0.05: left panel τrz; right panel |τ|.turing. The latter mode of propagation via localisation and symmetry breakingsuggests that the expanded axisymmetric dome is itself a stable configuration andthat a form of stress relaxation occurs via slow growth of the dome. Initially, Piimposed at the circular hole caused growth of the small invasion dome, (observedto be stable in the absence of further pressure increases). The pressure increase im-posed, Ptr−Pi, then causes growth of the dome, at the end of which Ptr is imposedover the larger transition dome radius, allowing a stress field of smaller shear stress.To test the plausibility of the above description, Fig. 4.13 plots invasion pressuresagainst rd for various H, assuming that the invading pressure is imposed on a hemi-spherical surface of radius rd (see Fig. 4.9b).We observe that the invading pressures at small rd attain a plateau to similarvalues to those for the hole. It seems that Pi increases to a maximum as rd in-creases, and then slowly reduces. Computations over a wider range of rd show133a) rd0 0.02 0.04 0.06 0.08 0.1P i01020304050607080H = 2H = 4H = 10H = 20H = 30b) rd0 0.1 0.2 0.3 0.4 0.5P i01020304050607080Figure 4.13: Invasion pressures, for a hemispherical invading dome: a) smallrd ; b) full range of rd .that Pi approaches the Poiseuille limit at large rd , (marked with broken horizon-tal lines for each H in Fig. 4.13). Consulting Fig. 4.5 it seems that the pressureincrease imposed experimentally, Ptr−Pi, is in the range 1−3. Increasing the in-vasion pressure by ∆P= 1−3, above that for rh = 0.01 in Fig. 4.13a, indicates thatthe increased invasion pressure will be balanced for an increase in dome size, ofsomewhere in the range rd = 0.025−0.075, over the experimental range of H. Weobserve that this range of dome radii approximately covers the range of observedtransition dome radii in Fig. 4.7. This suggests that the notion of the stress relaxingas the dome grows during transition is plausible.4.6 Summary and discussionWe have presented new results targeted at exposing the effects of the yield stress onthe pressure Pi required for one fluid to invade a column of fluid, through a smallhole. This setup was designed to simulate invasion into a well during primarycementing (or other operations when the wellbore fluid is stationary), for relativelylow porosity reservoirs where pores may be considered isolated. Water was usedin place of gas (to eliminate buoyancy and capillary effects) and the invading fluidwas a Carbopol gel. Approximately 100 experiments were performed, repeatedlycovering 3 different Carbopol concentrations and various heights of column. Thiswas supplemented by a computational study.For low column heights H ∼ 1 (Hˆ ∼ Rˆ) there is little effect of H, but as H134increases we observe a steady approximately linear increase in Pi over the experi-mental range. Dimensionally, Pi measures the invasion over-pressure (i.e. pressureabove the hydrostatic pressure in the fluid column), scaled with the yield stress.Thus, our results show approximately linear increase in invasion over-pressurewith Hˆ. Also the invasion over-pressure increases linearly with the yield stress,as follows simply from the scaling adopted.The experiments show significant variability, hence the repetition at each heightand Carbopol concentration, despite considerable evolution in the experimentalprocedure to reduce this. For example, we have carefully implemented protocolsfor fluid preparation, tank filling, eliminating invasion hole plugging, etc..., andhave eliminated buoyancy, capillary, static and atmospheric pressures via our ex-perimental design. The remaining variability stems from the fact that we are mea-suring an isolated onset event in a dynamically evolving process, and that this eventis identified phenomenologically by the appearance of a minuscule invasion dome.A number of interesting stages have been observed during the experiments:mixing, invasion, transition, fracture and arrest. The passage from invasion totransition pressure seems to represent elastic-plastic yielding close to the invasionhole. The initial growth of the transition dome and then slowing of growth (incases where the fracture does not start immediately) suggests a relaxation of thestress field, due to the overpressure now being applied over a larger area. We haveseen that the expanding dome interface can be either relatively smooth or granular.This does not appear to have any bearing on the stability of the transition dome:either may be stable or unstable. Invasion and transition domes are approximatelyaxisymmetric. Fracture initiation and propagation represent a departure from sym-metry, probably due to either a local defect or a non-uniformity of the stress-field.The computational study has covered a range of invasion hole sizes and di-mensionless H. The computed invasion pressures follow similar qualitative trendsto the experiments but are themselves over-predicted. This is probably due to theideal visco-plastic (Bingham) law that we have implemented. The computationsalso reveal that the invasion pressures for small rh increase relatively slowly withH > 1. This is tied into the occurrence of a local dome-like region of yielding closeto the hole, i.e. the invasion overpressure first causes fluid to yield and recirculatelocally within a dome shape, that appears analogous to those observed during the135transition stage.As H is significantly increased, for small rh, the increase in Pi eventually gen-erates sufficient stress to reach the walls of the cylinder. After this the invasionprocess changes from a local phenomena to a global one, in which resistance ofthe fluid occurs on the scale of the cylinder. More specifically, there remains anO(1) region close to the hole within which the magnitude of the deviatoric stressescomponents are all significant. The stresses generally decrease away from the hole,but at sufficiently large H yielded fluid does extend to the wall. Above this localnear-hole region, the fluid adopts essentially a Poiseuille profile, with linearly de-creasing shear stress from centre to wall. Hoop and extensional stresses, which areimportant in the near-hole region, decay in the Poiseuille region as H is increased.We further observe that the invasion pressure increase proceeds in parallel tothe Poiseuille prediction (Pi = 2H) both for small rh at sufficiently large H andfor larger rh. For the latter there is no range of H for which yielding is local andisolated, instead the Poiseuille regime is entered immediately. Returning to theexperimental results, the invasion pressure increases at a rate that is significantlybelow Pi = 2H. This, the hole size and the phenomena observed, all suggest thatour experiments are fully in the local regime of invasion. The initial domes areapproximately axisymmetric and apart from more complex constitutive behaviourthe computational and experimental results present a coherent picture.Regarding the cementing process, a few aspects appear relevant. Firstly, thecomplex behaviour of Pi with H and secondly the change from local to non-localinvasion. Here we neglect effects of any filtercake that may have formed duringdrilling. Although H is very large in the wellbore setting, pore size is also verysmall. Taking a typical annular gap as the global scale, pore sizes in the sub-micron range are very likely to invade locally (rh < 0.0001 even for large H). Thiswill also depend in a significant way on the porosity, which can also be interpretedas rh, i.e. at larger porosity values pores can no-longer be considered as isolated:local invasion domes influence adjacent pores. Our computed results suggest thatin the cylindrical geometry this happens for rh ' 0.1.Secondly, a different perspective on local/non-local yielding comes from theobservations of smooth and granular surfaces in our transition domes. Here thescale of the granularity appears to be in the range of 1− 100 µm, which may be136related to the Carbopol gel microstructure and its stress response on this scale. Inthe well, cement slurries are in reality fine suspensions and pore sizes in tight rocksextend down to the scale of the suspension microstructure, (indeed in such rocksit is not uncommon to consider Knudsen effects on porous media flows). Thus,local invasion on the scale of the slurry microstructure is likely to be the norm incementing and this requires separate study using real cements.Thirdly, the influence of the yield stress in linearly increasing invasion over-pressure is useful, if also anticipated. If the invasion is local then the critical pointis that it may occur at significantly lower pressures than those predicted from non-local analyses, e.g. predicting flow in the annular gap. If however yielding is local,with recirculation of fluids close to the pore opening, many other effects may in-fluence the potential and mechanism for annular fluid to exchange with the porefluid, and the timescale for that exchange, e.g. buoyancy, capillarity, diffusion, os-motic pressure etc. These effects have not been studied here. In the same context,note that here we have tried to eliminate local variability in our experiments, butin any cement placement process (and in the drilling process) the wellbore is over-pressured. Thus, yield stress fluid is forced into the pores, filtercake/skin typicallyforms close to the borehole, and these effects may in practical situations be themain influence of the yield stress on the actual invasion stage.Finally, in the above context - if we are to assume that local invasion occurs andthat local gas streams coalesce-an interesting fluid mechanics problem to considerwould be the propagation of a large gas stream upwards through an inclined annu-lus of stationary yield stress fluid. Although in our simple setup we have shownthe relevance of Poiseuille flow-like bounds, in inclined eccentric annuli the gaspath is less predictable and may exchange/by-pass in situ fluids as it rises up theannulus.4.7 Concluding remarksIn this chapter, the invasion of water into Carbopol was studied. In this study, forthe sake of simplicity, we neglected the effect of interfacial tension and buoyancyforces by using iso-dense and miscible fluids. However, in a real gas migrationproblem, this assumption is not necessarily valid. Therefore, a more general study137should be performed to investigate the role of these parameters. On top of that,according to our scaling method used in this chapter, the viscosity, or the density,of the invading fluid shouldn’t have any effect on the invasion pressure. In order toconfirm the validity of this argument and its generality, more experiments shouldbe performed by using fluids with different range of viscosities and densities. Wetherefore continued the invasion experiments by using a range of different invadingfluids to investigate the role of buoyancy, surface tension, viscosity, as described inChapter 5.138Chapter 5Onset of Miscible and ImmiscibleFluids invasion into a ViscoplasticFluid 1In the first part of this study Chapter 4 (see [123] also), we considered water as theinvading fluid and discovered a surprisingly complex invasion process and exploredthe variation in invasion pressure with both yield stress and height of column. Inthis chapter, we explore a range of different invading fluids. First we stay with mis-cible liquids and consider glycerin solutions of 2 different concentrations invadinginto a Carbopol column. The glycerin solutions are both more viscous and denserthan water; similar Carbopol concentrations were used as in Chapter 4. Secondly,we performed experiments with a silicon oil of identical density to the in-situ Car-bopol, involving both interfacial tension and higher viscosity. Finally, we testedair as the invading fluid: larger interfacial tension, lower viscosity and significantbuoyancy.We simulate fluid invasion into a gelled cement slurry using an explained lab-oratory experiment in Chapter 4. This process is relevant to the construction of oiland gas wells, in which a tall column of cement suspension must resist fluid inva-sion through a combination of static pressure, yield stress and interfacial tension.1A version of this chapter has been published in Physics of Fluids journal [157].139Sufficiently over-pressured fluids may enter from the surrounding rock leading tofailure of the well integrity.Here we model the cement suspension using a Carbopol solution (yield stressfluid), and apply different over-pressured invading fluids through a centrally posi-tioned hole at the bottom of the circular column. We study water, glycerin, siliconoil and air as invading fluids, in order to delineate the effects of yield stress, inter-facial tension and column height on fluid invasion. We find that invasion is easiestfor miscible fluids, which penetrate locally at significantly lower invasion pres-sures than immiscible fluids. Viscosity affects this process by retarding the initialdiffusive mixing of the fluids, which tends to weaken the gel locally. More vis-cous invading fluids require larger invasion pressures and result in larger invasiondomes. The silicon oil penetrated in the form of a slowly expanding dome, resistedat the walls of the column: effectively by a Poiseuille flow above it in the Car-bopol. Invasion pressures were significantly larger than for the glycerin solutions.The largest invasion pressures were, however, found for air, which is influencedapproximately equally by interfacial tension and yield stress.An outline of our chapter is as follows. Section 5.1 presents a simple dimen-sional analysis that serves well as a framework within which to interpret our results.The experimental method is outlined in Section 5.2 along with the properties andrheology of invading and invaded fluids. Our experimental results are presented intwo sections: miscible fluids (Section 5.3) and immiscible fluids (Section 5.4) Thechapter ends with a brief discussion about the findings of this study in Section 5.5.5.1 Dimensional analysis of invasionAs discussed in Section 4.1 here we study the process of invasion of fluids intoa column of yield stress fluid, in an apparatus similar to that used previously inChapter 4. Our fluid domain consists of a height Hˆ of yield stress fluid (density ρˆ ,yield stress τˆY ), filling a cylindrical container of radius Rˆ. On top of the yield stressfluid is a height HˆC− Hˆ of water (density ρˆw = ρˆ), open above to the atmosphere.The sealed base of the column has a single invasion hole of radius Rˆh, positionedcentrally. The invasion hole is connected by a tube to a reservoir of invading fluid(density ρˆi), the surface of which is at a height Hˆi above the base of the cylinder.140In this chapter we adopt the practice of denoting dimensional variables with the ·ˆsymbol and dimensionless variables without.Let us first consider the static forces acting prior to invasion. The gage pressureat the base of the column is ρˆ gˆHˆC and that just inside the invasion hole is Pˆh =ρˆigˆHˆi. If Pˆh > ρˆ gˆHˆC the pressure differential pushes the invading fluids into thecolumn. In the absence of interfacial tension or a yield stress, invasion occursdirectly whenPˆh > ρˆ gˆHˆC. (5.1)If the invaded fluid possesses a yield stress but is miscible, we may expect theiryield stress to resist the deviatoric stresses generated by the excess pressure in(5.1). Therefore, we might expect that invasion only occurs for:Pˆh > ρˆ gˆHˆC +PY τˆY , (5.2)where PY is a dimensionless coefficient, which may be a function of the other di-mensionless groups. Similarly, if the fluids are immiscible, with interfacial tensioncoefficient σˆ we might expect an additional resistive stress ∝ σˆ/Rˆh, and henceextend the model as:Pˆh > ρˆ gˆHˆC +PY τˆY +PitσˆRˆh, (5.3)where again Pit is a dimensionless coefficient which may be a function of other di-mensionless groups. The additive form assumed in (5.3) is justified by consideringthe limiting cases of either zero yield stress or zero surface tension.To reduce the number of variables, we scale stresses with the yield stress τˆYand lengths with the column radius Rˆ. The dimensionless invading hole pressuredifferential Ph is simply:Ph =Pˆh− ρˆ gˆHˆCτˆY=(ρˆiHˆi− ρˆHˆC)gˆτˆY, (5.4)and the smallest value of Ph above which invasion occurs is called the invasionpressure, Pi. There are two dimensionless geometric ratios: H = Hˆ/Rˆ and rh =Rˆh/Rˆ. If we have a yield stress and interfacial tension present, there is an additional141dimensionless stress balanceCaY =RˆhτˆYσˆ, (5.5)where CaY might be termed a yield capillary number.From (5.1)-(5.3) we see that Pi = 0 in the absence of either yield stress orinterfacial tension effects. In the case of a miscible fluid invading a yield stresscolumn, we know that invasion occurs for:Ph > Pi = PY (H,rh,CaY ) : CaY = ∞, (5.6)Here we assumed that PY depends only on the height H of fluid with a yield stress,and not on HˆC/Rˆ. The point is that, if a significant amount of fluid invade, theyield stress fluid must yield and displace all the way to the surface of the column.Thus, the resisting force scales with the yield stress acting over a surface area thatis proportional to Hˆ (not HˆC).Computations in Chapter 4 reveal that for small rh the yield stress effects areessentially localised around the hole and increase only gradually with H. However,for larger rh or sufficiently large H the yield stress resists invasion primarily at thewalls of the cylinder. This Poiseuille flow mode is easily calculated:PY,Pois(H) = 2H(=2piRˆHˆ τˆY[Pˆh− ρˆ gˆHˆC]piRˆ2),i.e. the differential pressure is spread over the height of the yield stress fluid andresisted by the yield stress acting at the wall. Thus, we find thatPY (H,rh,∞)→ PY,Pois(H), as rh→ 1, H→ ∞. (5.7)Although PY is simple to calculate, in our previous miscible fluid invasion experiments[123]we observed that yielding and invasion occur locally. On including interfacial ten-sion, we see that (5.3) becomes:Ph > Pi = PY +PitCaY: Pit = Pit(H,rh,CaY ). (5.8)PY and Pit determine the contribution of yield stress and interfacial tension to the142invasion pressure. The above description is useful mainly as a framework in whichto view our experiments. Although simplistic, even in this description there isuncertainty about whether PY and Pit depend on CaY , or whether we can decouplecapillary and yield stress effects. Secondly, the analysis is a static balance of forces,in other words it describes the instantaneous tendency to invade. In the case ofmiscible fluids we have already seen in Chapter 4 that invasion is a multi-stagetransient process. Hence, there are various limitations which we intend to explorein this chapter.5.2 Experimental description5.2.1 SetupWe used the same simple experimental setup as explained in Chapter 4 to explorethe flow discussed in Section 5.1. The yield stress fluid is placed to height Hˆ insidea long plexiglas cylinder of radius Rˆ and length Lˆ , with sealed base. Generally, inour experiments Hˆ < Lˆ and the space above the yield stress fluid is filled to heightHˆC with water so that the total height of liquid (and static head) in the cylinder isconstant between experiments, regardless of height of the yield stress fluid. Theinjection point for the invading fluid is via a hole of radius Rˆh positioned centrallyin the base of the cylinder. The same experimental method explained in Chapter 4 isapplied here. The invading fluid initially fills a tube connected to a reservoir that isplaced on top of a motorized scissor jack, i.e. a manometer design; the parametersexplained here and in Section 5.1 are shown in the schematic setup; see Fig. 5.1.The pressure exerted by the column of invading fluid Pˆh is given by the differencein static pressures of the two columns. Atmospheric pressure is balanced and inthis static configuration the invading pressure can be countered only by yield stresseffects and (potentially) surface tension effects.An image processing technique is applied here to determine the position andthe size of domes and bubbles. Firstly, the intensity of the background is subtractedfrom the images in order to remove noise and other background effects. Then themaximum of the second derivative of the intensity is found to detect the edges ofthe objects. All of the images and the time-stamps are then analyzed using this143LaserLaser sheetGate valveMotorized scissor jackInvading fluid reservoirYield stress fluid𝐻" 𝐻"#2.500𝑅%&𝐻"'Figure 5.1: Schematic of the experimental setup.H2O G45 G58 R550ρˆ [g/cm3] 1 1.06 1.12 1.065µˆ [cP] 1 4.9 9.8 125Table 5.1: Physical properties of the invading liquids.technique, implemented in MATLAB. Further post-processing is carried out asneeded to estimate radii, volumes and front speeds.5.2.2 Fluids usedThe in-situ fluid is the same Carbopol EZ-2 (Lubrizol) that we used for water in-jection in Chapter 4 and the preparation method and rheology measurement is dis-cussed in detail in Section 4.3.1. Two set of fluids, miscible and immiscible, areused as invading fluids. All of them are Newtonian and the experimental sequencesperformed were designed to reveal the effect of each single parameter: viscosity,density difference and interfacial tension. The 3 miscible fluids which are injectedinto Carbopol are: water (H2O), glycerin 45 % (G45), and glycerin 58% (G58).Two immiscible fluids are used: Rhodorsil oil (BLUESIL FLD 550 from BluestarSilicones; R550) and air. The densities and viscosities of these liquids are given inTable 5.1. On injecting the R550, the objective is to isolate the effect of interfacialtension on invasion. Consequently, for these experiments the Carbopol density isincreased by weighting with glycerin, to match that of the invading fluid.1445.3 Invasion of miscible liquidsAs discussed in Section 5.1, at its simplest description, the onset of invasion of amiscible fluid is expected to be characterised by Pi = PY , which depends only onH and rh. In reality, a hole geometry is characterized by more than a single di-mensionless parameter rh. To eliminate variability, for our experiments the samecylinder and hole were used throughout. To explore the invasion process we var-ied Hˆ and τˆY for each invading fluid: 3 Carbopol concentrations, 0.15, 0.16 and0.17%wt, and 5 heights of yield stress fluid, H = Hˆ/Rˆ = 2, 5, 10, 15, 20. In-cluding repetitions for consistency we performed around 60 experiments for eachfluid.The experimental procedure followed for G45 and G58 is the same as for water,as described at length in Chapter 4. Five stages of the experiment are identified: (i)Mixing; (ii) Invasion; (iii) Transition; (iv) Fracture; (v) Arrest.5.3.1 Mixing and invasion stagesThe experiment starts with the invading fluid slightly overbalanced, but with nodetectable invading motion. We slowly increase the applied pressure by raisingthe invading fluid reservoir (increasing Pˆh). The increase rate is 5µm/s, whichamounts to ≈ 3Pa/min, which is sufficiently slow for fluid heights to equilibratethrough the tubing. In the initial mixing stage no displacement is evident. A smalldiffuse/mixed region forms near the injection hole (of size ∼ Rˆh and thickness∼ 0.1mm). Formation of this region after two miscible fluids have been in contactfor a while is not surprising and we attribute this to molecular diffusive processes.The applied pressure ramp continues to increase as we enter the invasion stage.Here the diffuse interface continues to grow, but we also observe a miniscule domeforming near the centre of the hole/interface. The dye in the dome is bright ratherthan diffuse, indicating the first instance of penetration. However, the dome doesnot grow further if we stop increasing the applied pressure and the size of the domeappears to change in proportion to the size of the applied pressure. Both featuressuggest that the response of the Carbopol to the initial invasion is elastic.The small dome indicates the start of penetration and the invasion stage refersto the entire duration of this period where elastic deformation takes place. During145this stage, as the pressure ramp increases the mini-dome expands and the elas-tic strain experienced by the Carbopol increases. Eventually we reach a yieldingpoint for the Carbopol which we term the invasion point, which we interpret me-chanically as the change in the Carbopol behaviour from elastic to visco-plasticdeformation. In the experiment, the invasion point is not measured from deforma-tion, but is signalled by the dome becoming unstable and oscillatory. At this pointwe stop increasing the pressure ramp and record the current value of Pˆh, whichis the invasion pressure, i.e. the pressure at which the invaded fluid yields plasti-cally. Please note that invasion point is beginning of transition stage. For reader’sconvenience we use invasion pressure instead of transition pressure through thischapter.Comparing G45 and G58 with our earlier experiments using water, a first ob-servation is that the mixing and invasion happens significantly later for the glycerinsolutions than for the water. Typically the mixing stage starts after 30 minutes forG45 and G58 whereas it is less than 10 minutes for water.The second set of observations concern invasion pressures. The recorded inva-sion pressures, Pˆh, are scaled via (5.4) and used to give an experimental measure-ment of the dimensionless invasion pressure Pi = PY (H,rh,∞); see (5.6). These areplotted against the scaled height of the column of Carbopol, H, in Fig. 5.2. Theinvasion pressures of G45 and G58 at each height are marked with a green circleand blue square, respectively. It is observed that the pressure differential Pi of theinvading fluids increases with an increase in the height of the invaded fluid column.In dimensional terms, the invasion pressures evidently increase with the yield stresstoo. The increase in the Pi against H is noticeable in Fig. 5.2, but is significantlyslower than the Poiseuille flow gradient PY,Pois(H) = 2H. This suggests that forG45 and G58, as for water before Chapter 4 the stress field and yielding associatedwith invasion are strongly localized.We note that it is only the oscillation of the invasion dome and associatedchange in light intensity that signals the end of the invasion stage. Both mea-sures can be slightly subjective and this is the reason for the relatively large errorbars in Fig. 5.2. Comparing the invasion pressures for G45 and G58, although theinvasion pressures for G45 appear consistently larger, this difference falls withinthe range of the error bars, and we infer that the invasion pressures of these two1460 5 10 15 2051015202530Figure 5.2: Invasion pressures of miscible fluids against height of the Car-bopol column:  G58; • G45; N water from Chapter 4.glycerin solutions are close to each other.However, there is a marked difference between the glycerin data and that ofwater. Although the increase with H is approximately the same, the Pi is signifi-cantly smaller for water. This contradicts the dimensional analysis of Section 5.1.The cause of the increased measured Pi is straightforward: as remarked earlier themixing and invasion stages last significantly longer for G45 and G58. All the ex-periments start with zero pressure differential (Pˆh = 0) and the pressure ramp rate isidentical. Therefore, a larger pressure overbalance is achieved before the invasionpressure is attained for a longer duration.The root cause of the later mixing and invasion stages appears to be relatedto molecular and/or physicochemical processes. Molecular diffusivity generallydecreases with viscosity. Our camera resolution is insufficient to observe the na-ture of the diffusive process, e.g. whether it is relatively homogeneous throughthe Carbopol or does it diffuse slowly through gel particles and quickly along theboundaries? In either case, it seems reasonable to expect that the glycerin solutionsdiffuse slower into the Carbopol, leading to the extended mixing and invasion stagetimescales. We might also hypothesize that the diffusive penetration locally weak-147H2O G45 G58Volume [mm3] 527 ± 262 1716 ± 439 1341 ± 437ˆ˙γmax [s−1] ∼ 8 ∼ 110 ∼ 85Table 5.2: The volume of domes before fracture/splashing and the maximumstrain rate of Carbopol during expansion of the dome.ens the Carbopol gel via a dilution effect. This hypothesis would be consistent withthe glycerin solutions requiring more time for the initial mini-dome to emerge sig-nalling onset of invasion. Osmotic processes may also play a significant role here,depending on the invading fluid. Although we may hypothesize regarding thesemolecular/physicochemical causes, investigation requires a simpler experimentalsetup.5.3.2 From transition to ruptureWhen the invasion point is reached, the pressure ramp is stopped, signalling thestart of what we have termed the transition stage. During the transition stage thedome growth may be initially arrested (dormant phase), perhaps responding to thesuddenly constant driving pressure, but then enters a period of steadily acceleratinggrowth. As commented, the invaded fluid appears to have yielded locally, allowingflow rather than elastic response. The small invasion dome increases to a maximumsize before rupturing.The maximal sizes of dome can be estimated from the image processing andare tabulated in Table 5.2. The maximum volume of the glycerin domes, beforerupture is significantly larger than that obtained from water earlier in Chapter 4.The volume of the glycerin domes are 2-4 times those of water. This can be at-tributed to the fact that a larger pressure difference is required for the glycerin toinvade into the Carbopol than for water.The image processing also allows us to estimate the volumetric flow rate intothe dome, Qˆd , and from the dome radius Rˆd we can make a crude estimate of thelocal strain rate due to the dome expansion, ˆ˙γmax, using ˆ˙γmax = Qˆd/Rˆ3d , from valuesjust before rupture; see also Table 5.2. These values are surprisingly high for thelarger glycerin domes, but also significant in the water invasion. Certainly this148Figure 5.3: Recirculatory vortices formed inside a glycerin dome duringtransition stage. Here Carbopol concentration is 0.15% wt and H =15.2confirms that dome growth during the transition stage coincides with viscoplasticyielding of the Carbopol surrounding the dome, as hypothesised earlier.We also observe secondary flows developing in the larger glycerin domes asthey expand. An example is shown in Fig. 5.3, in which we see both recirculatoryflows and evidence of entrainment and dispersion of Carbopol by the secondaryflows (see the darker contours following the recirculation). These recirculatoryvortices are not observable in the smaller glycerin domes nor were they observed inthe previous water invasion results, although our visualization technique improvedin our later experiments. Panoramas of representative domes observed during glyc-erin invasion are shown in Figs. 5.4 and 5.5. It appears that the flow pattern insidethe dome is one of radial expansion, when sufficiently small, becoming recircula-tory when the dome is large enough.The circulatory flow is driven by the vortex formed during expansion and webelieve that viscous stresses are primarily responsible for the Carbopol entrain-ment. Formation of similar circulation patterns inside a polymer droplet sediment-ing in a miscible and lighter fluid is reported by Ref. [158] and secondary flowswithin droplets of differing viscosities are found in classical Stokes flow solutions.There appears to be no coherent trend in terms of size/shape variations witheither H or the Carbopol concentration for the panoramas of G45 and G58 domesshown in Figs. 5.4 and 5.5, and this was also noted for our earlier water invasionexperiments explained in Chapter 4. Indeed the only consistent trends observed are149C=0.15%, H=5C=0.15%, H=18.8 C=0.16%, H=2C=0.16%, H=2C=0.15%, H=10C=0.16%, H=5C=0.16%, H=5  C=0.16%, H=15.2C=0.17%, H=18.8 C=0.17%, H=5C=0.15%, H=15.2C=0.15%, H=19.2C=0.17%, H=2C=0.16%, H=15.2C=0.16%, H=19C=0.17%, H=2C=0.17%, H=15.3C=0.15%, H=5C=0.17%, H=19C=0.17%, H=10C=0.16%, H=18.4C=0.15%, H=15.2C=0.17%, H=5C=0.16%, H=2C=0.15%, H=10C=0.15%, H=5C=0.16%, H=105 mmFigure 5.4: Illustrations of transition dome shapes observed during G45 inva-sion for a range of different Carbopol concentrations, C, and heights, H.Snapshots for a given C and H set correspond to different experiments,repeated to reduce variability of the data.the larger sizes, of glycerin compared to water and of G45 compared to G58. Wealso observe a slight slumping of the larger glycerin domes compared to the waterdomes, probably due to density differences.We might now consider why there is such evident variability in dome size. Afirst reason we believe relates to the subjectivity of identifying the transition/in-vasion pressure (discussed earlier and evidenced by the large variability and errorbars in Fig. 5.2). Secondly, it is apparent that the transition stage represents at leasttwo competing effects. From the static computations in Chapter 4 we have seenthat imposing the invasion pressure on the surface of a dome as opposed on thehole surface, results in significant relaxation of the stress field, local to the inter-150C=0.16%, H=15.2C=0.15%, H=5C=0.16%, H=19.2 C=0.16%, H=10C=0.17%, H=2.2C=0.16%, H=2.2C=0.17%, H=15.2C=0.15%, H=2C=0.16%, H=5C=0.17%, H=10C=0.15%, H=10C=0.16%, H=10C=0.16%, H=19.2C=0.15%, H=15.2C=0.17%, H=19.2C=0.16%, H=15.2C=0.16%, H=5C=0.16%, H=2C=0.17%, H=5C=0.15%, H=19Figure 5.5: Illustrations of transition dome shapes observed during G58 inva-sion for a range of different Carbopol concentrations, C, and heights, H.Snapshots for a given C and H set correspond to different experiments,repeated to reduce variability of the data.face. Thus, at constant Pˆh, as the dome grows the (static) deviatoric stresses withinthe Carbopol should be reduced. On the other hand we have seen that the domegrowth accelerates to the point of rupture, suggesting weakening of resistance. Wehave also seen for large domes relatively rapid recirculations and large strain rateestimates. These signal that at the final stages of the transition, domes are progres-sively unstable, again promoting variability.5.3.3 Fracture and arrestThe rupture that signals the end of transition is observed in different ways. Forsmaller domes we often see the initial formation of a small finger (or fingers) thatextend out from the dome into the Carbopol, of dimension 1− 10 µm. These ir-regular features, just evident at the pixel scale, grow/burst into a dyke-like fracture.The larger domes suddenly burst, with no warning irregularity, rupturing along acurve rather than pointwise, but again forming a dyke-like fracture. These dykes151c)a) b)Figure 5.6: Examples of fracture and arrest stage in: a) H2O, b) G58 c) G45propagate in the Carbopol in one or several sheets. Water dykes as shown in Chap-ter 4 tend to rise up in the tube; see Fig. 5.6a. By comparison, the glycerin dykestend to rotate around the tube and rise up only very slightly (see Fig. 5.6b and c),which can be attributed to the higher density of glycerin to Carbopol.Another difference between glycerin and water dykes is in their morphology.The fracture planes of glycerin are thicker than those of water and there is lessfingering/tip splitting evident at the front; see Fig. 5.6. Thickness variation couldbe explained using the fact that glycerin has higher viscosity, so there will be ahigher pressure drop when passing through a channel of the same size as the waterfracture. For fixed driving pressure the glycerin optimizes its pathway by creatingthicker fracture planes.1525.4 Invasion of Immiscible FluidsTwo immiscible fluids, rhodorsil oil (R550) and air, were injected into the Car-bopol column. Physical properties of R550 are listed in Table 5.1. As discussed inSection 5.1, the onset of invasion of an immiscible fluid is expected to be a functionof H, rh and CaY . Experiments were performed to investigate the effects of H, bya similar procedure as for the miscible liquids, and of CaY by considering differentinvasion fluids.The experimental protocol had to be modified for invasion of the immisciblefluids. Interfacial tension prevents the mixing regime from occurring, (again con-firming its molecular diffusive origin). Thus, we do not see the diffuse layer andsmall penetration emerging, nor the oscillatory motion of the mini-dome signallingthe “invasion point”. Although the invading fluid behaviour is different, we expectthe Carbopol to respond mechanically in a similar way, transitioning from elasticto visco-plastic deformation during invasion. In order to allow sufficient “delaytime” for yielding to be observed, a small modification has been made. The ap-plied pressure is initially increased continuously (at a similar rate to the miscibleexperiments) until a point below which invasion is not expected to happen, whichis known approximately from the previous experiments. During this initial period,no invasion is observed. Then a stepped ramp pressure profile is imposed. At eachstep the pressure increase is arrested for around 2 minutes, then it is resumed atconstant rate (for a height increase of ≈ 2 mm, or ≈ 20 Pa), then repeated. Anexample is shown in see Fig. 5.7. The relatively large steps in Pˆh are imposed sincethe invasion pressures are significantly larger.Calculating CaY requires a representative value for the interfacial tension be-tween our fluids. This is non-trivial for yield stress fluids as many of the usualmethods for measuring interfacial tension are not suitable, i.e. are affected or evendominated by the yield stress. However, in recent years different researchers havesucceeded to measure a surface tension coefficient for Carbopol-air[159–161] us-ing 2 different methods and with some consistency: σˆ ≈ 50− 66 mN/m. Thevariation arises due to measurement protocols, depending on extension/compres-sion and elastic effects; see the discussion in [161]. A rule of thumb suggested byBoujlel and Coussot [159] is that the σˆ ≈ 10% less than the value for the inter-153Figure 5.7: A stepped ramp pressure profile for increasing the applied pres-sure during immiscible fluids injection.stitial liquid in the gel, comparing against 66 mN/m against a reference value forair and pure-water of 72 mN/m. Evidently, the interstitial liquid is not pure waterand it has been suggested that the true surface tension is simply that of the intersti-tial liquid. Other recent bubble rise experiments[162] measured σˆ(= 73.4 mN/m),using a standard ring-type tensiometer. For R550, which is a clear light siliconoil, such data is not available. The product data sheet[163] gives σˆ = 24.5 mN/mas the surface tension and oil-water interfacial tensions are typically in the range20−40 mN/m. This is a commercial product with unknown compositions, whichhampers application of theoretical methods for estimating σˆ . In the following weshall assume σˆ ≈ 18−30 mN/m.5.4.1 Invasion and propagation of Rhodorsil oilRhodorsil oil (R550) is very viscous and slightly more dense than Carbopol. Toisolate the effect of surface tension on the invasion pressure, the density of theCarbopol solution is increased to match that of rhodorsil oil, by using a low con-centration glycerin solution. This gives similar yield stresses as for the misciblefluids studied earlier.It is observed that R550 creates a dome immediately after penetrating into154Figure 5.8: Example invasion of R550 into a density matched Carbopol so-lution.the column of yield-stress fluid and then it expands very slowly; see Fig. 5.8. Ittakes some hours until it displaces around 5−8 cm of the column of the fluid; seeFig. 5.9.Interestingly unlike the miscible invasion, as here there is no localization of theinvasion nor any mixing, the steady increase of dome volume results in displace-ment upwards of the Carbopol column. The invasion pressures of R550 againstthe height of yield stress fluid are shown in Fig. 5.10. As we see, Pi increasesapproximately linearly with H. Assuming that the slope of the linear increase isdetermined by the Poiseuille flow, it appears that (5.8) should approximate the in-vasion pressures reasonably and consequently we have fitted the data to:Pi = PY,Pois+PitCaY≈ 2H +31.11(±10.89), (5.9)with the fitted line (2H+31.11) also sketched in Fig. 5.10. The Poiseuille increaseis remarkably evident.For our ranges of yield stress and estimated range for the interfacial tensionof R550, we calculate CaY ∈ [0.054,0.159]. This range of CaY in turn suggeststhat Pit ∈ [2.27,3.20]. Supposing that the initial penetration were characterizedby a hemispherical initial dome of the hole radius, then Pit = 2, so this range ofestimated Pit seems at least reasonable.155Figure 5.9: An example of flow developement after injecting R550 into theCarbopol. Here concentration of Carbopol is 0.15% wt and H = 10.0 5 10 15 2020406080100120140160Figure 5.10: Invasion pressures of immiscible fluids against height of theCarbopol column:  R550, • Air.156Figure 5.11: A typical sequences of bubble formation in Carbopol. In thisexample, concentration of Carbopol is 0.16% wt and H = 10.5.4.2 Air invasionThe second immiscible fluid was air. Compared to R550 the interfacial tension wasincreased by a factor of 2-3, but also a far smaller viscosity and density. A typicalsequence of an invading air experiment is shown in Fig. 5.11. We see that the on-set is quite localised, with no mixing and an intial (approximately) hemisphericalexpansion. However, relatively rapidly the effect of buoyancy appears to domi-nate and the radius of the invading air column in fact contracts from a maximum,then pinches off from the hole and rises up as a bubble. Our intention was not tostudy bubble rise, so for now we concentrate on invasion pressure. It must be men-tioned that the invasion/bubble growth and pinch off all occur at the same imposedpressure, i.e. once invasion starts the process continues through to detachment rel-atively rapidly. Evidently, as the bubble expands surface tension effects decreaseand buoyancy also contributes to this accelerating process.The scaled invasion pressures of air are plotted against the height of the columnin Fig. 5.10. The results reveal that the invasion pressure of air is not stronglydependent on the height of the column. There is in fact a mild increase in Pi withH apparent. This increase is of the order of that for water and glycerin (see Fig. 5.2earlier), but is not significant compared to the error bar here. The weak dependenceis consistent with localised yielding of the fluid around the initial invasion dome.The massive increase in Pi above the values for R550 suggests dominance bythe capillary force. If we assume a splitting: Pi = PY +Pit/CaY , between yieldstress and capillary effects as in (5.8), we may estimate each contribution. As afirst estimate of the capillary term Pit/CaY , we might simply multiply that of R550by the ratio of interfacial tensions, i.e. CaY is 2-3 times smaller for air than forR550, (within the limits of our estimates and data). This leads to a mean value for157Pit/CaY in the range 60-95, whereas the average value for Pi Fig. 5.10 is around125.8 (marked by the broken line).If the above estimates are reasonable, this suggests that the remainder PY shouldhave mean value in the approximate range 30 to 60, which we note is significantlylarger than that determined in our miscible invasion experiments (PY ∈ (8,25) inFig. 5.2), which also showed a localised invasion. Potentially, this effect may alsobe due to interfacial tension acting on a much smaller scale, which weakens the gelin a miscible invasion and allows localised initial invasion on the 1−10µm scale.In order to estimate the potential contribution of the yield stress, we have per-formed a sequence of invasion experiments. After the initial invasion bubble prop-agates, we stop the flow, reduce Pˆh, wait for 5 minutes and repeat the invasionexperiment.A number of authors have studied bubble propagation experimentally in Car-bopol and other yield stress fluids[128, 162]. Damage due to elastic and/or thixotropiceffects is common[128, 164] and unless the fluid is treated between bubbles thepropagation is affected significantly, e.g. in [162] to ensure repeatability the Car-bopol column is remixed after every bubble and a waiting time imposed before thenext experiment.Here we leave the column of Carbopol undisturbed and repeat the invasionexperiment a second and third time. The values of invasion pressures obtainedduring these repeat tests are shown in Fig. 5.12. As clear from this figure Pidecreases dramatically in the second and third tries. The decrease in Pi is reducedbetween 2nd and 3rd invasions. This significant decrease in the invasion pressurecan be attributed to the time dependency of the rheology. Clearly the structure ofthe fluid is damaged after the first invasion and 5 minutes rest is not long enough forCarbopol to recover its initial structure. This is consistent with our own experienceand that of [131].As there is no reason to expect a change in interfacial tension, we attribute theobserved drop in Pi to a drop in the yield stress contribution PY . The size of the dropbetween initial and third invasion spans the range 30 to 60, discussed above as anestimate for PY , i.e. confirming that PY is significantly increased over the misciblefluids cases.The size and velocity of bubbles in the 2nd and 3rd invasion experiments was1581 2 3406080100120140160180Figure 5.12: The invasion pressure (Pi) of the air versus the number of repeatexperiments made into the same Carbopol.also measured. As shown in Fig. 5.13, there is a significant decrease in the sizeof the ensuing bubbles. This suggests that rheology of the Carbopol has decreasedand a lower buoyancy force is enough to overcome it. In general we observe thatit takes less time for the bubbles to detach and rise up through the Carbopol. Theincrease in the velocity of the bubbles in the 2nd and 3rd experiments is shown inFig. 5.14. Again this confirms the reduced resistance of the Carbopol against thebubbles.Figure 5.15 shows the speed of the bubble plotted against the bubble volume,only for the first bubble. At first glance it appears that small bubbles rise faster.However, the smaller bubbles are also found for the Carbopol with the lowest con-centration: the higher concentrations have larger yield stresses and larger effectiveviscosity, both of which retard motion despite the larger volume.5.5 Summary and discussionWith the new results presented in this chapter, the invasion process qualitativelyand invasion pressures quantitatively for a full set of fluids, involving stresses gen-erated by any of buoyancy, viscosity and capillary forces which are present to vary-ing degrees, are demonstrated. In combination with those obtained from Chapter 4,the results can be distilled into the following points:1591 2 30123456 103Figure 5.13: The volume of the bubbles versus the number of repeat experi-ments made into the same Carbopol.1 2 300.511.522.5 102Figure 5.14: The velocity of the bubbles versus the number of repeat experi-ments made into the same Carbopol.Our initial dimensional analysis ignores miscibility effects and suggests that theinvasion pressure Pi should depend on (H,rh,CaY ). Here rh does not vary withinour experiment and we have split Pi into a yield pressure part and an interfacialtension term (scaled with 1/CaY ). Our results show that this splitting is problematicunless miscibility effects are better understood, but otherwise serves as a usefulconceptual framework.Our tests with glycerin solutions, comparing against the water invasion stud-ies of Chapter 4 reveal the following: (i) the glycerin solutions have a slower1602 2.5 3 3.5 4 4.5 510345678910111213Figure 5.15: Rising velocity of the bubbles against their volume.mixing/invasion stage (longer times before transition); (ii) invasion pressures areconsequently higher; (iii) after transition the invading domes grow more rapidlyand are significantly larger than those for water at the point when fracturing initi-ates. These differences are attributed to the larger viscosities of the glycerin, whichslows diffusive/mixing processes. On the other hand, there are many similarities tothe water invasion studies of Chapter 4. The invasion stages are similar and inva-sion is a strongly localised process. The increase in Pi with H is far slower than thePoiseuille flow limit (2H). The transition point also appears here to be associatedwith elastic-plastic yielding of the Carbopol.We found no significant effect of the density difference on the invasion pres-sure, which is consistent with our dimensional analysis. Two glycerin concentra-tions were used (G45 and G58). It was noted that the invasion pressures and domesizes were larger with G45 than G58, whereas the larger concentration would beexpected to have the lower diffusivity. This was therefore unexpected, althoughthese differences lie within the error bars of the data, and we have seen no obvi-ous explanation. Another phenomenon noted in the larger glycerin domes was therecirculating secondary flows which appears to entrain Carbopol into the invadingfluid. These flows were not observed in the smaller domes. Following invasionand rupturing of the dome, the glycerin solutions propagate in dyke like fracturesthat are thicker than those for the less viscous water and which slump towards thebottom of the invasion column (presumably under gravity). Our measured invasion161pressures do have relatively large error bars. Part of this is due to the experimentalmethodology and a degree of subjectivity in judging when the transition pressureis attained. The invasion pressure measured is not easily defined and could corre-spond to an upper bound on the Pi value corresponding to the elastic to (elasto-)viscoplastic transition, i.e. because the dome is still increasing while the pressureramp is stopped. The yielding process is itself only idealized as occurring at asingle pressure/stress and clearly our apparatus is not a well-designed mechanicaltest for yielding. Other artifacts may be present and also some Carbopol gels arecharacterized by a time-dependent behavior (see e.g. [165] and [166]). Abovethe yield stress, the fluidization of the material is reached at a time which dependson the shear rate and the yield stress (see Divoux et al. [32]). Thus, the pressure(ramp) increase rate may have some effect here.Our experiments with immiscible fluids involved a density matched silicon oil(R550) and air. It was immediately apparent that interfacial tension had a largeeffect on invasion. Our experimental protocol was changed to increase the invasionpressure in larger steps than for the miscible fluids. Mixing was eliminated as wasthe occurrence of an initial micro-invasion dome. Instead, once invasion startedwe observe a clear interface expanding into the Carbopol column. In the case ofthe R550 the interface evolved as a hemispherical dome, slowly filling the bottomof the column. The invasion pressure was significantly larger than for the glycerinsolutions and was non-local in that it was resisted by yielding at the walls of thecolumn, with Pi increasing as the Poiseuille flow ∼ 2H.The air also did not mix and had no micro-invasion dome. However, the inva-sion was localised in that buoyancy dominated after any significant influx, leadingto the invading fluid stretching upwards into a long bubble, which eventually de-tached and propagated to the surface. Thus, resistance of invasion at the walls didnot occur and Pi was approximately constant with H. Also Pi was much largerfor air than for R550. Further analysis of this increased Pi suggested that about50% of Pi is directly attributable to interfacial tension and the remainder to yieldstress. However, the yield stress is only able to resist effectively because there isno mixing/diffusion.Finally we consider the implications of our results for the industrial process ofwell cementing, which motivated the study. First, we see that invasion of misci-162ble fluids is least affected by either the yield stress or the height of the column,e.g. water invasion. The invasion process is strongly localized and retarded by anincreased viscosity.Interfacial tension has a significant role to play in conjunction with the yieldstress. The interfacial tension prevents local diffusion and weakening of the gelstructure with subsequent invasion on this scale. This allows the yield stress toresist on macroscopic scale. Thus, for example we might expect additives that in-crease the yield stress of the slurry to be effective in preventing oil invasion. Theprincipal difference between air and R550 is dominated by buoyancy: the initial in-vasion is similar and dome like, but air injection migrated upwards which preventedthe dome from growing outwards to the wall. We might expect a similar effect ina well. Thus, although the gas invasion pressure may be significantly higher thanthat for water or oil, it seems to be largely independent of H: once entered intothe annulus reservoir gas will channel upwards through the cement slurry, ratherthan displace it. We suggest that effective cement additives for gas invasion wouldneed to force invading gas pockets to expand locally and displace the entire annu-lar cross-section. In this way rheological contributions to Pi that scale with H canbe realised. For example an additive that quickly develops viscoelastic propertiesin the liquid phase of the cement (i.e. downhole bubblegum), is likely to be moreeffective at preventing invasion than a pure yield stress effect alone.5.6 Concluding remarksWe have presented results obtained from our experimental study of the invasion ofmiscible and immiscible fluids into a static column of yield stress fluid in Chapter 4and Chapter 5. We observed that the interfacial tension has significant effect onboth the invasion process and invasion pressure. We find that the pressures requiredto initiate miscible fluid invasion are significantly lower than those triggering theinvasion of immiscible fluids. The reason lies in the fact that for miscible fluidsthere is an initial diffusive mixing stage which facilitates their local penetration.Viscosity potentially retards this stage: more viscous invading fluids require largerinvasion pressures. In miscible fluid invasions, the stresses relax immediately afterthe invasion, but then the invading dome experiences a brittle yielding/fracturing163with viscous fingering of the propagating invasion front. Whereas, the immisciblefluids penetrate in the form of a slowly expanding dome, resisted at the walls ofthe column: effectively by a Poiseuille flow above it in the Carbopol column. Thepressure required to initiate gas invasion is influenced approximately equally byinterfacial tension and yield stress.164Chapter 6Summary and conclusionThis thesis has addressed in detail two flow problems. Firstly, we have studied den-sity stable and unstable displacement of yield stress fluids along a vertical channel,using computational and analytical methods. Secondly, we have studied the pro-cess of invasion of miscible and immiscible fluids into a tall column of yield stressfluid, mainly experimentally but supplemented with dimensional analysis and somecomputation.In this chapter we review the main findings and contributions made to under-standing of each flow: both the scientific results and industrial implications. Thethesis closes with a discussion of limitations (Section 6.3) and suggestions for fu-ture research directions in this area (Section 6.4).6.1 Buoyancy effects on micro-annulus formation:Newtonian-Bingham fluid displacements in verticalchannelsThis problem has been studied in Chapter 2 and Chapter 3, for density stable anddensity unstable configurations, respectively.6.1.1 Scientific contributionsWe may view the chief overall contribution of the thesis in this area as complet-ing the study of displacement flows of a yield stress fluid by a Newtonian fluid165along a vertical channel. Iso-dense displacements were studied in [54, 55], whichintroduced the concept of the static residual wall layer in a viscoplastic fluid dis-placement flow, established that it was a dynamically stable structure and noted itsrelevance for the phenomenon of micro-annulus formation in primary cementing.Since most cementing flows do exhibit density differences, the iso-dense as-sumption in [54, 55] is a serious limitation to applicability. In this thesis we haveexposed this clearly, as our results show that buoyancy can have significant effectson residual layer thickness and on the flow stability. Although simplified to onlya Newtonian fluid displacing a Bingham fluid, the problems considered do allowus to gain intuition into the comparative effects of changes in buoyancy, viscous,yield and inertial stress contributions, via different dimensionless balances, and theeffects of viscosity ratio. In this way, we cover the most practical effects.• Buoyant displacements of a Bingham fluid by a Newtonian fluid along avertical channel were studied, leading to understanding of the thickness andlong-time behaviour of the residual layer. The flow is effectively governedby 4 dimensionless parameters: the Newtonian Bingham number (BN), theviscosity ratio (m), the Reynolds number (Re), and modified Froude number(Fr). The computational studies performed varied these parameters overwide ranges (while remaining in observably laminar regimes). Reynolds andFroude numbers were not varied independently, but instead were selectedto give discrete values of χ∗ = 2Re/Fr2 (signed depending on the densitydifference). This represents the ratio of buoyancy stress to viscous stressof the displacing fluid, which appears in parallel multi-layer flows of the 2fluids in vertical channels. The parameter χ∗ was used for density stableflows and replaced with χ =−χ∗ for density unstable flows.• For density stable flows the parametric effects are as follows: increasing mincreases the residual layer thickness h and increasing χ∗ decreases h. Thusas expected, increasing buoyancy or the displacing fluid viscosity improvesthe displacement. Counter-intuitive however, is the decrease in h for increas-ing B = BN/m, meaning the yield stress of the displaced fluid. This effect ismost evident at small to moderate buoyancy: for large buoyancy the residuallayers are anyway diminished.166It has been shown that h is described very well as a function only of (B,χ∗/m),(see e.g. Fig. 2.12). Our interpretation of this is that B and (potentially)BN/χ∗ appear to control the first stage of frontal displacement, whereas thefilm draining process is largely governed by χ∗/m. Note that χ∗/m mea-sures the ratio of buoyancy stress to the viscous stress of the Bingham fluid(i.e. due only to the plastic viscosity), which is the main balance in any walllayer that is sheared.• The long time behavior of density stable displacements is described by h andhmax, the maximal static wall layer thickness. The latter is easily computedand hmax = hmax(χ∗,BN) (see Section 2.2). If hmax = 0 (which occurs forBN ≤ 6) the wall layer is moving and will eventually be removed. If insteadhmax > 0 we have 2 possibilities: either h < hmax and the wall layers arestatic, or h ≥ hmax and the wall layers are moving. If the wall layers aremoving and h ≥ hmax, it follows that h will decrease and approach hmax atwhich point it becomes static.• The major contribution and insight is in prediction of the residual layer be-haviour of density stable flows by classifying the displacement front be-haviours. Our displacement flows have shown 3 characteristic behaviours:frontal shock, spike and dispersive. The first interesting result from examin-ing our data is that when hmax > 0, we have only observed frontal shocks andspikes. Secondly, we have shown that frontal shocks appear to correspond toh < hmax, and spikes correspond to h(t)≥ hmax; see Fig. 2.15.On its own Fig. 2.15 is simply interesting, but we have also devised a way topredict the displacement front behaviour without carrying out a full 2D dis-placement simulation. This involves a simplified thin-film/lubrication modelof the displacement flow (see Section 2.3.2). The simplified displacementmodel also exhibits frontal behaviour in the same 3 categories as the 2Dsimulation, but the behaviour can be predicted by computing the flux func-tion instead of the full displacement. The comparison between front typesis shown in Fig. 2.21, showing that we may qualitatively predict the residuallayer behaviour of the 2D simulation very quickly from the simplified model.167• For density unstable displacement of a Bingham fluid with a Newtonianfluid we studied a similar range of dimensionless parameters, now withχ ∈ (0.1,500). The main contrast with the density stable flows was thatmany of the flows developed hydrodynamic instabilities.Three types of instabilities have been identified: frontal instabilities of Rayleigh-Taylor (RT) type, interfacial roll-waves of Kelvin-Helmholz (KH) type andviscous-controlled inverse bamboo and mushroom (IBM) morphologies. Eachclassification is interpreted loosely as in practice other physical effects arepresent. Also the flows we have simulated are not ideal multi-layer configu-rations designed for the study of individual instabilities.• For less significant buoyancy, say χ < 10, there is no evidence of instabilityand the flows are very similar to iso-dense and density stable, i.e. qualita-tively the flow behaviour changes smoothly as we cross χ = 0 (χ∗ = 0). Toget a feel for the significance of χ (or χ∗) note that the pressure gradientrequired for the displacing fluid to flow at unit velocity has size 12. Thus forexample, χ = 1 12 is not significant.• Instabilities arise at elevated values of χ . Increasing χ when the displacedBingham fluid is less viscous than the displacing fluid leads towards a counter-current flow and KH instabilities are observed in this scenario. This happenswith a combination of low m, weak BN and strong χ as it is shown in Fig. 3.7.IBM instabilities are generated when m > 1 and a two-layer structure is ini-tially formed during displacement. Since m > 1, these instabilities havea slow growth rate, i.e. controlled by the viscosity of the displaced fluid(panorama of the flow types observed for m > 1 is plotted in Figs. 3.11 &3.12). Within the IBM classification we have also observed an interestingregular patterning of static wall layers (footprinting); an example is shownin Fig. 3.6.RT instabilities are frontal instabilities and have been found independentlyof KH or IBM instabilities (which affect the layered flow behind the front).• The transition between stable and unstable regimes has been plotted in the(Fr,Re/Fr)-plane. It happens across a boundary of form: Re ∼ Frx, where168x is 2,8/5,4/5,2/5 depending on m and BN . The exponent x decreases withincreasing m values and slightly increases with BN (see Figs. 3.13- 3.17).Our study has been phenomenological in this aspect, so these transitions areintended more as a guide rather than as any serious attempt to curve-fit atransition curve. In terms of these parameters, although Re/Fr is useful toplot against, this balance does not represent the viscosity of the displacingfluid at all: including m and BN would be a potential improvement.• We have also explored static residual layers for density unstable flows. First,by assuming that fluid 2 is completely static, four flow regimes have beenidentified in our two layer model: 1- τxy decreases from zero in the center to-ward the wall, and the maximum absolute value will occur at the wall; 2- byincreasing χ further the slope of τxy is reduced in layer 2, but still the inter-facial and wall shear stress are negative; 3- by increasing χ further the wallshear stress becomes positive, but is still smaller than the interfacial stress;4- the wall shear stress is positive and bigger than the negative interfacialstress. These 4 regimes are plotted in Fig. 3.18.• The above classification was based purely on the stress distributions. Thefluid 2 layer next to the wall can be static or moving in all regimes, dependingon BN . However, the interesting fact is that the wall layer can be partiallyyielded in regimes 2 & 3, such that a layer near the interface yields and ismoved in the flow direction, but the wall layer remains static. We call thesepartially static wall layers.• For regimes 1-3 we have extended the concept of the maximal static walllayer to define hmax,a, where all of the fluid 2 layer is static. However, itis apparent that when the wall layer yields in regimes 2 or 3, and there is apartially static layer, potentially the partially static layer could be thicker thanthe fully static. This is because, by yielding and moving slowly downstreamthe fluid 2 layer contributed to the flow rate, hence reducing the pressuregradient and shear stresses.• The residual layer h left on the wall in our 2D simulations has also beenanalysed. This layer can be: 1- fully static; 2- partially static; 3- fully mov-169ing. We have seen that (see Fig. 3.19) fully moving layers can happen atall χ, BN and m values whereas partially static layers lie in regimes 2 & 3,as predicted by the 1D-model. For m ≤ 1, fully static layers for BN = 10can develop in regimes 1, 2 and 3 but for m > 1 they are distributed only inregime 2 & 3. For BN < 6, all the layers are moving or partially static exceptfor a few fully static layers which are very close to the yielding threshold,i.e. within numerical error.• As well as our stress-based analysis, we have solved the 2-layer model com-putationally, so that the stress profiles can now be considered as those actu-ally found as part of the velocity (and pressure gradient) solution, and notsimply inferred from the momentum balances and constraint of a static walllayer. Using this, it is observed that at any given set of (BN ,χ,m), the yieldsurface position yY varies with yi over a U-shaped curve which either has asingle minimum or is discontinuous with the broken middle part. This leadsto a simple calculation of the minimal yY and hence a maximal static layerhmax,p. The maximal static layer found satisfies hmax,p ≥ hmax,a, as it allowsfor partially static wall layers. The variation of static wall layer position withinterface position for a few cases is shown in Fig. 3.20.• The important results are that: (i) partially static wall layers can be formedfor BN < 6, unlike for the density stable case; (ii) as discussed above, thepartially static wall layers can for certain parameters yield a layer thicknesslarger than for a fully static wall layer. In agreement with 1D-model results,the thickness of wall layers obtained from 2D-computations in the fully staticregime are mostly less than hmax,p (see Fig. 3.23). A few cases have slightlythicker layers which may be due to the numerical procedure.• Similar to the results obtained for density stable displacements, the thicknessof residual layer increases with m and decreases with BN .• Finally we have explored a thin-layer version of the 2-layer 1D model, aswas done for density stable flows. Just as for the density stable case, the dis-placement front predictions from the 1D model are predictive of behaviour170on the 2D simulations. We are able to predict stable flows and also thosesusceptible to KH type instabilities (see Fig. 3.25).6.1.2 Industrial implicationsImplications and recommendations are difficult to make in absolute terms, as theentire process of primary cementing has many constraints. However, within thecontext of our results we can make a number of suggestions.Regarding density unstable displacement flows, we recommend these be avoided.In the first place, we have seen a wide range of hydrodynamic instabilities that re-sult in our density unstable simulations at large enough χ . Although conceivablythese might have been expected to improve the mud removal process, in practicewe have seen no observable benefit in terms of reduced h. In a cementing context,most probably m> 1 (i.e. wash displacing mud) and then any instabilities that groware IBM type, controlled by the mud viscosity and evolving slowly.Secondly, we have seen that density unstable displacement flows allow forstatic wall layers even below BN = 6. For large density differences, as we in-crease the layer thickness buoyancy acts to increase the shear stress in the mudlayer. This can result in small negative shear stresses near the wall, or even smallpositive shear stresses. The consequence is that a small yield stress can keep themud layer partially static at the wall.Combined, we see no mechanical benefit to density unstable displacementflows. This recommendation falls in line with current design practice for lami-nar displacements, e.g. as in Nelson and Guillot [5], and simply adds to underlyingreasons to chose fluids with a density hierarchy.Supposing now that we only consider density stable displacements. The fol-lowing are the industrially relevant results• The most important result is the simplest, that BN < 6 prevents a static layerfrom forming, even in the presence of stabilizing buoyancy. In dimensionalterms, this means that the wall shear stress of the displacing fluid (flowingon its own) should exceed the yield stress of the drilling mud. This conditionalso holds for other fluid types with or without a yield stress, i.e. provided171we can estimate the wall shear stress of the displacing fluid and the yieldstress of the drilling mud downhole.Although we feel this is the most valuable, partly also due to the simplicityof statement, we acknowledge that there are difficulties in application. In thefirst place the yield stress of the in-situ mud needs to be characterized andthat data made available to the service company at a design stage, so as toformulate the spacer fluid rheology. Daily mud reports from the rig need tobe consulted and this leads to a different concern, namely that of the poorrheological characterization of fluids at the wellsite, using standard 6 or 12speed viscometers, which are not ideal for yield stress estimation.Mud conditioning also plays an enormous role here. If the well is not circu-lated bottoms up a number of times before cement placement starts, the mudrheology may be far from that measured during the drilling phase preceding,thus estimating the yield stress for BN < 6 will be hard.For more general fluids this recommendation also requires use of some sortof computational design tool. At a minimum one needs to estimate wallshear stresses in eccentric annuli, estimate the eccentricity (stand-off) etc..This can be done using either standalone software already available in theindustry for calculating centralization and cementing hydraulics, or could beeventually embedded within a more sophisticated simulation software thatcomputes the annular displacement fully. Our remaining recommendationsassume that the service company or operator has some proficiency with thistype of calculation/simulation.• For cases where we potentially have a static layer, we might try to minimizehmax at design stage. We should bear in mind that increasing χ∗ reduceshmax. However, if increased χ∗ is not achieved by increasing the densitydifference, the operational alternative is to pump slower. However, now thereduction in viscous stress increases BN , which allows a larger hmax! Thus,optimization like this should be approached with caution, and note that hrather than hmax is to be reduced.• For cases where we potentially have a static layer (BN > 6), we may adopt172the following rubric to at least identify flow type. (i) Compute hmax(χ∗,BN).(ii) Analyze q(yi) to construct Vi(yi) and classify as either spike or frontalshock. (iii) If a frontal shock, expect a uniform static wall layer to formduring the displacement and to remain, with thickness h < hmax. (iv) If aspike, locally expect an initially moving wall layer, following passage of thedisplacement front, and eventually h→ hmax as t→ ∞.• In the primary cementing context, dispersive or spike type displacementsresult in moving residual layers. The timescale for these layers to be re-moved will increase with m (potentially also with B) and these mobile layersshould be regarded as potential sources of contamination. We have seen noevidence of instability in our simulations, but these are of limited duration.The velocity profiles in these draining layers have pseudo-plugs at the inter-face, i.e. yielded at the wall only but with a plug velocity that slowly variesalong the channel. Visco-plastic lubrication studies would suggest stability(e.g. [111]) but viscous-viscous theories (e.g. [112]) often show instabilitywhen the more viscous fluid abuts the wall. On the other hand, the frontalshocks, which produce a static layer h(∞)< hmax as soon as the front passes.These layers will be hydrodynamically stable and robustly remain. There isno contamination risk from interfacial mixing, but such layers are likely todry during cement hydration, forming a porous conduit. In this context, thespike displacements at large m appear to be the worst case: not only do wesuffer the contamination risk of mobile layers over a timescale ∝m, but alsothe layers at best thin towards h(∞)∼ hmax > 0, which remain on the walls.6.2 Invasion of fluids into a gelled fluid columnThis problem has been studied in Chapter 4 and Chapter 5 as a proxy for the processby which fluids invade from the formation into a column of cement slurry sittingstatic in the annulus at the end of placement.6.2.1 Scientific contributionsAs far as we know, this is the first study of this type, targeting fluid invasion underpressure into a yield stress fluid column. Over 200 experiments were performed173in our apparatus, repeatedly covering 3 different carbopol concentrations, variousheights of column and different invading fluid. This was supplemented by a com-putational study. The main points are as follows.• We have presented new results governing the invasion of fluids into a gelledstatic column of yield stress fluid. A dimensional analysis has been con-ducted for both miscible and immiscible cases. Ignoring the detail of mis-cibility effects, our analysis suggests that the invasion pressure Pi shoulddepend on (H,rh,CaY ) (column height, hole radius and yield capillary num-ber - all dimensionless). Here rh does not vary within our experiment. Wehave then split Pi into a yield pressure part and an interfacial tension term(scaled with 1/CaY ). Our results show that this splitting is problematic un-less miscibility effects are better understood, but otherwise serves as a use-ful conceptual framework. In any case, our results are discussed in terms ofthese dimensionless variables.• In our experiments we increase the pressure in the invading fluid, present ata small hole in the base of a tall reservoir of yield stress fluid. Our setup wasdesigned to simulate invasion into a well during primary cementing and toallow isolation of different stresses in simulating the invasion.• Our initial experiments used water as the invading fluid, to eliminate buoy-ancy and capillary effects. The invaded fluid was always a Carbopol gel.A number of interesting stages have been observed during the experiments(see Fig. 4.3): mixing, invasion, transition, fracture and arrest. The pres-sure measurements were targeted at invasion. The passage from invasion totransition pressure seems to represent elastic-plastic yielding close to the in-vasion hole. The early invasion mini-domes could be reversed (apart fromdiffusive effects) by reducing the hole pressure, indicating elasticity. Theinitial growth of the transition dome and then slowing of growth (in caseswhere the fracture does not start immediately) suggest yielding and then re-laxation of the stress field, due to the overpressure now being applied over alarger area.• The invasion pressures Pi versus H are plotted in Fig. 4.5. For low column174heights H ∼ 1 (Hˆ ∼ Rˆ) there is little effect of H, but as H increases weobserve a steady approximately linear increase in Pi over the experimentalrange. Dimensionally, Pi measures the invasion over-pressure (i.e. pressureabove the hydrostatic pressure in the fluid column), scaled with the yieldstress. Thus, our results show approximately linear increase in invasion over-pressure with Hˆ. Also the invasion over-pressure increases linearly with theyield stress from the scaling adopted.• The experiments show significant variability, hence the repetition at eachheight and Carbopol concentration, despite considerable evolution in the ex-perimental procedure to reduce this. For example, we have carefully im-plemented protocols for fluid preparation, tank filling, eliminating invasionhole plugging, etc., and have eliminated buoyancy, capillary, static and at-mospheric pressures via our experimental design. The remaining variabilitystems from the fact that we are measuring an isolated onset event in a dynam-ically evolving process, and that this event is identified phenomenologicallyby the appearance of a minuscule invasion dome.• We have seen that the expanding transition dome interface can be either rel-atively smooth or granular (examples are shown in Fig. 4.4). This does notappear to have any bearing on the stability of the transition dome, i.e. mean-ing whether fracture occurs with/without further pressure increase. Invasionand transition domes are approximately axisymmetric. Fracture initiationand propagation represent a departure from symmetry, probably due to ei-ther a local defect or a non-uniformity of the stress-field.• The computational study has covered a range of invasion hole sizes anddimensionless heights H. The computed invasion pressures, as shown inFig. 4.10, followed similar qualitative trends to the experiments but are them-selves over-predicted (by comparison). This is probably due to the idealvisco-plastic (Bingham) law that we have implemented. This ignores effectsof the mixing stage and has no elasticity.The computations also reveal that the invasion pressures for small rh increaserelatively slowly with H > 1. This is tied into the occurrence of a local175dome-like region of yielding close to the hole, i.e. the invasion overpressurefirst causes fluid to yield and recirculate locally within a dome shape. Thisappears analogous to the flows observed during the transition stage in ourexperiments.• As H is significantly increased, for small rh, the increase in Pi eventually gen-erates sufficient stress to reach the walls of the cylinder. After this the inva-sion process changes from a local phenomenon to a global one; the yieldingpattern in each case is presented in Figs. 4.11 & 4.12. In the global yielding,resistance of the fluid occurs on the scale of the cylinder, more specifically,there remains an O(1) region close to the hole within which the magnitude ofthe deviatoric stresses components are all significant. The stresses generallydecrease away from the hole, but at sufficiently large H yielded fluid doesextend to the wall. Above this local near-hole region, the fluid adopts essen-tially a Poiseuille profile, with linearly decreasing shear stress from centreto wall. Hoop and extensional stresses, which are important in the near-holeregion, decay in the Poiseuille region as H is increased.• We further observe that the invasion pressure increase proceeds in parallel tothe Poiseuille prediction (Pi = 2H) both for small rh at sufficiently large Hand for larger rh. For the latter there is no range of H for which yielding islocal and isolated, instead the Poiseuille regime is entered immediately.• Returning to the experimental results, the invasion pressures increases at arate that is always significantly below Pi = 2H. This, the hole size and thephenomena observed, all suggest that our experiments are fully in the lo-cal regime of invasions. The initial domes are approximately axisymmetricand apart from more complex constitutive behaviour the computational andexperimental results present a coherent picture.• Our tests with glycerin solutions, extended the range of tests with misci-ble fluids. Two glycerin concentrations were used (G45 and G58). Themain physical differences are increased density and viscosity, which in anidealised setting should not have any effect on the actual invasion pressure.There were many similarities to the water invasion studies. The different176invasion stages are similar and invasion is a strongly localised process, withinitial mixing, then an invasion minidome, then transition and a larger domegrows, etc. The increase in Pi with H is again far slower than the Poiseuilleflow limit (2H) (see Fig. 5.2). The transition point also appears here to beassociated with elastic-plastic yielding of the Carbopol.However, there were also significant differences. (i) The glycerin solutionshave a slower mixing/invasion stage (meaning a longer time before transitionstarts). (ii) As the hole pressure increase rates were the same, the invasionpressures are consequently higher than for water. (iii) After transition theinvading domes grow more rapidly and are significantly larger than those forwater, at the point when fracturing initiates. These differences are attributedto the larger viscosities of the glycerin, which slows diffusive/mixing pro-cesses.• With the above observations, it becomes clear that the initial stage of mix-ing is critical to invasion of miscible fluids. It was noted that the invasionpressures and dome sizes were larger with G45 than G58, whereas the largerconcentration would be expected to have the lower molecular diffusivity.This was therefore unexpected, although these differences lie within the er-ror bars of the data, and we have seen no obvious explanation. Possibly thismeans that the initial mixing stage, in which the invaded gel is apparentlyweakened, is not purely diffusive but also may have osmotic driven transportand other effects. We did not investigate further as our setup was not idealfor that type of experiment.• We found no significant effect of the density difference on the invasion pres-sure, which is consistent with our dimensional analysis. However, followinginvasion and rupturing of the dome, the glycerin solutions propagate in dykelike fractures that are thicker than those for the less viscous water and whichslump towards the bottom of the invasion column (presumably under theinfluence of gravity).• Our experiments with immiscible fluids involved a density matched siliconoil (R550) and air. It was immediately apparent that interfacial tension had a177large effect on invasion. Our experimental protocol was changed to increasethe invasion pressure in larger steps than for the miscible fluids. Mixing waseliminated as was the occurrence of an initial micro-invasion dome. Instead,once invasion started we observe a clear interface expanding into the Car-bopol column. In the case of the R550 the interface evolved as a hemispher-ical dome, slowly filling the bottom of the column. The invasion pressurewas significantly larger than for the glycerin solutions and was non-local inthat it was resisted by yielding at the walls of the column, with Pi increasingas the Poiseuille flow ∼ 2H, (see Fig. 5.10).• The air also did not mix and had no micro-invasion dome. However, the in-vasion was localised in that buoyancy dominated after any significant influx,leading to the invading fluid stretching upwards into a long bubble, whicheventually detached and propagated to the surface. Thus, resistance of inva-sion at the walls did not occur and Pi was approximately constant with H. Atypical sequence of bubble formation in Carbopol is shown in Fig. 5.11. AlsoPi was much larger for air than for R550. Further analysis of this increasedPi suggested that about 50% of Pi is directly attributable to interfacial tensionand the remainder to yield stress. However, the yield stress is only able toresist effectively because there is no mixing/difusion.6.2.2 Industrial implicationsRegarding the cementing process, a few aspects of our results are worth remarkingon, although we do not have specific industrial recommendations.• The complex behaviour of Pi with H and secondly the change from localto non-local invasion needs to be interpreted in the wellbore context. Evi-dently here we neglect effects of any filtercake that may have formed duringdrilling and so consider a clean porous wall. Although H is very large inthe wellbore setting, pore size is also very small. Taking a typical annulargap as the global scale, pore sizes in the sub-micron range are very likelyto invade locally (rh < 0.0001 even for large H). This will also depend in asignificant way on the porosity, which could also be linked to rh for a fixed178column radius. At large porosity values, individual pores can no-longer beconsidered as isolated: local invasion domes influence adjacent pores. Ourcomputed results suggest that in the cylindrical geometry this happens forrh ' 0.1.• A different perspective on local/non-local yielding comes from the obser-vations of smooth and granular surfaces in our transition domes. Here thescale of the granularity appears to be in the range of 1−100 µm, which maybe related to the Carbopol gel microstructure and its stress response on thisscale. In the well, cement slurries are in reality fine colloidal suspensionsand pore sizes in tight rocks extend down to the scale of the suspension mi-crostructure, (indeed in such rocks it is not uncommon to consider Knudseneffects on porous media flows). Thus, local invasion on the scale of the slurrymicrostructure is likely to be the norm in cementing. This requires separatestudy using real cements.• The influence of the yield stress in linearly increasing invasion over-pressureis useful, if also anticipated. If the invasion is local then the critical point isthat it may occur at significantly lower pressures than those predicted fromnon-local analyses, e.g. predicting flow in the annular gap. If however yield-ing is local, with recirculation of fluids close to the pore opening, manyother effects may influence the potential and mechanism for annular fluid toexchange with the pore fluid, and the timescale for that exchange, e.g. buoy-ancy, capillarity, diffusion, osmotic pressure etc. These effects have not beenstudied here.• In the same context, note that here we have tried to eliminate local variabilityin our experiments, but in any cement placement process (and in the drillingprocess) the wellbore is over-pressured. Thus, yield stress fluid is forcedinto the pores, filtercake/skin typically forms close to the borehole, and theseeffects may in practical situations be the main influence of the yield stresson the actual invasion stage.• Comparing different invasion fluids, we see that invasion of miscible fluids isleast affected by either the yield stress or the height of the column, e.g. wa-179ter/brine invasion. The invasion process is strongly localised and retardedonly by an increased viscosity.• Interfacial tension has a significant role to play in conjunction with the yieldstress. The interfacial tension prevents local diffusion and weakening ofthe gel structure, with subsequent invasion on this scale. This allows theyield stress to resist on macroscopic scale. Thus, for example we mightexpect additives that increase the yield stress of the slurry to be effective inpreventing oil invasion.• The principal difference between air and R550 is that air is dominated bybuoyancy: the initial invasion is similar and dome like, but the air injectionmigrated upwards as the size grew, which prevented the dome from growingoutwards to the wall. We might expect a similar effect in a well. Thus,although the gas invasion pressure may be significantly higher than that forwater or oil, it seems to be largely independent of H: once entered into theannulus reservoir gas will channel upwards through the cement slurry, ratherthan displace it.We suggest that effective cement additives for gas invasion would need toforce invading gas pockets to expand locally and displace the entire annularcross-section. In this way rheological contributions to Pi that scale with Hcan be realised. For example an additive that quickly develops viscoelasticproperties in the liquid phase of the cement (i.e. downhole bubblegum), islikely to be more effective at preventing invasion than a pure yield stresseffect alone.6.3 Thesis limitationsIn the first part of the thesis, the main restrictions have been the usual ones ofusing a simplified model. We list some of these limitations, which may also beconsidered as possible extensions.• Only vertical orientations considered.180• More rheological complexity can be introduced through consideration of twoHerschel-Bulkley fluids.• Some fluid combinations may be immiscible.• The two walls of the annulus are quite different and it could be interesting tolook at effects of a porous wall, possibly with influxes/losses.• In the case of a poorly conditioned mud, we might expect a gradient in yieldstress and viscosity - increasing towards the formation.• Three-dimensional effects will be present in at least some parts of an annu-lus.• For small diameter production casings the annulus is not particularly narrowand using a channel as a model might become invalid geometrically.In the second part of the thesis, limitations are not apparent in terms of whatwe have done. As with most lab experiments, it is relatively far from the actualinvasion of gas through a porous rock, but the simplification is what allows us tovisualize the invasion well. Obvious limitations are as follows.• The hole size is very large compared to a typical pore, so that real capillaryeffects are much larger than those in our experiment. We have not varied thehole size or shape.• Although buoyancy should not affect the actual invasion, it might be per-ceived as being more realistic to have the invasion hole on a sidewall• Having seen the complex stages of the invasion for miscible fluids, it is ap-parent that we are trying to measure the pressure at which elastic-plasticyielding occurs, which is an inherently transitional event. This may accountpartly for the large error bars in our invasion pressures. For the miscibleinvasion experiments in general we might question if there is a better wayto identify the point of invasion, or some other variation in procedure thatwould result in less experimental variability.181• Similarly, we have seen that miscible invasion has an important mixing stage.Our complex experiment is not ideal for studying this: instead a simple setupwith 2 fluids separated by a horizontal interface and emphasis on microscopyand visualization at the µm scale would be better.• Lastly, in the setup we have we could think of improvements to try to mea-sure strain in the Carbopol and eventually capture the yield surface as theinvasion progresses.6.4 Future directionsThe two projects have been quite different in scope, but both are research projectstargeted at development of fundamental physical understanding of underlying gasmigration processes.Some of the interesting areas for potential continuation of project 1 include thefollowing.1. Additional research to understand the temporal aspects of wall layer drainage,i.e. moving layers, and how this impacts displacement at different depths inthe cemented interval. For example, lower in the well the displacement flowlasts for significantly longer than further up near the top of the cementedinterval2. Consideration of how to integrate the results with existing models and simu-lations of the wellbore. In particular, at UBC and in a number of companies,there are annular displacement simulators which model the entire annulardisplacement of primary cementing. These models (e.g. [9]) calculate wallshear stresses throughout the well, but lose resolution in only dealing withgap-averaged quantities rather than variations across the gap, as we havestudied in the first part of this thesis. It would be of value to combine theseapproaches.3. It would be interesting to study inclinations other than horizontal and verti-cal as these occur in the well. However, it is less clear how these simplify to182a plane channel flow, as we have studied here. In particular, with density dif-ferences between fluids we should expect that there will be azimuthal flowsclose to the interface in any inclined annular displacement flow.4. It would be of interest to run lab-scale experiments directed at the formationof micro-annuli (static wall layers), to see if they are as robust as the modelssuggest. This could be done in an annular flow loop, which is available atUBC. Alternatively, a simpler 2D version could involve am inclinable Hele-Shaw cell apparatus.5. It could also be of value to extend the modelling studies to more complexfluids, e.g. two Herschel-Bulkley fluids, but this would be a significant gainin the number of dimensionless parameters to study. More interesting mightbe to include immiscibility/wetting effects within the models (relevant forsome cementing combinations).6. Finally, many of the hydrodynamic instabilities observed in the density un-stable flows have not been systematically studied for yield stress fluids. Thiscould be tackled using both the computational tools here and classical toolsfor stability analysis.In terms of interesting future research areas for continuation of the invasionstudies, we can consider the following.1. Having understood the importance of local invasion and miscibility using apolymer hydrogel (Carbopol), it would be interesting to see how well theseconclusions carry over to cement slurry invasion, i.e. a colloidal yield stresssuspension. The difficulty here is that we have relied heavily on visualiza-tion methods in our experiments, which will be ineffective in a standard ce-ment. This requires some thought and test experiments before moving ahead.A possibility could be to try laponite (transparent, clay-based, thixotropic),then possibly some newer transparent concretes (cement with fine transpar-ent aggregates), and finally cement slurries.2. It would be interesting to study the fracture/fingering stage observed in thelast stage of miscible fluids invasion. This problem can be tackled by con-183ducting a series of experiments in order to determine the rate of growth andthe maximum height that a fracture plane reaches (Arrest stage) and the ef-fect of rheology of the fluid on them.3. It would be of value to experiment with fluid rheologies, to see if we caninfluence gas invasion to become non-local, i.e. using a visco-elasto-plastic.This type of research could be experimental, testing different materials, butalso could be computational, i.e. identifying properties that could then betargeted in materials development.4. Our experiments to date have firstly, not considered variations in hole size.Secondly, we have operated at a relatively large-scale. It would be of valueto develop an apparatus that uses real core samples, or as an intermediatestage uses manufactured arrays of pores, so as to study invasion of differentfluid streams and how they interact.5. One interesting aspect of the computations performed was to see the varia-tion from local invasion to Poiseuille-like flow, as the hole radius increases.There is an interesting connection here between the two parts of the thesis,i.e. as rh→ 1, the invasion flows just become displacement flows in a pipe.This would be interesting to explore, both experimentally and computation-ally.6. Further up in the annulus, supposing that gas invasion results in a large gasstream/bubble, it would be interesting to study propagation along an annuluson the scale of the annular gap, e.g. at different inclinations. 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Pressure-driven flow of a suspension:Buoyancy effects. Int. J. Multiphase Flow, 24(1):105–130, 1998. → page239[210] J. F. Morris and F. Boulay. Curvilinear flows of noncolloidal suspensions:The role of normal stresses. J. Rheol., 43(5):1213–1237, 1999. → page 239203Appendix AA lubrication displacement modelHere we present more detail of the lubrication model used in Section 2.3.2. Thestarting point is the assumption that fluids 1 and 2 are separated by an interfacey = ±yi(x, t) that evolves in time and space. This is the infinite Pe limit of theNavier-Stokes model used in our 2D simulations, in which the concentration isreplaced by an interface separating the fluids. We assume the flow to be symmetricabout the channel centre and that the interface stretches along the x-axis into a long-thin flow, i.e. both the streamlines and interface have small aspect ratio δ  1; seeFig. A.1. With this assumption, standard thin-film/lubrication scalings lead to thefollowing system of equations:δ (1+At)ReDDtu = −∂ p∂x+∂∂yτ1,xy− ReFr2 +O(δ2), (A.1)δ (1−At)Re DDtu = −∂ p∂x+∂∂yτ2,xy+ReFr2+O(δ 2), (A.2)δ 3[1+φ(c)At]ReDDtv = −∂ p∂y+O(δ 2), (A.3)∂u∂x+∂v∂y= 0. (A.4)The velocity and tractions are continuous across the interface. Now we proceedto an axial two-layer model in the limit that δ → 0 with Re fixed, also assuming2040.5Newtonian  fluid  1Viscoplastic fluid  20hyˆi(xˆ, tˆ)xˆyˆgˆFigure A.1: Schematic of displacement geometry. Fluids 1 and 2 are sepa-rated by an interface y = yi(x, t).symmetry so that we may consider only half of the channel:ddyτ1,xy = − f , y ∈ [0,yi), (A.5)ddyτ2,xy = − f −χ∗, y ∈ (yi,0.5] (A.6)where f and χ∗ are the modified pressure gradient and the buoyancy number, re-spectively:χ∗ ≡ 2ReFr2, f ≡−∂ p∂x− ReFr2. (A.7)Here yi denotes the interface position. Boundary and interface conditions are:τ1,xy(0) = 0, (A.8)τ1,xy(yi) = τ2,xy(yi), (A.9)U(y+i ) = U(y−i ), (A.10)U(1/2) = 0. (A.11)Finally, the following flow constraint is satisfied:∫ 1/20Udy = 1/2. (A.12)As the displaced fluid is a non-Newtonian, the above system of equations isnonlinear. For completeness sake, we solve this system for the general case of205two Herschel-Bulkley fluids, at fixed χ∗, fixed yi and fixed rheological parameters.First consider f to be fixed. The shear stresses are equal at yi and are linear in y ineach layer. Thus, the shear stresses in each layer are expressed as a function of thewall shear stress τw and interfacial stress τi:τ1,xy =yyiτi, (A.13)τ2,xy =y− yi1/2− yi τw+1/2− y1/2− yi τi, (A.14)and we note thatτw =− f2 −χ∗(12− yi), τi =− f yi,so that f determines both τw and τi.The velocity in each fluid layer is obtained by using the constitutive laws ofeach fluid and integrating from the wall inwards:U2(y; f ,yi,χ∗) =∫ y1/2dUdydy =− 1m2J2,m2(y; f ,yi,χ∗) (A.15)U1(y; f ,yi,χ∗) = Ui+∫ yyidUdydy =Ui− 1m1 J1,m1(y; f ,yi,χ∗) (A.16)where Ui =U2(yi; f ,yi,χ∗). The functions Jk,m above are borrowed from [119] anddefined in §A.0.1. They may be evaluated algebraically and also integrated to givealgebraic expressions. Integrating the velocity, the flowrates in each layer are givenbyq1( f ,yi,χ∗) = Uiyi− 1m1∫ yi0J1,m1(y; f ,yi,χ∗) dy, (A.17)q2( f ,yi,χ∗) = − 1m2∫ 1/2yiJ2,m2(y; f ,yi,χ∗) dy. (A.18)These expressions are valid for 2 Herschel-Bulkley fluids, but here we set mk =1/nk = 1 for both displaced and displacing fluids, B2 = BN , B1 = 0, κ2 =m, κ1 = 1.The total flow rate is simply q1( f ,yi,χ∗)+ q2( f ,yi,χ∗), which can be shown toincrease monotonically with f . We iteratively solve for f such that the constraint(A.12) is satisfied. This then determines all other flow variables in the solution.206The main point of the velocity solution above is that u depends on (x, t) onlyvia the interface position yi(x, t), which itself satisfies the kinematic equation:∂yi∂ t+u∂yi∂x= v (A.19)The kinematic equation combines with the divergence free condition to give theinterface propagation equation (2.20).A.0.1 The functions Ik,p & Jk,pWe suppose that fluid 1 & 2 are Herschel-Bulkley fluids with scaled consistency,power law index and scaled yield stress κk, nk and Bk, respectively (k = 1,2), andset mk = 1/nk ≥ 1. The shear stress in each layer is defined linearly in terms of yand the end stress values of each layer [0,τi] and [τi,τw], respectively; see (A.13)& (A.14). Thus, the shear stresses depend on ( f ,yi,χ∗). The velocity gradient influid k is:dUdy(y) = sgn(τk,xy)(|τk,xy|(y)−Bk)mk+κmkk, (A.20)where (w)+ denotes the positive part of the function w.We now define the following integral expressions:I1,p(y; f ,yi,χ∗) = m1∫ y0(|τ1,xy|(y˜)−B1)p+κm1Hdy˜, (A.21)J1,p(y; f ,yi,χ∗) = m1∫ y0sgn(τ1,xy)(|τ1,xy|(y˜)−B1)p+κm11dy˜, (A.22)I2,p(y; f ,yi,χ∗) = m2∫ 1/2y(|τ2,xy|(y˜)−B2)p+κm22dy˜, (A.23)J2,p(y; f ,yi,χ∗) = m2∫ 1/2ysgn(τ2,xy)(|τ2,xy|(y˜)−B2)p+κm22dy˜. (A.24)These functions can be evaluated directly for p>−1. The method involves changeof the independent variable from y to τk,xy after which the integrals are evaluated interms of the end-point stresses in each layer.207The relevant algebraic expressions are:I1,p(y; f ,yi,χ∗) =yim1sgn(τ1,xy(y))(|τ1,xy|(y)−B1)p+1+(p+1)κm11 τi(A.25)J1,p(y; f ,yi,χ∗) =yim1(|τ1,xy|(y)−B1)p+1+(p+1)κm11 τi(A.26)I2,p(y; f ,yi,χ∗) =(1/2− yi)m2[sgn(τ2,xy(y))(|τ2,xy|(y)−B2)p+1+(p+1)κm22 (τi− τw)−sgn(τw)(|τw|−B2)p+1+ ](p+1)κm22 (τi− τw)(A.27)J2,p(y; f ,yi,χ∗) =(1/2− yi)m2[(|τ2,xy|(y)−B2)p+1+ − (|τw|−B2)p+1+ ](p+1)κm22 (τi− τw)(A.28)We integrate the functions Jk,p(y; f ,yi,χ∗) in order to evaluate the flow rates ineach layer:∫ yi0J1,p(y; f ,yi,χ∗) dy =yiI1,p+1(yi; f ,yi,χ∗)(p+1)τi, (A.29)∫ 1/2yiJ2,p(y; f ,yi,χ∗) dy =(1/2− yi)I2,p+1(yi; f ,yi,χ∗)(p+1)(τi− τw)−(1/2− yi)2m2(|τw|−B2)p+1+(p+1)κm22 (τi− τw). (A.30)Further details of these functions can be found in [119], and note that the expres-sions given here are simplified in the fluid 1 layer since τ1,xy = 0 at y = 0 fromsymmetry.208Appendix BFlow regime panoramasIn Chapter 3, we explored pictorially the different displacement flow regimes inthe (Re,χ) plane. Flow regimes for m = 0.1, 1, 10 are illustrated in Figs. 3.7-3.8,3.9-3.10,and 3.11-3.12, respectively. Each figure plots the displacement sequencesand classifications within the range Re ∈ [1,100]. Here, to give a complete picturewe present the flow regimes for m = 0.33, 3.As discussed in Chapter 3, for lower χ the flows are largely stable, but still wesee two distinct front types: dispersive and frontal shocks (similar to those foundin 2.3.1). For m ≤ 1 the instabilities are mainly of KH type, sometimes with RTfrontal instabilities (larger χ and significant Re). Flow regimes for m = 0.33 andBN = 1,5 are plotted in Figs. B.1 & B.2, respectively.For m > 1 the instabilities are largely of IBM type, with some RT at the front.The unstable flows occur with increasing Re at the larger values of BN . Figs. B.3& B.4 show the flow regimes for m = 3 and BN = 1,5, respectively.For BN = 10 there is very little instability, except at larger Re. Increasing BNtends to stabilize all the flows. In terms of stability classification we see littlequalitative difference between BN = 5, 10 (hence show only the results for BN = 5).209100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure B.1: Panorama of flow types observed for m = 0.33 and BN = 1. Markers indicate data position in (Re,χ) plane and flow classification: - stable,4 - KH,© - IBM; filled symbols - RT.210100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103``Figure B.2: Panorama of flow types observed for m = 0.33 and BN = 5. Markers as in Fig. B.1.211100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure B.3: Panorama of flow types observed for m = 3 and BN = 1. Markers as in Fig. B.1.212100 10110-110010110210320 30 40 50 60 70 80 90 10010-1100101102103Figure B.4: Panorama of flow types observed for m = 3 and BN = 5. Markers as in Fig. B.1.213Appendix CDevelopment of the experimentalsetupIt is worth commenting that the apparatus used and described in Chapter 4 & Chap-ter 5, was only arrived at by a process of continual improvement/refinement to en-sure repeatability. It is helpful to understand some of the choices and false directiontaken, i.e. to avoid the same mistakes for other researchers.• Our initial thought was to simply use an air line with pressure regulator (seeFig. C.1.a). Fine control of air pressure requires expensive medical-qualityregulators to attain 1 Pa scale precision, and most of the control is wastedon balancing atmospheric pressure. The manometer design is easier.• There are often debates about measurement of yield stress, i.e. how andto what precision? To minimize variations we kept to the same consistentpreparation of fluids throughout and fixed at just 3 concentrations (yieldstress values). Variability of the experiment is better controlled by chang-ing the height H of yield stress fluid and by varying the pressure imbalancethrough the reservoir height (see Fig. C.1.c).• We finally used an automated micro-meter scissor jack to lift the water col-umn slowly and precisely (see Fig. C.1.d). By this, the resolution of ourheight control system, in our final setup, is 0.1µm, which is equivalent to214a hydrostatic pressure of ≈ 0.001Pa. Moreover, this device enabled us toincrease the applied pressure in a similar/repeatable way in all of the experi-ments.• Although hole size is an interesting variable, we kept this constant for all ex-periments. For immiscible fluids, with a small hole capillary effects quicklybecome dominant. Also there were effects to avoid in the experiment, suchas blocking the hole with yield stress fluid.• Due to questions about potential slip at the walls, we applied a special coat-ing to the plexiglass container to avoid this. It was probably unecessary, aswe never saw any tendency for the invading fluids to migrate along the wall.• Initial experiments used ink instead of flourescent dye. The latter exposedmore features of the flow close to the invasion hole, but the ink was nice forvisualizing the post-invasion propagation (e.g see Fig. C.2).• We experimented initially with rectangular columns for the yield stress fluidcolumn, then when we found that the post-invasion propagation of the in-vading liquid was quite asymmetric, we decided to use a symmetric circularcylinder.• After the invasion occurs, the applied pressure drops since the height of in-vading fluid in the reservoir decreases as the fluid moves into the column ofthe invaded fluid. We increased the size of the reservoir to practically removethis effect and keep the applied pressure constant at the invasion hole.215i.Air is injected into carbopol. Injection pressure is controlled with pressure regulator.ii. Controlling injection pressure with manometer.iii. Invasion pressure is controlled by making difference in fluids level. The height difference is applied with lifting up the column of water by a scissor jack under the water tank.iv. Change the size of water tank and increasing the length ratio of the tank to 10.Laser Water-Fluorescent tank Laser level Carbopol- tube Carbopol container  Motorized lab jack Connector tube Monitoring Labjack from the PC c) d)b)a)Figure C.1: Development of experimental setup: a) Initially, a pressure regu-lator was used to inject air into Carbopol. b) The pressure regulator wasreplaced by a manometer setup. Pressure was applied by adding extrawater to its container to make difference between the fluid levels. c) Ascissor jack used to adjust the water column height. d) The apparatus isequipped with an automated scissor jack.216Figure C.2: The preliminary results obtained by injecting water (dyed with ablack ink) into Carbopol.217Appendix DPressure reduction in hydratingcement slurriesIn the problem of gas migration a key question is, where does the driving pressureunderbalance come from? In general, well operations preserve primary control ofthe well through control of fluid densities, i.e. wells are overbalanced, Yet, at leastearly stage gas migration cannot occur unless this is reversed. At least two theoriesare common.• One simple theory that has been proposed for gas invasion is as follows.Hydration leads to shrinkage, so that the slurry settles downwards in thewell. This downwards settlement is resisted by the yield stress of the fluid,hence reducing the static pressure in the cement column and allowing theformation (gas) pressure to exceed that in the annulus, leading to invasion.• A second more detailed theory is as follows. As the solid phase increases involume fraction through hydration, the cement suspension attains a percola-tion threshold where particles begin to join mechanically in a non-hydrodynamicway. As the solid fraction increases further, chemico-mechanical contactsbetween grains develop, with the remaining liquid slurry occupying inter-granular space in the developing porous media. In this case, can the solidphase become self-supporting, reducing the ability of the liquid phase to re-sist invasion from the gas within the formation?218To explore these concepts requires a suitably complex level of model, which weoutline in this appendix. For the first theory, we need macroscopic estimates of theevolution with time of both shrinkage and yield stress. For the second theory, amuch more detailed description must be developed.The work in this appendix is incomplete. It represents part review and partlyour attempts to put together a pragmatic but complete model of this complex pro-cess. We aim for a continuum level description, that bridges between models fornon-Newtonian suspensions and porous media flow, capturing the mechanics of thetransition between these. A pragmatic level of description of the cement hydrationchemistry must be included as this is responsible for changes in phase and the con-sequent transition from a slurry to porous solid. We aim to apply this model to along channel (section of the annulus) in which the cement slurry is placed at theend of pumping.Below in Section D.1 we give a targeted overview of the literature for mod-elling cement hydration, i.e. selective in being helpful building blocks towards theabove aims. In Section D.2 we give an overview of cement chemistry as relevantto the earlier stages of hydration: the induction period. On the macro-scale, thecement slurry in the early stages is often described rheologically as a Herschel-Bulkley fluid, i.e. shear-thinning yield stress fluid. As the solid phase grows, theremay be a transition from yield stress fluid to yield stress suspension, in which thesolid phase may settle. The constitutive law of the suspension are now a functionof its solid volume fraction (as well as potential compositional changes). Appropri-ate models for a Herschel-Bulkley fluid suspension are reviewed in Section D.4.1.Then, we derive the governing equations of the motion of a yield stress suspensionin Section D.4. We close this appendix with a discussion of the remaining parts tobe completed in Section D.5.D.1 IntroductionThe cement slurry is initially a fine colloidal suspension that changes on the micro-scale due to ongoing hydration reactions. As the solid phase increases in volumefraction, it attains a percolation threshold where particles (due to chemical bonds)begin to join mechanically in a non-hydrodynamic way. As the solid fraction in-219creases further, chemico-mechanical contacts between grains develop, with the re-maining liquid occupying the inter-granular space in the developing porous media.Therefore, we need a macroscopic framework for modeling the evolution of solidand liquid phase constituents during hydration, as they will then contribute to theevolution of stresses via constitutive relations.Thermo-mechanical aging of the concrete/cement during hydration has beenthe focus of many studies aiming to make a quantitative prediction of the cementstructure under thermal boundary conditions [167–172]. The framework of thesestudies is based on the model developed by Coussy [173] (known as Biot-Coussytheory) for a porous media saturated with pore water. In a nutshell, this model isbased on the macroscopic thermodynamics of porous continua in which energy isdissipated via three physical processes: thermo- and fluid mechanical work andbulk dissipation. The hydration reactions contribute to the bulk dissipation via anoverall property of the cement which is called “maturity” or “degree of hydration”.The porosity change was initially missing in these models, e.g. [167, 168], but laterit was improved by Gawin et al. [169, 170], Pesavento et al. [171] and Lecampion[172] by including its effect on the thermal stresses, both implicitly and explicitly.The explicit change was brought by applying the linear evolution of the poros-ity with the hydration degree [174, 175]. One of the earlier works modelling thethermo-mechanical aging of the cement is the study done by van Breugel [176].He developed an in-house software using the empirical relations obtained by ear-lier experiments to predict the quality of the cement (i.e. both morphology andstructure).As mentioned above, these studies have simplified the chemical reactions ofcement hydration and instead used an overall kinetic expression to describe thestructure of the cement by using a closure equation for the degree of hydration.There are several such models to predict the degree of hydration, which is thefraction of cement that has fully reacted with water. Some of these are as follows.1. Hydration reactions have been observed to follow a S-shaped reaction curve.Hence we can apply the Prout-Tompkins equation for estimating the degreeof reactions:ln(Γ1−Γ) = kt+C (D.1)220where Γ is degree of hydration, t is time, k & C are constants [177]. TheProut-Tompkins model is more closely related to autocatalytic reactions thannucleation-growth reactions with overlapping volumes [178].2. Degree of hydration can be calculated from the ratio of the heat releasedup to a specific time (t) to the total heat that would be released when thehydration is complete [169, 179].Γ=Qhyd,tQhyd(D.2)3. Avrami [180] divided the hydration kinetics into three stages: (1) an induc-tion period at early times, (2) a nucleation and growth period at intermediatetimes, and (3) a diffusion-controlled period at later times and proposed thefollowing model:− ln(1− (Γ−Γ0) = [k(t− t0)]m (D.3)where Γ is the degree of reaction at time t and k is a rate constant for anucleation-controlled process. The constants Γ0 and t0 define the degree ofreaction and time at which the nucleation and growth kinetic regime begins.The exponent m = [(p/s)+q], where: p = 1 for 1D growth (needles/fibers),p = 2 for 2D (sheets/plates), p = 3 for 3D isotropic growth (sphere); s =1 for interface or phase-boundary-controlled growth, s = 2 for diffusion-controlled growth; q = 0 for no nucleation (nucleation saturation), q = 1 forcontinuous nucleation at a constant rate.Next, as we are interested in structural change that starts when the cementslurry is still a suspension, and consequently fluid stresses and viscous dissipationwithin the fluid phase are important, we need to relate degree of hydration to thesolid-fraction. The physical description is too simplified if we use a linear relationbetween the degree of hydration (obtained from one of the above models) to thesolids fraction in the cement slurry, for two reasons:• The effect of chemical retarders is neglected.221• In a vertical geometry as the hydrostatic pressure varies by depth, the effectof porosity/solids becomes more pronounced, i.e. through settling and theeffects compressive stresses. Therefore, it is advisable to include the effectof the potential heterogeneity of the cement structure.There are a large number of papers in the scientific and engineering literaturethat have tried to mathematically model the chemical reactions and species trans-port within setting cement. One of the first models was derived by Billingham andCoveney [181], Preece et al. [182], to mathematically describe the induction pe-riod by using a clock reaction model. Preece et al. [183] applied this model andby including the major component of cement, tricalcium silicate, they evaluatedthe length of the induction period and the species concentrations at the end of thisstage. Later the authors have added more cement ingredients to their model, in-cluding gypsum and another chemical retarder; see [184]. Here they have solveda system of advection-diffusion equations to evaluate the thickening time in thepresence of a chemical retarder.D.2 An introduction to the chemistry of the CementThe main components of the cement are: C3S (tricalcium silicate or aelite), C2S (di-calcium silicate or belite), C3A (Aluminate) and C4AF (Ferite). These componentsreact at different rates and play different roles in five stages of the cement hydration[5]: (I) Preinduction period, (II) Induction period, (III) Acceleration period, (IV)Deceleration period, and (V) Diffusion period.At the first step, in the so-called pre-induction stage, C3S and C3A dissolve andrelease Ca2+ and OH−1 ions into the mixture water. After dissolution, the mixedwater is no longer pure H2O and the concentration of ionic species increases rapidlyin the mixture water. A layer of hydration products forms around the cement parti-cles in a less than one minute and separates them from the pore solution. This stagecan be influenced by different many additives, to control the length of this stage anddelay the “initial set”. Therefore, the rheology of the cement does not change sig-nificantly. Also in the operational context, pre-induction will occur locally duringthe pumping phase. Following this stage, a period of low activity starts whichinvolves mostly reactions of the silicate phases, the so-called induction period or222stage (II).The transition between stages I & II, from a fast activity to a low-activity, hasbeen explained via three theories:1. Formation of a hydrate layer around the particles which acts as a diffusionbarrier.2. Formation of an electrical double-layer which intercepts the passage of ionsinto the solution.3. Supersaturation of the solution with respect to CH.In reality all 3 processes may combine together in slowing hydration, i.e. the in-duction phase. The first and second theories demand microscopic estimates ofmorphological evolution of the cement particles, which is beyond the focus of ourstudy. We will assume that the induction period starts on the basis of the thirdtheory, and later can consider ways to modify to account for either of the first twotheories.According to the CH theory, the reduction in the rate of reactions is causedby the formation of CH gel which covers the surface of the grains and limits theaccess of C3S to the water, consequently the reactions become slow. Once the con-centration of the CHg reaches the supersaturation level, it precipitates as CHc andC3S starts to react rapidly again which ends the induction period. For operationalreasons, gypsum (a common retarder) is usually added at first to the cement tolengthen the low inactivity period and prevent early gelation. Gypsum works byreleasing some silicate ions into the mixture, which poison CH nuclei and form CHgel.By the end of the induction period, nucleation and growth starts. This periodis divided into two stages: acceleration and deceleration. The reaction rate in thesestages is typically dependent on the temperature and volume fraction. It increasesrapidly up to a maximum (and remains at it for less than 24 hours) and then de-creases to less than half of its maximum and transitions more gradually into thedeceleration period. During nucleation and growth, the nuclei of C-S-H and CHprecipitate and accumulate in the pore solution. CH crystallizes from the solu-tion, CSH fills the pore water spaces and crystalline ettringites continue to grow223Aluminate*phaseCalcium0Silicate*phaseRetarder*(Gypsum)DeccelerationAccelerationHydration*shellEttringite rods*Pre9induction Induction<Outer*C0S0HInner*C0S0H(I)< (II)< (III)< (IV)<Initial<set Final<setFigure D.1: Schematic of cement undergoing hydration process including: I)Pre-induction, II) Induction, III) Acceleration, IV) Deceleration stages.as needle shaped solid structures. All this leads to the development of a cohe-sive network i.e. a porous structure, where the solid volume fraction reaches thepercolation threshold, φ = φm, signalled by an increase in the cement gel strength.At the end of the nucleation and growth stage, the cement structure consistsof a small internal pore space filled with pore solution (the remaining mixture wa-ter), and unreacted cores of the cement particles enclosed by a continuous layerof the hydration product. The larger pores are called capillary pores. Finally, hy-dration completes by outward diffusion of the dissolved ions of the cement andtheir precipitation into the capillary pores, or by inward diffusion of water to reachthe unreacted cement cores. As the layer of hydration product around the cementcores gets thicker, this process slows and becomes diffusion dominated [167, 168].A schematic of hydration process and structure evolution is depicted in Fig.D.1D.3 Chemical kineticsAs explained in Section D.2, after the cement has been in contact with water fora few minutes, a series of fast grain boundary dissolution reactions occur. As aresult, the Ca2+ and OH− ions are released into the solution and the first-stage ofCSH, which is a gel layer, surrounds the cement grains. The reactions then slowdown and hydration continues slowly during the induction period (II) which is afew hours long. After the induction period, reactions resume at a higher rate upuntil the porous structure of the cement forms. This induction period is, in fact, a224transition stage after which slurry begins to thicken and ultimately converts into aporous medium. In this section, we outline a prospective model, which includesthe kinetics of the cement during this transition.The silicate phase, C3S, together with the aluminate phase, C3A , which isimportant at the beginning of cement setting and early strength development, com-prise the largest part of the cement slurry. C3A is the most reactive ingredient andmay lead to the unfavourable early setting of the cement. Therefore, Calcium sul-fate or gypsum, CaSO4, is added initially to it to prevent “flash set”. We include allof these species and assume that the solid phase is composed of those three maincomponents: C3S,C3A and CaSO4. The reactions start with the following rapidboundary reactions:(1) C3S+3H2O→ 3Ca2++4OH−1+H2SiO2−4 rˆ1 = kˆ1Cˆ31 ˆ¯C1(2CˆCH,sat −Cˆ12)(2) C3A+6H2O→ 3Ca2++4OH−1+2Al(OH)−14 rˆ2 = kˆ2Cˆ61 ˆ¯C2(1−ˆ¯C4ˆ¯C4,sat)(3) CaSO4→Ca2++SO2−4 rˆ3 = kˆ3 ˆ¯C3and the following bulk reactions happen:(4) H2SiO2−4 +32Ca2++OH−+H2O→ (1− ε)CSHg+ εCSHc rˆ4 = kˆ4Cˆ1Cˆ6Cˆ3/25 Cˆ7(5) CSHg+CSHc→ 2CSHc rˆ5 = kˆ5(Tˆ )Cˆ11 ˆ¯C5where kx and rx are respectively the rate constant and reaction rate of reactionnumber x. Cˆi and ˆ¯C j is the concentration of species number i and j in the liquidand solid phase respectively. The properties of each species in solid phase is shownwith a ·¯ accent. We use a stoichiometric law for formulating the reaction rates. Therate of the reaction of C3S, r1, decreases as the concentration of calcium hydroxide,CHg, increases.As mentioned in Section D.2, silicate ions poison the CH nuclei to not precipi-tate until the liquid phase supersaturates with respect to the CH ions and the nucleibecome stable and precipitate. This can be explained mathematically by using a“clock reaction” mechanism. We model the induction period and the artificial in-duction, which is generated due to the presence of gypsum, [181–184] by using225this mechanism as follows:(6) Ca2++2OH−→ (1− ε)CHg+ εCHc rˆ6 = kˆ6Cˆ26 ˆ¯C6(7) CHc+CHg→ 2CHc rˆ7 = kˆ7(Tˆ )Cˆ12 ˆ¯C6(8) CHc+H2SiO2−4 →CH p rˆ8 = kˆ8Cˆ7 ˆ¯C5(2CˆCH,sat −Cˆ12)+In the clock reaction model, CHc ions are assumed to be “clock reactants”. Theirproduction is “inhibited” by the silicate ions until the liquid phase becomes super-saturated with respect to these ions. In a “clock reaction” the product is consumed,by an “inhibitor”, much faster than it is produced, up until the inhibitor is com-pletely consumed. As the concentration of CHg exceeds 2CCH,sat , or the gypsumdiminishes, the concentration of CHc dramatically increases due to the autocat-alytic reaction (7).According to the theory of “thermal explosion”, self-acceleration, or an auto-catalytic reaction, stems from temperature increase. Hydration reactions are gen-erally exothermic and the heat evolved during the course of the reaction raises thetemperature of the reacting mixtures, thereby accelerating the reaction rates, lead-ing to further heating and further reactions [185]. Therefore, we assume that therate constants of the autocatalytic reactions are a function of the temperature usingArrhenius equation [186]:kˆx(Tˆ ) = kˆx(Tˆ0)Ax e− EˆRˆ( 1Tˆ− 1Tˆ0)(D.4)where Ax, E, and R are a constant, the activation energy and the universal gasconstant, respectively, which are specific to each reaction. Tˆ is temperature and Tˆ0is the initial temperature. kˆx(Tˆ ) is the rate constant of the reaction number x at T .It is known that chemical retarders including a sulphate phase, add a so-calledartificial induction phase to the hydration process. This artificial induction periodwhich prevents the flash set in the cement can also be described by a clock reaction,226as follows:(9) 2Al(OH)−4 +6Ca2++4OH−+3SO2−4 +26H2O→ (1− ε)ettg+ εettc rˆ9 = kˆ9Cˆ261 Cˆ32Cˆ46Cˆ65Cˆ23( ˆ¯C3,0− ˆ¯C3)+(10) ettg+ ettc→ 2ettc rˆ10 = kˆ10(Tˆ ) ˆ¯C4Cˆ9(11) ettc+R→ ett p rˆ11 = kˆ11Cˆ4 ˆ¯C4(12) Ca+R→CaR rˆ12 = kˆ12Cˆ4Cˆ5when gypsum is over(13) 4Al(OH)−4 +6Ca2++8OH−+15H2O→C2AHc8 +C4AHc1rˆ13 = kˆ13Cˆ131 Cˆ6Cˆ65Cˆ43(ˆ¯C3− ˆ¯C3,0)+Here C3A is a fast reactant and quickly reacts with the calcium ions . They producecrystalline ettringite, needle shaped crystals, which cover the grain surface. As aresult, further reactions stop and the cement sets very fast. With the aid of sulphateions, liberated from the reaction of the gypsum reaction (3), the produced ettringiteis mostly is in gel form and deposits on the grain surface (C3A). Ettringite gel isimpermeable to water and reaction (2) cannot proceed rapidly until the inhibitor(which is a retarder, see (11)), is completely consumed and reaction (10) becomesavailable. Therefore, the C3A lasts longer and cement setting is delayed. Hereplease note that we have assumed that all of the chemical reactions are elementaryreactions. Hence we include those also which are believed to create intermediateproducts e.g ett p,ettg. ˆ¯C3,0 is the initial value of the gypsum. Thanks to the heavyside function in rˆ13 and rˆ9, once the retarder is over, reaction (13) substitutes forreaction (9). Reaction (13) mainly serves to signal the transition from the inductionperiod to the acceleration period.The rate of production and consumption of each species in solid and liquidphase can respectively be found using:d ˆ¯m jdtˆ= Mˆ jΣ8i Bi j rˆi j = 1,2, ..,8 (D.5)dmˆidtˆ= MˆiΣ13j Bi j rˆ j i = 1,2, ..,13 (D.6)where Mˆ j is the molar mass of each species and the Bi j are the stoichiometriccoefficients. By substituting the reaction rates and the stoichiometric coefficients,227i Ions in liquid phase Mass rate per molar mass1 H2O −3rˆ1−6rˆ2− rˆ4−26rˆ9−15rˆ132 SO2−4 −rˆ3−3rˆ93 Al(OH)−4 2rˆ2−2rˆ9−2rˆ134 R −rˆ11− rˆ125 Ca2+ 3rˆ1+3rˆ2+ rˆ3−3/2rˆ4− rˆ6−6rˆ9− rˆ12−6rˆ136 OH− 4rˆ1+4rˆ2− rˆ4−2rˆ6−4rˆ9−8rˆ137 H2SiO2−4 rˆ1− rˆ48 CaR rˆ129 ettg (1− ε)rˆ9− rˆ1010 ett p rˆ1111 CSHg (1− ε)rˆ4− rˆ512 CHg (1− ε)rˆ6− rˆ713 CH p rˆ8Table D.1: The index of the species and the rate of consumption/productionof each in liquid phase.j Ions in solid phase Mass rate per molar mass1 C3S −rˆ12 C3A −rˆ23 CaSO4 −rˆ34 ettc ε rˆ9+ rˆ10− rˆ115 CSHc ε rˆ4+ rˆ56 CHc ε rˆ6+2rˆ7− rˆ87 C2AHc8 rˆ138 C4AHc1 rˆ13Table D.2: The index of the species and the rate of consumption/productionof each in solid phase.the mass rate of each species (per its molar mass) is evaluated from the relationslisted in Tables D.1 & D.2.D.3.1 Continuum approachWe take a continuum approach and assume that there are N cells in the domain, andall of the cells initially contain an equal portion of solid and water phase: 1N (mˆw,0+228Vw,oVc,oVshVuwVgVucVhc VcVna) Initial situation b) During hydrationFigure D.2: Cement compartment before and during the hydration process[175].mˆc,0). To find out the volume of cement slurry occupied with voids or chemicalshrinkage while conserving mass of each phase, we assume that the compartmentsof each cell are comprised as follows (see [175]):Vˆt = Vˆuc+Vˆhc+Vˆg+Vˆuw+Vˆsh. (D.7)Here Vˆuc,Vˆhc,Vˆg,Vˆuw,Vˆsh are the volume fraction of untreated cement, hydrated ce-ment, gel space which may be filled with gel water, unreacted water and shrinkage.We have used the same approach for volume categorization and similar notations tothose introduced by Powers and Brownyard [187]. The breakdown of the cementduring hydration is illustrated in Fig. D.2.229Each of these volumes can be calculated usingVˆuw = mˆw,0/ρˆw− mˆg/ρˆg− mˆn/ρˆn volume of unreacted waterVˆhc = mˆu,c/ρˆc+ mˆn/ρˆn volume of hydration productsVˆg = mˆg/ρˆg volume of gel spaceVˆuc = mˆc,0/ρˆc,0− mˆu,c/ρˆc volume of unreacted cementVˆs = Vˆt −Vˆc−Vˆhc−Vˆg−Vˆuw volume of shrinkage of the system(D.8)where ρˆc and ρˆuc are the density of reacted and unreacted cement, ρˆn is density ofreacted water (products of the reaction which are in liquid phase), ρˆw is density ofwater, and ρˆg is density of adsorbed water.ρˆn =13∑i=1ρˆiCˆi, ρˆc =8∑j=1ρˆ j ˆ¯C j,Note that ρˆn and ρˆc are a function of (sˆ, tˆ).Here mˆc, mˆg, mˆn, mˆc,0, mˆw,0 are the mass of reacted cement, adsorbed water,reacted water, and initial mass of unreacted cement and water respectively. Thevariables mˆc, mˆn can be obtained by integrating the sum of mass of each phase, D.1& D.2, over time,ˆ˙mn =13∑i=1ˆ˙mi, ˆ˙mc =8∑j=1ˆ¯˙m jAdsorbed water mˆg is also called evaporable/gel water [175] which is the ac-cumulated water, either in the pores or adsorbed water on the solid surfaces. Theamount of this water is measured and reported in experimental studies [187, 188]and found out that it is a function of relative humidity and the amount of reactedcement. According to Powers and Brownyard [187] the pore space is saturatedwith four water layers covering the hydrated cement and it is at least covered withone layer of water. Finally, from comparing the fitted values from this predictionand experimental values, they found out that the amount of gel water is approxi-mately equal to the reacted /non-evaporable/ chemically combined water. We usethis approximation and assume that mˆg = mˆn.Note that depending on the amount of water in the system shrinkage poresmay/not be filled with water. If we assume that reactions happen under sealed230conditions, due to chemical shrinkage void spaces form in the structure. Otherwise,if extra water becomes provided to the system from boundaries, it fills the chemicalshrinkage volume.After we have calculated the amount of the mass of solid and liquid phase ateach time step, the solid volume fraction can be obtained viaφ =mˆcνˆc+ mˆucνˆucmˆc,0νˆc,0+ mˆw,0νˆw,0(D.9)where νˆ is the specific volume of each species [175].D.3.2 Governing EquationAs is discussed in the previous section, pre-induction is dominated by boundarydissolution and probably also takes place during cement mixing/placement. There-fore, the pre-induction stage is used mainly to provide initial conditions. We maysimplify by assuming that advection doesn’t play any role in the reactions. Induc-tion is a low activity period which is diffusion dominated, so we scale the transportequation with a diffusion time scale. The species in the liquid phase are transportedaccording to:D(Cˆi (1−φ))Dtˆ=−∇ˆ. jˆi+ ˆ˙mi/Mˆi︸ ︷︷ ︸fˆ (Cˆi)(D.10)(D.11)where jˆi is the concentration flux. For a binary mixture, Fick’s law reduces tojˆi =−Dˆi j∇Cˆi (D.12)We scale the concentration of the reactants with their initial values and productswith the maximum values that can be achieved when cement sets. These values aremeasured experimentally and reported in [187]. The diffusive time is the time scaleof the problem. By assuming the length scale of the problem at this stage is the231radius of the particles, Rˆi, we can find∂Ci∂ t+Pe∂Ci∂ r=1r∂∂ r(r2∂Ci∂ r)−Φ2i f (Ci) (D.13)wherePe =uˆRˆiDˆ, (D.14)Φ2i =fˆ (Cˆi)Rˆ2iDˆ. (D.15)Here Pe is Pe´clet number and Φ2i is the Thiele modulus, which describes the ratioof diffusion time scale to the reaction rates.For the solid phase, the constituent species are not affected by advective ordiffusive transport, so they obey a simple ODE:φ∂C j∂ t= ˙¯m j (D.16)Hence, we can calculate the concentration of each species in all of the cells. Oncethe solid volume fraction becomes high enough such that buoyancy forces on thetypical cement particles become significant, settling starts and we must can evolu-tion of the suspension using the model explained below in §D.4.D.4 Herschel-Bulkley extended suspension balancemodel (SBM)Here we advance a continuum model for the evolving slurry and its motions. Thisrequires a rheological model that links with the usual descriptions of the pumpedslurry (i.e. a Herschel-Bulkley fluid), and allows for a multi-phase description thatcan account for settling and dispersion effects.For the rheology, we adopt the recently developed a general framework forshear-thinning and yield stress fluid suspensions Chateau et al. [189], Coussot et al.[190], Mahaut et al. [191], Ovarlez et al. [192, 193], Vu et al. [194], which has been232validated at least partially. The bulk suspension viscosity ηˆ , is decomposed as:ηˆ = ηˆ fηr(φ), (D.17)where ηˆ f is referred to as the liquid phase viscosity and ηr(φ) is the dimensionlessrelative viscosity, modelled e.g. by the Krieger-Dougherty law:ηr(φ) =[1− φφm]−2.5φm, (D.18)or close variant; see [195–197]. Apart from conceptual simplicity, there exist gen-eralisations to particles of different shapes e.g. rods/fibres, see [198]. Here φm de-notes the maximal packing fraction. The relative viscosity is further decomposedinto ηr(φ) = ηp(φ)+ 1, where ηp(φ) is called the particle phase viscosity. Thecombination ηˆ fηp(φ) usually appears as part of the solids phase stress. Note thatηp(φ)∼ φ as φ → 0, and ηp(φ)→ ∞ as φ → φm.In the liquid phase, the particles act to amplify effects of a bulk shear rate im-posed on the suspension, i.e. since the particles themselves do not deform. Thisamplification can be crudely estimated as being related to the inter-particle separa-tion and has been modelled as:ˆ˙γloc =[ηr(φ)1−φ]1/2ˆ˙γ, (D.19)which is verified reasonably well by experimental results [191, 192] and agreeswith theoretical considerations, [189]. When the strain rate ˆ˙γ is imposed on thesuspension, ˆ˙γloc is felt by the liquid, and assuming a Herschel-Bulkley type closure:ηˆ f ( ˆ˙γ,φ) = ηˆ f ,0( ˆ˙γloc(φ)) = κˆ0[ηr(φ)1−φ](n−1)/2ˆ˙γn−1+τˆY 0ˆ˙γ[ηr(φ)1−φ]−1/2. (D.20)This can be interpreted as assuming that the liquid within the suspension obeysa Herschel-Bulkley type law, but with φ -dependent consistency and yield stressdefined by:κˆ = κˆ0[ηr(φ)1−φ](n−1)/2τˆY =τˆY 0[ηr(φ)1−φ]1/2 . (D.21)233Of course, when ηr(φ) is included, the effect of increasing φ is to increase theeffective viscosity.D.4.1 Suspension transportWe adopt a continuum approach in which variables are interpreted as being volumeaveraged over a suitably chosen local averaging volume, but not time-averaged.Variables specific to solid or liquid phases are phase-averaged, defined using thecharacteristic function of the phase, the local instantaneous variable (e.g. solidsphase velocity) and a suitable smoothing or weighting function g; see e.g. [199].The suspension mass and momentum balances are as follows:∂ ρˆ∂ tˆ+ ∇ˆ · [ρˆsφ uˆp+ ρˆ f (1−φ)uˆ f ] = 0, (D.22)DDtˆ[ρˆuˆ] = bˆ+ ∇ˆ · Σˆ. (D.23)where uˆp and uˆ f are the solid and fluid phase-averaged velocities, bˆ and Σˆ arethe volume averaged suspension body force and stress tensor, respectively; see[200]. Finally φ is the solids volume fraction. The suspension stress Σˆ is oftendecomposed into individual contributions from both fluid and solid phases:Σˆ=−pˆ f I+ τˆ f + Σˆp (D.24)The material derivative in (D.23) isDDtˆ=∂∂ tˆ+ uˆ · ∇ˆ,where uˆ is the volume averaged suspension velocity.These equations are obtained by phase-averaging the individual equations forsolid and liquid phases, then summing to eliminate the inter-phase terms. Thus, wealso need to consider conservation equations for one phase, taken here as the solids234phase:∂∂ tˆ[φρˆs]+ ∇ˆ · [φρˆsuˆp] =−sˆ f , (D.25)ρˆpDpDtˆ[φ uˆp] = ∇ˆ · Σˆp+ Σˆi∫Sin · σˆg dS+ bˆ. (D.26)The mass conservation equation has a source term −sˆ f which denotes the increaserate of solids (mass per unit time per unit volume), which comes from the chemicalspecies models. The same term appears in the fluid phase mass conservation equa-tion, cancelling on summing to give (D.22). On the left-hand side of (D.26), Dp/Dtˆdenotes the material derivative using uˆp for the advective term. The first term onthe right hand side is the particle stress, i.e. particle contribution to the suspensionstress. The second term is the volume-averaged traction on the particle surfacesand the last term is the average body force. We have neglected any momentumsource term contributed to the reactions.We can note that the volume-averaged velocity uˆ is no longer divergence-free(due to the reactions). Instead we have∇ˆ · uˆ = sˆ f[1ρˆ f− 1ρˆs]−φ[ DpρˆsDtˆρˆs]− (1−φ)[ D f ρˆ fDtˆρˆ f]. (D.27)Solids phase diffusion and dispersionDiffusive and dispersive effects combine with the averaged forces acting on thesolids phase to distribute the particles. The solid phase mass conservation equation(D.25) is typically manipulated to give a transport equation for evolution of φ .Note that the phase-averaging adopted preserves the instantaneous dynamics ofeach configuration, so that (D.25) has no diffusive flux.At least two general approaches have been taken to model particle phase dif-fusion: (i) the diffusive flux approach of Leighton & Acrivos [201], modified byPhillips et al. [202]; (ii) the Suspension Balance Model (SBM) of Nott & Brady[203]. We follow the SBM approach of Brady and co-workers, [203]. The relative235velocity (uˆr = uˆs− uˆ f ) is substituted into (D.25) to give:∂φ∂ tˆ+ ∇ˆ · [φ uˆ] = −∇ˆ · [φ(uˆp− uˆ)]− sˆ fρˆs −φ[ DpρˆsDtˆρˆs]= −∇ˆ · [φ(1−φ)uˆr]− sˆ fρˆs −φ[ DpρˆsDtˆρˆs]= ∇ˆ · [φ(1−φ)Mˆ(ηˆ f , dˆp,φ)fˆD]− sˆ fρˆs −φ[ DpρˆsDtˆρˆs], (D.28)where Mˆ denotes the particle mobility and fˆD is the phase-averaged particle dragforce. The usual approach now is to consider (D.26) in the limit of small inertia,whereby fˆD may be substituted into (D.28), apparently leading to the divergence ofthe remaining terms on the right-hand side of (D.26).More recently [204] and [205] have shown that the the volume-averaged trac-tion (2nd term on the right-hand-side of (D.26)) may be expressed as the sum ofa particle-averaged force and the negative divergence of the particle stress. In thissense, the particle stress does not contribute to the particle momentum equation.The only terms on the right-hand side of (D.26) are the external body force andthe particle-averaged force, which consists of the hydrodynamic forces, contacttraction forces and inter particle forces. This questions the foundation of the SBMmodel in attributing the shear-induced particle migration to the divergence of par-ticle phase stress. However, [205] showed that the particle-averaged force can beexpressed in terms of an interphase drag force and a divergence of a stress ten-sor which includes the effects of hydrodynamic, contact and inter particle forces.Consequently, the form of particle phase momentum equation used in the SBM ap-proach is correct, but using conventional closures for the interphase drag force andthe stress tensor is partly in error. This suggests that further research is required todetermine the correct stress tensor closure, which remains to be done.Despite the issues raised above, we continue to follow the classical SBM ap-proach in developing our model. One advantage of this is that the particle tem-perature is included in the SBM approach, which itself is useful for modellingnon-Stokesian particle effects. We replace (D.26) with the solid momentum equa-236tion:ρˆsφDpDtˆuˆp = fˆB+ fˆD+ ∇ˆ · Σˆp (D.29)The phase-averaged forces acting on the particles contribute to 2 terms: fˆB rep-resenting the net solids phase body force and fˆD representing the hydrodynamicforces on the solids phase. As discussed earlier, except in exceptional situationscontact forces may be neglected. The phase-averaged net solids phase body forceis:fˆB = φ [ρˆs− ρˆ f ]g. (D.30)The phase-averaged drag force fˆD is modelled via a hindered settling closure:fˆD =−3φρˆ fCD(Rep,l)4h(φ)dˆp|uˆr|uˆr. (D.31)We adopt the same framework as Ovarlez and co-workers. In [193] the authorsstudy particle settling in yield stress fluids, perpendicular to the main direction ofshear. They advocate using a Newtonian hindering function h(φ):h(φ) =1−φηr(φ). (D.32)to modify the single particle settling speed. They identify two limits (a plasticregime and a viscous regime) in each of which they use a drag law incorporatingµˆ f = ηˆ f ( ˆ˙γ,φ), as defined in (D.20). A quite similar usage of ηˆ f for fitting a dragcoefficient is described in [206]. The drag coefficient CD(Rep,l) = 24/Rep,l only tocover the Stokesian regimes.The Reynolds number is based on ηˆ f ( ˆ˙γ,φ) and |uˆr| i.e.Rep,l =ρˆ f |uˆr|dˆpηˆ f. (D.33)Inverting (D.31), with the drag coefficient, leads straightforwardly to a relationuˆr = −Mˆ(φ , ˆ˙γ, |fˆD|)fˆD. This relationship for the relative velocity is valid for aslong as the particles are settling within the liquid phase. However, above a certainpercolation threshold φp it is assumed that the solid phase begins to form a porousnetwork. For φ > φp the solid phase velocity becomes zero and uˆr = −uˆs. Thus,237we modeluˆr = −dˆ2ph(φ)18ηˆ f φ fˆD, φ ≤ φp,− kˆp(φ)ηˆ f , φ > φp.(D.34)The transition above is to a Darcy-type regime for φ > φp, modelled via the per-meability kˆp(φ). We assume that this transition will be relatively smooth as thepercolation process is continuous. Thus for example:kˆp(φp) =dˆ2ph(φp)18φp,and we expect that the resistance increases at larger φ , i.e. kˆp(φ)≤ dˆ2ph(φ)/(18φ).Returning to the SBM derivation, it is assumed that the left-hand side of (D.29)is relatively small, (which would hold e.g. in conditions where the flow is steady,rectilinear and fully developed). In this case, we may write:fˆD ≈−[fˆB+ ∇ˆ · Σˆp], (D.35)and the solids mass conservation equation becomes:∂φ∂ tˆ+ ∇ˆ · [φ uˆ] =−∇ˆ · [φ(1−φ)Mˆ(φ , ˆ˙γ, |fˆD|)(fˆB+ ∇ˆ · Σˆp)]− sˆ fρˆs −φ[ DpρˆsDtˆρˆs],(D.36)The particle stress tensor is usually modelled by the expression:Σˆp =−ΠˆZ+ ηˆ fηp(φ) ˆ˙γ, (D.37)see e.g. [207]. The second term in (D.37) is the particle shear stress term. The bulkrate of strain tensor for the suspension is ˆ˙γ = ˆ˙γi j. Note that the bulk suspensionstrain rate isˆ˙γ =[123∑i, j=1ˆ˙γ2i j]1/2.The first term in (D.37) is a product of the particle pressure Πˆ and the tensor Z,through which we account for normal stress differences. It is argued that a reason-238able approximation to the tensor Z is:Z = λ1 0 00 λ2 00 0 λ3where the directions x1 and x2 are in the plane of shear, (here this would be locallyaligned with the mean flow along the fracture). According to the scaling, λ1 +λ2 +λ3 = 3 and from [208], a common choice for shear flow is λ1 = λ2 = 2λ3 =6/5. Other choices can be made if there is specific knowledge of the normal stressdifferences.Here the SBM approach improves upon diffusive flux models by introducingthe granular or particle temperature Θˆ≥ 0, which is the root mean square fluctuat-ing velocity of the particle phase:Θˆ=13〈(uˆp−〈uˆ〉p) · (uˆp−〈uˆ〉p)〉. (D.38)The particle temperature is used in an alternative closure expression for the solidsphase stress, see e.g. [203, 207–209, 209, 210], in particular eliminating the effectsof the vanishing strain rate at the channel centre. The expense of doing this is thatanother variable is introduced.However, in simple flows we can often simplify and solve for/eliminate Θˆ. Incementing flows it is unclear if Θˆ will be significant, except perhaps in he case ofstrong settling, or perhaps othe fluid re-arrangement in the annulus. The approachof [203] is to replace the bulk strain rate with a term proportional to the particletemperature. It is known that in a homogeneous suspension,ˆ˙γ2 ∝ dˆ−2p Θˆ,and from this we model the particle pressure asΠˆ= ηˆ f dˆ−1p Θˆ1/2 p(φ), (D.39)239where p(φ) is given by:p(φ) = 2√3kφφ 1/2(ηp)ξ .An evolution equation for Θˆ is derived in [203] as a simplified form of themechanical energy balance for the particle phase:3ρˆsφC(φ)DpΘˆDtˆ= Σˆp : ˆ˙γ+4√3ηˆ f dˆ−2p β (φ)Θˆ1/2|uˆr|−12ηˆ f dˆ−2p α(φ)Θˆ− ∇ˆ · qˆ.(D.40)where qˆ is a particle heat flux vector. The functions C(φ), α(φ) and β (φ) aredimensionless, as defined in [203]. There are slight differences with [203] due toour preference to work with the particle diameter rather than radius, and a factor of1/3 missing in [203] from the usual definition of particle temperature common ine.g. kinetic theories. The heat-capacity term C(φ) is not specified, as it is not usedin the Stokesian regime and for flows that are steady/developed this term would berelatively small. The functions α(φ) and β (φ) are specified semi-empirically byconsidering different limits of the model; see [203] and subsequent modifications.The field qˆ represents the phase-averaged fluctuating component of the solids phasedissipation, i.e.qˆ =−〈Σˆ′p · (uˆp−〈uˆ〉p)〉,which is modelled via a Fourier-type “heat flux” law:qˆ =−3ηˆ fκΘ(φ)∇ˆΘˆ. (D.41)The above represents a first model of the cement slurry.D.5 Future workIn this appendix we have reviewed a range of modes in bringing together the nec-essary pieces to model cement hydration. So far however, we have not proceededto analyse or solve the model equations. This is a major undertaking.In section Section D.2, a chemical model that describes the chemical reactionswhich the mixture of water-cement undergoes up until the thickening occurs, was240introduced. This process is exothermic and releases a great deal of heat which leadsto a significant acceleration in a number of the reactions. In addition, the cementedannuli, which is focus of this study, is usually exposed to temperature variationsalong the well. Therefore, we need also include the energy equation within ourmodel. We also need to properly formulate boundary and initial conditions fordifferent model scenarios of relevance.241

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