Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Yield stress fluid flows in uneven geometries : applications to the oil & gas industry Roustaei, Ali 2016

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2016_may_roustaei_ali.pdf [ 30.02MB ]
Metadata
JSON: 24-1.0300446.json
JSON-LD: 24-1.0300446-ld.json
RDF/XML (Pretty): 24-1.0300446-rdf.xml
RDF/JSON: 24-1.0300446-rdf.json
Turtle: 24-1.0300446-turtle.txt
N-Triples: 24-1.0300446-rdf-ntriples.txt
Original Record: 24-1.0300446-source.json
Full Text
24-1.0300446-fulltext.txt
Citation
24-1.0300446.ris

Full Text

nizly strzss uiy ofis in unzvzn gzomztrizsOvpplixvtions to thz oil ; gvs inyustrybyAla RomslaeaBSc Mechanical Engineering, Sharif University of Technology, 2007MSc Mechanical Engineering, Sharif University of Technology, 2010a thesis submitted in partial fulfillmentof the requirements for the degree ofYoxtor of ehilosophyafthe faculty of graduate and postdoctoralstudies(Mechanical Engineering)The Mfanersaly of Bralash Colmebaa(Vafcomner)May 2()6c© Ala Romslaea, 2()6VwstrvxtWe study a set of yield stress fluid flows in channels with geometric non-uniformities, motivated by theoretical aspects and industrial applications.Methodology is primarily computational and we try to analytically investi-gate as much as possible. Theoretical interest arises from the self-selectionphenomenon, meaning that the original flow geometry is modified by thefluid itself. This occurs due to yield stress and is accomplished via stagnantzones of the fluid attached to the boundary of original geometry. Industrialmotivations stem from oil/gas well construction operations: primary andsqueeze cementing and hydraulic fracturing. In all we have drilling mud,cement or a gelled fluid which exhibit yield stress. Specifically, we modela washout along the well as a non-uniform channel and extensively studyflows through it. This is an enlarged segment of the well where the wellboreis washed out or collapsed. The main industrial concern is the residual mudleft in the washout after primary cementing which weakens the hydraulicsealing function of the cement.Self-selection has been analytically studied for duct flows [168, 169], andnot much in 2D flows. Chapter 2 is a study of self-selection in wavy walledchannels as a model for smooth non-uniform channels. We find similarresults to [168, 169], however a complete understanding eludes us. Chapter 3looks at the flow of Bingham fluid in fractures. We study the limits ofvalidity of Darcy approach first and then focus at the minimal pressuredrop required to mobilize the fluid in fracture. We demonstrate knowingself-selection properties can greatly improve approximations here. Chapters4-6 are step by step investigation of the flows in washout, from Stokes toiiinertial and finally displacement flow. In Chapter 4 we show self-selectionin Stokes flow of washout and use it to estimate the residual fluid in thewashout. We study the effects of inertia on it in Chapter 5, illustrating onlyfinite amount of inertia would help in better displacement of the mud whichis counter intuitive. Chapter 6 is a preliminary study of the displacementflow and we report some interesting observations.iiierzfvxzThe contents of this thesis are the results of the research of the author, AliRoustaei during the course of his PhD studies at UBC, under the supervisionof professor Ian Frigaard. The following papers have been published and/orare in progress• Roustaei, A., and I.A. Frigaard. “The occurrence of fouling layers inthe flow of a yield stress fluid along a wavy-walled channel.” Journalof Non-Newtonian Fluid Mechanics 198 (2013):109-124.This paper was co-authored with I.A. Frigaard and I did the imple-mentation of the code, running, data compilation and contributed toanalysis of the results. I. A. Frigaard supervised the research.• Roustaei, A., A. Gosselin, and I.A. Frigaard. “Residual drilling mudduring conditioning of uneven boreholes in primary cementing. Part1: Rheology and geometry effects in non-inertial flows.” Journal ofNon-Newtonian Fluid Mechanics 220 (2015): 87-98.This paper was co-authored with I.A. Frigaard and I did the imple-mentation of the code, running, data compilation and contributed toanalysis of the results. A. Gosselin–intern student–implemented thekoch semi-fractal shape and assisted with running some of the compu-tations. I.A. Frigaard supervised the research.• Roustaei, A., and I.A. Frigaard. “Residual drilling mud during con-ditioning of uneven boreholes in primary cementing. Part 2: Steadyivlaminar inertial flows.” Journal of Non-Newtonian Fluid Mechanics226 (2015): 1-15.This paper was co-authored with I.A. Frigaard and I did the imple-mentation of the code, running, data compilation and contributed toanalysis of the results. I.A. Frigaard supervised the research.• Roustaei, A., T. Chevalier, L. Talon and I.A. Frigaard “Non-Darcyeffects in fracture flows of a yield stress fluid” under review.This paper was co-authored primarily with I.A. Frigaard and L. Talon.I and I.A. Frigaard ran the main computations and analysis for thepaper and T. Chevalier and L. Talon contributed to the affine fracturecomputations and collaborated on some aspects of the analysis.• A. Maleki, S. Hormozi, A. Roustaei, and I. A. Frigaard. “Macro-sizedrop encapsulation.” Journal of Fluid Mechanics 769 (2015): 482-521.This paper was the outcome of A. Maleki’s masters thesis research. Ihelped in the implementation of his code, initial computations of theencapsulation flows and to validation/verification studies. The paperwas authored primarily by the rest of authors. I read the draft andprovided comments and participated in some of the analysis. I. A.Frigaard supervised the research.vivwlz of ContzntsVwstrvxt C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C iierzfvxz C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C ivivwlz of Xontznts C C C C C C C C C C C C C C C C C C C C C C C C C C C viaist of Figurzs C C C C C C C C C C C C C C C C C C C C C C C C C C C C C fliVxknowlzygmznts C C C C C C C C C C C C C C C C C C C C C C C C C C flflflF Introyuxtion C C C C C C C C C C C C C C C C C C C C C C C C C C C C F1.1 Industrial motivations . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Primary cementing . . . . . . . . . . . . . . . . . . . . 21.1.2 Plug cementing for abandonment . . . . . . . . . . . . 61.1.3 Squeeze cementing and other remedial tasks . . . . . . 71.1.4 Hydraulic fracturing . . . . . . . . . . . . . . . . . . . 81.1.5 Other fracture-like geometries . . . . . . . . . . . . . . 101.2 Yield stress fluids . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 Thixotropy . . . . . . . . . . . . . . . . . . . . . . . . 131.2.2 The yield stress debate . . . . . . . . . . . . . . . . . . 141.2.3 Governing equations . . . . . . . . . . . . . . . . . . . 161.2.4 Lubrication paradox . . . . . . . . . . . . . . . . . . . 201.3 Variational principles for yield stress fluids . . . . . . . . . . . 221.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . 281.4.1 Methods that use the exact Bingham model . . . . . . 28vi1.4.2 Regularization methods . . . . . . . . . . . . . . . . . 341.4.3 Selected applications of computational methods . . . . 371.5 Related work on self-selection . . . . . . . . . . . . . . . . . . 401.5.1 Self-selection of flow geometry in duct flows . . . . . . 411.5.2 Non-uniform channels and self-selection . . . . . . . . 471.6 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Flow of v Binghvm uiy in wvv– wvllzy xhvnnzls C C C C C C 532.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.2 The wavy-walled channel . . . . . . . . . . . . . . . . . . . . 582.2.1 Numerical solution method . . . . . . . . . . . . . . . 602.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.3.1 Channels with small  . . . . . . . . . . . . . . . . . . 632.3.2 Channels with  ∼ df(1) . . . . . . . . . . . . . . . . 682.3.3 Parameter regimes . . . . . . . . . . . . . . . . . . . . 702.3.4 Characteristic features of the fouling layer forW ≳ df(1) 752.3.5 Complexity of the Newtonian limit . . . . . . . . . . . 842.4 Discussion and summary . . . . . . . . . . . . . . . . . . . . . 883 Flow of –izly strzss uiys vlong frvxturzO przssurz yropprzyixtion vny ow onszt C C C C C C C C C C C C C C C C C C C C NE3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.2.1 Fracture geometries . . . . . . . . . . . . . . . . . . . 973.2.2 Computational overview . . . . . . . . . . . . . . . . . 983.2.3 Example results . . . . . . . . . . . . . . . . . . . . . 993.3 Darcy-law estimates . . . . . . . . . . . . . . . . . . . . . . . 1023.4 The limit of no flow . . . . . . . . . . . . . . . . . . . . . . . 1093.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 1123.4.2 The critical limit from a variational method . . . . . . 1143.4.3 Short fractures . . . . . . . . . . . . . . . . . . . . . . 1193.4.4 Long fractures with no fouling . . . . . . . . . . . . . 1223.4.5 Intermediate fractures with partial fouling . . . . . . . 124vii3.4.6 Affine fractures . . . . . . . . . . . . . . . . . . . . . . 1353.5 Discussion and summary . . . . . . . . . . . . . . . . . . . . . 140I htokzs ow through wvshouts C C C C C C C C C C C C C C C C C FII4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.2 A simplified washout model . . . . . . . . . . . . . . . . . . . 1484.2.1 The model problem, gz = 0 . . . . . . . . . . . . . . . 1504.2.2 Washout geometries . . . . . . . . . . . . . . . . . . . 1524.2.3 Computational overview . . . . . . . . . . . . . . . . . 1534.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1564.3.1 Examples results . . . . . . . . . . . . . . . . . . . . . 1574.3.2 Parametric variations . . . . . . . . . . . . . . . . . . 1614.3.3 Flowing area . . . . . . . . . . . . . . . . . . . . . . . 1654.3.4 Pressure drop . . . . . . . . . . . . . . . . . . . . . . . 1684.4 Discussion and summary . . . . . . . . . . . . . . . . . . . . . 1715 Inzrtiv zffzxts in thz ow of –izly strzss uiy in wvshout C FL35.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.2 The simplified washout . . . . . . . . . . . . . . . . . . . . . . 1785.3 Computational method . . . . . . . . . . . . . . . . . . . . . . 1815.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.4.1 Flow rate effects . . . . . . . . . . . . . . . . . . . . . 1885.4.2 Varying Hz . . . . . . . . . . . . . . . . . . . . . . . . 1975.4.3 Effects of washout shape . . . . . . . . . . . . . . . . . 1985.5 Discussion and summary . . . . . . . . . . . . . . . . . . . . . 2026 Displvxzmznt ow C C C C C C C C C C C C C C C C C C C C C C C C C 2E56.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.2 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2096.3 Numerical method for interface tracking: VOF . . . . . . . . 2126.4 Numerical method for the solution of the Navier-Stokes equa-tion with an interface . . . . . . . . . . . . . . . . . . . . . . 2166.5 Validation with previous steady computations . . . . . . . . . 2206.6 Preliminary computational results . . . . . . . . . . . . . . . 221viii6.6.1 Discussion and summary . . . . . . . . . . . . . . . . . 244L hummvr– vny xonxlusions C C C C C C C C C C C C C C C C C C C C 25E7.1 Results, insights and contributions from the individual chapters2507.1.1 Bingham fluid flow along wavy channels (Chapter 2) . 2507.1.2 Fracture flow and critical pressure drop (Chapter 3) . 2537.1.3 Washout flows at gz = 0; Chapter 4 . . . . . . . . . . 2567.1.4 Inertial flow in washouts; Chapter 5 . . . . . . . . . . 2587.1.5 Displacement flow; Chapter 6 . . . . . . . . . . . . . . 2597.2 Context of the thesis . . . . . . . . . . . . . . . . . . . . . . . 2637.2.1 Scientific context . . . . . . . . . . . . . . . . . . . . . 2637.2.2 Industrial context . . . . . . . . . . . . . . . . . . . . 2677.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2707.3.1 Rheological idealizations . . . . . . . . . . . . . . . . . 2707.3.2 Experimental study . . . . . . . . . . . . . . . . . . . 2707.3.3 Computations . . . . . . . . . . . . . . . . . . . . . . . 2717.4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . 271Biwliogrvph– C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C 2LIV bvxroBsizz yrop znxvpsulvtion C C C C C C C C C C C C C C C C C 2NMA.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298A.2 Physical problem . . . . . . . . . . . . . . . . . . . . . . . . . 301A.2.1 The plane Poiseuille solution jc (y) . . . . . . . . . . 306A.2.2 Inertial effects and the droplet flow . . . . . . . . . . . 306A.2.3 Why large droplets yield the plug . . . . . . . . . . . . 308A.3 Small-slender droplets: h(x) ∼  ≪ 1 . . . . . . . . . . . . . . 310A.3.1 O(1) solution in the yielded region . . . . . . . . . . . 313A.3.2 O() solution in the yielded layer . . . . . . . . . . . . 314A.3.3 Size of the yield surface perturbation from yyN0 andeffects of inertia . . . . . . . . . . . . . . . . . . . . . 316A.4 Order unity iso-dense droplets: h(x) ∼  ∼ 1 . . . . . . . . . 319A.4.1 Computational algorithm and benchmarking . . . . . 320A.4.2 Effects of droplet spacing . . . . . . . . . . . . . . . . 323ixA.4.3 Encapsulation and failure . . . . . . . . . . . . . . . . 324A.4.4 Perturbation of pressure . . . . . . . . . . . . . . . . . 329A.4.5 Maximum size of encapsulated droplets . . . . . . . . 331A.5 Encapsulation with fluids of different densities . . . . . . . . . 333A.6 Droplet encapsulation in a pipe . . . . . . . . . . . . . . . . . 335A.6.1 Poiseuille flow solution . . . . . . . . . . . . . . . . . . 337A.6.2 Slender drop . . . . . . . . . . . . . . . . . . . . . . . 338A.6.3 Examples of encapsulation and failure . . . . . . . . . 340A.7 Discussion and summary . . . . . . . . . . . . . . . . . . . . . 343B homz notzs on xomputvtions C C C C C C C C C C C C C C C C C C 35EB.1 Convergence and mesh adaptation . . . . . . . . . . . . . . . 350B.2 Defining yield surfaces . . . . . . . . . . . . . . . . . . . . . . 352B.3 Ribbons of unyielded region . . . . . . . . . . . . . . . . . . . 354xaist of FigurzsFigure 1.1 Primary cementing and static mud in washout region . . 4Figure 1.2 DeepWater Horizon explosion at Gulf of Mexico, April2010. One of the main causes was poor primary cementing 6Figure 1.3 Squeeze cementing, used for correction of primary cement.Right: showing the schematic of equipment for squeeze.Left: a closer view to fracture flow, courtesy of GKE en-gineering . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 1.4 Plug cements are used to persistently isolate necessaryzones for abandonment, Figure courtesy of GKE engineering 8Figure 1.5 Depiction of fracturing, photo from US EnvironmentalProtection Agency [3] . . . . . . . . . . . . . . . . . . . . 9Figure 1.6 The velocity profile and yielded/unyielded fluid in channelflow of a yield stress fluid . . . . . . . . . . . . . . . . . . 18Figure 1.7 Lubrication paradox example: flow of yield stress fluid ina slowly varying channel with pseudo plug region . . . . . 21Figure 1.8 An example of mesh adaptation strategy suggested bySaramito and Roquet to improve accuracy of capturingyield surfaces. Very fine, highly anisotropic mesh can beobserved around yield surfaces. Figure from the Galleryof Rheolef library [213] . . . . . . . . . . . . . . . . . . . . 32Figure 1.9 The Bingham model (dashed black line) and the typicalcurves for Papanastasiou (red), Bercovier (green) and bi-viscosity regularizations . . . . . . . . . . . . . . . . . . . 35xiFigure 1.10 The original Fig 5a in paper by Burgos et al. [44]. Circlesshow the analytic yield surface and other lines are yieldsurfaces from Papanastasiou regularization with increas-ing value of m. The main difference is that the convexityof regularized yield surface is reversed compared to ana-lytical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 1.11 Original Fig 3 in paper [168], showing the stagnant zoneVWX which exists in the corner point W and the convexityof the boundary . . . . . . . . . . . . . . . . . . . . . . . 41Figure 1.12 Flow of viscoplastic fluid in a square pipe. As shown in[168] the yield surface of static regions has the shape ofarc of a circle . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 1.13 Original Fig. 8 in paper [169], showing how the boundaryof duct can be arbitrarily changed without changing thesolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 1.14 Original Fig. 12 of paper [169]. Showing arbitrary sub-domain ! inside the main duct domain. The flow in maindomain has stagnant zone dg1g2. This is sufficient con-dition for hatched area of subdomain ! to be stagnant . . 46Figure 2.1 Schematic of the flow geometry and boundary conditionsΩ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 2.2 Showing a mesh adaptation cycle for the case of (h =1P  = O125P W = 10) . . . . . . . . . . . . . . . . . . . . . 62Figure 2.3 Effects of increasing amplitude h on speed (left panel)and pressure (right panel), for  = O05, W = 10. From topto bottom, h = (O01P O25P 1P 2P 4). The same color scale isvalid for each h. The dark lines in the left panel are thestreamlines, gray regions in right are plugs. . . . . . . . . 63Figure 2.4 More detailed results from the case  = O05, W = 10,h = 4. Left panel, from top to bottom shows (pP xxP xy).Right panel from top to bottom shows (xxP yyP ‖‖) re-spectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 65xiiFigure 2.5 Effects of increasing amplitude h on speed (left panel)and pressure (right panel), for a bigger  than Fig 2.3, = O25P W = 10. From top to bottom, h = (O25P O5P 1).The same color scale is valid for each h. The dark linesin the left panel are the streamlines, gray regions in rightare plugs. . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 2.6 Panorama of plug shapes for W = 1 (top) and W = 2(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 2.7 Panorama of plug shapes for W = 5 (top) and W = 10(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 2.8 Appearance and evolution of mushroom shape plug andstress field (‖‖) at the narrowest part of channel for (h =1P  = O25) and W = (20P 10P 5P 2) (from left to right);gray regions are plugs (figure zoomed on narrowest partof channel). . . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 2.9 Plot of flow regimes in (hP ) plane for different Binghamnumbers, a) W = O1, b) W = 1, c) W = 10, d) W = 100.Markers of flow regimes: (▲) intact central plug+fouling,(♦) broken central plug+fouling, (□) broken central plug-no fouling, (◦) intact central plug-no fouling. . . . . . . . 74Figure 2.10 Phase plot of all computations (≈500 points), Markersof flow regimes: (▲) intact central plug+fouling, (♦) bro-ken central plug+fouling, (□) broken central plug-no foul-ing, (◦) intact central plug-no fouling, equations hn =M039U2+M55U+M07U2+3M21U+M15, hs =M054U2+1M03U+M31U2+5M67U+M66represent fittedcurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 2.11 Positions y = yp of the two yield surfaces in the widestpart of the channel: a)  = O5P W = 2; b)  = O25P W = 10;c)  = O125P W = 10; d)  = O05P W = 10. . . . . . . . . . . 77xiiiFigure 2.12 comparison of wavy wall flow ( = O5P h = 1P W = 2)with a triangular and rectangular walls, left panel: speed,right panel: pressure, The same color scale is valid foreach column. The dark lines in the left panel are thestreamlines, gray regions in right are plugs. . . . . . . . . 78Figure 2.13 Ratio of yielded fluid, !, plotted against W˜: a)  = 1P 0O5P 0O25;b)  = 0O125P 0O05P 0O025. . . . . . . . . . . . . . . . . . . 80Figure 2.14 Detail of the yielded layer at the widest part of the channelfor (h = 4P  = O05P W = 10), (see also Fig 2.3). Left panel,from top to bottom shows (jxP e ). Right panel from topto bottom shows (xyP yy). . . . . . . . . . . . . . . . . . 81Figure 2.15 The parameter v(W˜P !)y0 from our computed data withfouling layers. The symbols are for different , followingthe key in Fig. 2.13. . . . . . . . . . . . . . . . . . . . . . 83Figure 2.16 Streamlines and unyielded regions for a) Newtonian flow(h = O65P  = O5) where recirculation has just begun; b)same channel but now with W = O01; c) same W as b) butnow h is increased to h = 0O75; d) same channel as c) butW is increased to W = O04. . . . . . . . . . . . . . . . . . . 85Figure 2.17 Streamlines and unyielded regions for (h = 2O5P  = O5)and a) Newtonian flow b) W = O001 c) W = O05 d) W = O09e) W = O12 f) W = O13 . . . . . . . . . . . . . . . . . . . . 87Figure 3.1 a) Schematic of the fracture geometry showing dimen-sional parameters; b) schematic of the wavy-walled di-mensionless geometry, with lower wall shifted to the rightby  a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 3.2 Computed examples of speed |u| & streamlines (left col-umn), and pressure p with (gray) unyielded plugs (rightcolumn), at H = 1, a = 10, W = 2. Top row: triangularfracture profile,  = 0; middle row: wavy fracture profile, = 0; bottom row: wavy fracture profile,  = 0O5. . . . . 101xivFigure 3.3 Computed examples of speed |u| & streamlines (left col-umn), and scaled pressure pRW with (gray) unyielded plugs(right column), at H = 1, a = 10,  = 0, wavy fractureprofile. Top row: W = 5; middle row: W = 10; bottomrow: W = 100. . . . . . . . . . . . . . . . . . . . . . . . . 102Figure 3.4 Computed examples of speed |u| & streamlines (left col-umn), and pressure p with (gray) unyielded plugs (rightcolumn), at W = 2, a = 10,  = 0O5, wavy fracture profile.Top row: H = 3; bottom row: H = 5. . . . . . . . . . . . 103Figure 3.5 Computed examples of the speed |u| & streamlines with(gray) unyielded plugs, atH = 2, a = 20, for a symmetricaffine fracture. Top row: W = 1; middle row: W = 10;bottom row: W = 100. . . . . . . . . . . . . . . . . . . . . 104Figure 3.6 Relative error in predicted pressure drops between numer-ically computed and lubrication approximation (3.12), forW = 0O1 in wavy fracture: a) H = 2 and varying a; b)a = 10 and varying H. For both figures: black circle = 0, red square  = 0O25, blue cross  = 0O5. . . . . . . 106Figure 3.7 a) Ratio of pressure drops computed numerically (∆pa )and from the lubrication approximation (∆pL): W = 1(circles), W = 5 (squares), W = 10 (diamonds); filledsymbols from triangular fracture profile, hollow symbolsfrom wavy profile;  = 0. b) Relative error in predictedpressure drops between numerically computed and lubri-cation approximation (3.12), for a ≥ 10 and W ≥ 1 in awavy fracture: black circle  = 0, red square  = 0O25,blue cross  = 0O5. . . . . . . . . . . . . . . . . . . . . . . 108xvFigure 3.8 a) Consistent errors in average pressure gradient predic-tion using (3.12), for H = 2, a = 2: black = wavy,red/white = triangular; circles = numerical, squares =lubrication approximation. b) relative error for the datain a) (squares); relative error using the lubrication ap-proximation (3.12) with the fracture width replaced byyield surface position (diamonds). c) Ratio of relativeerrors (lubrication approximation using yield stress posi-tion vs lubrication approximation using fracture width):H = 2, a = 2 in black; H = 5, a = 20 in red. Note thatthe multiple points displayed at the same W correspondto different values of  . . . . . . . . . . . . . . . . . . . . 110Figure 3.9 Colormap of the speed with superimposed streamlines ina relatively short fracture H = 1O5P a = 4: increasingW = 10P 100P 1000 (from left to right). Unyielded regionsare shown in gray. . . . . . . . . . . . . . . . . . . . . . . 113Figure 3.10 Colormap of the speed with superimposed streamlines ina relatively short fracture H = 0O1P a = 20: increasingW = 10P 100P 1000 (from top to bottom). Unyieldedregions are shown in gray. . . . . . . . . . . . . . . . . . . 114Figure 3.11 Colormap of the speed with superimposed streamlines ina relatively short fracture H = 2P a = 20: increasingW = 10P 100P 1000 (from top to bottom). Unyieldedregions are shown in gray. . . . . . . . . . . . . . . . . . . 115Figure 3.12 Limiting behaviour asW →∞ for the geometries of Figs. 3.9-3.11: a) dy plotted against W; b) ∆eRa plotted againstW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 3.13 Viscous dissipation, plastic dissipation and flow rate, v(u∗P u∗),j(u∗) andf(u∗) for the 3 geometries illustrated in Figs. 3.9-3.11, from left to right. . . . . . . . . . . . . . . . . . . . . 118xviFigure 3.14 Limiting behaviour as W →∞ for 5 short fracture geome-tries: a) shear layer thickness y0; b) dyv − dy. Data isshown for: ◦PH = 1P a = 3), (□PH = O5P a = 2), (△PH =O25P a = 4), (∗PH = 3P a = 2), (+PH = 4P a = 4); recallthat dyv = 1 for short fractures. . . . . . . . . . . . . . . 121Figure 3.15 Convergence of dyv −dy as W →∞ for H = 1, a = 20. . 123Figure 3.16 Comparison of computed velocity (solid line) with theasymptotic approximation at: a) 2xRa = 0O2; b) 2xRa =0O4; c) 2xRa = 0O6; d) 2xRa = 0O8, (as marked by verticalbroken lines in the colormap). Main figure shows the col-ormap of speed with streamlines and unyielded regions ingray; H = 2P a = 20P W = 10000. . . . . . . . . . . . . . . 127Figure 3.17 Colormaps of xx (unyielded regions in gray) for W =1000. Top panels: H = 0O5, a = 40, comparisons areshown with the predictions of (3.35) at values 2xRa =0O25P 0O5P 0O7, as marked with broken lines on the col-ormap. Lower panel: H = 2, a = 20, comparisons areshown with the predictions of (3.35) at values 2xRa =0O45P 0O6P 0O8 . . . . . . . . . . . . . . . . . . . . . . . . . 129Figure 3.18 a) Variation of dy(xy ) from (3.37) for: a = 20, H =0P 1P OOO8 (black); a = 6, H = 0P 0O25P 0O5OOO2 (red).b) Variation of dyv (computed by maximizing dy(xy )in (3.37)) with a for H = 1 (black) and H = 4 (red).Broken lines indicate the limit of dyv with no fouling. . . 133Figure 3.19 a) Variation of dyv, computed by maximizing (3.37). b)Variation of the value of xy , that maximizes (3.37), scaledwith aR2. c) Variation ofdyv, approximated as the largestvalue of dy from our 2D computations, typically at W =104. d) Absolute relative error between a & c. . . . . . . . 134Figure 3.20 a) Variation of dyv, computed by maximizing (3.41). b)Variation of the value of xy , that maximizes (3.41), scaledwith aR2. c) Absolute relative error between a & our dyvcalculated from the 2D computations. . . . . . . . . . . . 136xviiFigure 3.21 Computed examples of speed |u| & streamlines for a sym-metric affine fracture. Parameters are: H = 2, a = 20,W = 100P 1000P 10000, from top to bottom, with (gray)unyielded plugs. . . . . . . . . . . . . . . . . . . . . . . . 137Figure 3.22 Computed examples of speed |u| & streamlines for a frac-ture formed from two different affine surfaces. Parametersare: H = 2, a = 20, W = 100P 1000P 10000, from top tobottom, with (gray) unyielded plugs. . . . . . . . . . . . . 138Figure 3.23 Computed examples of speed |u| & streamlines for a frac-ture formed from two identical affine surfaces, shifted.Parameters are: H = 2, a = 20, W = 100P 1000P 10000,from top to bottom, with (gray) unyielded plugs. . . . . . 139Figure 3.24 Computed Oldroyd number as function of the Binghamnumber for the three different affine fractures of Figs. 3.21-3.23. Blue squares: the two surface are symmetrical. Redcircles: the two surfaces are uncorrelated. Black lozenges:the two surfaces are identical, shifted laterally with a con-stant gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Figure 4.1 Geometry of the washout in a section along the annulus. . 148Figure 4.2 Dimensionless computational domain and boundary con-ditions for this study. . . . . . . . . . . . . . . . . . . . . 152Figure 4.3 Four different categories of symmetric washout geometry,shown for x Q 0: a) sudden expansion-contraction (squarewave); b) triangular wave; c) sinusoidal wave; d) semi-fractal, based on the Koch snow flake. All washout shapesare fully determined by (hP ). . . . . . . . . . . . . . . . . 153Figure 4.4 Streamlines superimposed on colourmaps of the fluid speedfor different types of washout:  = 0O25, W = 5. Columnsare h = (0O5P 1P 2P 4) from left to right and unyielded re-gions are shown in gray. . . . . . . . . . . . . . . . . . . . 158xviiiFigure 4.5 Streamlines superimposed on colourmaps of the fluid speedfor different types of washout:  = 0O25, W = 100. Columnsare h = (0O5P 1P 2P 4) from left to right and unyielded re-gions are shown in gray. . . . . . . . . . . . . . . . . . . . 159Figure 4.6 Components of the stress for different types of washout: = 0O25, W = 5, h = 4. Columns show xx, xy, yy,p respectively, from left to right. The color key for eachcolumn is same shown at the bottom. Unyielded regionscolored with gray. . . . . . . . . . . . . . . . . . . . . . . 161Figure 4.7 Components of the stress speed for different types of washout: = 0O25, W = 100, h = 4. Columns show xx, xy, yy,p respectively, from left to right. The color key for eachcolumn is same shown at the bottom. Unyielded regionscolored with gray. . . . . . . . . . . . . . . . . . . . . . . 162Figure 4.8 Unyielded regions of the flow (shown in black) for differ-ent types of washout at W = 5. For each type of washoutcolumns show  = (O5P O25P O1P O05) from left to right respec-tively and rows are h = (O5P 1P 2P 4) from top to bottomrespectively . . . . . . . . . . . . . . . . . . . . . . . . . . 163Figure 4.9 Unyielded regions of the flow (shown in black) for dif-ferent types of washout at W = 100, for each type ofwashout columns show  = (O5P O25P O1P O05) from left toright respectively and rows are h = (O5P 1P 2P 4) from topto bottom respectively . . . . . . . . . . . . . . . . . . . . 164Figure 4.10 Onset of stationary fluid at the upper wall for: a) trian-gular wave; b) sinusoidal wave. Dark symbols indicatestationary fluid at the upper wall. . . . . . . . . . . . . . 166Figure 4.11 Comparison of static region for triangular and wavy washout(solid black/red) with the corresponding square washout(dashed). Only washout enlargement section is shown.Flowing are correction is shown in gray. Rows are h =(1P 2P 3) from top to bottom, for the two left columns(WP ) = (2P O5) and two right columns (WP ) = (10P O25) . 167xixFigure 4.12 Corrected flowing area of wavy and triangular washoutsagainst the square washout flowing area as the washoutgets deeper (increasing h). Left plot (WP ) = (2P O5)and right plot (WP ) = (10P O25). Markers are ◦) Wavywashout and ▽) linear washout. The dashed blue lineshowsVYN∞(WP ), the limiting flowing area of square washout168Figure 4.13 Pressure drop reduction (compared to uniform channel)for the square washout: a) W = 1; b) W = 10. In bthe broken red line indicates the analogous results for thewavy washout. . . . . . . . . . . . . . . . . . . . . . . . . 170Figure 4.14 Averaged computed pressure p¯(x) (square washout) com-pared with the pressure in the uniform channel (brokenline): a) (hP PW) = (O5P O25P 10); (hP PW) = (O5P O1P 10). . . 170Figure 4.15 Computed pressure gradient reduction for large h, com-pared to Poiseuille flow uniform pressure gradient (for thesquare washout) . . . . . . . . . . . . . . . . . . . . . . . 171Figure 5.1 Geometry of the washout in a section along the annulus. . 175Figure 5.2 Dimensionless computational domain and boundary con-ditions for this chapter. . . . . . . . . . . . . . . . . . . . 181Figure 5.3 Decay of the velocity residual ‖un+1 − un‖L2R∆t for thewashout (hP ) = (4P 0O25) at increasing W = (0P 1P 2P 8):(a) gz = 2000; (b) gz = 500. . . . . . . . . . . . . . . . . 185Figure 5.4 Validation of our code with published results from thelid-driven cavity flow. Plots show the horizontal veloc-ity component, j(0O5P y), along the centre of the cavity.a) Comparison with Ghia et al. [107] for gz = 1000,W = 0. b) Comparison with Syrakos et al. [224] andwith Vola2003 et al. [241] for gz = 1000, W = 10. . . . . . 185xxFigure 5.5 Convergence of the entrance velocity profile u(−0O5−1 −lxP y) to the plane Poiseuille solution as the entrance lengthlx = lw increases. The black dotted line shows the planePoiseuille flow: (gzPWP hP −1) = (200P 2P 10P 10) and en-trance lengths are lx = (0P 2P 5P 10) indicated by markers(+P •P□P△), respectively. . . . . . . . . . . . . . . . . . . 186Figure 5.6 Colourmaps of the flow speed with streamlines and pres-sure compared for periodic washout (two left columns)with washout that has entrance length (two right columns).Parameters are (hP PW) = (2P 0O25P 5) andgz = 0P 50P 100P 200,from top to bottom. The gray regions indicate unyieldedfluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Figure 5.7 Colormaps of the flow variables, restricted to the washoutregion for (hP PW) = (1P 0O25P 5). At the top of the figurethe columns show speed, pressure and xx; the columnsat the bottom of the figure show xy, xx and yy, fromleft to right. The rows show gz = 0P 50P 100P 200P 500,from top to bottom. The gray regions indicate unyieldedfluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Figure 5.8 Effects of increasing flow rate at (hP ) = (1P 0O25) andgz = 10P 50P 100P 200P 500 from top to bottom, at fixedHz = 500. Left: velocity magnitude and streamlines;right pressure with unyielded regions shown as gray. . . . 191Figure 5.9 Average pressure along the channel (black) compared withthe Poiseuille flow pressure (broken line) as the flow rateincreases. (hP −1) = (1P 4) and gz = 50P 100P 200P 500with fixed Hz = 500. The vertical dotted lines marks thewashout start and end. . . . . . . . . . . . . . . . . . . . . 192xxiFigure 5.10 Pressure drop offset due to washout with increasing gzat fixed Hz = 500. The solid line shows the pressure dropand the broken line shows the normalized value with thepressure drop, normalized with the Poiseuille flow pres-sure drop over the length of washout. a) (hP ) = (1P 0O25)and b) (hP ) = (1P 0O1). . . . . . . . . . . . . . . . . . . . 193Figure 5.11 Flowing area (black, +) and displaced area (red,©) vari-ations with increasing gz at fixed Hz = 500. (a) (hP ) =(1P 0O25) (b) (hP ) = (2P 0O1) (c) (hP ) = (1P 0O1) . . . . . . 194Figure 5.12 Effects of increasing flow rates at fixed Hz. The inertialeffects are controlled by gz = 50P 100P 200P 400 (topto bottom) and geometric variables are (hP ) = (1P 0O25).The gray areas denote unyielded fluid. Left column: Hz =200, right column: Hz = 1000. . . . . . . . . . . . . . . . 196Figure 5.13 Flowing area (black, +) and displaced area (red,©) varia-tion with increasing gz for (hP ) = (1P 0O25) and differentfixed Hedstro¨m numbers: (a) Hz = 200 (b) Hz = 1000 . . 197Figure 5.14 Variation of the x-component of velocity u(xP y) with in-creasing h at a) x = −0O25−1, b) x = 0, c) x = 0O25−1for a square washout: (gzPWP ) = (100P 5P 0O1). . . . . . . 198Figure 5.15 Variation of the x-component of velocity u(xP y) with in-creasing h at a) x = −0O25−1, b) x = 0, c) x = 0O25−1for a sinusoidally wavy washout: (gzPWP ) = (100P 5P 0O1). 199Figure 5.16 comparison of x-component of velocity u(xP y) at a) x =−0O25−1, b) x = 0, c) x = 0O25−1 for (gzPWP hP ) =(100P 5P 30P 0O1), between three different washout shapes.The markers are: □, expansion-contraction/square washout;△, linear variation/triangular washout; ©, wavy sinu-soidal washout. . . . . . . . . . . . . . . . . . . . . . . . . 200xxiiFigure 5.17 Plot of viscous (solid line) and plastic (broken line) dissi-pation functionals for (gzPWP ) = (100P 5P 0O1) and differ-ent types of washouts. The markers are: □, expansion-contraction/square washout; △, linear variation/triangu-lar washout; ©, wavy sinusoidal washout. . . . . . . . . . 201Figure 6.1 Problem domain, geometrical dimensions and boundaryconditions used for displacement flow . . . . . . . . . . . 211Figure 6.2 Volume fractions values assigned to each cell in VOF . . . 214Figure 6.3 A 1D domain and its finite volume discretization. Figurecourtesy of C. Olivier-Gooch . . . . . . . . . . . . . . . . 215Figure 6.4 Checker-board pattern of pressure field due to violationof Inf-Sup condition. . . . . . . . . . . . . . . . . . . . . . 218Figure 6.5 The original element types used for displacement flows a)velocity, b) pressure, c) multiplier, and d) color spaces . . 219Figure 6.6 The modified elements for Inf-Sup compatibility. a) ve-locity, b) pressure, c) multiplier, and d) color spaces. Thered line shows rectangle divided to two triangular elements.219Figure 6.7 The elements set with discrete divergence free property.a) velocity, b) pressure, c) multiplier, and d) color spaces. 220Figure 6.8 x-velocity profiles u(xP y) at three sections of the washout(left to right): x = −aR4P 0P aR4. The displacementsolver (red circles) is compared with the steady inertialsolutions of the flow computed using FreeFEM++ withmesh adaptation (solid lines). The rows are gz = 10, allwith Hz = 500. . . . . . . . . . . . . . . . . . . . . . . . . 221Figure 6.9 x-velocity profiles u(xP y) at three sections of the washout(left to right): x = −aR4P 0P aR4. The displacementsolver (red circles) is compared with the steady inertialsolutions of the flow computed using FreeFEM++ withmesh adaptation (solid lines). The rows are gz = 50, allwith Hz = 500. . . . . . . . . . . . . . . . . . . . . . . . . 222xxiiiFigure 6.10 x-velocity profiles u(xP y) at three sections of the washout(left to right): x = −aR4P 0P aR4. The displacementsolver (red circles) is compared with the steady inertialsolutions of the flow computed using FreeFEM++ withmesh adaptation (solid lines). The rows are gz = 100, allwith Hz = 500. . . . . . . . . . . . . . . . . . . . . . . . . 222Figure 6.11 x-velocity profiles u(xP y) at three sections of the washout(left to right): x = −aR4P 0P aR4. The displacementsolver (red circles) is compared with the steady inertialsolutions of the flow computed using FreeFEM++ withmesh adaptation (solid lines). The rows are gz = 100, allwith Hz = 500. . . . . . . . . . . . . . . . . . . . . . . . . 223Figure 6.12 x-velocity profiles u(xP y) at three sections of the washout(left to right): x = −aR4P 0P aR4. The displacementsolver (red circles) is compared with the steady inertialsolutions of the flow computed using FreeFEM++ withmesh adaptation (solid lines). The rows are gz = 500, allwith Hz = 500. . . . . . . . . . . . . . . . . . . . . . . . . 223Figure 6.13 Colormaps of x for H = 2P a = 4P b = 0O1 and gz =10P 50P 100P 200 from top to bottom. The flow is fromleft to right and the times are t¯ = 4P 4P 4P 4 . . . . . . . . 224Figure 6.14 The speed colormap for H = 2P a = 4P b = 0O1 andgz = 10P 50P 100P 200 from top to bottom (the sameflows as Figure 6.13). . . . . . . . . . . . . . . . . . . . . . 225Figure 6.15 Colormaps of x for H = 2P a = 4P b = 9 and gz =10P 50P 100P 200 from top to bottom and times are t¯ =4P 4P 4P 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Figure 6.16 Colormaps of x for H = 2P a = 4P b = 27 and gz =10P 50P 100P 200 from top to bottom and the times aret¯ = 4P 4P 4P 4. . . . . . . . . . . . . . . . . . . . . . . . . 227Figure 6.17 Colormaps of x for H = 2P a = 10P b = 9 and gz =10P 50P 100P 200 from top to bottom and times are t¯ =5P 5P 5P 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 228xxivFigure 6.18 Colormaps of x for H = 2P a = 10P b = 27 and gz =10P 50P 100P 200 from top to bottom and the times aret¯ = 5P 5P 5P 5. . . . . . . . . . . . . . . . . . . . . . . . . 229Figure 6.19 Colormaps of x for a = 20 and the rest of parametersthe same as Figure 6.18. Flow demonstrates a steady fulldisplacement. . . . . . . . . . . . . . . . . . . . . . . . . . 230Figure 6.20 Colormaps of x for H = 2P a = 20P b = 3 and gz =10P 50P 100P 200 from top to bottom and snapshot timesare t¯ = 6P 6P 6P 6. . . . . . . . . . . . . . . . . . . . . . . 232Figure 6.21 Colormaps of x for H = 2P a = 20P b = 0O3 and gz =10P 50P 100P 200 from top to bottom and times are t¯ =6P 6P 3O42P 1O5. . . . . . . . . . . . . . . . . . . . . . . . . 232Figure 6.22 Colormaps of x for H = 2P a = 10P b = 0O1 and gz =10P 50P 100P 200 from top to bottom and times are t¯ =5P 5P 3O7P 1O84. . . . . . . . . . . . . . . . . . . . . . . . . 234Figure 6.23 The evolution of x for H = 2P a = 20P gz = 100P b = 0O3 235Figure 6.24 The stagnant fluid circled in a) is displaced by yielding,and can be tracked in b). Parameters are H = 5P a =10P gz = 100P b = 0O1 . . . . . . . . . . . . . . . . . . . 236Figure 6.25 Progress of x for a stable displacement. H = 5P a =20P gz = 10P b = 0O3. . . . . . . . . . . . . . . . . . . . . 237Figure 6.26 Colormaps of x for H = 2P a = 10P b = 0O3 and gz =10P 50P 100P 200 from top to bottom and times are t¯ =5P 5P 5P 1O3. . . . . . . . . . . . . . . . . . . . . . . . . . . 238Figure 6.27 Colormap of the speed forH = 2P a = 10P gz = 100P b =0O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238Figure 6.28 Comparing the effect of increasing H on final shape of thedisplacement. Colormaps of x are shown. Left columnH = 5 and right column H = 10. Rows from top tobottom: gz = 10P 50P 100 and a = 10P b = 0O3. . . . . . 239Figure 6.29 The effect of increasingH in unstable flows. a = 20P gz =200Pb = 0O3 and from top to bottom H = 2P 5P 10. . . . . 243Figure 6.30 Colormap of x for H = 10P a = 20P gz = 200P b = 27 . 243xxvFigure 6.31 Colormaps of x for H = 10P a = 20P b = 0O3 and fromtop to bottom gz = 10P 50P 100. . . . . . . . . . . . . . . 244Figure 6.32 Colormaps of x for increasingb = 0O1P 0O3P 3P 9P 27 fromtop to bottom and H = 10P a = 20P gz = 10. Times aret¯ = 7O4P 9O4P 8O9P 11O5P 10 respectively . . . . . . . . . . . 245Figure 6.33 Colormaps of x for increasingb = 0O1P 0O3P 3P 9P 27 fromtop to bottom and H = 2P a = 10P gz = 10. Times aret¯ = 5 for all. . . . . . . . . . . . . . . . . . . . . . . . . . . 246Figure A.1 Geometry of the encapsulated droplet train . . . . . . . . 302Figure A.2 Contours of the m-component of velocity obtained by theperturbation solution; H = 0O2,  = 0O1 and W = 1: a)leading order j0(mPn ); b) corrected velocity j0(mPn ) +j1(mPn ). Uncorrected yield surface position yy(m) markedwith dashed line and corrected yield surface yg (m) markedwith solid line. Pressure gradient for both solutions is alsoplotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316Figure A.3 Position of the yield surfaces yg (m) (solid line) and yy(m)(dashed line) for a range of different (elliptical) dropletsat W = 10. Top row a = 0O5, bottom row a = 0O75 andcolumns left to right H = 0O05, 0O1 and 0O2. Note thatyyN0 is equal to the values of yy(m) at two ends. Due tothe adopted scaling, all elliptic droplets are mapped to acircle with radius of H. . . . . . . . . . . . . . . . . . . . 318Figure A.4 Mesh adaptation cycles for the case of W = 20, H = 0O1and a = 0O5, where we start with an initial mesh scale,y0 ≈ 0O02: a) cycle 1, 3165 elements; b) cycle 2, 12689elements; c) cycle 3, 21425 elements; d) cycle 4, 37290elements. After the 4th cycle the typical mesh sizes are:in the plug: 0O02; near the yield surface: 0O001 × 0O004;near the wall: 0O002. . . . . . . . . . . . . . . . . . . . . . 322xxviFigure A.5 Code Validation: Error of numerical results for channelPoiseuille flow compared to the exact solution; © : W =10, ♢ : W = 20 and △ : W = 50. a) velocity and b) pressure322Figure A.6 Example of the effects of droplet spacing for W = 20,H = 0O5 and a = 0O6: a) l = 1O625; b) l = 1O8; c) l = 2O5.The colour map shows the range of speeds in the flow andgrey areas show unyielded fluid. . . . . . . . . . . . . . . . 324Figure A.7 Variation in the pressure drop upstream of the dropletplotted versus a scaled spacing between droplets. a) W =10, ▽: H = 0O45, a = 0O6; ©: H = 0O55, a = 0O15;□: H = 0O05, a = 0O75; T: H = 0O25, a = 0O25 and ♢:H = 0O375, a = 0O5 b)W = 20, ▽ : H = 0O5, a = 0O6; ©:H = 0O65, a = 0O2; □: H = 0O1, a = 0O9; T: H = 0O25,a = 0O25 and ♢: H = 0O375, a = 0O5. The broken linesshow the pressure drop for Poiseuille flow. . . . . . . . . . 325Figure A.8 Speed distribution (u) as the droplet gets larger. Rowsrepresent constant Bingham number; H = 0O2 over leftcolumn (slznyzr yrop) and a = 0O5 over right column(fvt yrop). a) W = 5 and a = 0O55, 0O6, 0O65, 0O675 and0O7; b)W = 5 and H = 0O3, 0O34, 0O375, 0O4 and 0O425;c)W = 20 and a = 0O8, 0O85, 0O9, 0O925 and 1;d) W = 20and H = O5, 0O55, 0O575, 0O61 and 0O625; e)W = 50 anda = 0O95, 1, 1O05, 1O15, 1O2 f)W = 50 and H = 0O6, 0O65,0O69, 0O725 and 0O75 . . . . . . . . . . . . . . . . . . . . . 326Figure A.9 Stress distribution as the droplet gets larger. W = 20,H = 0O2 and ;a = 0O8, 0O85, 0O9, 0O925 and 1. a) speed;b)pressure; c)xy; d)xx; e)xx = −p + xx and f)yy =−p+ yy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328Figure A.10 Stress distribution as the droplet gets larger. W = 20,a = 0O7 and H = 0O5, 0O55, 0O575, 0O61 and 0O625. a)speed; b)pressure; c)xy; d)xx; e)xx and f)yy . . . . . . 330xxviiFigure A.11 Variation of pressure perturbation parameter ϕp with sizeof the droplet for different Bingham numbers (◦ : W = 5,□ : W = 20 and △ : W = 50); a) drops with varyinglength and constant height (H = 0O2) and b) drops withvarying height and constant length (a = 0O5). In eachsubplot the largest value of a or H corresponds to a casewith a yielded plug. . . . . . . . . . . . . . . . . . . . . . 331Figure A.12 Maximum size of encapsulated droplets, computed for fivedifferent W. ©: W = 5; □: W = 10; △: W = 20, ♢:W = 50 and T: W = 200 . . . . . . . . . . . . . . . . . . . 332Figure A.13 W = 20 (a) Variation of maximum height (maximum Hwhich does not break the plug) for two different length ofdrop □: a = 0O4 and©: a = 1O75 with density difference(). (b) Maximum size of drop for for three different .©:  = 20; □:  = 0 and △:  = −20. . . . . . . . . . . 334Figure A.14 Stress distribution as the droplet becomes heavier: W =20, a = 0O9 and H = 0O2. From top to bottom in eachpanel:  = −10, −1, 0, 1 and 10. a) speed; b) pressure;c) xy; d) xx; e) xx and f) yy. . . . . . . . . . . . . . . 336Figure A.15 Stress distribution in the axisymmetric geometry as thedroplet gets longer: W = 20, H = 0O4 and a = 1O2, 1O25,1O3, 1O34 and 1O45. a) velocity; b) pressure; c) rz; d) zz;e) rr and f) . . . . . . . . . . . . . . . . . . . . . . . . 341Figure A.16 Stress distribution in the axisymmetric geometry as thedroplet height is increased: W = 20, a = 0O5 and ;H =0O4, 0O45, 0O5, 0O525 and 0O55. a) velocity; b) pressure; c)rz; d) zz; e) rr and f) . . . . . . . . . . . . . . . . . . 342Figure A.17 Maximum size of encapsulated droplets for four differentBingham numbers. ©: W = 5; □: W = 10; △: W = 20, ♢:W = 50 Included for comparison is a single curve showingthe maximum size of droplet in the channel geometry forW = 10 (broken line). . . . . . . . . . . . . . . . . . . . . 344Figure A.18 Encapsulation of drops with exotic shapes. . . . . . . . . 347xxviiiFigure B.1 Variation of residuals of strain rate multiplier (left) andvelocity field (right) with mesh adaptation. . . . . . . . . 351Figure B.2 The speed colormap with unyielded regions and the cor-responding mesh for different adaptation cycles. . . . . . 353Figure B.3 The variation of contours for stress. The solid gray is = y and the lighter shows  = y + O083. Parametersare H = 2P a = 20P W = 10000. . . . . . . . . . . . . . . . 354Figure B.4 The traction vector on the boundary of the unyielded re-gions for W = 2, H = 3, a = 10,  = 0O5 (top row inFigure 3.4). The bottom figure show a zoom on part ofribbon unyielded fluid which shows continuous tractionalong it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355xxixVxknofilzygmzntsThis research has been carried out at the University of British Columbia,supported financially by NSERC and Schlumberger through CRD project444985-12 and I appreciate this support. I also thank the department ofmechanical engineering of the University of British Columbia for giving methe excellence award.I am deeply grateful to my PhD supervisor, Professor Ian Frigaard.Preparation and submission of this thesis would not have been possible with-out Ian’s guidance, support and patience. The best I canb briefly describeIan is a sharp scientifically disciplined supervisor with a great sense of hu-mor who tries to pay attention to every aspect of his students, not only theacademic one. I really enjoyed being your student and I wish you all thebests. It would be my privilege to continue working with you.I would also like to thank my supervisory committee members, An-thony Wachs, Neil Balmforth and Carl Olivier-Gooch. Neil, I enjoyedyour asymptotic analysis course. Carl, your computational transport phe-nomenon courses was a complete journey from introductory to advancedtopics of finite volume method, which I liked much. Thank you Anthony foryour right to thz point advices which come from years of experience. Some-times just one sentence from you was the key to the solution of a problem.It was always a joy to talk with you.I thank all of our lab members, interns and everyone who I have workedwith during these years. Forgive me if I am not mentioning the names asthere would be too many people and I do not want to miss any, thank youall.xxxMother and father, you are at my heart and I have no meaning withoutyou. I do not know how to thank you for everything you gave me, yourendless love, kindness, support and whatever I can think of. I hope thisthesis can bring a small smile and feeling of satisfaction to you. Ali is stillyour small son and will remain so for ever.Last but not least, I want to thank my wife, my close companion inexperiencing every moment of this journey, especially the hard momentswhich were definitely much harder without you. We had some adversitiesin the early years of coming to Canada, however never forgot to love eachother and the time taught us how we can overcome anything together. Nowwe are looking at the horizon together, ready for the future with more loveand energy.Ali RoustaeiVancouver, BCJanuary 2016xxxiTo am big brohher Mahdi,I iieh you iere here . . .xxxiiChvptzr FIntroyuxtionMuch of the work presented in this thesis is motivated by flows that are foundin oil & gas well construction and later in well stimulation (although also oc-curring in other industrial and natural settings). The fluids are viscoplastic,meaning that they are viscous fluids with a yield stress. The geometries thatresult in these industrial flows have certain non-uniform features that resultin interesting static regions and also in limiting parameters that demarcateofi from noBofi, i.e. yielding.In this chapter we first explain industrial origins of the flows that westudy (§1.1) and hence part of the motivation for this thesis. Secondly,we review general features of yield stress fluids (§1.2). In §1.3 we give anoverview of the variational principles that underly viscoplastic fluid flows.Computational methods are discussed in §1.4. Section 1.5 reviews specificyield stress fluid flows that show similar geometric effects to those studiedhere, e.g. static regions and self-selection of the flowing domain. Finally,we close the chapter with an overview of the research objectives and thesisoutline (§1.6).FCF Inyustrivl motivvtionsSince the production of oil from Edwin Drake’s 1859 well near Titusville,Pennsylvania, the petroleum industry has grown tremendously. Drake’s well1can be considered a founder of modern commercial oil production practice:(a) it was drilled, not dug; (b) it used a steam engine [251] to power thedrilling; (c) there was a company associated with the drilling. Today oilrepresents the blood of the modern economy and many other industries likecommercial aviation, automotive, electronics, chemical and pharmaceutical,materials and others heavily depend on the availability of oil.This thesis considers single- and two-phase flows of viscoplastic fluidsflows in a variety of nonuniform geometries. The main characteristic ofthese fluids is that they deform only when the applied stress exceeds alimit ˆy called yield stress. Below ˆy they show rigid solid-like behaviour.This rheological behaviour can have an impact on many upstream oil & gasprocesses, which we now review.FCFCF erimvry xzmzntingWell construction is the most critical and technical part of oil & gas ex-traction engineering. Without a long life and trustworthy well, reservoirproduction is unlikely to reach its full potential. The key to such a wellis effective zonal isolation of different layers of the reservoir at the well.An innovation of Drake’s well was in drilling through a metal pipe (casing)to prevent collapse of the well and as a later pathway for producing oil.This idea is still employed and further improved by the process of primarycementing.Primary Cementing is a technique in well construction for placing cementslurry in the annular space between the metal casing and the borehole. Atypical primary cementing operation proceeds as follows. After the drillingis done, the casing is lowered into the well which is full of drilling mud. Thencement is pumped into the casing at the well head, cement goes all the waydown inside the pipe (casing), exits at the bottom, and displaces drillingmud as it moves back up the annulus. The cement has several roles [176]:1. The cement provides a hydraulic seal in the wellbore, preventing themigration of formation fluids in the annulus.2. The cement anchors and supports the casing.23. The cement protects the casing against corrosion, due to hot formationbrines and/or hydrogen sulfide (H2h)As well as cement, a number of other specially designed fluids (e.g. washesand spacers) are pumped ahead of the cement slurry to aid displacement/re-moval of the drilling mud, reduce cement contamination and to improve theeventual bonding to the outside of the casing and to the formation.Today’s oil wells can be over 10km long, with long horizontal sections.For these wells, having good hydraulic sealing provided by the cement through-out the well is critical to both safety and efficiency. A poor seal results inproduction loss and environmental effects. In extreme situations it maylead to loss of well control and to a blow out. A notorious example is theDeepwater Horizon blow out in the Gulf of Mexico at April 2010, whichcaused eleven deaths, sinking and explosion of the rig and massive oil spillin the Gulf. It was the largest environmental disaster in U.S. history [250].Later, investigations mentioned that not enough centralizers were used forthe cementing job [142], resulting in poor mud removal and leaving well inan unstable state.From a fluid mechanics perspective, primary cementing is a displace-ment flow. This is a term generally used for a two phase flow in which onephase (displacing fluid) is forcing another phase (displaced fluid) to moveout from its current location. Initially the displaced fluid is at rest, thedisplacing fluid is injected with pressure at inlet(s) and the displaced fluidstarts to exit from drain(s). Displacement flows are important in a wide va-riety of applications, including secondary and tertiary oil recovery, fixed bedregeneration in chemical processing, hydrology, and filtration [122]. Also inbio-medical applications (mucus [128] and biofilms [191]) and in food pro-cessing for cleaning of pipes and machinery [108]. An important parameterof these flows is the displacement efficiency, which is defined as the fractionof displaced fluid that is washed by the displacing fluid. A displacementefficiency of 1 is always desired as the goal is to completely evacuate thedisplaced fluid.Displacement flows like primary cementing are very rich in terms of phys-3MudWashoutCementflowCasingFigurz FCFO Primary cementing and static mud in washout regionical phenomena. This is due to having (at least) two fluids with differentdensity, viscosity, yield stress and other properties, all of can be differ. Buoy-ancy effects may be important and density and viscosity differences can causecomplex interfacial dynamics, e.g. Kelvin-Helmholtz and Rayleigh-Taylor in-stabilities. In the case of primary cementing, drilling mud, spacer fluid andcement slurries are often miscible with a weak molecular diffusion that leadsto high Pe´clet number in cementing flows. This means that over processtimescales a relatively sharp interface exists between fluids, provided thatno instability occurs.Another interesting issue with primary cementing flow which makes theprocess more complicated is that it consists of two different stages: (a) flowinside the casing (pipe geometry) when cement has not yet exited at thebottom and moves downward; (b) flow inside the annular gap, in which thefluids move upward. Having both downward and upward flow inevitablycauses density unstable flow in one direction which is not desired, due to4unstable mixing of the fluids. Usually for the downward displacement plas-tic plugs are used to separate the fluid stages. However, sometimes this isnot possible due to the operating conditions. The geometries are also verydifferent: the downward displacement geometry is a pipe of 3cm–50cm di-ameter; the upwards annular displacement can have an annular gap thatvaries between 2mm–5cm.In this thesis we study features of the narrow annular flows, both singlephase and displacement flow, and the focus is on the study of flow withinwvshouts. As mentioned, the annular gap is usually narrow with somestatistical variations in aperture due to drilling variability. However some-times, there are large variations in the gap width due to partial collapse ofwellbore during drilling. These regions are called washouts (Figure 1.1) andmay exist due to several reasons [176]:1. Soft and unconsolidated formations2. In-situ rock stresses3. Excessive jet velocity of fluids passing through the drillbit4. Chemical reactions5. Mechanical damage by the rotating drill stringSingle phase flow occurs just prior to cementing, when the drilling mud iscirculated around well to condition the mud. Displacement flows occur dur-ing mud removal/cement placement. Except for very shallow washouts, awashout always lowers the quality of primary cementing. This is because thecement fails to displace drilling mud in the depths of a washout section andmud stays attached to the wall after cementing. If static regions are con-nected longitudinally they can allow a path through which formation fluidsand gas can migrate to surface. Geometry is a key factor in the appearanceof static regions and washout geometries are unpredictable. Hence we needto systematically address the geometric diversity of washouts and the effectsthis has on the flow. This issue is examined in Chapter 4.5Figurz FC2O DeepWater Horizon explosion at Gulf of Mexico, April2010. One of the main causes was poor primary cementingFCFCG elug xzmznting for vwvnyonmzntCement plugs are set in wells sometimes during construction, e.g. to initi-ate directional drilling by providing a firm base for drill bit to turn (side-tracking), and always at the end of a well’s life in the abandonment process;see [176]. It is common to set a number of plugs to isolate producing zonesof the reservoir. Different regulations and methods exist worldwide. Theplacement process typically involes pumping cement down a narrow tube,displacing upwards the wellbore fluids. The geometry in the upwards flow-ing stage is thus again annular, similar to the primary cementing operation,although the annulus is no longer narrow.A recent trend in the North Sea has been to mill out the wellbore,through the casings and cement, so that the set plug fully isolates and sealsacross the steel and cement layers. These milled out sections are ∼ 100min length. The annular geometry of the milled out “window” is thus some-what similar to that of a deep and long washout in primary cementing. The6CementPackerPackerCasingPerforationCasing Formation Primary cement sheath Figurz FC3O Squeeze cementing, used for correction of primary cement.Right: showing the schematic of equipment for squeeze. Left: acloser view to fracture flow, courtesy of GKE engineeringconcern again is that the pumped slurry manages to effectively displace thewellbore fluids and bonds well to the rock formation.FCFCH hquzzzz xzmznting vny othzr rzmzyivl tvsksAfter primary cementing is performed, the hope is that a continuous sealhas been obtained along the whole length of the well. However, this doesnot happen always. Equally, sometimes later in the producing life of a wellthe cement seal may deteriorate (e.g. due to thermal or geophysical stresscycling), and perhaps crack. These defects are repaired by the squeezecementing process. In this operation the casing is typically perforated andisolated (Figure 1.3) with packers at the location of interest, before cementis forced into specific locations under an applied pressure.Although squeeze cementing usually refers to the phase after well com-pletion, i.e. to repair cracks/fissures or fill severe washouts, all in order toimprove productivity, other similar scenarios arise. During construction, ce-ment could be pumped into the rock formation to cure lost-circulation (also7Without Retaining “Platform”or a cement plug may be used as the “platform”. Figurz FCIO Plug cements are used to persistently isolate necessaryzones for abandonment, Figure courtesy of GKE engineeringpotentially linked to cement plugging). At the abandonment stage, somesqueezing is done in sections of the wellbore that have been perforated/frac-tured for production, i.e. to fill the fissures before the well itself is plugged(Figure 1.4).FCFCI Hyyrvulix frvxturingHydraulic fracturing is a well-stimulation technique used to boost oil andgas wells production. It is done by injecting a mixture of fluid and grains(called proppvnt) down the well at very high pressures (Figure 1.5). Thismixture enters into well perforations (small explosively initiated fractures)and fracture the rocks, resulting in an increased effective permeability whichallows hydrocarbon to flow easier. Common proppants are sand, ceramics oraluminum oxide. After the rock is fractured to the designed level, pressureis removed and the fracture closes on the proppant, squeezing the frac fluidaway. During this ofiwvxk stage the returning fluid is a mix of injectionfluid, formation brines, hydrocarbons and other well contaminants, all ofwhich are collected at the well head.Hydraulic fracturing is not a new technology, it has been used sinceat least 1950. However, there has been an increasing interest in recent8Figurz FC5O Depiction of fracturing, photo from US EnvironmentalProtection Agency [3]years, as conventional hydrocarbon resources are dwindling and productionfrom unconventional reserves such as shale gas and sandstone oil/gas hasbecome economically competitive with the aid of fracturing and extendedreach horizontal drilling. These resources are present in Canada, where alarge percentage of the reserves are held in sandstones. In British Columbiacurrent reserves are mainly natural gas in held in tight shales. The fracturingprocess is essential for developing these resources. There are also environ-mental concerns associated with fracturing. First, large amounts of waterare needed for fracturing, which is obtained from a mix of land and under-ground water. Usually the returned fluid (waste water) is treated and usedagain. However, during the process, frac fluid leaks off into the surroundingrocks. If not controlled properly, leak off can even be as large as %70 of9the initial volume [252]. This means that considerable fresh water will beneeded for each new well. Second, the leak-off fluid can contaminate un-derground water reservoirs used for drinking and agriculture, causing publichealth problems. Third, large fracturing jobs change the stress distributionin rock layers and hence seismic activity of the location. There has beenrecent concern that fracking may potentially trigger earthquakes.Many fracking operations simply use water with drag reducing agents,suspending the proppant (sand) in turbulent eddies. Viscous fluids are alsoused for fracking and these are basically water with the addition of differentchemicals and gelling agent which creates a yield stress in the fluid. Theyield stress helps to prevent proppant settling and gives better transportproperties. Although the yield stress is useful in this role, it also resultsin higher frictional losses. Also during the flowback stage the gelled fracfluid might get stuck in the fracture, reducing the permeability of fracture.Therefore the design of fluid properties must be carefully considered.In this thesis we consider the flow rate–pressure drop relation of a yieldstress fluid flowing along simple fractures. We specifically consider the prob-lem of the critical pressure drop necessary to start the flow in fracture, whichis also related to the invasion type flows discussed in squeeze cementing (seechapter 3).FCFC5 dthzr frvxturzBlikz gzomztrizsAs well as artificially occurring fractures, the flow of yield stress fluid innatural fractures is also important. Examples include drilling mud loss intoformations during drilling and various methods advocated for sealing Xd2sequestration reservoirs by injecting cement. In both scenarios we have theflow of a yield stress fluid into a porous media and due to yield stress thefluid invades only into a finite volume of the media for a given pressure. Thequestion here is that how far does the fluid invades? This also relates closelyto the problem of the critical pressure drop required to initiate flow along afracture (see chapter 3).10FCG nizly strzss uiysAs mentioned yield stress or viscoplastic fluids deform only after a certainstress limit ˆy is overcome. There are many examples of these fluids:1. Health/Cosmetics: gels, creams, toothpaste2. Foods: chocolate, yogurt, soft butter, cake batter, whipping cream,wet pasta O O O3. Natural and biological: Lava, mucus, some suspensions4. Industrial: Cement slurries, drilling mud, mine tailing, O O OThe simplest and most commonly used rheology model was first suggestedby Bingham [32], which corresponds (in 1D) to adding a yield stress to theNewtonian viscosity:ˆ = ˆy + ˆyuˆyyˆP ⇔ ˆy Q |ˆ | (1.1)yuˆyyˆ= 0P ⇔ ˆy ≥ |ˆ | (1.2)Here ˆ is the shear stress in a 2D parallel flow with velocity field uˆ(yˆ) inthe xˆ direction, ˆ is the plastic viscosity. The extension of this model to 3Dtensorial formulation was introduced by Prager [120, 195]:ˆij =(ˆ+ˆyˆ˙(uˆ))ˆ˙(uˆ)P ⇔ ˆy Q ˆ (1.3)ˆ˙(uˆ) = 0P ⇔ ˆy ≥ ˆ (1.4)Where uˆ is velocity vector, ˆij is the deviatoric stress tensor,ˆ˙(uˆ) = (∇uˆ+∇uˆg )Pis the strain rate tensor and the norms of strain rate and stress are definedas:ˆ˙(uˆ) =√12ˆ˙ij ˆ˙ij P ˆ =√12ˆij ˆij (1.5)11The locus of points with ˆ = ˆy represents the boundary of yielded and un-yielded regions and is called the yizly surfvxz. Prager also contributed to thevariational theory of viscoplastic fluid, especially minimization-maximizationprinciples for stokes flow which will be presented later. The Bingham modelin dimensionless1 form becomes:ij =(1 +W˙(u))˙ij P ⇔ W Q P (1.6)˙ij = 0P ⇔ W ≥ O (1.7)The dimensionless Bingham number W = ˆy lˆRˆjˆ is the ratio of the yieldstress to a typical viscous stress; lˆP jˆ are length and velocity scales respec-tively.In practice almost no fluid has the idealised behaviour of the Binghammodel. A more realistic model is the Herschel-Bulkley fluid which considersa power-law type of viscosity in the Bingham model. In dimensionless form:ij =(˙n−1 +W˙(u))˙ij P ⇔ W Q P (1.8)˙ij = 0P ⇔ W ≥ WO (1.9)For n Q 1 the fluid is shear thinning, meaning that the viscosity decreaseswith increasing shear rate. For n S 1 the fluid is shear thickening and theviscosity grows with shear rate. For n = 1 the model reduces to Binghamfluid.The Casson model is a more complicated and less common model that isbased on a structural model for the dynamic behavior of a solid suspension.It also includes a shear thinning effect.ˆij =(√ˆ+√ˆyˆ˙(uˆ))2ˆ˙ij P ⇔ ˆy Q ˆP (1.10)ˆ˙ij = 0P ⇔ ˆy ≥ ˆ O (1.11)1In thz thz“i“ vll ph–“ixvl quvntitiz“ fiith yimzn“ion vrz “hofin fiith hvtA vzxtor vnytzn“or zntitiz“ fiith woly vny “xvlvr vvluz“ u“ing normvl font12This model by also be generalised, replacing the square root with a power-law, i.e.ˆij =(ˆn +(ˆyˆ˙(uˆ))n)1Pnˆ˙ij ONote that the Bingham, Herschel-Bulkley and Casson models are dif-ferent only in the effective viscosity term. Hence, we expect the yieldingbehavior of the the models to be very similar.FCGCF ihiflotropyThe origin of the yield stress is related to the structure of the material.This means that the value of the yield stress may change with the state ofmaterial structure. When the rheology is time dependent in this way, we saythe the fluid shows thiflotropix behaviour. In fact many viscoplastic fluidssuch as muds, paints and food products show this behavior; see [65].Thixotropy generally widens the single yield stress value to a range be-tween a dynamic yield stress (ˆyw) and a static yield stress (ˆys). After afluid stops deforming, at the dynamic yield stress, the thixotropic structureof the material starts to form and the yield stress increases to its static limit,which might be significantly higher than the dynamic limit. If the materialis sheared again the structures is destroyed and the yield stress returns tothe dynamic value. Usually the time scale for reaching ˆys is much longerthan the reverse process, i.e. the structures are easily destroyed, but it takeslong time to heal back.Thixotropic yield stress fluid models usually make the yield stress (andother rheological parameters) dependent on a (scalar) structural variable. An evolution equation is introduced for this variable that contains bothconstruction and destruction terms. One simple and illustrative model is13from Houska [151], which takes the form (fixing n = 1):ˆy = ˆ0 + ˆ1 (1.12)ˆ = ˆ0 + ˆ1 (1.13)UUt+ uˆO∇ = vˆ(1− )− wˆˆ˙m (1.14)Here  ∈ [0 1] is the structure parameter, ˆ0P ˆ0 are the dynamic yieldstress and viscosity, ˆ1P ˆ1 are the thixotropic yield stress and viscosity, vˆ isthe buildup parameter, wˆ is the breakdown parameter and m an adjustableparameter. As you can see the yield stress and viscosity vary linearly with. Also we observe that when the fluid is at rest (uˆ = 0P ˆ˙ = 0) the structurebuilds up in an exponential way towards  = 1, which is the static structure.There are many different thixotropic models. In some of these destruc-tion of the structure is shear stress dependent, rather than ˆ˙-dependent; seee.g. [77, 78]. In other models the yield stress results wholly from thixotropy;see e.g. [68, 69, 202].FCGCG ihz yizly strzss yzwvtzNo matter whether analytical or computational methods are used for thesolution of viscoplastic flow, a key problem is to correctly identify unyieldedregions in the flow. The reason is that classical viscoplastic models giveconstitutive relations only for the yielded part of the flow. Inside unyieldedregions the only thing known is that the stress is below the yield stress andsatisfies the momentum equations. So it is not possible to solve the systemuntil we know the position of the yield surfaces, which themselves depend onthe stress field and are not known initially. The singular behaviour at theyield surface has motivated a lot of efforts in the viscoplastic communityto resolve the (yield surface) free boundary problem, in both analyticaland computational directions. The na¨ıve consideration of unyielded regionsresulted in the so called luwrixvtion pvrvyofl [146], where it was claimedthat it would be impossible to have true unyielded regions within flows withcomplex geometries, e.g. a varying width channel. This paradox will be14discussed in more detail later.The singularity of the effective viscosity has been a motivating force toquestion the actual existence of a rzvl yield stress. This was first raised byBarnes and Walters [24] who argued that the unyielded fluid is actually avery high viscous state at low shear rates and the known yield stress is athreshold at which abrupt change of viscosity occurs. In a later paper [23]Barnes gives evidence to support his viewpoint and asserts that the so called“yield stress” fluids are merely shear thinning with a very high viscosity atlow shear. Based on this idea, different rheological models have been pro-posed, e.g. [158], although such models were also proposed for computationalease much earlier. However, justification of Barnes’ claim needs strong ex-perimental evidence which is hard to obtain, as low shear rate experimentsare very delicate to do even with todays high precision rheometers. Oneimportant discovery with the help of modern equipment, is that the flowsat low stresses are not always steady; see [72]. Other complications such aswall slip and transient shear banding make low shear experiments hard tointerpret [70, 87]. There is much recent and ongoing work around this issue[50, 55, 71, 167, 184]. However it is not yet fully agreed how to define andmeasure a true yield stress.In practice even if a viscoplastic fluid is only highly viscous (and notreally unyielded), the timescale of the low shear flow is often so long that wemay consider the fluid as an ideal yield stress fluid. This is true for manyproblems in industry, such as those considered in this thesis. Thus, modelslike Bingham are still useful and effective. We will use this model for allthe work presented in this thesis, as it simplifies but still keeps the mainphysical effects that we are interested in studying.15FCGCH Govzrning zquvtionsThe Navier-Stokes equations govern the flow of a yield stress fluid. For anincompressible fluid with constant density the dimensionless equations are:gz(UuUt+ uO∇u)= −∇p+∇O + f (1.15)∇Ou = 0 (1.16)where the second equation is the conservation of mass, which has been re-duced to the divergence free condition for incompressible flows. Here thestresses are scaled with the viscous stress scale: (ˆjˆ0Raˆ) and the Reynoldsnumber is (/ˆjˆ0aˆRˆ), representing the ratio of inertial to viscous stresses.The dimensional values ˆP jˆ0P aˆP /ˆ are the plastic viscosity, velocity scale,length scale and fluid density, respectively. The source term f representsany external body force, e.g. typically gravity.Having the above equations and a rheological model e.g. Bingham, thesystem is closed and we should be able to solve the equations. Analyticalsolutions can be found for a number of situations like plane channel flow,or the lubrication approximation used for asymptotic solutions. For morecomplicated flows computation is used.eoiszuillz ow of Binghvm uiyTo introduce the problem of finding free boundaries in yield stress fluidflows we consider the plane Poiseuille flow of a Bingham fluid. The solu-tion of this flow of can be derived analytically. We consider the 2D in-finite channel with y = −1P 1 as the dimensionless locations of bottomand top walls, respectively; see Figure 1.6. No-slip boundary conditionsare is applied on these walls and we consider a steady fully developed flow.With these considerations the flow field and pressure gradient would beu = (u(y)P 0)P ∇p = (−fP 0) respectively, i.e. the flow is driven by the ap-plied pressure gradient f . The strain rate tensor and its norm becomes:˙xx = ˙yy = 0P ˙xy = ˙yx =yuyyP ˙ = |yuyy|O (1.17)16It follows that the only non-zero component of stress is xy and the Binghammodel reduces to:xy =1 + W∣∣∣∣yuyy∣∣∣∣ yuyy P ⇔ |xy| ≥ W; yuyy = 0P ⇔ |xy| Q WO (1.18)The momentum equations simplify to:0 =UxyUy+ fO (1.19)By integrating the above equation and knowing that xy(0) = 0 (from sym-metry) the stress distribution is found:xy = −fyO (1.20)From this solution we can instantly identify |y| ≤ ys = WR|f | as the un-yielded region of the flow. This part would be moving with a constantvelocity and is called a plug region as shown in Figure 1.6. Also we see thata minimum pressure gradient |f | = W should be applied to start the flow inthe channel. For |f | ≤ W the plug fills the entire channel and there wouldbe no flow. Using (1.18) and knowing that yuRyy ≤ 0 for the upper half ofchannel we get:yuyy−W = −fy for y ∈ [ys 1]P u(1) = 0P u(|y| Q ys) = upO (1.21)Integrating the above gives the velocity profile as:u(y) =y2 (1− ys)2 = upP ⇔ |y| ≤ ysPy2 [(1− ys)2 − (|y| − ys)2]P ⇔ |y| S ysO(1.22)Here up is the velocity of central plug which is the maximum velocity.Another formulation of the channel flow problem assumes a fixed flowrate(in place of fixing f). Here we would scale velocity with the mean velocity17unyielded regionyielded regionyielded regionxyy=1y=-1Figurz FC6O The velocity profile and yielded/unyielded fluid in channelflow of a yield stress fluidand then can then find ys by satisfying the flowrate constraint:∫ 10 uyy = 1.This leads to the Buckingham equation [42]:y3s − 3(1 +2W)ys + 2 = 0P (1.23)which has a single root ys ∈ [0P 1]. Having found ys, we can find f =WRys and the rest is as before. This formulation is useful e.g. for flow in anonuniform channel.If we look back to the solution procedure, it was possible to solve thisproblem analytically because we could obtain the stress distribution firstand then could use that to identify the unyielded region of the flow. Finallywith knowing ys we were able to integrate (1.22) over the appropriate range[ys 1] to obtain the velocity profile. In most of the problems it is not possibleto find the stress distribution first, hence we cannot find the yield surfacesand this is the main challenge.dthzr vnvl–tixvl solutions of –izly strzss uiyOther than Poiseiulle flow there are a few other simple cases like Couette flowand flow between two concentric cylinders that can be solved with the sameprocedure. The review paper of Bird et al. [201] gives analytical solutionsfor a variety of standard geometries. One of the rare 2D analytical solutionsis the shear flow in antiplane geometry which is derived by Burgos et al. [44],using a hodograph transformation. In addition to these, it is also possible tofind asymptotic solutions if careful considerations are taken. Examples here18are [17, 102, 246], all of which were able to construct physically consistentvelocity fields. Nevertheless in all of the mentioned work it is necessary tohave some knowledge of (or make assumptions about) the location of yieldsurfaces, so that the velocity field can be solved. This might be possiblefor simple geometries, but becomes very hard as soon as it gets a bit morecomplicated.A completely different direction for analytical investigation of yield stressflows is variational formulations, which turn to be very powerful. Here anintegral (or weak) formulation is used which is valid throughout the wholedomain of the flow. Hence, it can be used without knowing the position ofyield surfaces, contrary to the differential formulation. These formulationsusually do not give detailed information like the velocity profile, except inspecial cases. Instead through the use of functional analysis techniques,useful qualitative information can be obtained.The first step was made by Prager [196] who introduced two variationalformulations for Bingham fluid flow: velocity minimization and stress max-imization principles. Further development was made by Duvaut and Lions[92] who generalized the previous work and introduced variational inequali-ties for yield stress fluids. Glowinski and co-workers expanded this directionand used variational formulations for the development of robust numericalmethods based on convex optimization theory [99, 110, 111, 113]. Thesemethods are able to exactly handle the singularity of yield stress rheologicalmodels.The seminal work of Mosolov and Miasnikov [168, 169] are elegant ex-amples of the use of variational formulation. They studied the flow of yieldstress fluid inside ducts of arbitrary cross section shape. They gave qualita-tive predictions for the shape and location of yield surfaces and showed thatthe problem of finding the critical pressure drop necessary for the onset ofthe flow reduces to a geometric optimization problem. They derived exactvalues of critical pressure drop for several geometries like square, rectangleand others. This is explained more in Section 1.5.1.Another very useful equality for yield stress fluids is the classical me-chanical energy balance of the flow, relating viscous and plastic dissipation19of energy to the work done by external forces on the flow. Using it, an in-tegral expression of the minimum critical yield stress for flow stoppage canbe found. Also Glowinski [113] used it to show that certain transient flowsof yield stress comes to rest in a finite time, in contrast to Newtonian fluidsthat need infinite time for the kinetic energy to decay.FCGCI auwrixvtion pvrvyoflIn 1984 Lipscomb and Denn published a paper [146] which puzzled theviscoplastic community for a decade or so. Basically they argued that it isnot possible to have true unyielded regions for confined flows of yield stressfluids in any geometry which is more complicated than uniform channel.They were drawn to this conclusion by the study of lubrication type flows ofthe Bingham fluid in long-thin geometries, e.g. in molding. Lubrication/thinfilm scaling is a classical method of analysis for long-thin geometries and hasbeen successfully applied to many types of fluids, so there was no reason notto apply it for yield stress fluids. However it was giving contradictory resultswhen (na¨ıvely) used for Bingham fluid.To better understand this paradox let’s reconsider the flow of Binghamfluid in the channel of Section 1.2.3 but with a varying width of [−h(x) h(x)]where h(x) ≈ 1. Because the width is slowly varying we can assume the flowis mainly parallel (lubrication scaling). Then at each cross section of thechannel the zeroth order flow equations would be exactly the same as theplane Poiseuille flow. We can basically use (1.22) to obtain velocity profile.As the flow rate is constant through the channel we should use (1.23) withvarying Bingham number W(x) = yh(x)Rj0(x). Note that (1.22) predictsa plug at the centre of channel with velocity up. Apparently up is varyingalong the channel as W(x) is changing which means it is not possible tohave a true unyielded central plug moving with a constant speed. This isbasically the lubrication paradox argument by Lipscomb and Denn. Theterm pseudo-plug was coined for this region to make it distinct it from trueplug and a decade of ambiguity started. The Bingham model was quicklycharged with this problem and led many workers to consider regularized20Up(x)Pseudo plugFigurz FCLO Lubrication paradox example: flow of yield stress fluid ina slowly varying channel with pseudo plug regionviscoplastic constitutive models as the correct one.On the other hand the existence of true plugs at the points of symme-try of the flow geometry seems obvious, as the strain rate vanishes at thesepoints. Following this idea, Walton and Bittleston [246] built asymptoticsolutions with true plugs inside the pseudo plug region of axial flow betweentwo eccentric cylinders. The main point was to go to higher terms of asymp-totic expansion to get consistent stress fields which produce true plugs. Onedifficulty of resolving the paradox was that numerical verification of theseresults was very challenging. At the time most of the numerical solutionswere computed using regularization, which could not account for the exactBingham model. There was less awareness about the numerical methods de-veloped by Glowinski and coworkers [99, 110, 111, 113], that were relativelynew at the time. This, combined with lack of easily accessible hardwareand software led to a longer survival of the lubrication paradox. Szabo andHassager [225] were the first to compute the exact Bingham flow in theeccentric geometry of Walton and Bittleston and showed true plugs exist.Interestingly, they didn’t use the augmented Lagrangian methods of Glowin-ski, instead they started with a good guess of yield surfaces and then foundthe exact locations of them iteratively.Additional progress was made by Balmforth and Craster [17] who con-structed consistent higher order expansions for thin film flows of Binghamfluid. They used it for many geophysical related problems such as lava flowdown an inclined plane [20], iso-thermal lava domes [18] and non-isothermal21lava flows [19]. Frigaard and Ryan [102] considered a small amplitude wavywalled channel and again using higher order expansions they showed that atrue intact plug exists in the centre of channel for small enough amplitudes.Putz et al. [199] used the augmented Lagrangian method and numericallyverified results in [102]. They also showed that for larger amplitudes of wallthe central plug breaks into two parts, each around a symmetry point ofthe wall variation and constructed the associated asymptotic solution, withnumerical verification.In summary, with the help of all these works, the lubrication paradoxhas been successfully resolved. Once again the Bingham model has securedits position as the simplest physically sensible model of yield stress fluidsand we will be continuing to enjoy this simple but tricky model.FCH kvrivtionvl prinxiplzs for yizly strzss uiysAs mentioned in Section 1.2.3 weak formulations of yield stress fluid flowsare very useful. Prager [196] stated two variational principles for the Stokesflow of Bingham fluid: vzloxity minimizvtion and strzss mvflimizvtion. Theseprinciples can be generalized to the Herschel-Bulkley and many other rheo-logical models of generic type [133]:ij = (ϕ(˙) +y˙)˙ij P (1.24)provided the function ϕ(t) has specific properties.Let’s consider a sufficiently regular domain Ω in 2D or 3D. The boundaryis divided into two parts UΩtP UΩv where the stress and velocity are givenrespectively and also UΩt ∩ UΩv = ∅. The Stokes flow is governed by the22momentum equations, the continuity equation and the boundary conditions:∇O + f = 0 in k (1.25)∇Ou = 0 in k (1.26)On = in on UΩt (1.27)u = jv on UΩv (1.28)ij = (1 +W˙)˙ij(u) if W Q  (1.29)˙ij = 0 if W ≥  (1.30)where  = −pI +  is the total stress tensor, n is the unit normal vectorpointing to the outside of the flow domain, in is an imposed traction vectorand jv an imposed Dirichlet velocity. A kinematically admissible velocityfield is defined as one which satisfies the continuity equation (1.26) and theboundary condition (1.28) for the velocity. A statically admissible stressfield is defined as one which satisfies stress equilibrium equation (1.25) andthe boundary condition (1.27) for the traction. Note that the stress fieldcorresponding to a kinematically admissible velocity field may not satisfystress equilibrium (1.25). Accordingly the strain rate field correspondingto a statically admissible stress field does not necessarily satisfy continuityequation 1.26. Having these preliminaries we are ready.kzloxit– binimizvtion prinxiplzLet v∗ be a kinematically admissible velocity field for the problem (1.25)–(1.30). Excluding the case UΩv = ∅ then the unique solution u minimizesthe following functional:H(v∗) =12∫˙(v∗)2 yk +W∫˙(v∗) yk−∫f Ov∗ yk −∫SΩtinOv∗ yh (1.31)Note that the terms in this functional are not exactly those of the mechan-ical energy balance of the flow. It is half of viscous dissipation plus the23plastic dissipation, minus the power from the bodyforce and traction overthe boundary.htrzss bvflimizvtion prinxiplzLet i∗ be a statically admissible stress field and u the unique velocity fieldsolution for the problem 1.25–1.30. Excluding the case UΩt = ∅, then thestress solution maximizes the following functional:K(i∗) =∫SΩvi∗nOu yh −18∫(|∗ −W|+ (∗ −W))2 ykP (1.32)∗ =√12∗ij∗ij O (1.33)Here  ∗ is the deviatoric part of total stress i∗ and ∗ is its norm as defined.Note that the stress solution is unique only in the yielded regions of the flowand the second integral is actually zero over any unyielded regions where∗ ≤ W.As mentioned in Section 1.2.3, these variational principles are the foun-dation for a family of numerical algorithms that consider the exact Binghammodel without any modification. The augmented Lagrangian methods ofGlowinski et al. [99, 110, 111, 113] all use the strain minimization. Veryrecently Treskatis et al. [232] introduced another method using the stressmaximization formulation which shows promising results for a faster rate ofconvergence.Equivvlznxz of thz vvrivtionvl prinxiplzs for thz solutionAs noted in the previous sections the function H(v) is minimized by theactual velocity field solution and the function K(i) is maximized by theactual stress field solution. It can be proven [196] that these two functionsare equal for the solution field. In other wordsK(i∗) ≤ K(i) = H(u) ≤ H(v∗) (1.34)24Where uPi are solution velocity and stress fields and v∗Pi∗ are admissiblevelocity and stress fields respectively.kvrivtionvl inzquvlit–Duvaut & Lions [92] introduced variational inequalities for yield stress fluidflows. These include unsteady and inertial terms and are more general thanthe previous formulations for Stokes flow, (as considered above). FollowingDuvaut & Lions [92] we introduce the notation below for simplification:v(uPv) :=12∫˙ij(u)˙ij(v) yk j(v) :=∫˙(v) yk (1.35)w(uPvPw) :=∫uO∇vOw yk a(v) :=∫f Ov yk 〈uPv〉 :=∫uOv yk(1.36)The first 4 terms above represent the viscous dissipation, plastic dissipation(divided by W), inertial power and body force power, respectively. Consideru as the solution of the Bingham flow from differential formulation and vas a kinematically admissible velocity field, both divergence free. Then thetheorem finally shows (Duvaut and Lions [92]):gz[〈utPv − u〉+ w(uPuPv − u)] + v(uPv − u) +W[j(v)− j(u)]≥ a(v − u) +∫SΩt(v − u)Oin yh (1.37)Here UΩt is part of boundary which stress is given and in is traction vec-tor. This theorem has served for many numerical and analytical studies ofBingham fluid flow.kvrivtionvl zquvlit– =bzxhvnixvl Enzrg– wvlvnxz)Another useful integral relationship for yield stress fluid flows is simply themechanical energy balance of the flow. This is obtained by multiplying themomentum equation by the velocity and integrating. Using the notation25introduce in the above section:gz[〈utPu〉+ w(uPuPu)] + v(uPu) +Wj(u) = a(u) +∫ΩinOu yh (1.38)This equation has been used by Glowinski [113] to show that yield stressfluid takes finite time to rest when the left had side of (1.38) vanishes i.e.when the driving force stops injecting energy to sustain the flow. Huilgol[133] gives more results of this type.From this energy balance equation we can derive an expression for thecritical W = Wv at which the yield stress is large enough to stop the flow.For this purpose we can neglect the unsteady and inertial terms and the(1.38) simplifies as:v(uPu) = a(u) +∫ΩinOu yh −Wj(u) (1.39)Let L(u) = a(u) + ∫ΩinOu yh. Thus we can write:0 ≤ v(uPu) = j(u){L(u)j(u)−W}≤ j(u){Supv∈iNv ̸=0L(v)j(v)−W}(1.40)Here the space k is the space of all admissible velocity fields. Note thatv(uPu) and j(u) are always positive. So the expression within the bracketsshould inevitably be positive, which means:W ≤ Supv∈iNv ̸=0L(v)j(v)= Wv (1.41)This expression gives the critical W that would stop the flow. In practicethe problem is that we do not know how to evaluate Wv. However, wecan use other velocity fields (e.g. Newtonian flow with the same boundaryconditions) to obtain bounds for Wv or to give conditions for which thereshould be flow. An example is the paper by Dubash and Frigaard [91] whostudied conditions for moving and trapped bubbles in a viscoplastic medium.It is possible to get much better estimates of the Wv if we know the26approximate velocity field close to the stopping point. Numerical simulationcan help here and by replacing the velocity field in (1.41) by an approximatevelocity field, a good estimate of Wv can be found. Karimfazli et al. [141]have recently used this method for natural convection problems with yieldstress fluids. We will also use this method later to estimate the criticalpressure drop for the onset of flow in fractures.Xritixvl –izly strzss for eoiszuillz ow using vvrivtionvl mzthoyAs an example we may apply (1.41) for the Poiseuille flow in Section 1.2.3.Let us first consider that we do not know anything about the velocity profileclose to stopping. We insert the Newtonian solution into (1.41) and this givesus a lowzr wouny on the value of Wv. For the Newtonian solution:u(y) =f2(1− y2)P (1.42)L(u) =∫ 10fu(y) yy =f23P (1.43)j(u) =∫ 10|yuyy| yy = u(0)− u(1) = f2P (1.44)which gives us Waxwtonitn =2y3 as a lower bound for Wv S Waxwtonitn. Toimprove we need more information about velocity profile. Let us supposethat we suspect that close to the onset there would be a large plug in thecentre or the channel, moving with speed jp, separated from the walls bythin shear layers. Now we can approximate:L(u) =∫ 10fu yy ≈ fjpP (1.45)j(u) =∫ 10|yuyy| yy = jpP (1.46)and hence Wv ≈ yhphp = f , as the critical yield stress to stop the flow.We also know that this is actually the exact value, i.e. from the analyticalsolution. Note that we didn’t know the actual value jp, but the final resultfor Wv didn’t depend on it. This means (1.41) is scale independent. The27same procedure can be applied to more complicated scenarios. We will usethis idea for the critical pressure drop in fractures, considered in chapterChapter 3.FCI cumzrixvl mzthoysAs mentioned before in Section 1.2.2 the singularity of yield stress modelsis the main problem for both numerical and analytical investigation. Forsimplicity we use the Bingham model, but without the loss of generality. Innumerical simulation the different approaches that have been used can begrouped into 2 main categories:• Methods considering the exact Bingham model• Methods that modify the model to remove the singularity in the effec-tive viscosityWe review each of the groups.FCICF bzthoys thvt usz thz zflvxt Winghvm moyzlSzabo and Hassager [225] used a creative approach to compute the steadyflow of Bingham fluid between two eccentric cylinders. Basically they guesseda reasonable initial shape for the yield surfaces and solved the flow problemonly in the unyielded regions. The location of the yield surfaces was thencorrected based on the obtained solution. This was repeated iteratively untilconvergence. This approach has the following advantages:• It considers the exact Bingham model.• The shape of the unyielded regions can be obtained with very highprecision. They were able to resolve very sharp corners of the yieldsurface in [225], and showed that the stresses can be discontinuous onthese cornersSome disadvantages of this approach include:28• It relies on knowing a good initial guess for the location of unyieldedregions. Commonly we cannot guess these for non-trivial geometries.• There is no general equation for the evolution of the yield surface,so that the method used to update the yield surface position at eachiterate must be specially tailored to the problem.• Computational cost is high, even for steady flows several iterations ofsolving a full problem is needed.The first and second points above are quite restrictive. Although the resultsin [225] are impressive, there is also no theoretical basis to generalize themethod to more complex flows in a way that would guarantee convergenceof the iteration. In fact [225] is the only published paper using this kind ofidea, which was not apparently extended by the community.We should note here that Szabo and Hassager idea probably stems fromthe earlier work by Beris et al. [31] who studied the Bingham flow arounda sphere. They used regularization, however a nice way of detecting andadjusting yield surface was used.Vugmzntzy avgrvngivn vpprovxhzsOn the other hand, the methods developed by Glowinski and coworkers[99, 110, 111, 113] can handle the Bingham model exactly and can be usedfor any kind of geometry. The basic idea of these methods is to use thevariational formulations of the problem as discussed in Section 1.3, (mainlythe velocity minimization principle). We outline this here for convenience:H(u) =12v(uPu) +Wj(u)− a(u)−∫ΩtinOu yh (1.47)Where the functionals v(uPu)P j(u)P a(u) are defined in (1.35)-(1.36). Wesee the term j(u) which contains the plastic contribution is not differentiableat ˙(u) = 0, which are the unyielded regions. The singularity of the Binghammodel has thus shown itself here as well. This prevents the use of gradient-type methods for solution of the minimization problem. However thanks29to convex optimization theory we can still handle this situation if H(u)is sufficiently regular and convex, which is true here. In fact, Glowinskiformulates this optimization problem as the following more general type:minxf(Vx) + g(x) (1.48)Here V is a linear operator and fP g are convex functions. This is a genericform for a large range of problems in solid/fluid mechanics, statistics andsignal/image processing. Note that it is not required that fP g be differen-tiable.The idea used by Glowinski et al is to introduce y = Vx as a new variableinto the problem and then enforce the equality condition of y and Vx witha Lagrange multiplier. So the problem becomes:minxNyf(y) + g(x) suwjzxt to y = Vx (1.49)This becomes a saddle-point problem of Lagrange multiplier type, for theprincipal variables (xP y). Fortin and Glowinski [99] proposed a set of aug-mented Lagrangian methods for the solution of (1.49) that have becomewell known as ALG1-ALG4. The algorithms are quite similar, with ALG2perhaps the most popular method implemented for applications in engineer-ing, finance and other fields. It is also a popular algorithm for numericalsimulation of yield stress fluids, as we consider here.To explain ALG2 for Bingham fluid we consider the velocity minimiza-tion principle (1.47) which is stated in terms of velocity u. We set x = uas the principal variable in (1.49). The operator V in (1.49) is taken to be∇+∇g , which means the relaxed variable y = Vx would be the strain ratetensor of the associated velocity field and we use notation  for this. The30minimization functional for the Bingham fluid becomes:k (uPPi) =y(Tx)︷ ︸︸ ︷12∫2 yk +W∫ ykz(x)︷ ︸︸ ︷−∫f Ou yk −∫SΩtinOu yk+∫( − ˙(u)) : i yk + v2∫(˙(u)− )2 yk (1.50)The terms in the first row are just H(u) as (1.47), but with t(u) replaced byrelaxed variable . With this selection the functions f(Vx)P g(x) in (1.49) ofthe Bingham flow are shown. Note that this is not the only way of splittingfP g. The second row is the Lagrange multiplier term and (with v S 0) theaugmentation term, hence the name augmented Lagrangian. It is proventhat the optimal point of functional k (uPPi) is the same as the originalH(u). However, it depends on three variables instead of one. To handlethis, the minimization is performed separately on each variable, in fixedpoint format and then iterated. The final form of the popular algorithmALG2 of Glowinski is as follows. Start with initial value  = T = E (or abetter guess).htzp FO Minimization with respect to u reduces to a Stokes flow like systemwith addition of a right hand side term:−v∇O t = −∇p+∇O(i− v)P With given BC (1.51)∇Ou = 0 (1.52)htzp 2O Minimization with respect to  =0P ⇔ |i+ v t(u)| ≤ W(1− U|T+t _|)T+t _1+t P otherwise (1.53)htzp 3O Update of i, assuming that at each step, 0 Q / Q 1+√52 v is satisfiedi = i+ /( t − ) (1.54)31Figurz FCMO An example of mesh adaptation strategy suggested bySaramito and Roquet to improve accuracy of capturing yieldsurfaces. Very fine, highly anisotropic mesh can be observedaround yield surfaces. Figure from the Gallery of Rheolef li-brary [213]These 3 steps are continued until convergence is achieved. In many cases itseems that the choice / = v provides good convergence [109] and often wefollow it. The choice of optimal v is however problem dependent and thereis no easy way to find it in practice. We usually use 1 ≤ v ≤ 50.With this algorithm we can consider the exact Bingham model. However,due to meshing we may not be able to get nice sharp yield surface, e.g. asin Szabo and Hassager [225]. One remedy was suggested by Saramito andRouqet [206, 214]. They used anisotropic mesh adaptation and showed yieldsurfaces can be very accurately tracked with this strategy. An example forthe flow around a cylinder is shown in Figure 1.8. Our implementation ofALG2 very closely follows the works by Saramito and Rouquet [206, 214].In addition to ALG1-ALG4, there are other variations e.g. combiningregularization with multiplier for higher convergence rates or a multiplierlike algorithm which is based on the introduction of viscoplastic extra ten-32sor  and using projection. Further details can be found in the reviewarticle by Dean and Glowinski [80] and book chapter [112] by Glowinski andWachs. We have used ALG2 for all of our computations combined with theanisotropic adaptation strategy by Roquet and Saramito [206].All of the multiplier based methods discussed were developed about 3decades ago. In practice these methods work effectively, however the mainpractical issue is the slow convergence rate which is O(1R√k), where k is theiteration number. The reason for the slow convergence is that algorithmslike ALG2 are quite general in terms of the requirements imposed on thefunctionals (convexity only). From the existing theory we cannot get abetter convergence rate for such general problems.A very recent work from Treskatis et al. [232] has revisited the problemof Bingham fluid flow and proposed a method with O(1Rk) convergence.They note that the velocity minimization (1.50) is convex while the stressmaximization (1.32) is strictly convex: this property can be used to obtainhigher order of convergence. Strong convexity of a function f means thatthere exists  S 0 such that for (x1P x2) in the domain of f and t ∈ [0P 1]f(tx1 + (1− t)x2) ≤ tf(x1) + (1− t)f(x2)− 2t(1− t)|x1 − x2|2 (1.55)Instead of using velocity minimization Treskatis et al. use stress maximiza-tion and develop a variant of the FISTA method [27] for the solution ofBingham fluid flows. Their methodology has the following interesting prop-erties:• Considers the exact Bingham model.• There is no heuristic parameter like v in ALG2, which must be selectedand can significantly affect the convergence rate.• Gives convergence rates of O(1Rk).All of these properties makes this algorithm very exciting for use. Howeverwe didn’t have time to implement it as it was only very recently developed.As a final remark we say that algorithm ALG2 is one of the well knownset of algorithms under the name of Vltzrnvting Dirzxtion bzthoy of bultiB33plizr (VDbb), which have become extremely popular, due to applicationsin todays “hot topics”, such as: machine learning, data mining, statisticsand image processing [109].FCICG gzgulvrizvtion mzthoysThe methods described in the above section were efforts to keep the exactBingham model. These algorithms rely on fairly complex theories and areusually slow in practice. As we know, the main problem is the singularity ofthe effective viscosity in the Bingham model (1.6) at ˙(u) = 0. By lookingat the model (1.6) we can think of eliminating this singularity by adding asmall value ϵ, e.g.ij =(1 +Wϵ+ ˙(u))˙ij(u) (1.56)We can see that the modified rheological model becomes differentiable. Inother words, the singularity has been rzgulvrizzy. We can now use the reg-ularized model just like any generalized non-Newtonian fluid and solve theflow with available standard solvers. Several regularization models havebeen proposed.Bzrxovizr vny Englzmvn model from (1980); see [28]:ij =(1 +W√˙(u)2 + ϵ2)˙ij(u)O (1.57)BiBvisxosit– models, such as that from O’Donovan and Tanner [181] (1984):ij =(1 + U˙c )˙ij(u)P ⇔ ˙(u) ≤ ˙vP(1 + U˙(u))˙ij(u)P ⇔ ˙(u) S ˙vO (1.58)evpvnvstvsiou model from (1987); see [186]:ij =(1 +W˙(u)[1− z−m˙(u)])˙ij(u) (1.59)340 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.811.21.41.61.82|γ˙||τ|Figurz FCNO The Bingham model (dashed black line) and the typi-cal curves for Papanastasiou (red), Bercovier (green) and bi-viscosity regularizationsFigure 1.9 shows a sketch of how each of these models modify the Binghammodel.Physically, regularization replaces the unyielded fluid with a very highviscosity fluid, which means that there wouldn’t be any true unyielded re-gions with zero strain rate in the flow. This also makes the definition ofunyielded regions ambiguous in such computations. Of course one can take ≤ W as a definition of unyielded regions. However, the yield surfaces woulddepend on the value of regularization parameter ϵ (or m = 1Rϵ). This is themain disadvantage of regularization.One might intuitively expect from general continuity results that byletting ϵ → 0, the solution of the regularized model should converge to theexact Bingham model. Consideration of this limit by Frigaard and Nouar[101] shows that this intuition is only guaranteed for the velocity field and notnecessarily for the stress field. This means that the yield surfaces predictedby a regularized method may not converge to the exact Bingham model.This has been shown in a very nice paper by Burgos et al. [44]. Theyderive analytical solution for the anti-plane flow of a Herschel-Bulkley fluidand compare the analytic yield surface with those predicted by regularizationmodels such as Papanastasiou, Bercovier and bi-viscosity. They show that35Figurz FCFEO The original Fig 5a in paper by Burgos et al. [44]. Circlesshow the analytic yield surface and other lines are yield sur-faces from Papanastasiou regularization with increasing valueof m. The main difference is that the convexity of regularizedyield surface is reversed compared to analytical.some regularization methods (and parameters) predict a yield surface whichis convex toward the corner, in disagreement with the analytic result whichis concave. Figure 1.10 shows one example of the comparison in the originalpaper. With this shortcoming, one might think regularization methods arenot useful at all, however we can mention these positive points:• Easy to implement in standard Navier-Stokes solvers such as commer-cial software.• The convergence is faster, hence less computing time is needed.• If only the velocity field is of concern, it is guaranteed to converge tothe exact Bingham model.For many flows of industrial relevance e.g. in pipes, pumps, etc.. . . regularizationmethods are perfectly adequate and there is no real benefit to performing amore expensive computation for the same result.However, if accurate resolution of the stress field (i.e. yield surfaces) isimportant for the problem at hand, then multiplier based methods are abetter choice. One very important class of such problems is finding the36onset point of flow, as noted by Moyers-Gonzalez and Frigaard [171]. Forexample the minimum pressure drop to start the flow or the minimum yieldstress required to stop a particle from sedimenting. Another class, are thosedealing with computing finite time to rest of yield stress fluid flows, as notedby Glowinski and Wachs [112]. Superiority of multiplier methods is obviousin these cases. Industrial problems where multiplier based methods areadvantageous include those considered in this thesis, where we are directlyconcerned with finding unyielded zones and yield limits of flows.FCICH hzlzxtzy vpplixvtions of xomputvtionvl mzthoysWe give a brief chronological review of some of the computational studiesperformed using both regularization and multiplier methods. Definitely thisis not inclusive and the intention is to give a historical perspective of the ofadoption and evolution of the methods for different applications.Bzrxovizr vny Englzmvn =FC5L)This model was first used in analysis by Glowinski and co-workers, but theearliest computational application was for an inertial cavity flow; see [28]. Awell known application of the regularization (1.57) is in the study of Binghamflow around a sphere by Beris et al. [31]. Indeed it is one of the nicest workswith a regularization method. In this flow two plug zones appear at polesof the sphere and there is an envelope of unyielded fluid surrounding thesphere. They used a transformed mesh and carefully considered the yieldsurfaces via an iteration method. The result was very accurate shapes forthe yielded regions and the critical yield stress needed to keep the spherestatic in the fluid. They confirmed the quality of their results for the dragforce by examining the equality of the functionals H & K, introduced forvelocity minimization and stress maximization principles (Equation 1.34).Later Bercovier and Engleman used (1.57) for simulation of forming process[29]. Scott et al. [217] used (1.57) in the study of a 2:1 expansion, a stenosisand a bifurcation flow. Walton and Bittleston [246] solved the flow betweentwo eccentric cylinders using (1.57). Other examples of duct flows include37Taylor and Wilson [230] and Wang [247]. Nouar et al. [180] numerically andexperimentally considered the forced convection flows of Herschel-Bulkleyfluid between concentric ducts using (1.57).BiBvisxosit– moyzl =FC5M)O’Donovan and Tanner [181] introduced the bi-viscosity regularization (1.58)in 1984 and used it for study of squeeze flow. This model has been imple-mented in commercial CFD solvers, but has been used more often as ananalytical tool, e.g. Wilson [253] used (1.58) in an asymptotic study of thesqueeze flow.evpvnvstvsiou moyzl =FC5N)The most popular regularization model was introduced by Papanastasiou[186] in 1987. He used (1.59) to compute the flow in a diverging channeland also the extrusion flow of Bingham fluid. Ellwood et al. studied jetflows using (1.59); see [93]. A set of high quality computational studies ofthe flows involving interface dynamics have been conducted using (1.59), byTsamopoulos and co-workers. Direct tracking of the interface using meshmapping methods gave them accurate resolution of the flow. Examples are:spin coating flow [234], squeeze flow of a Bingham fluid [140, 221], transientdisplacement of visco-plastic fluids in expansion-contraction geometries withair [86] and a nice study of bubble rise in viscoplastic fluids [136].Fan, Phan-Tien and Tanner [95] used both (1.58) and (1.59) for the studyof advective mixing of Bingham flow between eccentric cylinders, with theinner one rotating. Alexandrou, Georgiou and coworkers have also used(1.59) for analyzing several industrial flows such as injection molding [6],and squeeze flow [7, 97].Mitsoulis and co-workers have extensively studied a wide range of in-dustrially related flows, all using (1.59). Examples are the Stokes flow ofa sphere in a tube [34], extrusion die flows [1, 143, 163, 190], duct flows[237], Bingham fluid flow around a cylinder [259], entry flow of a Binghamfluid in an expansion [164], squeeze flow [156, 165], fountain flow [161, 162],38toothpaste flow in extrusion dies with thixotropy [13] and dip coating flow[96]. The popularity of this regularization is due to it being very easy toimplement and due to the usage of the exponential term for regularization,giving a very sharp yet infinitely continuous approximation of the Binghammodel.Vugmzntzy avgrvngivn =Va) vpprovxhzsOn the other side, there is much less works using augmented Lagrangian(AL) methods. First published in 1976 in French editions, widespread un-derstanding waited until [99]. Various advanced texts have appeared fromGlowinski and co-authors (e.g. [98, 99] and others), most of which containbenchmark computations. Hurez et al. [134] analyzed die swell flow. Huigoland Pinazzi [132] considered the eccentric cylinders of Walton and Bittle-ston [246] and tried to verify their results using ALG2. This work had thepotential to support the idea that proper asymptotic solutions can resolvethe lubrication paradox. Unfortunately they didn’t make a comprehensivecomparison for the shape of the plug zones (i.e. the controversial issue) andjust showed a few velocity contour plots. This task was postponed to Putzand Frigaard [199], who compared AL and asymptotic solutions for lubri-cation flow in a long thin wavy channel. Roquet and Saramito [206, 214]introduced anisotropic mesh adaptation for computation of Bingham fluidsand thus achieved higher rates of convergence. Vola et al. [240] computedgravity currents of Bingham fluids. Weakly compressible yield stress flowsare studied by Vinay et al. [238, 239]. They modeled waxy crude oil as acompressible Bingham fluid and studied the pipeline restart process usingAL. Wachs [245] and Yu and Wachs [256], introduced a combined fictitiousdomain + AL algorithm for particulate flows of yield stress fluids. Finallysome recent papers show the increasing popularity of AL: Muravleva et al.[173] studied the time to rest of Bingham flows, Dimakopoulos et al. com-puted bubble rise flows in Bingham fluids [82].There are two studies that directly compare numerical results of aug-mented Lagrangian with regularization for the same flow. Putz and Frigaard39[199] consider lubrication flow in a long thin wavy channel. For this casean intact plug exists at the center of channel. They showed regularizationmethods can produce reasonable yield surfaces, but only if a proper yieldcriterion is used and the numerical errors should be carefully controlledwith respect to the regularization parameter, while augmented Lagrangianalways converges to correct result. The other paper by Dimakopoulos et al.[82] compares Papanastasiou regularization and augmented Lagrangian forthe steady bubble rise in yield stress fluid. They conclude that close to theyielding, augmented Lagrangian correctly captures asymptotic behavior ofthe drag coefficient while regularization fails.To wrap up, regarding the main two methods (regularization and aug-mented Lagrangian) we should say that both are useful and applicable inthe proper contexts. Regularization has been well established for the studyof many kinds of application flows for S 30 years. Augmented Lagrangianmethods have reached a mature state, however there is still room for appli-cation of AL to many flows of interest.FC5 gzlvtzy fiork on szlfBszlzxtionThis thesis largely deals with flows of yield stress fluids along channels withnon-uniform geometries. This is one type of flows that is directly connectedwith the various oil & gas related flows introduced in §1.1. Flows of vis-coplastic fluids have a unique feature which is the (possibility of) existenceof regions with static fluid attached to the walls of a flow region. For suchflows we can consider another geometry which is the same as initial one, butnow the yield surfaces of static regions in the original flow are replaced bysolid walls. Essentially the flow should be the same in these two geometries.The interesting mathematical interpretation is that the flow has szlfBszlzxtzyits flowing geometry and is neutral (to some extent) to changes we make inparts of the domain that are static. This is a specific aspect of the free-boundary problem. We will repeatedly come back to self-selection in thisthesis. Each chapter of this thesis has its own detailed related literature.However, here we try to give a centralized overview of works that are related40Figurz FCFFO Original Fig 3 in paper [168], showing the stagnant zoneVWX which exists in the corner point W and the convexity ofthe boundaryto flow in non-uniform geometries, stationary geometries and self-selection.FC5CF hzlfBszlzxtion of ofi gzomztry in yuxt ofisThe two seminal papers by Mosolov and Miasnikov [168, 169] study pres-sure driven flow of a Bingham fluid in ducts of an arbitrary cross-sectionalshape. These papers explain probably the class of non-trivial yield stressfluid flows which we understand analytically the best. Mosolov and Mias-nikov pioneered the use of variational principles. Also these papers give thebest understood results about the self-selection mechanisms of yield stressfluids, which gives insights into more complicated flows.Consider a fully developed flow of Bingham fluid in an arbitrary duct Ωin the (xP y)-plane. The boundary UΩ is assumed to be smooth, except atfinite number of points. We consider no-slip conditions to be satisfied on41UΩ, and the only non-zero velocity component is uz = u(xP y). Thus:˙xx = ˙yy = ˙zz = ˙xy = ˙yx = 0 (1.60)˙xz = ˙zx =yuyxP ˙yz = ˙zy =yuyy(1.61)˙ =√(yuyx)2 + (yuyy)2 = |∇u| (1.62)With these assumptions, only xz & yz are non-zero stress components andthe z-momentum equation is:UxzUx+UyzUy= −fP (1.63)where f is the constant pressure gradient in the z-direction. Mosolov andMiasnikov [168, 169] consider the velocity minimization principle for thisproblem:H(u) =12∫∇u2 +W∫|∇u| −∫fu (1.64)Similar to Section 1.3 they show that flow exists if and only if:f∫u ≤ W∫|∇u|P u(UΩ) = 0 (1.65)Which is written in the following form:∫u ≤ K∫|∇u|P u(UΩ) = 0P K = Wf(1.66)The above inequality is a pure functional analysis problem. The question iswhat are properties of the functions u(xP y) that satisfy the inequality? andwhat is the constant K?Through a detailed analysis of the above inequality Mosolov and Mias-nikov [168] give a number of interesting answers. We summarize the mainresults from their paper as follows.428F If u(xP y) is a smooth function with u(UΩ) = 0, then we have:∫u ≤ K∫|∇u|P K = sup!⊆ΩArea!Perimeter!(1.67)Where ! is an arbitrary sub-domain of Ω. This gives a critical pressuredrop WRK = fv. The necessary and sufficient value for having flow inthe duct is for f to exceed fv.82 There exist a sub-domain !∗ such that it gives the supremum value ofK in the above. The parts of U!∗ which are not on UΩ, should be arcsof circles that touch UΩ. In addition the convexity of the arc shouldbe towards the inside of the domain as shown in Figure 1.1183 If the domain Ω is p-connected, and y is the inner diameter of thedomain, then the constant K satisfies:y2≤ K ≤ 8py (1.68)Where the lower bound is exact in the inequality. The inner diameterof the domain is defined as mvx/(ePΩ), where e ∈ Ω is any pointin the domain and /(ePΩ) is the distance between e and boundary.This part gives lower and upper bounds for the critical pressure drop.Consider the biggest circle that fits in Ω completely. The lower boundstates that if the pressure drop is higher than what is needed to movethis circle (2WRy Q f), then flow definitely exists. The upper boundstates if f ≤ WR8py, then definitely there is no flow.8I The velocity distribution that minimizes (1.64) satisfies:∫Ωu ≤ 2K(fK −W)Area(Ω)P (1.69)which is a bound on the flow rate.85 Any local maximum of the solution u(xP y) is achieved in a domainwhich contains a circle of diameter WR8pf and does not contain a43raFigurz FCF2O Flow of viscoplastic fluid in a square pipe. As shown in[168] the yield surface of static regions has the shape of arc ofa circlecircle larger than 2WRf in diameter. Alsomvxu ≤(8pfW)2 2KArea(Ω).(fK −W) (1.70)The subdomain which contains the local maximum of u is called anucleus of the flow.86 For any pressure gradient that results in a positive flow, there existsat least one nucleus moving with constant speed, (i.e. such nuclei aresimply unyielded interior plug regions).All of the above results are very interesting. Using these we can computethe analytical critical pressure in many shapes. For example consider asquare pipe with sides of length v. If there are static zones, we know theywill be in the corners and will have the shape of an arc, say with radius r,as illustrated in Figure 1.12. According to result #2 we should find the rwhich maximizes the area to perimeter ratio:Wfv= K =bvxv2 − (4r2 − .r2)2.r + 4(v− 2r) (1.71)⇒ r = K = v2 +√.(1.72)Similarly the critical pressure drop can be computed for several other shapes.These results provide valuable insights about self-selection:441. The self-selection mechanism that yield stress fluid adopts is to max-imize the ratio of flow area to its perimeter. So self-selection can beformulated as a purely geometric problem.2. As a result of the above strategy, there will often be stagnant zonesat convex corners of the duct geometry. These are those corners forwhich a tangent arc will be inside the domain.In the second paper titled as “dn stvgnvnt ofi rzgions of v visxoBplvstixmzyium in pipzs” Mosolov and Miasnikov [169] focus on properties of staticzones and give more details and proofs:8L The boundary of stagnant zones are curved toward the unyielded do-main and the radius of any arc is not less than WRf . Boundaries ofnuclei are curved opposite (convex) to the unyielded domain with ra-dius of curvature less than WRf at each point. Thus, for example thereis a limiting f above which no fluid is trapped in corners.8M The yield surface of stagnant zones can approach the boundary at non-zero angle only at non smooth points of the boundary.8N Consider stagnant domain dVW in Figure 1.13 (original figure of thepaper), where  is the inner boundary of stagnant zone toward theflow. Assume arbitrary line a which divides dVW into two parts,then the solution of the flow will be the same if VdW is replaced bya. This is true also if the boundary of stagnant zone grows (shown atthe right) within some limits. Particularly if the growth is so big thata circle of diameter WRf fits into the expanded region then stagnantzone would be flowing.8FE Consider a sub-domain ! which completely lies in the main domainas shown in Figure 1.14. Then we will have u! ≤ uΩ, where u! isvelocity solution for sub-domain with the same conditions and uΩ isthe velocity of main domain. The contour dg1g2 shows the stagnantzone in the main domain. The interesting result is that for the flow in! we will have stagnant zone at least in the area shown by hatching.45Figurz FCF3O Original Fig. 8 in paper [169], showing how the bound-ary of duct can be arbitrarily changed without changing thesolution.Figurz FCFIO Original Fig. 12 of paper [169]. Showing arbitrary sub-domain ! inside the main duct domain. The flow in maindomain has stagnant zone dg1g2. This is sufficient conditionfor hatched area of subdomain ! to be stagnantThe theorems of these two papers represent the body of available ana-lytical knowledge about self-selection. Unfortunately the results are limitedto duct flows and do not necessarily extend to more general 2D flows suchas flow in non-uniform channels that we are interested in. There are some2D analytical works for very special cases, e.g. [14] that study local isolatedflow in a corner. Otherwise, we will have to use computations for 2D flows.However, at least they provide some intuitions to interpret the features ofself-selection for 2D flows. For example we will see later that static zonesshow up at the corners of an expansion-contraction channel, analogous to46duct flows. They also appear where the curvature of the channel wall is highand the yield surface of static region effectively reduces the curvature of thewall, again exactly the same mechanism shown by Mosolov and Miasnikovin ducts.Another interesting connection concerns how the outer boundary of stag-nant zones in 2D can be changed without affecting the flow? We saw thatfor ducts we can reduce the wall to any other line which is completely insidethe static zone. Perhaps a similar result should be true for 2D flows, as wecan pose the same stress and velocity field on the new boundary. But this isespecially important for the case of growth of the wall, i.e. as in expanding awashout. In channels, what are the restrictions on expanding the boundary?These are the questions that we try to investigate in considering washoutflows in primary cementing as the underlying industrial application.FC5CG conBuniform xhvnnzls vny szlfBszlzxtionThere are a number of previous works in the literature that consider non-uniform channel flows and are related to self-selection phenomena. Theworks include both numerical and experimental studies. Also there are a fewindustrial papers (SPE) that study the flow in model washout geometries andare related to the basic motivation of this part of the thesis. We review herespecific studies in which the appearance of stagnant zones and consequentself-selection are observed.The oldest study reporting static zones of the flow of a yield stress fluidin a fully 2D setting is by Abdali and Mitsoulis [1], who studied both 2Dand axisymmetric Stokes flows in a 4:1 contraction geometry. They wereinterested in this geometry due to extrusion die applications. Thus, mostof the paper discusses issues like entrance and exit correction, swell ratioand pressure drop.... However, they also mention that the “well-knownNewtonian vortex” in the corners becomes a “dead zone” even for smallyield stress values, and that its size grows by increasing the yield stress. Ina later work Mitsoulis and Huigol study the same flow in a wider range ofparameters, confirm the previous results and extend to the new range, [164].47Vradis and coworkers published two papers related to flow in a 1:2 ax-isymmetric expansion. In the first work [243] numerical simulation using thebi-viscosity model is performed for a variety of Reynolds and Bingham num-bers. They try to address the existence of stagnant zones at corners usingthis method, but finally state that no definite conclusion can be drawn. Intheir second work [115] higher quality simulations are performed and theyconduct experiments using a PIV method. They show that both numericaland experimental results show the existence of stagnant zones in the corner.As we have seen in 1D (duct flow), in 2D and in axisymmetric geometries,static zones show up in the corner regions. This has also been observedin 3D by Burgos and Alexandrou [8, 43] who investigated Herschel-Bulkleyflow in 1:2 and 1:4 three dimensional expansions.Jay et al. [138] study structure of the flow in 1:4 axisymmetric expansionof Herschel-Bulkley fluid, experimentally and computationally, for a rangeof Reynolds, yield stress and shear thinning index. Again stagnant zoneexists at corners for all range of parameters studied. However, dependingon the Reynolds and Bingham number there might be a vortex as well atthe corner region, which turns into a stagnant fluid zone by increasing theyield stress. They also nicely compared the shape of stagnant zone and vor-tex with experiments for a few cases. In a following study Jay et al. [139]consider axisymmetric contractions where, instead of an abrupt change, thecontraction takes place through an angled section. They show by decreas-ing the angle for fixed contraction ratio the stagnant zone becomes smallerand eventually disappears, i.e. this means that the contracting section isbecoming longer. More recently, de Souza Mendes et al. [79] conducted anexperimental and numerical study of the slow flow of Carbopol through anaxisymmetric expansion-contraction geometry. The static zones were nicelyvisualized by tracer particles and by taking long exposure time photos of theflow. With this technique the particles in stagnant zones are visible as shinydots while the flowing particles show a faint light path. They observe staticzones at corners in all range of parameters. When the length of expansion-contraction region is short these static zones are connected as one large piecewhich completely hides the walls of the test section from the flow. However48for longer lengths the flow touches the wall after a development region likeexpansion flows explained before.To recap, most of the computational and experimental studies use a sin-gular geometry, either sudden change in width or having corner points. Wewill try to study self-selection in a smooth sinusoidal channel in Chapter 2.FC6 ihzsis outlinzThe general class of petroleum industry problems that are closely relatedto this thesis were presented in above (e.g. §1.1). Broadly speaking, theseflows all involve of a yield stress fluid through a duct/channel with non-uniform cross section. In fact, applications of these flows are not at alllimited to well drilling/construction and hydraulic fracturing. Very similarflows are important and found in the food, pharmaceutical, health, cosmeticsand mining and industries, as well as in natural settings, e.g. mammaliandigestion and earthworms. Nevertheless, the need to understand these oiland gas industry flows remains the main motivation.Three themes run through the thesis: (i) Physical phenomena for whichthere is novelty in our analysis and/or understanding. (ii) Development ofnumerical and analytical methodologies for understanding these phenomena.(iii) Building our knowledge progressively of specific flows.Regarding (i) a pervading theme in each chapter is that of self-selectionof the flowing region. In mathematical terms, yield stress fluid flows are freeboundary problems, with the free boundary being the yield surface. Thoseparts of the flow for which the stress lies below the yield stress are not de-formed and (when attached to a wall) they may remain static. These staticfouling regions change the geometry of the fluid flow, which may in turnaffect physical features such as the pressure drop. The fouled material mayalso itself have an adverse effect on a process flow. Thus, better understand-ing of self-selection is key to understanding our target oil and gas industryflows.Regarding (ii) the methodological framework developed and used con-sists of: (a) Numerical computation of the flows, using the augmented La-49grangian method and mesh adaptivity (to define the yield surfaces), allwithin a finite element framework. (b) Dimensional analysis in order to re-duce the problem parameter space and to help with analysis of the results.(c) Some asymptotic analysis, for exploring yield limits. Important in (a) isto understand that while each numerical computation is relatively quick tocompute in 2D, we perform large-scale computations by investigating wideranges of parameters for each problem.Regarding (iii) we study two flows specifically. Chapters 2 & 3 studyflows along varying channels, extending existing knowledge to include deepvariations in width, the onset of fouling and eventually the limit of zeroflow. Chapters 4-6 comprise a progressively complex study of flow pastwashouts. Starting with Stokes flow we then include inertia and finally asecond displacing fluid. In more details, these chapters are as follows.Xhvptzr 2 considers the Stokes flow of Bingham fluid in a sinusoidal 2Dchannel. Here we focus on understanding the conditions when stagnantzones appear (i.e. the start of fouling and self-selection) in the channel.This is the first study that considers the self-selection mechanism in a2D smooth channel. All previous works have studied singular geome-tries such as expansions and contractions. Furthermore, this chapteris the extension of the previous works of Frigaard and Ryan [102] andPutz and Frigaard [199] to larger wall amplitude and shorter channellength which are non asymptotic ranges. Adding this last piece, a fullpicture of the Bingham fluid flow in sinusoidal walls emerges.Xhvptzr 3 studies the slow flow of a Bingham fluid along fractures. Weinvestigate the limits of applicability of the lubrication approximationand show that if self-selection is well understood then the valid rangeof lubrication can be extended to much wider limits of the geometricalparameters. For most of the study a simple wavy profile is used forfracture walls, thus extending Chapter 2. Also a few cases with morerealistic vffinz fracture geometries are considered. We are interestedin the pressure vs flow-rate relationship (effectively Darcy’s law) andhow non-linear effects for Bingham fluids affect Darcy’s law, i.e. via50nonlinearity of the constitutive equations vny via changing the flowdomain through self-selection. Another important question examinedhere is the minimum required pressure to start flow.Xhvptzr I turns to the study of self-selection in geometries relevant towashed out sections of an oil/gas well, during cementing operations.These channels consist of one straight wall and one non-uniform wall.The former represents the casing and the latter is the rock forma-tion/reservoir, which is irregular due to the washout. In reality thecementing geometry is a 3D annulus, but for simplicity we considera 2D longitudinal section along the axis. Naturally, the geometry ofwashout is very complicated and unpredictable. We therefore considerfour different classes of geometry for the washout. We observe thateventually self-selection gives very similar flow geometries, i.e. for suf-ficiently deep washouts and/or large yield stresses. Industrially this isvery useful to give a big picture of the worst mud-conditioning scenarioin the washout.Xhvptzr 5 extends the previous chapter by including inertia in the flowand looking how self-selection changes compared to the inertia-lesscase. This makes the study more relevant to the industrial application,but also makes the flows more complex. Thus we try to process theresults mostly to give practical insights. As inertial effects are thefocus of this chapter we use only the wavy wall shape for the washoutgeometry in this chapter.An interesting result is the existence of a critical Reynolds numbergzv at which the minimum amount of static mud in the washout isreached. Beyond gzv the amount of static mud actually increases.This is quite counter-intuitive to the common industrial perceptionthat the higher the Reynolds number, the better mud conditioningwill be. Also we observe that yield stress increases the stability limitof the flow, however we do not quantify this effect.Xhvptzr 6 starts to address the final problem relevant to washout in pri-51mary cementing, i.e. mud removal from the washout. This is a dis-placement flow and is very interesting theoretically as well as prac-tically. Since the number of dimensionless parameters increases dra-matically with two fluids, we study only the displacement flow of ayield stress fluid by a Newtonian fluid of the same density. This is thesimplest displacement flow with the possibility of static layers (repre-senting drilling mud that is not removed) on the walls. This servesas model for real cementing flows, giving insights into what new phe-nomena might need to be considered once a second fluid is introduced,and they might impact cementing. A sudden one-sided expansion-contraction channel is used for modeling the washout geometry, assuggested by the insights gained from the previous chapters.Xhvptzr L Wraps up the thesis, summarizing the main contributions, in-dustrially and scientifically. Limitations of the work are discussed anda number of suggestions are made for future directions of the research.52Chvptzr GFlofi of v Winghvm uiy infivvy fivllzy xhvnnzlsSlow flows of Bingham fluids through sinusoidal wavy-walled channels aredefined dimensionally by 3 dimensionless groups: (hP PW). These are theamplitude of the wall perturbation (h), the aspect ratio  (half-width tolength), and the Bingham number, W, denoting the ratio of yield stress totypical viscous stress. These flows are found to have stationary fouling layersat the wall in the deepest part of the channel, whenever the amplitude hexceeds some critical value hy . We have characterised the occurrence offouling and the main characteristics of fouling layers, by using Stokes flowcomputations extensively over the parameter space (hP PW).Fouling can occur over a range of channel aspect ratios  and progres-sively at larger h. Fouling begins at a value of h that varies primarilywith W: both necessary and sufficient conditions for fouling are given. Thelimit of large W appears to plateau to constant values of h, with the in-terpretation that for sufficiently shallow fixed geometries, as W →∞ (largeyield stress) some channels will never foul. At moderate W, for h S hy thefluid appears to self-select the flowing region, i.e. the shape of the foulinglayer. This phenomenon is partly understood in our analysis via selectionof a new length-scale for the flow in the widest part of the channel. Foulingat small W (generally in deeply wavy channels) coincides with the onset of53recirculation in Newtonian fluid flows.1GCF IntroyuxtionShear flow of Newtonian fluids along wavy walled ducts and over wavy topog-raphy has received significant attention over the years, e.g. [116, 178]. Insofaras duct flows are concerned, at least initially much of the engineering mo-tivation came from an interest in enhancing heat transfer, e.g. [185]. Otherinterest in such geometries comes from e.g. peristaltic pumping [45, 194] andpore scale modelling of porous media [258]. For non-Newtonian fluid flowsin these geometries a different set of applications are important.Here specifically we are interested in yield stress fluids and in the phe-nomenon of fouling, by which we mean that the fluid becomes stationaryin layers attached to the wall of a duct. Fouling occurs in a variety of in-dustrial flows (see [25, 51, 60, 129, 177, 219]) but is often associated withphysicochemical changes in the fluid in the flowing fluid, e.g. dried depositsof milk solids [61, 64], precipitation of asphaltene deposits [10, 94]. An al-ternative terminology sometimes used in specific cases is xhvnnzling. Herewe consider only the combination of rheological and geometric causes offouling. Neither effect alone is able to cause fouling. Flow of a yield stressfluid along a uniform plane channel or circular pipe exhibits maximal shearstress at the wall and unyielded fluid is found only in the centre of the duct.Equally, Newtonian fluids flowing in wavy walled channels do not form sta-tionary layers, although recirculatory regions may develop. Our use of theterm fouling for our flows is perhaps a little unconventional, as often foulingrefers to the build up of layers from the wall over time. However, one thelayers have formed their continued status at the wall is determined mostlyby mechanical considerations. Here we are concerned with the steady me-chanical problem of whether layers may remain static at the wall under givenflow and geometric conditions, rather than the actual formation process.In order for fouling to occur in a wavy-walled duct flow of a yield stress1A vzr“ion of thi“ xhvptzr hv“ vppzvrzy v“: AC gou“tvzi & IC [rigvvryC qihz oxxurrznxzof fouling lv–zr“ in thz ofi of v –izly “trz““ uiy vlong v fivv–-fivllzy xhvnnzlC7 J. Non-Newt. Flhid Mech.A 198:F0N{FGIA (G0FH)C54fluid it is intuitive that a sufficiently large amplitude of perturbation from auniform geometry is needed. Two important situations where such variationsare likely are: (i) in the processing of food pastes through machinery; (ii)in the construction of oil and gas wells. In the first case, where geometricnon-uniformities exist there is the risk of dead-zones appearing in the flow.As these represent sites for bacterial growth and a potential health hazardit is important to identify such zones, [236]. In oil and gas well drilling (seee.g. [41]) the well is drilled through a range of different geological strata.In soft or unconsolidated rock formations it is fairly common for the drilledwellbore to become uneven. This can depend on aspects of the drillingprocess (drill bit geometry and jetting), as well as on in-situ rock stresses,and on longer term effects due to prolonged contact of the rock with wellborefluids (e.g. chemical effects, swelling, etc). Drilling muds that are used inwell construction have a yield stress and are likely to become stationary indeep washouts. The consequences of this during drilling are not necessarilyproblematic but during the subsequent operation of primary cementing it isunlikely that the drilling mud will be displaced effectively from these regions;see e.g. [176].Evidently, problems such as (i) and (ii) above are ill-defined in termsof the range of potential geometries and process conditions. The objectiveof our study is to begin to systematically understand geometric effects onfouling with yield stress fluid flows. We take the simplest non-trivial case,of a Bingham fluid in Stokes flow along a channel with a sinusoidal wavy-wall. The geometry is described by a wavelength and amplitude of the walloscillation, and by the channel width, i.e. 3 parameters. Inertia is eliminatedfor ease, as in any case stationary flows are clearly non-inertial locally, andthe Bingham rheology is certainly simplified compared to that of most yieldstress fluids. Nevertheless, we can expect to gain insight into the mainqualitative effects of fouling.Wavy walled channel flows of Bingham fluids have been studied previ-ously in [102, 199]. Although physically analogous to that studied here thefocus was quite different. These studies looked at the asymptotic limit ofsmall aspect ratio (long wavelength channels with small amplitudes), and in55particular sought to clarify the behaviour of the central plug region as theamplitude of the wall perturbation was increased from zero. In [102] it wasshown that for sufficiently small perturbations of the wall, the flow retainedits characteristic rigid central plug. In [199] the focus was on breaking of thecentral plug region as the wall perturbation amplitude was increased. Thisanalysis was still focused at the long wavelength limit and showed that afterthe plug breaks the centre of the channel is occupied by a mix of rigid plugsand low shear extensional pszuyoBplug regions. However, the amplitudes ofwall perturbation were still modest in [199]. Here we continue these studiesby looking firstly at a range of different wavelength wall perturbations andalways at large amplitude wall perturbations, sufficient to allow the fluid tobecome stationary in the widest part of the channel.The phenomenon of fouling of yield stress fluid flows due to geometricfeatures has been identified before. Perhaps the oldest study is that of [168]who consider generically the conditions needed for stationary wall regions inone-dimensional duct flows. For example, in flow of a Bingham fluid alonga duct of square cross-section, if the driving pressure gradient is too smallthen static regions form in the corners of the duct. Below a critical valueof the pressure gradient (see [168]) the corner regions join with the centralplug region and the entire flow becomes stationary. Computed examples ofthis type of phenomenon are given in [131, 171, 214].Moving to multi-dimensional velocity fields, flow of yield stress fluidsthrough an expansion-contraction has been studied experimentally and com-putationally in [79, 174, 175]. In [79] Carbopol solutions were pumpedthrough a sudden expansion/contraction, i.e. narrow to wide to narrow pipe.A range of flow rates are studied and the yield surfaces are visualised nicelyby a particle seeding arrangement. Although at fairly low Reynolds num-bers, inertial effects are evident in the results via asymmetry of the flow (notpresent in the accompanying computations). There is a significant qualita-tive difference with the results from our study, presumably resulting fromhaving the abrupt change in diameter. For all the experiments shown, stag-nant regions first appear in the corners of the expansion (rather than atthe centre of the widest part) and for all results shown there are stagnant56regions (whereas we shall see an onset value in terms of the amplitude).Comparisons are made between experimental and numerical results, whichare at least qualitatively in agreement. The computations in [79, 174] usea finite volume method and a form of viscosity regularization to computesolutions. In [175] qualitatively similar computations are carried out, butusing an elasto-viscoplastic model. We must also recognise that a similarphenomenon of fouling in sharp corners occurs in the flow of yield stress flu-ids through a sudden expansion (or contraction). For example, in [164] bothplanar and axisymmetric expansion flows are studied, over a wide range ofBingham and Reynolds numbers, showing significant regions of static fluidin the corner after the expansion. As already commented, the main geomet-ric difference with the above studies is that the change in channel geometrythat we consider is smooth, so that fouling occurs first in the widest part ofthe channel, driven by extensional stress gradients.Other related flows include peristaltic pumping, which has been studiedrecently by [235] for Herschel-Bulkley fluids. It is worth noting that al-though geometrically similar, typically in models of peristaltic flow the wallitself is displaced in and out, shearing the fluid. Thus, it is not clear howyield stress effects manifest at all in such flows. Instead of fouling, thereis the phenomenon of fluid trapping in which a region of recirculating fluidappears, translating with the wall wave speed. This is largely driven bykinematics and the results in [235] suggest that the effects of rheology onthis phenomenon are rather minimal.An outline of this chapter is as follows. Below in §2.2 we describe thephysical problem and give a brief overview of the solution method and itsimplementation. Results are presented in §2.3, covering the main featuresof fouling layers as the three main dimensionless parameters are varied,with conclusions drawn from approximately 500 computations. The chaptercloses with a brief summary of results and discussion.57GCG ihz fivvyBfivllzy xhvnnzlThe problem we consider is very much as described in [102, 199]: the two-dimensional (2D) Stokes flow of a Bingham fluid in a periodically wavywalled channel, periodic in x, as illustrated in Fig. 2.1. We adopt the con-vention of denoting dimensional variables with a hvt symbol, e.g. ·ˆ, and workprimarily with dimensionless variables. The dimensionless Stokes equationsare:0 = −UpUx+UUxxx +UUyxyP (2.1a)0 = −UpUy+UUxxy +UUyyyP (2.1b)0 =UuUx+UvUyP (2.1c)where u = (uP v) is the velocity, p is the pressure and ij is the deviatoricstress tensor. The constitutive laws are:ij =(1 +W˙(u))˙ij ⇐⇒  S W (2.2a)˙ij(u) = 0⇐⇒  ≤ WP (2.2b)where˙ij(u) =UuiUxj+UujUxiP u = (uP v) = (u1P u2)P fl = (xP y) = (x1P x2)Oand ˙,  are the norms of ˙ij , ij , defined as˙ =√12∑ij˙2ij and  =√12∑ij2ij O (2.3)The single dimensionless number appearing above is the Bingham num-ber, W:W ≡ ˆl Yˆˆjˆ0P (2.4)where ˆl is the yield stress and ˆ is the plastic viscosity of the fluid; see582DˆLˆHˆ∆PˆLˆvˆ = 0τˆxx = 0vˆ = 0vˆ = 0τˆxx = 0ΩFigurz 2CFO Schematic of the flow geometry and boundary conditionsΩ.[196]. The mean channel half-width is Yˆ and the mean flow velocity inthe x-direction is jˆ0. In (2.1) & (2.2), lengths have been scaled with Yˆ,velocities with jˆ0 and stresses with the viscous stress: ˆjˆ0RYˆ.The channel geometry is as depicted in Fig. 2.1. The upper wall is aty = yw(x):yw(x) = 1 +h2(1− cos 2.x)OThe parameter  = YˆRaˆ reflects the aspect ratio between the channel half-width and the dimensionless wavelength aˆ of the wall variation. The pa-rameter h = HˆRYˆ is the maximal depth of the wall perturbation.2 Moreprecisely, the unperturbed channel is considered to have uniform half-width1 and the perturbed channel has half width varying from 1, at x = 0, to(1 + h) at x = 1R2, with periodicity −1. We consider the flow as steadyand periodic and consider only a single period of the channel. Further, aswe consider a Stokes flow, in place of periodicity conditions we may in-fer symmetry about the channel centreline and maximal/minimal widths.Therefore, only one quarter of the channel is considered; see Fig. 2.1. At2ihz yznition of  i“ iyzntixvl fiith thvt in pF0GA FNNrC ihz yznition of h hzrz yiffzr“from thvt in pF0GA FNNrA fihzrz h fiv“ yznzy “o thvt thz “xvlzy xhvnnzl hvlf-fiiyth vvrizyfrom (F− h) to (F + h)C59the upper wall we impose no-slip conditions:u(xP yw(x)) = v(xP yw(x)) = 0P (2.5)and at the channel centreline, symmetry conditions:xy(xP 0) = v(xP 0) = 0O (2.6)Symmetry conditions are imposed at x = 0 and x = 1R2:xx(xP y) = v(xP y) = 0P at x = 0P 1R2O (2.7)Finally, as we have scaled with the mean velocity, the mean pressure gradientin the x-direction must be adjusted to ensure that∫ yw(x)0u(xP y)yy = 1O (2.8)GCGCF cumzrixvl solution mzthoyThe finite element method is used for discretizing the Stokes problem (2.1)-(2.8). Due to symmetry of the flow and to save computational time onlya quarter of the channel is considered, as illustrated in Fig. 2.1. A limitednumber of computations were also carried out in the development stage,to verify the symmetry of the flow. The algorithmic approach we haveused is the augmented Lagrangian (AL) method described by Glowinskiand co-workers; see [99, 110, 111]. The chief advantage of AL over viscosityregularization is that the exact form of constitutive relation is used and thealgorithm gives truly unyielded regions with exactly zero strain rate. As theultimate goal of this chapter is to study the onset of fouling layers (which arestationary unyielded regions), it is important to distinguish true unyieldedregions of the flow from those where the strain rate is merely small. This isthe main motivation for using the AL method.Very briefly, in AL the proper variational formulation of a Binghamfluid flow is used; see e.g. Duvaut & Lions [92]. This formulation contains60non-differentiable terms which hamper the use of conventional optimiza-tion methods for solving the variational form. To circumvent this problemtwo Lagrange multipliers (ij P iij) are introduced into the variational for-mulation and the global optimization problem is split into three separateoptimization problems, with respect to uP ij and iij . These are solvedsequentially until acceptable convergence is achieved. The minimizationproblem for u (step 1) reduces to a classical weak form of Stokes flow. Theminimization problems for ij & iij (steps 2 & 3) reduce to explicit algebraicupdates on each discrete component (node or element, according to the im-plementation). The Lagrange multipliers (ij P iij) have physical meanings:they represent the strain rate tensor and an admissible deviatoric stress ten-sor of the flow, when the algorithm has converged. Regions with ij = 0can be easily identified as unyielded regions of the flow. The details of thenumerical method are explained in [199].The AL was implemented within the excellent code ghzolzf [213], aC++ FEM library by P. Saramito and colleagues in Grenoble. For a goodexplanation of the details of the algorithm we refer reader to the papers bySaramito & Roquet [206, 207, 214]. The special feature of this implementa-tion is that the meshing is adaptive, according to the positions of the yieldsurfaces (identified via the Hessian of the strain rate). A typical computa-tion is started with a nearly uniform triangular mesh in which the top wavywall is divided to 100 (O125 ⩽ ) or 150 ( Q O125) edges. For each stepof the refinement cycle the computations are iterated up to the point thatmvx{∥∥un − un−1∥∥ P∥∥n − n−1∥∥} ⩽ 10−6, or that a maximum number ofiterations (4000) is reached. At the end of the computation a new meshwith denser cells around the yield surface is generated and the computationis repeated for the new mesh on the new cycle. An example of the meshesgenerated in a typical refinement cycle is shown in Fig. 2.2. Typically weuse four mesh adaptation cycles. The yield surfaces are clearly identified inthe last adaptation. BAMG performs the meshing [36, 37]. On any compu-tation the maximum number of mesh points was limited to 40000, due toavailable hardware. The details of mesh adaptation strategy can be foundat [206]. The code was validated with a uniform channel flow, for which61Initial mesh: 3460 cells First adaptation: 4598 cellsSecond adaptation: 5869 cells Fourth adaptation: 9228 cellsFigurz 2C2O Showing a mesh adaptation cycle for the case of (h =1P  = O125P W = 10)the analytical solution is available. Other benchmark computations can befound in [199, 206].GCH gzsultsAs discussed in §2.1 the aim of this chapter is to study the onset and charac-teristics of fouling layers in the widest part of the channel. For fixed (WP ),these appear to arise as the depth h exceeds some critical value hy . Here westart by presenting a range of typical flow results that illustrate the main fea-tures of this phenomenon. In §2.3.1 we examine numerical results computedfor small , for which we find fouling layers. We examine the asymptoticsolutions from [102, 199], to see if they are able to predict hy . Section 2.3.2considers channels of order 1 aspect ratio. We then give a broad overview ofthe types of flow observed, in terms of (WP P h); see §2.3.3. Characteristicsof the shape of the yield surfaces when fouling occurs are explored in §2.3.4,for W ∼ df(1), focusing on self-selection of the flow regime. Interestingfeatures of the Newtonian limit W → 0 are considered in §2.3.5.62Figurz 2C3O Effects of increasing amplitude h on speed (left panel) andpressure (right panel), for  = O05, W = 10. From top to bottom,h = (O01P O25P 1P 2P 4). The same color scale is valid for each h.The dark lines in the left panel are the streamlines, gray regionsin right are plugs.GCHCF Chvnnzls fiith smvll Figure 2.3 shows a typical progression in the solution and plug regions as his increased from zero in a channel of relatively small aspect ratio ( = O05).The small aspect ratio puts the computations in the range of the analysis of[102, 199] for small enough h. For h = 0O01 we can see that the unyieldedplug remains intact, being slightly wider in the narrower part of the channelthan in the wider part, as observed in [102]. The velocity field is only slightlyperturbed from the uniform plane Poiseuille flow and the streamlines arepseudo-parallel.At larger h the single central plug breaks, due to growth of extensional63stresses. The flow consists of two true plugs connected by an extensionalpseudo-plug region. Outside the plug/pseudo-plug regions we find a shearedregion extending to the wall for h = 0O25P h = 1. This breaking process andflow structure was studied in [199]. As h increases the speed in the widerpart of the channel decreases. This induces extensional stress gradients andalso lowers the pressure drop through the wider part of the channel. Thecombination of these allows the appearance of the fouling layer, at hy Q 2,as evidenced by the results for h = 2 (the exact value of hy could be founditeratively). The fouling layer grows thicker as h increases. We also observethat as the single plug breaks in two, the plug region in the narrow part of thechannel is initially wider than in the wide part. However as h increases, theplug in the widest part of the channel grows wider, mimicking the reducedpressure drop.In Fig. 2.4 we examine the last case from Fig. 2.3 (h = 4) in more detail,showing the various stress fields and the strain rate. There are a number ofinteresting features. Firstly, the pressure is by far the largest component ofthe stress in the narrower part of the channel, up until approximately the po-sition where the fouling layer appears at on the upper wall. From this pointand into the widest part of the channel, the extensional stress has compa-rable size to the pressure. Indeed we observe that yy is approximately zerobetween the larger plug and the fouling layer, suggesting that the pressure isbalanced with yy(= −xx) in this region. We also observe that xy appearsto only just exceed the yield stress (W = 10) in this region. This is the keydifference between this flow and the shear flow in a plane channel, where|xy| increases linearly until the wall. Leaving the region between the plugand fouling layer, yy remains approximately independent of y, suggestingthat xy is approximately independent of x (from the Stokes equations).Note however that near the upper wall, as we enter the expansion and thefluid is yielded, there is a thin layer of higher stress gradients.For sufficiently small  we may expect the asymptotic solutions of [102,199] to be valid. A natural question is to ask if these solutions are ableto predict the fouling layer? The general asymptotic structure of the solu-tions is to have two outzr regions matched in an innzr layer close to the64Figurz 2CIO More detailed results from the case  = O05, W = 10,h = 4. Left panel, from top to bottom shows (pP xxP xy). Rightpanel from top to bottom shows (xxP yyP ‖‖) respectively.pseudo-yield surface y = yy(x). The outer region that extends from theplug/pseudo-plug to the channel wall is of interest here, since this is wherethe fouling layer first appears as h increases. The lubrication approximationin [102, 199] considers an outer solution in the yielded region that is of form:p ∼ p0(x) + p1(x)OOO (2.9)u ∼ u0(xP y) + u1(x) + OOOP (2.10)The x-dependency in u0 comes only via the channel width yw(x). The65leading order velocity is:u0(xP y) =up(x)P y ∈ [0P yy(x)]up(x)[1− (y − yy(x))2(yw(x)− yy(x))2]y ∈ (yy(x)P yw(x)](2.11)where the pseudo-plug speed is:up(x) =W2yy(x)(yw(x)− yy(x))2O (2.12)The position of the pseudo-yield surface, yy(x) = ywR, where  = (W∗), isthe single root of the cubic Buckingham equation:23 − (3 + 6W∗)2 + 1 = 0P (2.13)for which  S 1, and W∗ = Wy2w. On finding the root, (see Fig. 2 in [102]),we can determine p0 up to an additive constant from the pressure gradient:yy =W−p0Nx O (2.14)The deviatoric stress components are:xy ∼ u0Ny +Wsgn(u0Ny) + u1Ny +d(2)P (2.15)xx ∼ 2(1 +W|u0Ny|)u0NxP (2.16)The leading order shear stress term can also be represented as: xy ∼−p0Nxy + d(). The first order velocity is given in [102, 199] and givesthe first order shear stress and pressure gradient.We can see from the above that it is impossible for the leading ordersolution to exhibit fouling. The leading order pressure gradient is negativeand the leading order shear stress satisfies xy ∼ yp0Nx + d(). This is theusual Poiseuille flow solution resolved on a channel of width 2yw(x). Theonly way in which the stress can fall below the yield stress at the outer wallis if the higher order terms become extremely large.66However, we can at least examine whether the trends towards foulingare present in the higher order terms. If we consider the widest part of thechannel where xx = 0 then the occurrence of a fouling layer depends only onxy. Instead of decreasing monotonically, it is necessary for xy to increasenear the wall. The axial momentum balance (2.1a) is valid at x = 1R2, andwe see that since p has no y-dependency up to second order in , an increasein xy is only possible ifSSxxx Q 0 in the yielded region.The x-dependency of the leading order velocity u0 enters only throughthe channel width, so thatUu0Ux=Uu0UywdywdxOWe see that the x-derivative of xx will contain terms with both 1st andsecond derivatives of yw(x). Since x = 1R2 is a maximum of yw(x), onlythe second derivative terms are non-zero at x = 1R2:(UUxxx)∣∣∣∣x=1P2= 2([1 +W|u0Ny|]d2ywdx2Uu0Uyw)∣∣∣∣x=1P2O (2.17)The effective viscosity is positive and the second derivative of yw is negativeat the maximum, so that the sign of SSxxx depends on:Uu0Uyw=UupUyw[1− (y − yy)2(yw − yy)2]+ 2up(y − yy)(yw − yy)3[(yw − y) UyyUyw+ (y − yy)](2.18)The first term above is always negative as up decreases in a wider channel.In contrast, the second term is always positive as yy increases with yw. Wealso note that the first term, vanishes at the wall whereas the second termis positive at the wall, but vanishes near the pseudo-yield surface yy. Itfollows that at the maximal channel width (x = 1R2), SSxxx S 0 close tothe pseudo-yield surface yy but changes sign so thatSSxxx Q 0 close to thewall yw. Therefore, the physical mechanism that would allow xy to increasenear the wall and a fouling layer to develop is present in the asymptoticsolution.67The mechanism by which the fouling layer emerges is via a change inthe streamwise gradient of the extensional stresses, as we approach in theupper wall. It is worth noting that the extensional stress effect representedin (2.17), although of the required sign, is also rather weak. The secondderivative of yw is of second order in . Therefore, it is unlikely to overcomebulk pressure gradient effects unless the pressure gradient is also significantlyreduced in the wider part of the channel, as we have observed in Fig. 2.4,i.e. it seems that reduction in the pressure gradients is also necessary for afouling layer to emerge. Finally, we need to note that the emergence of afouling layer in the asymptotic solution would also mean the breakdown ofthe scaling assumptions that are behind the approximation.GCHCG Chvnnzls fiith  ∼ Of())Fouling layers also occur in channels of larger aspect ratio , for which theasymptotic theories are not valid and which have not to our knowledge beensystematically studied. An example of a shorter channel is shown in Fig. 2.5for (WP ) = (10P 0O25). Perhaps surprisingly, we observe that fouling occurswhen the central plug region is still intact. Also we can see a large variationin the plug width, between the narrowest and widest part of the channel. Ata larger depth (h = 1) the central plug breaks, but this is after the onset offouling. An interesting feature of the pressure field is the region of negativepressure in Fig. 2.5 near the upper wall when fouling occurs. Essentially thefluid is being sucked away from the wall.Related to the negative pressure, we observe that there are two compet-ing influences that generate extensional strain rates (and stresses). Firstly,there is a bulk geometric effect in that the channel widens with x, so thatwe expect SuSx Q 0 as the flow rate is fixed. This in turn suggests xx Q 0.On the other hand, the velocity at the upper wall is zero, whereas in the ab-sence of fouling we expect a positive axial component of velocity away fromthe wall. Therefore, acceleration away from the wall within the undulationimplies SuSx S 0 and hence xx S 0. These effects are reversed as we movefrom the widest part of the channel to the narrowest. The distribution of68Figurz 2C5O Effects of increasing amplitude h on speed (left panel) andpressure (right panel), for a bigger  than Fig 2.3,  = O25P W =10. From top to bottom, h = (O25P O5P 1). The same color scaleis valid for each h. The dark lines in the left panel are thestreamlines, gray regions in right are plugs.69xx inferred by the geometric/kinematic competition just discussed suggeststhat SSxxx Q 0 in the upper part of the channel, which is needed in (2.1a)in order for xy to increase with y, which must occur at the onset of fouling.GCHCH evrvmztzr rzgimzsApproximately 500 cases have been computed, covering a significant rangeof (WP hP ). Here we present the main features of the results, in terms ofa qualitative description of the flows. Figures 2.6 & 2.7 show the evolutionof unyielded regions with changing (hP ), each at fixed Bingham numbers:W = (1P 2P 5P 10). Several interesting points can be observed. First, forthe limit of short channels ( ∼ df(1)) once h is large enough to have asignificant fouling layer, the flowing region seems to become approximatelyinvariant with h. It does not seem that sufficient extensional stresses aredeveloped in order to break the central plug. The fouling layer appears tochoose its own geometry of yielded fluid.For longer channels (zOgO  ≤ O25) increasing the unevenness h createshigh extensional stresses close to the inflow which can break the centralplug, as we see for  = 0O05 in Figs. 2.6 & 2.7. As h increases further we seean interesting phenomenon close to the entrance, where the plug effectivelydisappears. In fact, the plug is still there, but very reduced. Indeed, asymmetry and continuity argument establishes that there should always bea true plug at the centre of the narrowest part of the channel. At largerW we are able to follow the evolution of the narrow section plug for theseintermediate . For example, for (WP hP ) = (5P 1O5P 0O25) we see a triangularmushroom shvpz plug, connected to (0P 0) by a thin strand of unyieldedplug. Figure 2.8 explores this evolution in more detail. As h increases or Wdecreases the thickness of the stvlk of the mushroom decreases to below themesh resolution.As the channel gets longer (smaller ) extensional stresses grow and thecentral plug is broken even for large W, unless the case of h ≪ d() isconsidered; see [102]. On breaking, there appears to be a form of stressrelaxation at the narrow part of the channel, as the plugs actually grow in70Figurz 2C6O Panorama of plug shapes for W = 1 (top) and W = 2(bottom)71Figurz 2CLO Panorama of plug shapes for W = 5 (top) and W = 10(bottom)72Figurz 2CMO Appearance and evolution of mushroom shape plug andstress field (‖‖) at the narrowest part of channel for (h = 1P  =O25) and W = (20P 10P 5P 2) (from left to right); gray regions areplugs (figure zoomed on narrowest part of channel).size and the mushroom shape is lost. However, small  alone does not itseems prevent fouling, which still occurs for sufficiently large h and W.Qualitatively, four different regimes have been observed: the centralplug may be broken or unbroken, a fouling layer may be present or ab-sent. From a practical perspective, we might be more concerned with thefouling layer than the central plug being broken. The results presented inFigs. 2.6 and 2.7 suggest that the product h as a relevant geometric cri-teria for detecting the onset of fouling layer (at fixed W). Dimensionally,note that h = HˆRaˆ, which is rather natural as an indicator of fouling (seethe schematic in Fig. 2.1). Figure 2.9 plots the 4 different types of flow forW = (O1P 1P 10P 100) and in each case attempts to delineate fouling regimesfrom non fouling regimes by a curve of constant h. The value of h appearsto decrease monotonically with W, i.e. at larger W and fixed aspect ratio we expect a lower value of hy , as is intuitive. Equally, looking at Fig. 2.9we see that our results confirm that if, for example, (WP hP ) has a foulinglayer then larger values of (WP hP ) will also have fouling layers. We shallsee later that there is some complexity as W → 0, but for W ∼ df(1) thisconclusion holds true.Although a reasonable approximation is given by the curves of constanth in Fig. 2.9, this is not completely accurate as a prediction: there is also730 0.2 0.4 0.6 0.8 100.511.522.533.54δhhδ = .34a)0 0.2 0.4 0.6 0.8 100.511.522.533.544.55δhhδ = .19b)0 0.2 0.4 0.6 0.8 100.511.522.533.54δhhδ = .1c)0 0.2 0.4 0.6 0.8 100.511.522.533.54δhhδ = .06d)Figurz 2CNO Plot of flow regimes in (hP ) plane for different Binghamnumbers, a) W = O1, b) W = 1, c) W = 10, d) W = 100. Markersof flow regimes: (▲) intact central plug+fouling, (♦) brokencentral plug+fouling, (□) broken central plug-no fouling, (◦)intact central plug-no fouling.74some small independent variation with  (or equivalently h). This is evidentfrom Fig. 2.9 where in each case some points can be spotted in the wrong sideof the constant h curves. Nevertheless, the characterization of the foulingtransition as occurring for h = f(W) is very close. To explore this furtherFig. 2.10 shows all of our computational results (≈500 different cases) plottedin the (WP h)-plane. We see that there is not a single boundary separatingfouling flows from non-fouling flows, but instead there is a narrow rangeof h at each W over which we transition. In other words, we can identifytwo critical curves in our results. The curve hs = g(W) gives suffixizntconditions for there to be a fouling layer. The curve hn = f(W) givesnzxzssvry conditions for there to be a fouling layer. Between these two curveswe find both fouling flows and non-fouling flows, with slight dependency onh and .The sufficient and necessary curves have been fitted, as indicated inFig. 2.10, by the following two functions:g(W) = hs =0O054W2 + 1O03W + 0O31W2 + 5O67W + 0O66P (2.19)f(W) = hn =0O039W2 + 0O55W + 0O07W2 + 3O21W + 0O15; (2.20)evidently f(W) ≤ g(W). The above functions coincide approximately asW → 0. This limit is closely related to the onset of recirculation for aNewtonian fluid, as we explore later in §2.3.5. Both expressions (2.19) &(2.20) approach constant plateaus (with constant offset) as W → ∞. Forany fixed h this confirms the insights from the asymptotic analysis that forsufficiently small  there should be no fouling layer. However, although ournumerical data supports the idea of a constant plateau, we have not exploredthe range of very high W.GCHCI Chvrvxtzristix fzvturzs of thz fouling lvyzr forB ≳ Of())We have seen that at fixed W ∼ d(1) a fouling layer appears at h = hy andremains for h S hy . It was observed that the shape of the upper plug region750 20 40 60 80 10010−210−1100BhδhδshδnFigurz 2CFEO Phase plot of all computations (≈500 points), Markersof flow regimes: (▲) intact central plug+fouling, (♦) brokencentral plug+fouling, (□) broken central plug-no fouling, (◦)intact central plug-no fouling, equations hn =M039U2+M55U+M07U2+3M21U+M15,hs =M054U2+1M03U+M31U2+5M67U+M66represent fitted curves.and central plug, although initially sensitive to h S hy , becomes increasinglyfixed. In particular, at the widest part of the channel (x = 1R2) the widthof yielded fluid between the two plugs appears to become constant as hincreases. This is illustrated in Fig. 2.11 for a number of different values of(WP ). Particularly for larger values of  the positions of the yield surfacesbecome constant rapidly for h S hy .To some extent the above feature is expected. Once there is a significantfouling layer filling the undulation in the wall, changes in h have only aminor effect on the geometry of the yielded flow region. In other wordsthe flow selects the shape of the yielded region close to the widest partof the channel, independent of the depth of the fouling region behind the760.5 1 1.5 2 2.5 3 3.50.20.40.60.811.21.41.6ypha)0 2 4 6 8 100.20.40.60.811.21.41.61.8yphb)0 5 10 15 200.811.21.41.61.822.22.42.6hypc)0 5 10 15 20 25 301.522.533.544.55yphd)Figurz 2CFFO Positions y = yp of the two yield surfaces in the widestpart of the channel: a)  = O5P W = 2; b)  = O25P W = 10; c) = O125P W = 10; d)  = O05P W = 10.yield surface.3 This self-selection of the flow region is further illustrated inFig. 2.12. Although the channel shape is significantly different the yieldedflow region selected in each case is quite similar.If the idea of self-selection of the (yielded) channel shape is correct,we might expect some kind of similarity scaling of the flow variables tobe recovered in the widest part of the channel. Let us denote the width3htrixtl–A thi“ yz“xription vppliz“ onl– to B ∼ OS(F) vny fiz fiill zflplorz B ≪ F lvtzrC77Figurz 2CF2O comparison of wavy wall flow ( = O5P h = 1P W = 2)with a triangular and rectangular walls, left panel: speed, rightpanel: pressure, The same color scale is valid for each column.The dark lines in the left panel are the streamlines, gray regionsin right are plugs.78of the central plug by ypv and the position of the second yield surface byy = ypv+2y0, i.e. the width of yielded layer at the widest part of the channelis 2y0. Once h S hy and the shape of the fouling layer becomes established, itfollows that Yˆ is no longer the appropriate length-scale for the flow. Insteadwe may expect Yˆ(ypv + 2y0) to be the relevant length-scale. In Stokes flowthe solution is generally dependent only on the Bingham number, so nowwe redefineW˜ = W(ypv + 2y0) =ˆl Yˆ(ypv + 2y0)ˆjˆ0O (2.21)In terms of the new length-scale, the dimensionless ratio of yielded to un-yielded fluid in the channel of maximum width Yˆ(ypv + 2y0) is !:! =2y0ypv + 2y0O (2.22)i.e. (1 − !) is the scaled central plug width. Figure 2.13 plots ! againstW˜ for each of our computations that have a fouling layer. As expectedfrom the above discussion, at each  the computed ! collapse onto a curve,dependent only on W˜. At W˜ = 0 we have ! → 1 and !(W˜) asymptotes to aconstant plateau at large W˜ (as far as our data suggest). The dependenceon  is rather weak and interestingly, the largest yielded ratios are observedat intermediate . We have no insights into why this might be.We now focus on the yielded region between central plug and foulinglayer, in an effort to gain insight into the flow structure. Figure 2.14 showsthe same solution as in Fig. 2.4, (parameters:  = O05, W = 10, h = 4),but zoomed in close to the yielded part of the flow. Qualitatively similarsolutions are obtained for other parameters. There is considerable symmetryin the flow, about the centreline of the yielded layer, evident in both velocityand stress fields.Most interesting is the observation that yy ≈ 0 throughout this region.Note from Fig. 2.4 that the net pressure drop along the channel is approx-imately 80 for these parameters, yet this is largely focused in the narrowpart of the channel. The consequence of yy ≈ 0 is that p ≈ yy = −xx atleading order. We also note that the main variation xy appears to be with79a)0 50 100 1500.10.20.30.40.50.60.70.80.91δ = 1δ = 0.5δ = 0.25B˜ωb)0 100 200 300 4000.20.30.40.50.60.70.80.91δ = 0.025δ = 0.05δ = 0.125B˜ωFigurz 2CF3O Ratio of yielded fluid, !, plotted against W˜: a)  =1P 0O5P 0O25; b)  = 0O125P 0O05P 0O025.y, across the yielded region. We now consider the form of the solution basedon these observations. First let us shift to a coordinate system based at themidpoint of the yielded region at the widest part of the channel and scaledwith y0:x¯ =x− 12y0P y¯ =y − ypx− y0y0O (2.23)We rescale the stresses, pressure and Bingham number also with y0: ¯ij =y0ij , p¯ = y0p, W¯ = y0W.At x¯ = 0, the computational results suggest a shear stress distributionof form: ¯xy = −W¯[1 + v(1− y¯2)], for some v. But there is also a weaker x¯-dependency. Since the yielded region is observed to become narrower awayfrom the widest part of the channel, and due to the expected symmetry atx¯ = 0, we assume the following form:¯xy = −W¯[1 + v(1− y¯2 − F (x¯))] (2.24)where F (x¯) is a positive even function of x¯, vanishing at x¯ = 0. The pressure80Figurz 2CFIO Detail of the yielded layer at the widest part of the chan-nel for (h = 4P  = O05P W = 10), (see also Fig 2.3). Left panel,from top to bottom shows (jxP e ). Right panel from top tobottom shows (xyP yy).and extensional stress must adopt the forms:p¯ = W¯vy¯[x¯+12yFyx¯](2.25)¯xx = −W¯vy¯[x¯− 12yFyx¯]O (2.26)With this form of stress field, the yield surfaces are where: ¯2xy + ¯2x2 = W¯2,i.e. where[1 + v(1− y¯2 − F (x¯))]2 + v2y¯2[x¯− 12yFyx¯]2= 1O81We observe that firstly only y¯2 is determined by this equation and secondlythat the yield surface positions are even functions of x¯, as suggested byFig. 2.4.The constant parameter v we may determine via the solution at x¯ = 0.Since ¯xx = 0, we find that:UuUy¯= −vW¯(1− y¯2)P ⇒ u(0P y¯) = vW¯[1− y¯ + y¯3 − 13]OThe speed of the central plug is upluz = u(0P−1) = 4vW¯R3 and the parameterv is given by the condition that the mean flux through the half-channel is 1:1y0= y¯pvu(−1) +∫ 1−1u(y¯)yy¯ =4vW¯3[y¯pv + 1]P (2.27)where y¯pv = ypvRy0. Therefore,v =0O75y0W¯(1 + y¯pv)⇒ v(W˜P !) = 0O75y0W˜(1− 0O5!)O (2.28)Figure 2.15 plots the parameter v(W˜P !)y0 for all our computed data withfouling layers. The data collapses in the form v ≈ f(W˜)Ry0.Potentially now we might try to construct the function F (x¯) and to findthe yield surface positions y¯ = ±y¯(x¯). The yield surface position is slavedto the form of F (x¯) by the condition the yield stress is attained at the yieldsurface:0 = [1 + v(1− [y¯(x¯)]2 − F (x¯))]2 + v2[y¯(x¯)]2[x¯− 12yFyx¯]2− 1O (2.29)Two conditions are obvious for F (x¯). Firstly, at any x¯ ̸= 0, the velocity mustattain the plug speed upluz = 4vW¯R3 at y¯ = −y¯(x¯). The velocity gradientis:UuUy¯=(1− W¯¯)¯xyOSince, ¯ = [¯2xy + ¯2xx]1P2 and ¯xy are even functions of y¯, it follows that[u(x¯P y¯)−u(x¯P 0)] is an odd function of y¯. Taking into account the symmetry82101 10210−310−210−1100101B˜a(B˜,ω)d0Figurz 2CF5O The parameter v(W˜P !)y0 from our computed data withfouling layers. The symbols are for different , following thekey in Fig. 2.13.of u, we can write the condition on the plug speed as:∫ w¯(x¯)0(1− W¯¯)¯xy yy¯ = −2vW¯3O (2.30)We note that this is effectively a first order differential equation for F (x¯) asthe first derivative of F (x¯) appears in ¯xx.A second condition on F (x¯) comes from mass conservation: all of thefluid must flow within the region bounded by the fouling yield surface aty¯ = y¯(x¯). Suppose that we have F (x¯) and y¯(x¯) satisfying (2.29) and (2.30).83Then, in order to conserve the volume flux along the channel, we need:1y0=∫ w¯(x¯)−y¯pc−1u(x¯P y¯) yy = u(x¯P−y¯(x¯))[y¯pv + 1− y¯(x¯)] +∫ w¯(x¯)−w¯(x¯)u(x¯P y¯) yy¯=4vW¯3[y¯pv + 1− y¯(x¯)] + 2y¯(x¯)u(x¯P 0) +∫ w¯(x¯)−w¯(x¯)[u(x¯P y¯)− u(x¯P 0)] yy¯=4vW¯3[y¯pv + 1− y¯(x¯)] + 2y¯(x¯)2vW¯3+ 0 =4vW¯3[y¯pv + 1] (2.31)Remarkably, this condition is already satisfied. Therefore, it suffices to at-tain the true plug speed at y¯ = −y¯(x¯) in order to satisfy the flow constraint.This is essentially due to the symmetry properties of the assumed solution.We have not proceeded further than this. Although in principle onecould substitute from (2.29) for y¯(x¯) and then unravel (2.30) to produce thefirst order differential equation for F (x¯), it does not appear straightforwardto solve. Both F (x¯) and the first derivative vanish at x¯ = 0, suggestingsome indeterminacy. Note also that the equations are still effectively pa-rameterised by y¯pv and W¯, which are not determined. Although results suchas Fig. 2.15 suggest strong similarity properties these are derived zmpirixvllyfrom the computation, rather than determined v priori. It is still unclearhow y¯pv and W¯ are determined. Possibly, having determined the solutionand shape of the central plug region, one could formulate a momentum bal-ance on the plug (one condition) and impose continuity of velocity with thepseudo-plug region immediately preceding (second condition).GCHC5 Complzflity of thz czfitonivn limitThe results shown so far have not considered the Newtonian limit, W →0. The precise Newtonian flow has not to our knowledge been studied forh ∼ df(1). However, Pozikridis [193] studies a similar flow with one wavywall and one plane wall, using a boundary element method. We can inferthat our results, when both walls have symmetric undulations, should beat least qualitatively similar to those of [193]. Pozikridis finds instead ofstationary regions that recirculatory vortices appear for sufficiently large84a) b) c) d)Figurz 2CF6O Streamlines and unyielded regions for a) Newtonian flow(h = O65P  = O5) where recirculation has just begun; b) samechannel but now with W = O01; c) same W as b) but now h isincreased to h = 0O75; d) same channel as c) but W is increasedto W = O04.h. The critical values, say (h)a , now depend on . The limit  → 0finds (h)a ≈ 0O56 and this decreases, asymptoting to a constant lowerlimit at large ; e.g. for  ≈ 1, (h)a ≈ 0O28. Pozikridis [193] also compareshis computed results against a first and second order perturbation theoryprediction, neither of which produce quantitatively good predictions. Forh S (h)a , when the undulation is sufficiently deep, the Newtonian flowcan have secondary vortices forming. An analogy can be formed betweenthese flows and the classical corner vortex flows studied earlier by Moffatt[166]. As with Moffatt’s vortices the length-scale and intensity of the eddiesdecay significantly with each successive vortex.From the perspective of fouling and trapping fluid, we can consider recir-culation as analogous to the onset of fouling, although the trapped Newto-nian fluid is not static. At the onset of recirculation we do have xy = xx = 0at the wall in the widest part of channel, suggesting that in the same geome-try a Bingham fluid W ≈ 0 would have a fouling layer appearing. Indeed, wecan consider the Newtonian solution as the zero-th order term in a small Wperturbation expansion for the Bingham fluid solution. Figure 2.16 exploressome of the features of small W flows at  = 0O5. First of all, Fig. 2.16a85shows the Newtonian onset of recirculation, for h ≈ 0O65, where we observea very small vortex close to the upper wall. It is worth commenting that (atleast for  = 0O5) the critical value of (h)a appears close to that in [193].This suggests that for Stokes flow the effects of the shape of the oppositewall are not critical in determining onset of recirculation.As expected, the deviatoric stresses are close to zero at onset and we seethat for a small W = 0O01 there is a small static layer in the same geometry(Fig. 2.16b). On increasing h to h = 0O75 we get the flow of Fig. 2.16c, inwhich the fouling layer has not survived! Instead we observe a vortex witha plug close to the centre. Initially this may look strange, but note thata zero strain rate is associated with the centre of the vortex, not with thepoint on the wall. As we increase h past its critical value for recirculationxy becomes positive at the wall in the Newtonian solution. Although |xy|is still much less than the bulk flow stresses ( ˆhˆ0Wˆ), it can be sufficient tobreak the yield stress ˆl , i.e. provided W ≪ 1. On increasing the yield stresssufficiently (W = 0O04), we get Fig. 2.16d where the fouling layer is presentagain.As in [193], as we increase h a second vortex appears in our computedNewtonian flow, as illustrated in Fig. 2.17a. Presumably these are also simi-lar to Moffatt’s vortices, [166]. Figures 2.17b-f show the same flow geometrybut for an increasing sequence of W. Even for W = O001 the second vor-tex is replaced by a fouling layer (Fig. 2.17b). It is worth recalling thatalthough both length and intensity of the Moffatt vortices decreases expo-nentially with successive generations, the intensity decrease is significantlylarger (typically by a factor ∼ 103). The deviatoric stress would follow asimilar scaling and thus, it is hardly surprising that even for W = O001 thereis no motion.The transition in the primary vortex in Figs. 2.17b-f is also interest-ing. The small plug region at the vortex centre grows progressively with W(Figs. 2.17b-d). A small plug grows out from the stagnation point at thewall as both vortex plug and fouling layer approach each other (Fig. 2.17e).Finally, the yield stress is sufficient (W = 0O13) to extinguish flow even inthe primary vortex (Fig. 2.17f).86a) b) c) d) e) f)Figurz 2CFLO Streamlines and unyielded regions for (h = 2O5P  = O5)and a) Newtonian flow b) W = O001 c) W = O05 d) W = O09 e)W = O12 f) W = O13In conclusion, the Newtonian limit of this problem is rather interesting.Theoretically in a sharp-cornered wedge, we would expect to find an infinitenumber of vortices and hence an infinite number of small centre-vortex plugsshould be possible as W → 0. However, as the vortex intensity decays ex-ponentially these will be difficult to capture computationally. Although wewere only able to compute 2 generations of vortices, with a better resolutionand a sharper/deeper wall undulation, we can expect to find further gener-ations. With care, it is likely that the small W solutions could be computedwith a second vortex (or perhaps approximated by a perturbation method).Finally, it is emphasized again that these flow regimes only happen for smallW. When W =/ˆyˆhˆ0PWˆ∼ df(1) the yield stress is of the order of the bulkflow viscous stresses. The viscous stresses in the secondary flow zone arean order of magnitude smaller than the bulk flow viscous stresses, meaningfor W ∼ df(1) insufficient stress is generated by the secondary flow and therecirculation zones become static.87GCI Yisxussion vny summvryIn this initial chapter of the thesis, we have shown that slow flows of Binghamfluids through wavy-walled channels develop fouling layers if the amplitudeof the wall perturbation h exceeds some critical value hy . The fouling layerappears first at the widest part of the channel and is apparently unrelatedto whether or not the central plug region is broken or intact. The mainmechanism responsible for the onset of fouling is the influence of the geom-etry on the extensional stresses. At the widest part of the channel we findthat xx increases with x when close to the central plug, but xx decreaseswith x close to the wall upper wall. This effect allows xy to increase withy close to the upper wall, which is necessary for the fouling layer to form.We have systematically studied the results of approximately 500 com-putations, largely covering the parameter space of (WP hP ). Qualitatively,4 types of flow can be distinguished (fouling or no fouling; broken plug orintact plug) and we have presented a new understanding of where thesedifferent regimes can be found. Perhaps surprisingly, for  ∼ df(1) intactcentral plugs are more prevalent that in the limit  → 0. In the lattercase, extensional stresses break the central plug before fouling occurs. Pro-vided W ∼ df(1), if fouling occurs at (WP hP ), then larger values of the 3parameters also have fouling layers.Regarding onset of fouling, our data suggests that this occurs at a valueof h that varies mainly with W, but there is also some weak variation withh and . We have been able to characterise our results by two curves: hs =g(W) and hn = f(W), describing sufficient and necessary conditions forfouling, respectively. The fitted expressions are given in equations (2.19) &(2.20), valid within the range of our data W ≲ 102. These fitted expressionssuggest that at large W, there are constant values of h (independent ofW) describing sufficient and necessary conditions for fouling. This has theinteresting consequence that certain geometries will not have fouling layersin the widest part of the channel, even as W →∞. This interesting limit isstudied more explicitly in Chapter 3, as it corresponds to the limit of zeroflow along the channel: a fact hidden by the scaling adopted here.88At moderate W, as h is increased beyond hy , further changes to the shapeof the fouling layer are minimal. The fluid effectively selects the shape ofthe flowing region. Computations with different shapes of wall perturbationcan tend to give very similar fouling layer shapes, at least in the widestpart of the channel. The flow here is characterised by yy ≈ 0. We haveattempted to understand the solution structure and selection mechanism,but a complete understanding eludes us. Re-scaling the flow variables withthe width of the flowing fluid in the widest part of the channel shows a highdegree of self-similarity in determining the yielded flowing region and theplug speed, (Figs. 2.13 & 2.15), but still leave undetermined parameters.The limiting case of W → 0 has also been examined. The flow analogy forthis limiting case is with the onset of recirculation for a Newtonian fluid. Forsmall enough W, as h increases beyond hy , we can find cases of recirculationof the Bingham fluid. On increasing h still further a series of decayingvortices can be found for W = 0. The intensity of the Newtonian vorticesdecays rapidly with each successive vortex. Thus, for W S 0 we have foundthat secondary vortices tend to become stagnant even for very small W.Qualitatively, we might expect very similar results for other commonyield stress fluid models (Casson, Herschel-Bulkley, etc). The geometryconsidered here has a high degree of symmetry and there are many otherwavy-walled variants that merit attention. For example, for well cementingapplications, one wall planar and one wall uneven would correspond to awashout geometry (see Chapter 4). In porous media/fractures & cracks, itmight be interesting to offset the sinusoidal variations to produce tortuouschannels and/or to consider more complex geometries, avenues pursued tosome extent in Chapter 3. On top of these interesting avenues, the effectsof including inertia in computations of flows along such geometries may besignificant (see Chapter 5) and eventually many industrial applications inthe oil & gas industry involve displacement by another fluid (studied inChapter 6).89Chvptzr HFlofi of yizly strzss uiysvlong frvxturzO przssurz yropprzyixtion vny ofi onsztThere are many situations where non-Newtonian fluids flow through porousmedia, narrow fractures or cracks. A common way of approximating thistype of flow is to build a closure law based on geometrically simpler struc-tures. Thus, fibre bundle and similar lubrication/Hele-Shaw approximationsare frequently used to derive laws that are extensions of the classical Darcylaw. In considering only Newtonian fluids, non-Darcy effects can arise fromgeometrical complexity and are often corrected via a variety of methods.In non-Newtonian fluids, these same complexities still remain, but one en-counters additional non-Darcy effects. Firstly, the nonlinear nature of theconstitutive law naturally results in flow laws that mimic nonlinear filtra-tion. The coupling of geometry to flow is generally more complex in thesenonlinear relationships than for Newtonian flows. A second important effectin considering yield stress fluids occurs in fouling of the pore-space/fracture,i.e. in regions of the domain in which the fluid remains stationary, attachedto the walls, as explored in Chapter 2. This can be viewed as a novel typeof non-Darcy effect, the study of which is the focus of this chapter.We explore firstly the limits of validity of standard lubrication/Hele-90Shaw approximations to the flow laws, showing that the range of geometriesfor which the approximation is effective is indeed limited to small aspectratios HRa. Included in this part of the study is the effect of the onsetof fouling, at intermediate yield stresses (captured in the Bingham numberW). Fouling leads to d(1) errors in predicting the flow-rate vs pressure droprelationship, due to self-selection of the flowing domain. We then show howthis can be corrected for if the fouled geometry is known. Next we focus onthe critical limit of zero flow as the yield stress is increased (equivalently thepressure drop decreased). We derive a theoretical characterization of thislimit, explore limiting asymptotic behaviours in short and long fractures,and derive a simplistic toy model that is able to approximate the computedcritical limit remarkably effectively over a wide range of fracture aspectratios. For more complex fractures we make less progress, but may inferthat the limiting process could be approximated effectively provided thefouling phenomenon is better understood.1HCF IntroyuxtionMany complex fluids used in industrial applications exhibit yield stress be-haviour, e.g. polymers, colloidal suspensions, foams, cement slurries, paints,muds, heavy oil [66], and flows are actively studied from theoretical, rheo-logical and experimental perspectives; [21, 67]. Here we are concerned withthe flow of such fluids in natural rocks, both in porous media and alongfractures. Interest in the flow characteristics of non-Newtonian fluids inthese settings dates back at least 50 years, e.g. [187, 215]. Much of the earlywork is summarised in the text by [22], and particularly the large body ofSoviet-era research.Yield stress fluid interest initially stemmed from the production of heavyoils that exhibit a limiting pressure gradient (nonlinear filtration) and morerecently from enhanced oil recovery operations, e.g. [188]. Sealants are rou-tinely injected into brickwork to block the spread of moisture in old buildings1A vzr“ion of thi“ xhvptzr hv“ wzzn “uwmittzy for puwlixvtion to J. Flhid Mech. inDzxzmwzr G0F5 vny i“ unyzr rzvizfi: AC gou“tvziA iC ChzvvlizrA aC ivlon & IC [rigvvryCqcon-Dvrx– zffzxt“ in frvxturz ofi“ of v –izly “trz““ uiyC7C91(a damp-proof course). Cement injection into porous media is advocated asa means of sealing Xd2 storage reservoirs. Cement slurries and drilling fluidsflow along uneven narrow eccentric annuli in the primary cementing process.In the squeeze cementing process oil & gas wells are repaired by injectingcements and other thin slurries into thin cracks; see [176]. Injection of yieldstress fluids has also been proposed as a potential method for porosime-try by [11]. In hydraulic fracturing operations, some frac fluids used havea yield stress (designed in order to enhance proppant transport). At theend of hydraulic fracturing, the flowback phase attempts to clean the gelledfluids from the proppant laden fracture. Thus, we see that many differentindustrial processes are concerned with these flows.Consequently, the determination of a constitutive law to relate the flowrate to the applied pressure in porous or fractured materials has been thesubject of many investigations in the past, e.g. [4, 15, 16, 52, 56, 57, 62,187, 215, 223, 227]. However, this remains a challenging and sometimescontroversial issue; see [35]. Based on experimental observations [4, 53, 57,187], the common constitutive law proposed in the literature between theflow rate and the applied pressure has the following form:fˆ ∝ (∆eˆ −∆eˆv)nPwhere fˆ is the mean flow rate, n is the Herschel-Bulkley power law indexand ∆eˆv is a minimal limiting pressure drop, below which there is no flow.However, theoretical models and direct simulations [56, 59, 210, 212,228, 229] have revealed that the presence of heterogeneities in the porousmedium can induce a complex dependence of the flow rate with the pressure.Since each flow path experiences different heterogeneous pore throats, theircritical opening pressures are different. As a result, the disorder of theporous media induces a hierarchy in the flow paths that open, leading to anon-trivial relationship between the flow rate and the applied pressure drop;see [59, 228].A rather simple approach to model the flow in porous media is the so-called “fibre bundle model” [56, 75], where the material is considered as a92succession of parallel tubes, each with a uniform throat (i.e. cross-sectionalarea). This model has the advantage of being easily solved because theconstitutive equation and the critical pressure is known in each throat. Themajor drawback is of course that it neglects the influence of heterogeneityalong each flow path, which can be non-trivial. It is therefore crucial tobetter investigate flows along a single channel with spatially varying width(or equivalently a tube of varying cross-section for 3D porous media), inorder to better understand the flow in both porous media and in rough 2Dfractures.As with the Newtonian case, the most classical approach is to assumea lubrication or Hele-Shaw approximation [88, 106, 154, 205], i.e. the meanflow is driven by uniform pressure gradient along the varying gap. Thisapproach has been long exploited in the modelling of laminar primary ce-menting flows, e.g. [33, 189]. After some algebra involving the solution ofthe Buckingham-Reiner equation, the flow rate can the be related to thelocal width of the gap. In a single rough channel [227] show that the criticalpressure is proportional to the harmonic mean of the gap width and the flow-pressure relationship is related to a series of power-means. In non-uniformHele-Shaw geometries the critical pressure is local and the fluid flows prefer-entially, e.g. in the eccentric annular cementing flows of [33, 189] the widestpart of the annulus can flow while the narrowest part is stationary.While the Hele-Shaw/lubrication approach can be straightforwardly ap-plied to randomized varying gap widths, it has been long recognised that todo so may result in significant error where yield stress fluids are concerned.First of all, one faces the usual geometric restrictions of slow geometric vari-ation that are likely to be invalid for many fractures. Secondly, as demon-strated by [146], straightforward application of Hele-Shaw/lubrication scal-ing arguments leads to an apparent contradiction in the leading order flowsolutions. Although this inconsistency has long since been resolved in differ-ent geometries [17, 199, 246], the leading order stress fields exhibit an d(1)deviation from those of the na¨ıve Hele-Shaw/lubrication approach. Thisdiscrepancy is due to extensional stresses that develop within the (initiallyunyielded) central plug eventually causing it to break; see [102, 199].93While plug-breaking occurs for relatively modest heterogeneities thatwould normally allow for lubrication approaches to retain validity, largeramplitude heterogeneity leads to the emergence of so-called “fouling layers”at the walls of the duct, in which residual fluid is held stationary by theyield stress. This geometric effect has been observed scientifically in variousexperimental and computational/analytical studies [35, 58, 79, 208, 209], aswell as being an everyday observation for those who spread butter/margarineon Ryvita or adhesive on the back of ceramic bathroom tiles. [208] demon-strate that as the heterogeneity amplitude increases at fixed yield stress,fouling occurs beyond a critical amplitude and give an empirical predic-tion for the onset of fouling; (see Chapter 2). Such predictions are howeverstrongly dependent on the specific geometry, e.g. in the abrupt expansion of[79] fouling occurs first in the corners, whereas for the smooth geometries of[208], fouling is first found in the deepest part of the channel.The most important result, from the perspective of predicting flow rate-pressure drop relationships, is that fouling results in self-selection of and(1) variation of the flowing geometry [209], which should therefore havean d(1) effect on such relationships. However, such phenomena have notbeen systematically studied from the perspective of flow rate-pressure drop.This is one of the objectives of this chapter. In particular, we have theobjective of understanding and quantifying the flow onset problem, i.e. ina given section of an uneven fracture what is the critical pressure drop atwhich the fluid begins to flow? Other objectives are to examine the ranges offracture amplitude to fracture length for which lubrication/Hele-Shaw typescaling remains valid and to begin to explore how to apply these results tofracture geometries of high complexity.A plan of the chapter is as follows. In §3.2 following we present the modelproblem and fracture geometries that we consider in this chapter, scale thecorresponding equations, give an overview of the computational method andpresent example results. This is followed in §3.3 by a systematic compari-son of the flow rate-pressure drop relationships from the computations, withthose from a na¨ıve lubrication approximation, including some methods toimprove the approximations in the situation where fouling occurs. Section943.4 focuses on analysis of the critical limit of zero flow, both from mathemati-cal and computational perspectives. This is largely focused at characterizingthe limiting flows in simple fracture geometries, although insights are alsogained from these results for more complicated and realistic fractures. Thechapter closes with a discussion of the main results and future directions.HCG boyzl sztupFigure 3.1 shows the flow geometry and notation used in the current study.We assume a fracture of nominal minimal width 2Yˆ, with walls located atyˆ = ±[Yˆ + nˆ±(xˆ)], where 0 ≤ nˆ±(xˆ) ≤ Hˆ. Both walls and the flow areassumed to be periodic in xˆ with period aˆ. The periodic fracture cell isassumed to be filled with a yield stress fluid (assumed a Bingham fluid forsimplicity), that is flowing slowly in Stokes flow. The Stokes equations aremade dimensionless using Yˆ as length-scale. As we wish to understand howDarcy law type behaviour is modified geometrically and rheologically, we areinterested in the mapping from flow rate (mean velocity) to pressure drop(and vice versa).In a fracture of varying width the flow rate fˆ is constant and the pressuregradient varies along the flow direction. An imposed flow formulation willbe adopted. Let jˆ0 denote the mean velocity along the fracture, defined atthe minimum fracture width, i.e. jˆ0 = fˆR2Yˆ, which is used to scale thevelocity. The shear stresses are scaled with ˆjˆ0RYˆ, where ˆ is the plasticviscosity of the Bingham fluid. Any static pressure component is subtractedfrom the pressure, before also scaling with ˆjˆ0RYˆ. The resulting equationsare:0 = −UpUx+UUxxx +UUyxyP (3.1)0 = −UpUy+UUxxy +UUyyyP (3.2)0 =UuUx+UvUyP (3.3)where u = (uP v) is the velocity, p is the modified pressure and ij is the95Figurz 3CFO a) Schematic of the fracture geometry showing dimen-sional parameters; b) schematic of the wavy-walled dimension-less geometry, with lower wall shifted to the right by  a.deviatoric stress tensor. The scaled constitutive laws are:ij =(1 +W˙(u))˙ij ⇐⇒  S W (3.4a)˙ij(u) = 0⇐⇒  ≤ WP (3.4b)where˙ij(u) =UuiUxj+UujUxiP u = (uP v) = (u1P u2)P fl = (xP y) = (x1P x2)Oand ˙,  are the norms of ˙ij , ij , defined as˙ =√12∑ij˙2ij and  =√12∑ij2ij O (3.5)96A single dimensionless number appears above, the Bingham number, W:W ≡ ˆl Yˆˆjˆ0P (3.6)represents the competition between the yield stress ˆl and the viscousstresses. Two other geometric groups characterize the fracture shape: thedimensionless length a = aˆRYˆ, and the maximal out of gauge depth,H = HˆRYˆ; see Fig. 3.1.No-slip conditions are satisfied at the upper and lower wallsu = 0P at y = 1 + y+(x) and y = −1− y−(x)P (3.7)where y±(x) = nˆ±(xˆ)RYˆ. At the ends of the fracture we impose periodicity:u(−aR2P y) = u(aR2P y)P ij(−aR2P y) = ij(aR2P y)P p(−aR2P y) = p(aR2P y) + ∆pP(3.8)2 =∫ 1+y+(x)−1−y−(x)u1(xP y) dyO (3.9)Here ∆p denotes the frictional pressure drop, which is part of the solutionand is determined by satisfying the flow rate constraint (3.9).HCGCF Frvxturz gzomztrizsThe dimensionless fracture wall shape y±(x) satisfy the bounds: 0 ≤ y±(x) ≤H = HˆRYˆ. We consider 2 generic simplified fracture geometries (sinusoidalwalls and triangular walls) and a more complex affine geometry. An exampleaffine geometry is shown schematically in Fig. 3.1a and the sinusoidal wavyfracture in Fig. 3.1b. The sinusoidal fracture widths are given by:y+(x) =H2[1 + cos(2.xRa)]P y−(x) =H2[1 + cos(2.[xRa−  ])]P (3.10)with  ∈ [0P 1] denoting a phase shift of the lower wall relative to the upper,i.e. the lower wall is translated  a to the right. The triangular geometry is97defined analogously.In many applications it has been observed that fracture roughness maydisplay self-affine correlations, [38, 155, 216]. A surface yˆ(xˆ) is self-affine ifthe probability to find the increment ∆yˆ after a distance ∆xˆ displays thescaling invariance:p(∆yˆP∆xˆ) = p(∆yˆP ∆xˆ)Pwhere  is any scaling factor and  is called the Hurst exponent. An impor-tant property of self-affine surfaces is that all the moments scale with thelength of measurement as:bn(xˆ) = (〈|yˆ(xˆ+ xˆ0)− yˆ(xˆ0)|n〉)1Pn ∝ |xˆ− xˆ0|For the present chapter, we have generated three types of fracture. In thefirst type, the two walls y± are independently generated with a Hurst ex-ponent  = 0O5 using a Fourier transform method (see [212, 226]). Bothsurfaces are scaled to have min y± = 0 and max y± = H. In the secondtype, we used y+ = y−. In the third type, we used y+ = −y−.HCGCG Computvtionvl ovzrvizfiNumerical solution of viscoplastic fluids poses a unique challenge which isthe singularity of the effective viscosity in the constitutive equation (3.4),where ˙ → 0, (unyielded regions). Aside from such points, the Binghamfluid is simply a generalized Non-Newtonian fluid with an effective viscosity = (1 + U˙(u)). A common work around to the singular effective viscosityis to simply rzgulvrizz the viscosity, introducing a small parameter ϵ ≪1, such that the effective viscosity scales like ϵ−1 as ˙ → 0. Many suchregularizations are possible. It can be shown that as ϵ → 0 the velocitysolution will converge to that of the exact Bingham fluid, but the stressfield may not; see [101]. Consequently we may not infer the correct shapeof yield surface from  = W, e.g. [247].Instead we use the augmented Lagrangian (AL) method [99, 110], which98uses the proper variational formulation of the problem as formulated forexample by [92]. The AL method introduces two new fields: the strainrate  and stress i, in addition to the velocity and pressure (uP p) of theoriginal problem. In this way it relaxes the convex but non-differentiablevelocity minimization problem to an associated saddle point problem. Thesaddle point problem is solved iteratively. Each iteration consists of 3 Uzawasplit steps, to update (unP pn), n and in respectively. These iterations arerepeated until mvx{|n − ˙(u)n|L2P |un − un−1|} ≤ 10−6 is satisfied, or amaximum number of iterations (here 10000) is reached. The Pi fields ofthe AL method converge respectively to the strain rate and deviatoric stresstensors of the exact Bingham flow.We have implemented the AL method using the freely available FreeFEM++finite element environment [118]. Our algorithm is based on that in [214]with the addition of a flowrate constraint. To satisfy the inf-sup condition,Taylor-Hood (e2− e1) elements are used for velocity and pressure. Lineardiscontinuous elements e1w are used for  and i fields, to follow compati-bility conditions between the velocity space and strain/stress spaces. Bothvelocity and pressure are implicitly solved and the system matrix is factor-ized once for all iterations. To improve the accuracy of the solution whilekeeping reasonable runtime, five cycles of the anisotropic mesh adaptation[36] is used. A typical computation starts with size of 8000-10000 meshpoints. A new mesh is generated based on a metric computed from the cur-rent solution. We use the dissipation field as the metric which results in finermesh around the yield surface. After the fifth cycle the mesh may containup to 150,000 mesh points and the yield surfaces can be clearly identifiedby a much finer local mesh. More details of the numerical algorithm arepresented in [208, 209] and skipped here for conciseness.HCGCH Eflvmplz rzsultsWe have computed over 2000 flows, covering a wide range of H, a, W, forboth triangular and wavy fracture profiles with different  . In additionwe have computed a smaller number of flows in affine fracture geometries.99Before analyzing specific features of the flow, we present some examples toillustrate qualitative features of our results.Figure 3.2 shows results from a relatively modest fracture geometry andyield stress rheology (H = 1, a = 10, W = 2). The top two rows show thetriangular and wavy profiles for  = 0 (symmetric fracture). The flow isevidently symmetric about the centreline and widest part of the fracture.Unyielded plug regions are found at the symmetry points in x and closeto the fracture centreline. There is very little difference, qualitatively orquantitatively, between triangular and wavy profiles. The bottom row showsthe wavy profile at  = 0O5, where the lower wall is out of phase with theupper wall. Here the differences are quite significant. The velocity fieldappears to adapt smoothly to the wavy geometry, but no longer has thefastest fluid in the centre of each cross-section. Instead the fastest travellingfluid moves at larger radius of curvature, while the plug regions are displacedinto each bend. Due to symmetry effects we observe a rather spectaculareffect on the pressure field: the displaced central plug regions are joined bythin strands of unyielded fluid. It appears that the pressure is discontinuousacross these strands, which is possible. Note however that the tractionvector, defined by the normal to these strands, is continuous.Figure 3.3 explores the effects of increasing the yield stress on the flow ofFig. 3.2 (H = 1, a = 10,  = 0). The flow of course remains symmetric and,since the flow rate is fixed, we only observe a relatively small change in thestreamlines (left column) as W is increased. This however masks the changesthat are occurring in the pressure and stress fields. Increasing W results ina widening of the plugs in both narrow and wide parts of the fracture (rightcolumn). The magnitude of the pressure field also increases with W: the flowrate is fixed and the pressure gradient therefore needs to overcome the yieldstress everywhere along the fracture to ensure flow (hence pRW is shown toaid comparison). For W ≈ 10 we observe that a region of stationary fluidemerges at the upper and lower walls, in the deepest part of the fracture.We refer to this as a fouling layer. The fouling layer grows in size as Wincreases. Growth of the fouling layer has effects on the pressure drop alongthe fracture and is an important part of the yield limit that is attained as100Figurz 3C2O Computed examples of speed |u| & streamlines (left col-umn), and pressure p with (gray) unyielded plugs (right col-umn), at H = 1, a = 10, W = 2. Top row: triangular fractureprofile,  = 0; middle row: wavy fracture profile,  = 0; bottomrow: wavy fracture profile,  = 0O5.W →∞, both of which are studied later; see §3.4.Figure 3.4 explores the effects of increasing H on the flow illustrated inthe last row of Fig. 3.2 (W = 2, a = 10,  = 0O5). The flow asymmetryis preserved as H is increased, together with the interesting unyielded fluidstrands. The main observation is that, although the yield stress is main-tained constant, the region of fouled fluid in the deepest parts of the fractureincreases markedly with H. The increase in fouling has the interesting effectof reducing the tortuosity of the flowing region of fluid. This self-selectionof the flowing area is a unique effect of the yield stress.Finally we show an example from an affine fracture at H = 2, a = 20,for increasing W; see Fig. 3.5. We observe that even for W = 1 much of101Figurz 3C3O Computed examples of speed |u| & streamlines (left col-umn), and scaled pressure pRW with (gray) unyielded plugs(right column), at H = 1, a = 10,  = 0, wavy fracture profile.Top row: W = 5; middle row: W = 10; bottom row: W = 100.the small scale roughness of the fracture wall is smoothed out by fouledimmobile fluid. This effect increases with W as also the size of plug regionswithin the flowing region increases. Unyielded flowing plugs are observed atthe symmetry points of the flow. We can observe a slight asymmetry in thepositions of the flowing plug regions, although for all practical purposes thevelocity field is symmetrical about the x-axis. This asymmetry is due to theunstructured mesh, which is not constrained to be symmetric.HCH YvrxyBlvfi zstimvtzsIn a uniform channel of dimensionless width 2(1+h) the dimensionless pres-sure gradient is found from the constraint of fixed flow rate. This amounts102Figurz 3CIO Computed examples of speed |u| & streamlines (left col-umn), and pressure p with (gray) unyielded plugs (right col-umn), at W = 2, a = 10,  = 0O5, wavy fracture profile. Toprow: H = 3; bottom row: H = 5.103Figurz 3C5O Computed examples of the speed |u| & streamlines with(gray) unyielded plugs, at H = 2, a = 20, for a symmetric affinefracture. Top row: W = 1; middle row: W = 10; bottom row:W = 100.104to solving the Buckingham-Reiner equation:1 =W(1 + h)23ϕ(1− ϕ)2(1 + ϕR2) (3.11)for the dimensionless parameter ϕ ∈ [0P 1], which denotes the ratio of yieldstress to wall shear stress. We note that ϕ depends only on the parameterW(1 + h)2, which can be interpreted as an appropriately modified Binghamnumber, i.e. the mean velocity is reduced by 1R(1 + h) due to mass conser-vation, and the minimal width is amplified by (1 + h). Having found ϕ bysolving (3.11) numerically, the pressure gradient is computed from:∣∣∣∣UpUx∣∣∣∣ = W(1 + h)ϕO (3.12)In the case that the fracture has slowly varying width in x, it is naturalto expect that the uniform channel solution gives a leading order approxi-mation. The flow rate is the same at each x along the fracture, so we simplycompute ϕ(x) from (3.11) using the varying width:2(1+ h(x)) = 2+ y+(x) + y−(x)P ⇒ h(x) = 12[y+(x) + y−(x)]O (3.13)We then evaluate the pressure gradient from (3.12). For example, for thewavy channel we have:h(x) =H4[2 + cos(2.xRa) + cos(2.[xRa−  ])]OAdopting this procedure, we compute the lubrication pressure drop alongthe fracture ∆pL, using (3.11) & (3.12). This may be compared directlywith the numerical pressure drop ∆pa , from the finite element solution. Weevaluate the accuracy of the lubrication approximation using the relativeerror in predicted pressure drops:Relative error =2|∆pa −∆pL|∆pa +∆pLO105a) L0 2 4 6 8 10RelativeError10-1100b) H1 2 3 4 5RelativeError10-1100Figurz 3C6O Relative error in predicted pressure drops between nu-merically computed and lubrication approximation (3.12), forW = 0O1 in wavy fracture: a) H = 2 and varying a; b) a = 10and varying H. For both figures: black circle  = 0, red square = 0O25, blue cross  = 0O5.Figure 3.6 shows typical variations in relative error as both H and a arevaried, for relatively small W = 0O1. For this small value of W the velocityfield is very close to that of a Newtonian fluid as are the relative errors inpressure drop. We observe that the relative error decreases for fixed H as aincreases and for fixed a as H decreases. The relative errors appear largestfor  = 0O5, when the sinusoidal walls variations are out of phase. Thisparticular effect is due to tortuosity.For Newtonian fluids various authors have proposed corrections to Darcy’slaw, based on improved representations of the geometric effects. For exam-ple, [258] have used an effective fracture width, derived by calibrating withthe analytical solution from a sinusoidal variation. They have then gener-alized this approach somewhat to more general planar fractures. A slightlydifferent approach is followed by [106], who essentially use the fracture wallgeometry to define a centreline of the fracture and consequently the localfracture width. This new fracture width is used in the classical lubricationapproximation, integrated along the fracture length. This approach does106have the advantage of addressing tortuosity directly, i.e. the increase in flowpath length, but is impractical in rough fractures as it relies on differenti-ating the fracture geometry. In general we can say that the main efforts tocorrect geometric effects on Darcy’s law for Newtonian fluids do not involvefluid rheology. This is for the simple reason that the geometry and rheologydecouple for a Newtonian fluid, due to the linearity of the Stokes equations.We may infer from general continuity results as W → 0 that any of thesecorrection methods could be extended perturbatively into the weakly non-linear regime of 0 Q W ≪ 1; indeed this is a relatively straightforward butlaborious algebraic exercise to so.We turn instead to moderate values of W. Figure 3.7a shows the ratioof pressure drops along the fracture: ∆paR∆pL, for W = 1P 5P 10 and at = 0. Both wavy and triangular fracture shapes are plotted. We observethat the computed pressure drop exceeds the lubrication pressure drop. Theratio approaches 1 as HRa → 0. Note that multiple computations in ourdata set have the same values of HRa. Taking now W ≥ 1 and a ≥ 10, weplot in Fig. 3.7b the relative error for the wavy fracture, grouped by phaseshift  . We observe that the relative errors are numerically of size ∼ HRaover all parameters. Note that at the same  and HRa different data pointscorrespond to different W. Interestingly, the smallest errors are found for = 0O5, i.e. out of phase (reversing the trend at small W). Possibly thisresults from some form of cancellation of errors when averaged over a fullwavelength (due to the phase shift). Analogous results are found for thetriangular fracture profile. This leads us to suppose that for quite generalgeometries with HRa ≪ 1, using (3.12) will give a reliable approximationto the pressure drop; roughly speaking, errors of 10% or less are found forHRa Q 0O05.We now explore a slightly different yield stress effect. We have seenin Fig. 3.3 that the flow domain may self-select. From our earlier work[208, 209] we expect to find this phenomena for sufficiently large HRa andW. The occurrence of a significant unyielded plug region in the deep parts ofthe fracture (called fouling layers) clearly changes the flow geometry, makingthis is a non-Darcy flow effect that is unique to yield stress fluids. In the107a) H/L10-3 10-2 10-1∆pN/∆pL11.11.21.31.41.51.61.71.8b) H/L10-3 10-2 10-1RelativeError10-610-410-2100Figurz 3CLO a) Ratio of pressure drops computed numerically (∆pa )and from the lubrication approximation (∆pL): W = 1 (cir-cles), W = 5 (squares), W = 10 (diamonds); filled symbols fromtriangular fracture profile, hollow symbols from wavy profile; = 0. b) Relative error in predicted pressure drops betweennumerically computed and lubrication approximation (3.12), fora ≥ 10 and W ≥ 1 in a wavy fracture: black circle  = 0, redsquare  = 0O25, blue cross  = 0O5.case that we have fouling layers, the yield surface forms one boundary of theflow domain. This surface is defined as a contour of constant  = W, andis coincidentally a material surface when u = 0 within the plug. Imposingthe condition u = 0 on the yield surface and solving only within the flowingregion of the fracture gives the same velocity solution. This suggests that areasonable way of approximating the pressure drop through such a fracturewould be to replace y±(x) in (3.13) with the yield surface positions of theboundary of the fouling layer, denoted say y = ±yyN±(x), i.e.h(x) =12[min{y+(x)P yyN+(x)}+min{y−(x)P yyN−(x)}]O (3.14)To illustrate this approach, we take a fracture geometry H = 2 & a = 2,for which the lubrication approximation (3.12) should not provide a good108approximation. The 2D computed mean pressure gradients and those ap-proximated from (3.12) are shown in Fig. 3.8a for both triangular and wavyprofiles, over  = 0P 0O125P 0O25P 0O5, and for increasing values W ≥ 1. Bothmean pressure gradients increase with W, as might be expected, and we ob-serve a consistent and significant error for all parameters. The relative erroris shown in Fig. 3.8b for the same data (squares) and we see this is ∼ 1 overall parameters. Also in Fig. 3.8b (diamonds) are the results of predictingthe pressure drop using (3.12) but taking the modified fracture width from(3.14), i.e. where there is a fouling layer we take the yield surface positioninstead of the fracture wall position. Physically, as W increases the foulinglayer progressively fills in and smoothes the fracture wall variations. Thus,as W increases the geometry of (3.14) resembles a geometry more suited toa lubrication approximation. On using (3.14) we see a significant decreasein the relative error, so that at large W equation (3.12) again leads to areasonable approximation of the pressure drop, but with the inconvenientcaveat that we must first know the extent of the fouling region in order tomake this prediction! Nevertheless, this is an interesting and unusual flowin that increasing the non-Newtonian nature of the fluid leads to improvedapproximation.Finally, we must note that the improvement in approximation is geom-etry dependent. For a fracture that is anyway relatively long and thin, asW → ∞ we may either: (i) have no fouling at all, or (ii) have a foulinglayer that does not fill the entire wall profile at large W. Characteristics ofthe limit of large W relate to the limit of no flow along the fracture, whichwe study below in §3.4. Fig. 3.8c illustrates the reduction in relative error,using (3.14) compared to (3.13), for two specific fracture geometries. We seethat short wavelength fractures are most affected by using (3.14).HCI ihz limit of no ofiWhen yield stress fluids are pushed through a pipe or duct, it is well knownthat a certain critical pressure gradient must be exceeded in order for flowto occur. For example, in pipe of diameter Yˆ the pressure gradient must109a) B100 101 102∣ ∣ ∣ ∣∂p∂x∣ ∣ ∣ ∣100101102b)B100 101 102RelativeError10-210-1100101c) B100 101 102RatioofRelativeErrors00.20.40.60.81H = 5, L = 20H = 2, L = 2Figurz 3CMO a) Consistent errors in average pressure gradient predic-tion using (3.12), for H = 2, a = 2: black = wavy, red/white =triangular; circles = numerical, squares = lubrication approxi-mation. b) relative error for the data in a) (squares); relative er-ror using the lubrication approximation (3.12) with the fracturewidth replaced by yield surface position (diamonds). c) Ratioof relative errors (lubrication approximation using yield stressposition vs lubrication approximation using fracture width):H = 2, a = 2 in black; H = 5, a = 20 in red. Note thatthe multiple points displayed at the same W correspond to dif-ferent values of  .110exceed 4ˆl RYˆ. Critical pressure gradients are known for many other ductcross-sections shapes, following the seminal work of [169, 170]. Here the fullytwo-dimensional nature of the flow along the fracture complicates things, butphysical intuition still suggests that a critical pressure drop is required inorder to initiate flow.One approach to finding the critical pressure drop would be to computeflows at successively large pressure drops, until flow initiates. However, thedimensionless formulation we have adopted appears to prevent this, sincewe have scaled with a mean velocity so that (3.9) is always satisfied. Thus,an alternative scaling is needed to study this limiting flow directly. Supposetherefore that we impose a fixed pressure drop ∆eˆ along the fracture. Wethen define a velocity scale jˆ∗ to implicitly balance with the shear stress,i.e.∆eˆaˆ=ˆjˆ∗Yˆ2⇒ jˆ∗ = ∆eˆ Yˆ2aˆˆOThe viscous stresses and the modified pressure are scaled with ˆjˆ∗RYˆ, thevelocity is scaled with jˆ∗ and lengths again with Yˆ. This results in the samesystem (3.1)-(3.4b), with all variables now designated with a ∗ to denote thedifferent scaling. In place of W in the constitutive laws (3.4a) & (3.4b) wenow have:dy ≡ ˆl Yˆˆjˆ∗=aˆYˆˆl∆eˆP (3.15)representing the balance between the yield stress and the imposed pressuredrop. We refer to dy as the Oldroyd number. The boundary conditions areagain:u∗ = 0P at y = 1 + y+(x) and y = −1− y−(x)Pu∗(−aR2P y) = u∗(aR2P y)P∗ij(−aR2P y) = ∗ij(aR2P y)Pp∗(−aR2P y) = p∗(aR2P y) + aPi.e. now the pressure drop is known (with average gradient −1 along thefracture); the flow rate must be calculated. Our physical intuition about111requiring a finite pressure drop to initiate flow along the fracture translatesinto the belief that there will be a critical value, say dy = dyv, that sep-arates flowing and static fractures. This notion will be made more precisebelow in §3.4.2.As in either formulation described, the flow is a Stokes flow with aunique velocity solution, we expect that the solutions may be mapped toone-another. Equivalence of the 2 formulations is established straightfor-wardly by re-scaling, from which we deduce:dy = aW∆p= Wq∗P 2q∗ =∫ 1+y+(x)−1−y−(x)u∗1(xP y) dyP (3.16)i.e. 2q∗ is the areal flow rate from the imposed pressure formulation, whichis equivalent to the inverse of the mean pressure gradient, aR∆p, computedin the imposed velocity formulation.Although it is quite possible to compute u∗ from the imposed pressureformulation and then vary dy to study the limit of no flow, numerically thisis less well-conditioned than using the fixed flow rate formulation. Sincethe computational method is iterative and has tolerances imposed for con-vergence, it proves easier to impose a tolerance on an iteration for whichu ∼ d(1), than where u∗ → 0. The identity (3.16) shows that it is feasibleto work in this way. Using the imposed flow formulation we find ∆p aspart of the solution and monitor convergence of dy = aWR∆p→ dyv as weincrease W. We also see that if a critical stress balance is achieved, u∗ → 0(i.e. as dy→ dyv), then implicitly q∗ ∼ W−1 as W →∞.HCICF EflvmplzsWe now examine 3 example sequences where we take increasingly large Wat fixed geometry, to understand the limiting behaviour. For simplicity wefix  = 0. Figure 3.9 shows a relatively short fracture (H = 1O5P a = 4) atW = 10P 100P 1000. Even at W = 10 the majority of the flow is unyielded,with stationary fouling regions filling the deep parts of the fracture andan intact plug moving along the centre. These regions are separated by a112Figurz 3CNO Colormap of the speed with superimposed streamlines ina relatively short fracture H = 1O5P a = 4: increasing W =10P 100P 1000 (from left to right). Unyielded regions are shownin gray.thin shear layer that extends between the narrowest parts of the fracture,widening slightly at the deepest parts. As W increases the width of thissheared layer is reduced, albeit slowly.Next we consider a relatively long fracture with small HRa, (H =0O1P a = 20); see Fig. 3.10. The flow has unyielded plug regions at thenarrowest and widest parts of the fracture, but no static fouling layers inthe deepest parts of the fracture. The plug in the widest part is movingslower than that in the narrowest part and these two plugs remain sepa-rated as W is increased.Lastly, we consider a more intermediate geometry (H = 2P a = 20); seeFig. 3.11. As may be expected the limiting process at large W is qualitativelysomewhere between the previous 2 examples. The narrowest and widestparts of the fracture have moving central plug regions, but there is also astatic fouling layer in the deepest part of the fracture, extending betweensay x ∈ (−xy P xy ). The fouling layer and central plug are separated by ashear layer, that decreases slowly in width as W increases (similar to theshort fracture). The length xy appears to approach a constant value as Wincreases. Similar to the long fracture, the narrow and wide plugs remainseparated at large W, moving at different speeds.113Figurz 3CFEO Colormap of the speed with superimposed streamlinesin a relatively short fracture H = 0O1P a = 20: increasingW = 10P 100P 1000 (from top to bottom). Unyielded regionsare shown in gray.Figure 3.12 plots the variation of dy and ∆eRa, with increasing W, forthe 3 geometries illustrated in Figs. 3.9-3.11. We see that in all cases thecomputed dy appears to asymptote to a constant value at large W. Thisvalue denotes the critical limit dyv. Interestingly, the short and long frac-tures appear to asymptote to dyv ≈ 1, whereas the intermediate geometryasymptotes to a value that is significantly larger. These examples are typicalof the behaviour found over the whole range of our results.HCICG ihz xritixvl limit from v vvrivtionvl mzthoyWe now define the critical limit more precisely. Using the imposed pressureformulation has some advantages for this, as we may use standard variationaltechniques (as in e.g. [198]) to consider the zero flow limit. The solution u∗114Figurz 3CFFO Colormap of the speed with superimposed streamlinesin a relatively short fracture H = 2P a = 20: increasing W =10P 100P 1000 (from top to bottom). Unyielded regions areshown in gray.satisfies:0 ≤ v(u∗Pu∗) = −dy j(u∗) + a∫ 1+y+(x)−1−y−(x)u∗1(xP y) dy= −dy j(u∗) +∫ LP2−LP2∫ 1+y+(x)−1−y−(x)u∗1(xP y) dy dx = dy j(u∗) +f(u∗)O(3.17)115100 101 102 103 1040.20.40.60.811.21.41.6BOd  H = 1.5, L= 4H = 0.1, L= 20H = 2, L= 20100 101 102 103 104100101102103104B∆P/L  H = 1.5, L= 4H = 0.1, L= 20H = 2, L= 20Figurz 3CF2O Limiting behaviour as W → ∞ for the geometries ofFigs. 3.9-3.11: a) dy plotted against W; b) ∆eRa plottedagainst W.The functionals v(u∗Pu∗) and j(u∗) denote respectively the viscous andplastic dissipation rate functionals:v(u∗Pv∗) ≡∫ LP2−LP2∫ 1+y+(x)−1−y−(x)12˙ij(u∗)˙ij(v∗) dy dxP u∗Pv∗ ∈ Vj(v∗) ≡∫ LP2−LP2∫ 1+y+(x)−1−y−(x)˙(v∗) dy dxP v∗ ∈ V Pand note that we have extended the velocity integral at the inflow to theentire fracture:f(u∗) ≡∫ LP2−LP2∫ 1+y+(x)−1−y−(x)u∗1(xP y) dy dxPas the flow rate is identical through each cross-section. Continuing thisanalysis:0 ≤ v(u∗Pu∗) ≤ −j(u∗)[dy− supv∗∈VNv∗≠0f(v∗)j(v∗)]≡ −j(u∗) [dy−dyv] P(3.18)116where V denotes the space of admissible velocity solutions. The critical valueof dy is thus formally defined as:dyv = supv∗∈VNv∗̸=0f(v∗)j(v∗)O (3.19)Following a similar procedure to [198] we can show thatv(u∗Pu∗) ∼ d([dyv −dy]2) as dy→ dy∗−v P (3.20)dyj(u∗) ∼ f(u∗) ∼ d(dyv −dy) as dy→ dy∗−v P (3.21)j(u∗) ≳ v(u∗Pu∗)[dyv −dy] ≥ 0 as dy→ dy∗−v O (3.22)Figure 3.13 shows the computed values of v(u∗P u∗), j(u∗) and f(u∗) asW is increased, for the 3 geometries illustrated in Figs. 3.9-3.11. We seethat v(u∗P u∗) does indeed converge much faster than j(u∗) and f(u∗), as isimplicit in the above bounds. Note that the d([dyv − dy]2) in (3.20) is alower bound on the decay rate of the viscous dissipation and we can see thatthe decay rate is indeed faster than quadratic. Equation (3.22) ensures thatthe plastic dissipation decays to zero at least one order slower and (3.21)ensures that the limiting balance is between the plastic dissipation and flowrate. We again observe this in Fig. 3.13. This in turn can be used to arguethat the supremum in (3.19) is in fact achieved by the solution. Indeed, ifone knows the distribution of the limiting velocity solution, this can be usedto estimate dyv by inserting in (3.19). Note that the size of the limitingvelocity is not important in this determination as (3.19) is scale invariant.For computing the velocity, especially numerically, it can be more conve-nient to work with d(1) quantities, e.g. in tracking convergence. In movingbetween formulations it is necessary to rescale velocities with q∗. Thus,to evaluate dyv by inserting the limiting solution u∗ into (3.19), we mayinstead work directly with u in the large W limit, i.e.dyv ∼ limU→∞f(u)j(u)= limU→∞2aj(u)P (3.23)117100 102 10410−410−2100B  a(u⋆, u⋆)j(u⋆)Q(u⋆)100 102 10410−410−310−210−1100B  a(u⋆, u⋆)j(u⋆)Q(u⋆)100 102 10410−410−2100B  a(u⋆, u⋆)j(u⋆)Q(u⋆)Figurz 3CF3O Viscous dissipation, plastic dissipation and flow rate,v(u∗P u∗), j(u∗) and f(u∗) for the 3 geometries illustrated inFigs. 3.9-3.11, from left to right.118on noting that f(u) = 2a in the fixed flow rate formulation. In the nextsections we will estimate j(u) in order to derive approximations to dyv.HCICH hhort frvxturzsIn short fractures we have seen that increasingly narrow sheared layers sep-arate fouling layers and the moving plug. As the plug remains intact asW → ∞, from mass conservation we may assume that the plug velocityup ∼ 1. There is an evident symmetry in the shape of the sheared re-gions, which we assume can be reasonably approximated by boundaries:y = ±[yv ± y0(WP x)], in upper and lower sheared layers respectively. Herey = ±yv denote the central positions of the sheared layers at x = 0. Toleading order we may expect that within the sheared layers:˙(u) ∼∣∣∣∣UuUy∣∣∣∣ ∼ d(upy0)OOn integrating first with respect to y and then along the sheared layer wefind a leading order contribution to j(u), of size ∼ upa, from each shearedlayer. Therefore, we find that j(u) ∼ 2upa ∼ 2a, and hence that dyv ∼ 1,as observed.We may extend this analysis to examine the convergence of dy→ dy−v .Let us suppose that y0(WP 0) ∼ W−k as W →∞. Examination of the velocityprofile at x = 0 reveals that u(0P y) shows a variation across the sheared layerthat is approximately cubic in y ∓ yv. This leads to the approximation:u(xP y) ≈ up2− 3up2[(y − yvy0(WP x))− 13(y − yvy0(WP x))3]Pin the upper sheared layer. In fact as the ends x = ±aR2 are approached thecubic approximation changes to quadratic, but at the same time y0(WP x)decreases, so that contributions from these regions are smaller. It can beobserved that the above profile integrates in y to satisfy the flow rate con-straint exactly with up = 1 (and analogously in the lower shear layer). The119leading order components of the shear rate are:˙xy ∼ UuUy≈ − 32y0[1−(y − yvy0)2]P ˙xx ∼ 2UuUx≈ 3y′0y0[(y − yvy0)−(y − yvy0)3]OAs the sheared layers are relatively long and thin we may assume that|y′0(WP x)| ∼ y′0(WP 0)Ra≪ 1, and therefore find in the upper layer:˙(u) ≈ 32y0[1−(y − yvy0)2]1 + 4[y′0]2[(y−ycw0)−(y−ycw0)3]2[1−(y−ycw0)2]21P2∼ 32y0[1−(y − yvy0)2]+3[y′0]2y0[(y − yvy0)2−(y − yvy0)4]+d([y′0]4)(3.24)and a similar contribution from the lower layer. Integrating over the 2sheared layers gives:j(u) ∼ 2a+ 85∫ LP2−LP2[y′0]2 yxPand therefore as W →∞:dy ∼ f(u∗)j(u∗)=f(u)j(u)∼ 1− 45a∫ LP2−LP2[y′0]2 yx ∼ 1−d(1a2W2k)O (3.25)Therefore, we have dyv − dy ∼ a−2W−2k and since also dy = Wq∗, asW →∞ we have: q∗ ∼ 1RW −d(a−2W−2k−1). Returning now to (3.21):f(u∗) = 2aq∗ ∼ 2aRW −d(a−1W−2k−1) ∼ d(dyv −dy) ∼ d(a−2W−2k)OThis implies that k ≤ 1R2 is the maximal convergence rate for the width ofthe sheared layer in short fractures, (recall y0(WP 0) ∼ W−k).Following on similar lines, for short fractures, we may deduce that v(u∗Pu∗) ∼120102 103 10410−1m = −1/3Bd0102 103 10410−2m = −2/3B1−OdFigurz 3CFIO Limiting behaviour as W → ∞ for 5 short fracture ge-ometries: a) shear layer thickness y0; b) dyv − dy. Datais shown for: ◦PH = 1P a = 3), (□PH = O5P a = 2),(△PH = O25P a = 4), (∗PH = 3P a = 2), (+PH = 4P a = 4);recall that dyv = 1 for short fractures.a(q∗)2Ry0, as W →∞. From the bound (3.20) we deduce that:aW−2+k ∼ a(q∗)2Ry0 ∼ v(u∗Pu∗) ∼ d([dyv−dy]2) ∼ d(a−4W−4k)P =⇒ 5k ≤ 2P(3.26)i.e. k ≤ 0O4. Finally, we examine (3.22):aW−1 ∼ aq∗ ∼ j(u∗) ≳ v(u∗Pu∗)[dyv −dy] ∼a(q∗)2y0a−2W−2k∼ a3W3k−2 =⇒ k ≥ 1R3O(3.27)To summarise, as W → ∞, theoretical arguments imply that dy →1 − d(W−2k) and y0 ∼ W−k for some k ∈ [1R3P 0O4]. Figure 3.14 exploresthe same convergence from our numerical results, taken from 5 differentshort fracture geometries, each as W is increased. It appears that in facty0 ∼ W−1P3 with dy→ 1−d(W−2P3), in accordance with this analysis.121HCICI aong frvxturzs fiith no foulingIn a long symmetric fracture we commonly observe true plug regions inthe widest and narrowest parts of the fracture, separated by pseudo-plugregions. For a ≫ 1 we can expect that the flow is pseudo 1D and that thelubrication approximation should give a reasonable estimate of the pressuregradients and flow velocity. Indeed this has been verified in §3.3. Therefore,we may use the lubrication pressure gradient to evaluate the limiting dy.At large W we may expand (3.11) in series form:ϕ ∼ 1−√2(1 + h)W1P2+23(1 + h)2W+√29(1 + h)3W3P2+ OOOO (3.28)Thus, the total pressure drop is∆p =∫ LP2−LP2|UpUx| yx =∫ LP2−LP2W(1 + h)ϕyxP∼ W∫ LP2−LP211 + h+√2(1 + h)2W1P2+d(W−1) yxPdy =Wa∆p∼ a∫ LP2−LP211 + hyx+√2W1P2∫ LP2−LP21(1 + h)2yx+d(aW−1)P∼(1a∫ LP2−LP211 + hyx)−1 1− √2W1P21L∫ LP2−LP21(1+h)2yx1L∫ LP2−LP211+h yx+d(W−1) O(3.29)The first term above is clearly dyv, and can be derived by effectively inte-grating the yield stress along the slowly varying wall. For example, for thewavy interface we find the expression:dyv =√1 +H +1− cos(. )8H2OReferring back to Fig. 3.10 and the limiting behaviour illustrated in Fig. 3.12,we note that typically dyv S 1 for this flow regime. The condition for having122100 101 102 10310−1 m = −1/2BOdc−Od  Odc = 1.2274Figurz 3CF5O Convergence of dyv−dy as W →∞ for H = 1, a = 20.no fouling layer seems to require a small ratioHRa. Provided this is satisfied,dyv is in fact independent of a. In the example shown, since we have smallH it appears that dyv ≈ 1, but this need not be the case.From the analysis leading to (3.29) we see that dyv − dy ∼ W−1P2as W → ∞, i.e. faster than convergence with W for the short fractures.Comparing with the bounds (3.20)-(3.22) we deduce that:q∗ ∼(Wa∫ LP2−LP211 + hyx)−1+d(W−3P2)P j(u∗) ∼ 2aW+d(W−3P2)P asW →∞Pand the following boundv(u∗Pu∗) ≲ d(W−3P2)P as W →∞OFigure 3.15 illustrates the convergence rate of dyv −dy as W →∞ for oneof our computed “long” fractures. The W−1P2 scaling is verified.123HCIC5 Intzrmzyivtz frvxturzs fiith pvrtivl foulingIn the previous 2 subsections we have seen that for both short and long frac-tures, it is possible to estimate the critical limit using (3.12). For sufficientlyshort periodic fractures the flow yields at the narrowest width, which meansthat (3.12) can be used, taking the modified fracture width from (3.14),which amounts to h(x) → 0 as W → ∞. For long enough fractures, nofouling occurs and (3.12) is applied directly. However, as we have seen inFig. 3.11 for intermediateHRa the limit W →∞ results in only a limited por-tion of the fracture being fouled, say x ∈ [−xy P xy ] for a symmetric fracture( = 0). So far it is unclear: (i) how to predict dyv for such intermediatefractures; (ii) how is the fouling length xy determined; (iii) what determinestransitions from nominally short to intzrmzyivtz to long fractures?One feature observed in e.g. Fig. 3.11 is that for intermediate HRa asignificant portion of the fracture length remains yielded in the limit W →∞. Indeed the limiting solutions appear to result in true unyielded regionsonly in the narrowest and widest parts of the fracture (approximately x ∈[−xy P xy ]), with a sheared pseudo-plug region in between. The pseudo-plugregion is of course necessary, since the true plug speeds in wide and narrowparts of the fracture are different. This type of region arises in many visco-plastic flows.ihz xlvssixvl pszuyoBplug proxzyurzAnalysis of the pseudo-plug region stems from the studies of [246] and [17]that resolved the so-called lubrication paradox of [146]. We examine thisapproach as offering a potential description of the pseudo-plug observedhere. In this approach, the plug velocity is derived from the Buckingham-Reiner equation (3.11), which is approximated at large W by (3.28). Fromϕ we deduce the yield surface position at y = ±yl (x) = (1 + h(x))ϕ:yl (x) ∼ 1 + h(x)−√2W1P2+23(1 + h(x))W+√29(1 + h(x))2W3P2+ OOOO (3.30)124Interestingly, the thickness of the high shear layer is independent of h(x) toleading order at large W, simply following the wall profile. The pseudo-plugvelocity to leading order is calculated as:upN0 ∼ 11 + h[1 +√23(1 + h)W1P2− 169(1 + h)2W+d((1 + h)−3W−3P2)](3.31)As for the thin shear layers observed in short fractures, we deduce that ˙xy ∼upN0W1P2 within the sheared layer. The size of ˙xx comes from differentiatingupN0 with respect to x, i.e.|˙xx| ∼∣∣∣∣yhyx∣∣∣∣ ∼ 1aPsince the x-dependency is through both h(x) and yl (x), both derivatives ofwhich scale with whwx at large W (we assume H ∼ d(1).In the sheared layers, the leading order shear rate varies linearly with y:UuUy∼ ˙xy = ˙xy +W ∼ W(1− yyl)Pso that the velocity profile is parabolic in y. It becomes apparent that thescaling arguments that have been made (i.e. assuming |˙xy| ≫ |˙xx|), breakdown as y → y+l and more specifically for:y − yl ∼ yhyxW−1 ∼ 1WaOThe above is the usual argument for emergence of a pseudo-plug region,within which the strain rates are of comparable size. Since, within anypseudo-plug the main cause of straining will come from the geometricalchanges we may assume that |˙xy| ∼ |˙xx| ∼∣∣whwx∣∣ ∼ 1Ra. Therefore, withinthe pseudo-plug we expect that:ij =[1 +W˙]˙ij ∼ W ˙ij˙[1 +d(1Wa)]O125At leading order therefore we have that  = W in the pseudo-plug.For large a, the x-momentum equation at leading order in the pseudo-plug still does not include xx. It follows that:xy ∼ −W y1 + h(x)P xx ∼ −sign(h′(x))W√1− y2(1 + h(x))2P (3.32)noting that yl (x) ∼ 1+h(x). Equally we may deduce that to leading order:˙xx ∼ − 2h′(x)(1 + h(x))2P ˙xy ∼ −y|˙xx|√y2 + (1 + h(x))2P (3.33)which may be integrated to give the velocity distribution across the pseudo-plug:u ∼ upN0 + |˙xx|[√2(1 + h(x))−√y2 + (1 + h(x))2]ONote that under the assumption of near-parallel flow, i.e. a → ∞, bothstrain rates scale with |˙xx| ∼ HRa and consequently the correction to theleading order velocity from the variation across the plug is also ∼ HRa.In Fig. 3.16 we make a comparison between the predicted pseudo-plugx-velocity and that from the 2D computations. We can see that the predic-tion is not particularly good, in terms of the shape of the velocity profiles.Note that we have not carried out a matching procedure here [199], whichwould soften the corners of the pseudo-plug x-velocity. The shape of thecorrected velocity, across the pseudo plug, does not capture the variationsin the computed velocity field.eszuyoBplug rzgionsOne reason for the poor performance of the classical pseudo-plug procedureabove is that the limit we consider at zero flow is the critical limit of zeroflow as W →∞, for reasonably large a. However, it is not necessarily closeto the asymptotic limit a→∞, which is required for the preceding analysisto be valid. The stress distribution found as a → ∞ is shear-dominated,even within the pseudo-plug. Thus, for example xx → 0 in (3.32) as theyield surface is approached, as the solution must match with that in the1260.1 0.2 0.3 0.4−2−1012uxya) 0 0.1 0.2 0.3 0.4 0.5−2−1012uxyb)0 0.2 0.4 0.6−2−1012uxyc) 0 0.2 0.4 0.6 0.8−2−1012uxyd)Figurz 3CF6O Comparison of computed velocity (solid line) with theasymptotic approximation at: a) 2xRa = 0O2; b) 2xRa = 0O4;c) 2xRa = 0O6; d) 2xRa = 0O8, (as marked by vertical brokenlines in the colormap). Main figure shows the colormap ofspeed with streamlines and unyielded regions in gray; H =2P a = 20P W = 10000.127thin shear-layer close to the wall. This is not found to be the case here atintermediate a.On the other hand, parts of the scaling of the previous section are correct.Firstly, the velocity is observed to vanish in a relatively thin layer close to thewall. Secondly, since the strain rate is ˙ ≪ W within the pseudo-plug, theapproximation  ∼ W holds throughout the pseudo-plug. In place of (3.32),the following is found to give a reasonable representation of the shear stressfield within the pseudo-plug:xyW∼ −sign(y)√1−[f0|h′(x)|2 +√1− 2]2(3.34)xxW∼ −sign(h′(x))[f0|h′(x)|2 +√1− 2] (3.35)where  = yR(1+h(x)). Note that f0h′(x) approximates a normalized xx asthe walls are approached. Setting f0 = 0, the distribution of (3.32) is recov-ered, but generally we find that f0 ≈ 1 within the pseudo-plug. Figure 3.17compares (3.35) with the computed stress distributions across the pseudo-plug for 2 different intermediate geometries, in both cases illustrating thegood agreement.io– moyzl for zstimvting j(u) vs W →∞We expect that the intermediate fractures have asymptotic behaviour asW →∞ that is intermediate between that of the short fractures and the longfractures with no fouling. Let us therefore consider fractures with relativelylarge a and H ∼ d(1) so that h′(x) is relatively small. For simplicitywe consider  = 0. The flow rate constraint determines f(u) = 2a andtherefore, to estimate dy in the limit W →∞ it suffices to estimate j(u). Wesuppose that the limiting flows are divided into 2 distinct regions. Firstly,in the wider part of the fracture we assume that there are fouling layersthat fill the deepest parts of the fracture for x ∈ (−xy P xy ). Secondly, forx ̸∈ (−xy P xy ) we assume that there are no fouled regions at the walls of thefracture.In the unfouled regions the mean velocity in the x-direction is simply128−1500 −1000 −500 0 500−1.5−1−0.500.511.5τxxya) −1000 −800 −600 −400 −200 0 200−1.5−1−0.500.511.5τxxyb) −1000 −800 −600 −400 −200 0 200−1.5−1−0.500.511.5τxxyc)−1000 −500 0 500−3−2−10123τxxya) −1000 −500 0 500 1000−3−2−10123τxxyb) −2000 −1000 0 1000−3−2−10123τxxyc)Figurz 3CFLO Colormaps of xx (unyielded regions in gray) for W =1000. Top panels: H = 0O5, a = 40, comparisons are shownwith the predictions of (3.35) at values 2xRa = 0O25P 0O5P 0O7,as marked with broken lines on the colormap. Lower panel:H = 2, a = 20, comparisons are shown with the predictionsof (3.35) at values 2xRa = 0O45P 0O6P 0O8129u¯ = 1R(1 + h(x)). In these regions, as we approach the walls, the velocitydrops from u ≈ u¯ to zero over a thin layer of d(W−1P2). This thin layergives a contribution to j(u) of approximate sizejs(u) ≈ 4∫ LP2xf11 + h(x)yxPwith the subscript denoting the shear layer. Further away from the wall,in the pseudo-plug, we assume that variations from the mean velocity aredriven by axial variations in the fracture shape, and that u ≈ u¯. Assumingu ≈ u¯ suggests that:UuUx≈ − h′(x)(1 + h(x))2⇒ v ≈ d(h′(x))Pand hence we expect the next order of terms approximating u to also havesize d(h′(x)). It follows that within the pseudo-plug:˙(u) = |˙xx|[1 +(˙xy˙xx)2]1P2∼ 2|h′(x)|(1 + h(x))2[1 +(˙xy˙xx)2]1P2O (3.36)The contributions to j(u) for x ̸∈ (−xy P xy ), within the pseudo-plug is there-fore:jpp(u) ≈ 4∫ LP2xf2|h′(x)|(1 + h(x))2∫ 1+h(x)0[1 +(˙xy˙xx)2]1P2yy yxNote that since we have assumed  = 0 we have exploited symmetry in bothx and y to simplify this expression. To calculate the strain rates we assumethat (3.34) & (3.35) approximate the stresses in the pseudo-plug. The ratio130of the strain rates is then given by that of the stresses. Therefore, we find:∫ 1+h0[1 +(˙xy˙xx)2]1P2yy = (1 + h)∫ 101f0|h′|2 +√1− 2 y=.21 + h1 +√f0|h′|≈ (1 + h).2[1 +d(|h′|1P2)]Pjpp(u) ≈ 4.∫ LP2xf|h′|(1 + h)11 +√f0|h′|yx = 4. ln(1+h(xy ))[1+d(|h′|1P2)]OSecondly, we consider x ∈ (−xy P xy ), where only a thin layer of fluid issheared. The contribution jy (u) to j(u) from the fouling region comes fromthe thin shear layers, at y ≈ ±[1+h(xy )]. The plug velocity is approximatelyup ≈ [1 + h(xy )]−1 and it therefore follows that:jy (u) ≈ 4xy1 + h(xy )[1 +d((y′0)2)Pwhere again y0(xPW) is taken to represent the width of the sheared layer(see the analysis of the short fracture). To summarise, if xy is known thenj(u) is approximately:j(u) ≈ js(u) + jpp(u) + jy (u)≈ 4∫ LP2xf11 + h(x)yx+ 4. ln(1 + h(xy )) +4xy1 + h(xy )+ OOOwhere the additional terms are of smaller order in either 1Ra or 1RW. Con-sequently, as W →∞ we approximate:dy(xy ) ∼ f(u)j(u)≈aR2∫ LP2xf11 + h(x)yx+xy1 + h(xy )+ . ln(1 + h(xy ))(3.37)In general we expect that the strain rate is minimized, amongst all con-131straints. If we regard xy as being selected in this way, the actual xy is deter-mined by maximizing dy(xy ) with respect to xy , thus giving dyv. Note thatthe first 2 terms in the denominator of (3.37) effectively interpolate betweenthe limiting dyv derived for short channels (xy → aR2) and that for longchannels with no fouling (xy → 0). The 3rd term in (3.37) also vanishes asxy → aR2, so that the expression for short channels is contained in (3.37).Figure 3.18a shows examples of the variation of dy(xy ) computed from(3.37) for a = 20, H = 0P 1P OOO8 (black), and a = 6, H = 0P 0O25P 0O5OOO2(red). The maximum of dy(xy ) is attained either at an endpoint or at thezero of:xy = .(1 + h(xy ))P (3.38)which is straightforwardly computed. For the examples shown, for a = 6the maximum is dyv = 1, attained at xy = aR2, for all H. As (1+h(xy )) ∈[1P 1+H], we see that the above equation has a root only for a S 2.. Sincewhen this is not satisfied we find dyv = 1, which is the short fracture limitwe adopt the conditiona Q 2.P (3.39)as our definition of short fractures. At larger a we see that there is amaximum in (0P aR2) and compute this numerically. As a→∞ we find that2xyRa→ 0 since xy → (1+H).. The 2nd and 3rd terms in the denominatorof (3.37) then become d(1Ra) smaller than the first, which converges to dyvfor the long fracture limit without fouling. Figure 3.18b shows the variationof dyv with a for 2 values of H, computed as the maximum of (3.37). Fora satisfying (3.39) we see dyv = 1, increasing smoothly to the asymptoticvalues for no fouling at large a, marked with the broken lines.We have computed dyv from this toy model, by maximizing (3.37) overxy . Figure 3.19a & b plot respectively dyv and xy as functions of (HPa).We see that the limiting behaviour for both short and very long fractures isrepresented in dyv. As (HPa) both increase the predicted dyv also increasesaway from 1. The computed values of xy are close to aR2 for short fracturesand approach zero only for very long fractures (Fig. 3.19b). Figure 3.19cpresents a contour plot of the largest values of dy attained in our compu-132a) 2xf/L0 0.2 0.4 0.6 0.8 1Od(xf)0.60.811.21.41.6 H = 8, L = 20H = 2, L = 6b)L100 102 104Odc11.21.41.61.822.22.4H = 1H = 4Figurz 3CFMO a) Variation of dy(xy ) from (3.37) for: a = 20, H =0P 1P OOO8 (black); a = 6, H = 0P 0O25P 0O5OOO2 (red). b) Varia-tion of dyv (computed by maximizing dy(xy ) in (3.37)) witha for H = 1 (black) and H = 4 (red). Broken lines indicatethe limit of dyv with no fouling.tations at each (HPa), (typically computed for W = 104), which may beregarded as a computed estimate of dyv. Comparing with Fig. 3.19a we seethat the qualitative trends are extremely well represented by this approxima-tion and the quantitative agreement is good for either smaller H or smallera, but deteriorates as both parameters increase. This is of course not sur-prising as the approximation is based on the strain rates being driven mostlyby variation in u¯ with x, and we have neglected correction terms involvingf0|h′| (valid for large a). Figure 3.19d plots the relative error between thecomputed dyv from our 2D computed results and that approximated frommaximizing (3.37). We see that the relative error is Q 10%, close to theaxes and increases to approximately 15% over much of the (HPa) parame-ter space explored. Evidently, once both H and a are much larger than 1the geometry represents a large vacuous space rather than a fracture.133a) H0 5 10L510152025303540455011.522.5b) H0 5 10L510152025303540455000.20.40.60.81c) H2 4 6 8 10L102030405011.522.5d) H2 4 6 8 10L10203040500.050.10.150.20.250.3Figurz 3CFNO a) Variation of dyv, computed by maximizing (3.37). b)Variation of the value of xy , that maximizes (3.37), scaled withaR2. c) Variation of dyv, approximated as the largest valueof dy from our 2D computations, typically at W = 104. d)Absolute relative error between a & c.Vn improvzy to– moyzlAlthough the relative errors in Fig. 3.19d are reasonable, it is possible toimprove the estimate of j(u). From consideration of Fig. 3.16 and similar,it is evident that the pseudo-plug behaviour is more complex than thatgiven by the proposed jpp(u). The simplest improvement is to include the134neglected f0|h′| term, i.e.jpp(u) ≈ 4.∫ LP2xf|h′|(1 + h)11 +√f0|h′|yxO (3.40)The stress approximations (3.34) & (3.35) can be matched with the com-puted stresses at each x in the pseudo-plug, approximating f0(x) from thestress values close to the wall. Interestingly, this procedure suggests thatf0(x) = constant throughout the pseudo-plug region. Inspection of manycomputations suggests that f0 ≈ 1.We may now repeat the same procedure as for the toy model. We findxy by maximizing dyxf , now given by:dy(xy ) ∼ f(u)j(u)≈aR2∫ LP2xf11 + h(x)yx+xy1 + h(xy )+ .∫ LP2xf|h′|(1 + h)11 +√f0|h′|yxO (3.41)This results in dyv and xy as illustrated in Fig. 3.20a & b, respectively. Weobserve a general increase in dyv and a small shift in xy . The relative errorwith the computed 2D results is however diminished; see Fig. 3.20c, nowtypically remaining below 10% for most of the parameter space.HCIC6 Vffinz frvxturzsFinally we present some examples of the limiting process in affine fractures.Three different styles of affine fracture are generated for intermediateH = 2,a = 20. Figures 3.21-3.23 present the sequence W = 100P 1000P 10000 foreach case, plotting the speed, streamlines and unyielded plug regions. Ingeneral we observe that the small scale fracture roughness is always fouled.As W increases a larger fraction of the fracture becomes immobilised. Theboundaries of the flowing part of the fracture are formed by arc-like surfacesthat span between locally narrow points of the fracture wall. The radius ofcurvature of these surfaces increases with W.135a) H0 5 10L510152025303540455011.522.5b) H0 5 10L510152025303540455000.20.40.60.81c) H2 4 6 8 10L10203040500.020.040.060.080.10.120.140.160.18Figurz 3C2EO a) Variation of dyv, computed by maximizing (3.41). b)Variation of the value of xy , that maximizes (3.41), scaled withaR2. c) Absolute relative error between a & our dyv calculatedfrom the 2D computations.136Figurz 3C2FO Computed examples of speed |u| & streamlines for asymmetric affine fracture. Parameters are: H = 2, a = 20,W = 100P 1000P 10000, from top to bottom, with (gray) un-yielded plugs.It is notable that as W increases, the self-selected flowing channel ge-ometry consists of a series of joined segments along which the streamlinesare approximately parallel. In such sections shear components evidentlydominate. These parallel segments are connected by angular convergingand diverging segments, in which extensional stresses and strain rates willbe significant. The limiting process as W →∞ appears to result again in acombination of thin yielded shear-layers and non-vanishingd(1) pseudo-plug137Figurz 3C22O Computed examples of speed |u|& streamlines for a frac-ture formed from two different affine surfaces. Parameters are:H = 2, a = 20, W = 100P 1000P 10000, from top to bottom,with (gray) unyielded plugs.regions. Although the evident complexity of the self-selection and limitingprocesses so far elude a simple explanation, there is some hope that thesecomponent structures can be understood. Thus, if one can first understandthe geometric features of flow self-selection (meaning the fouling process andorientation of flowing part of the fracture, generating tortuosity ) it shouldbe possible to estimate the critical pressure drops (or yield stresses).More quantitatively, we may evaluate dyv for these geometries from the138Figurz 3C23O Computed examples of speed |u|& streamlines for a frac-ture formed from two identical affine surfaces, shifted. Param-eters are: H = 2, a = 20, W = 100P 1000P 10000, from top tobottom, with (gray) unyielded plugs.computed 2D solution. Figure 3.24 shows convergence of dy → dyv asW →∞. The symmetric fracture has the smallest dyv, and also appears tohave the more constricted flow apertures. The other 2 geometries appearto compensate increasing tortuosity (smaller dy) with an larger effectivechannel width (larger dy).139100 101 102 103 1040.60.811.21.41.6BOdFigurz 3C2IO Computed Oldroyd number as function of the Binghamnumber for the three different affine fractures of Figs. 3.21-3.23. Blue squares: the two surface are symmetrical. Redcircles: the two surfaces are uncorrelated. Black lozenges: thetwo surfaces are identical, shifted laterally with a constant gap.HC5 Yisxussion vny summvryIn this chapter we have investigated the two-dimensional flow of a Binghamfluid along an uneven fracture. The work has 2 principal foci. Firstly,to determine numerically the flow rate-pressure drop relationship in suchgeometries (the appropriate Darcy law), to understand better the limits ofsimple approximations that are presently used in a somewhat vy hox manner.Secondly, we explore the limiting values of pressure drop (equivalently yieldstress) at which non-zero flows are first found. This question has criticalconsequences for the study of flow onset in pressure driven porous media140flows, i.e. selection of the critical initial path, as well as helping providepractical estimates for invasive sealing of porous media/channels, i.e. howfar will a sealing fluid penetrate under a given driving pressure?The initial part of the chapter has extensively studied geometric effectsin idealised fractures (periodic with wavy or triangular profiles) of dimen-sion (HPa). Strict application of lubrication/Hele-Shaw type approxima-tions to the Darcy law has been shown to be limited to HRa ≪ 1, as maybe expected. As a rule of thumb, errors of 10% or less appear to requireHRa ⪅ 1R20. For geometries outside of these limits, flow approximations arevulnerable to similar effects to Newtonian fluids, e.g. developing tortuosityetc. One significant difference between Newtonian and Bingham (or othergeneralised Newtonian) fluids is in the coupling of geometrical and rheolog-ical parameters in the flow law. In the Newtonian fluid literature there aremany efforts to improve and extend Darcy-law type estimates to these situ-ations. Here we have resisted the temptation to develop analogous methods.It is clear that some of these methods would be effective, especially in thelimit of low W and for HRa increasing away from the lubrication limit, butinflicting this algebraic misery on the reader is left to other researchers.Moving to larger W or for HRa ̸≪ 1 a more serious limitation of lubrica-tion type approximations to the Darcy law emerges. Unyielded fluid appearsat the wall in deeper parts of the fracture (fouling layers). Fouling layersprovide an d(1) adjustment of the flowing area and hence d(1) errors inthe predictions of lubrication type approximations. We have demonstratedthat these errors can be reduced significantly, by adopting the yield surfaceof the fouling layers as the new fracture wall and applying the usual lubri-cation approximation. Therefore, approximating the flow-law presents noparticular problem provided the fouling layers are known. Unfortunately,this is not the case as the fluid self-selects the flowing area.Previous work has addressed the question of onset of fouling in idealizedgeometries with d(1) variations [208, 209]; (see Chapter 2 & Chapter 4).Smaller scale roughness also seems to readily foul, although to our knowledgethis has not been quantified. At intermediate W estimating the degree offouling appears difficult, as a general problem for which one would like to141specify a simple predictive closure relationship. However, numerical solutionas here is an effective tool even for the very complex affine geometries, andcan be easily extended to more general yield stress fluid models. The otherparameter range where fouling can be predicted is that in which W → ∞and the flow stops.A significant part of the chapter has considered the limit of no flow W →∞, which via a rescaling can be characterized with the Oldroyd number:dy ≡ aˆYˆˆl∆eˆP (3.42)directly representing the balance between applied pressure and resistingyield stress. Above a critical value dyv there is no flow. On the theo-retical side, we have formally characterized dyv as a limiting ratio of flowrate to plastic dissipation, shown that this limit is attained by the velocitysolution, and provided general bounds on convergence of the viscous dissi-pation, plastic dissipation and flow rate, in the limit dy → dy−v . We havethen studied the approach to dyv numerically and asymptotically for sim-pler symmetric fracture geometries, deriving the leading order behaviour inthe cases of both short and long-thin fractures.For short fractures we find that dyv = 1: at large W the flow consists ofan intact central plug region separated from the walls (and fouling layers) bya progressively thinning shear layer. This limit is analogous to the flow in auniform channel, for which also dyv = 1. Short fractures, for the purposesof this limiting process, are defined as a Q 2.. For long-thin fracturesno fouling occurs and the lubrication approximation is valid, leading to areadily calculable dyv S 1.Intermediate fractures are more interesting. The central plug region hereis broken into 2 parts (in wide and narrow parts of the fracture) and remainsso as W →∞. The plug in the widest part occupies a length approximatelyequal to that of the fouling layer (defined as 2xy ), whereas the size of plugsin the narrowest part of the fracture remains of d(1). Between the two plugregions we find a slightly sheared pseudo-plug region. The pseudo-plug joinsto the wall in a narrow layer of high shear and a similar high shear layer142separates fouling layers and the central plug in the wide part of the fracture.Based on these observations, we have proposed a rather simple toy modelto describe the limiting flows. In each of the above regions we are able toestimate the strain rate and hence construct an approximation to the plasticdissipation that depends only on the single parameter xy , the half-length ofthe fouled layers. Via minimizing the plastic dissipation at fixed flow rate weare able to estimate both xy and to calculate the limiting dyv. The relativeerrors in this crude approximation are typically ⪅ 10− 15%, allowing us toeffectively predict critical pressure for the onset of flow.We have also performed several simulations in self-affine fracture asshown in Figs. 3.21-3.23. Although these geometries deserve a far more com-plete study, we may infer some trends from our current results. Firstly, bothpseudo-plugs and the fouling layers are present in the affine fractures. Sec-ondly, the limiting zero flow behaviours again appears to be characterizedby geometries that simplistically are componentwise constructed of shearlayers, pseudo-plugs and fouled regions. This leads us to the conclusion thatto understand the flow one needs to understand how the fouling layer isself-selected. It seems for instance obvious that it acts as a filter for thesmall scales - the roughness problem. It is however unclear how the larger-scale heterogeneities are filtered. Similarly to the wavy fractures, the finalselected flowing channel seems to have a rather constant width, at least insections.Finally, we should observe that the critical Oldroyd number frequentlyis quite far from that obtain by the lubrication approximation, where itis derived from the harmonic mean of the fracture width. The result has astrong consequence if one want to investigate onset/channelization in porousmedia or in a rough Hele-Shaw cells (2D fractures). As seen, the range ofwavelengths for which the na¨ıve lubrication approximation is valid is limitedand its usage will incur d(1) errors.143Chvptzr Ihtokzs ofi through fivshoutsIn this chapter we begin to study the industrial flows that arise in theprimary cementing of oil and gas wells, in cases where the well has a washedout section. This narrow annular geometry is idealized by studying flows ina longitudinal 2D section. We perform a detailed computational study of theflow of a Bingham fluid along a narrow channel, with one locally uneven wall(the washed out section of an oil well about to be cemented). By studying awide range of different washout geometries, of differing shapes characterizedby dimensionless height h and length a, we discover that the flows exhibita type of self-similarity in the limit of large h and W. More specifically,in this limit we find that regardless of the washout geometry, the area ofthe channel that contains moving fluid is the same for each a. Essentially,uneven and distant parts of the washout become full of static fluid, below theyield stress, while the flow self-selects its own unique geometry. The washoutgeometry with the largest flowing region within the washout, appears to bethe square wave. We show how a simple correction can be calculated thatallows one to predict the flowing area for other washout geometries. Lastly,we examine the effect on the pressure drop of different washout geometries.The flows considered in this chapter are Stokes flows, which might corre-spond to the breaking of circulation in the well, i.e. low initial flow rate. Thisrestriction is to enable us to study a simplified initial problem with only 3dimensionless groups, while addressing some of the complexity of the geom-144etry (which is typically unknown in a washout). Subsequently, in Chapter 5we consider inertial flows and later in Chapter 6 we consider later in thecementing process when the drilling mud is displaced by another fluid.1ICF IntroyuxtionA crucial part of the primary cementing operation [176] is the removal ofdrilling mud from the annular space between casing and formation. Thisoccurs during the fluid displacement (or cement placement) phase of theprocess, which is itself a complex flow not fully understood; see e.g. [33,49, 189]. Prior to the fluid displacement phase, most service companies andoperators recommend to pre-circulate the well, by pumping the drilling mudfrom the bottom to the top of the well at least once (“circulating bottoms-up”).The circulation phase has two main purposes. Firstly, drilled cuttingsand other solids still in the well, which may have settled as the casing isrun in to the borehole, are cleared from the flowpath. Secondly, the circula-tion serves to shear the mud and hence condition it prior to displacement.Depending on the operational circumstances and the type of drilling mud,the mud may have been static in the borehole for a period of hours beforecirculation. Over this time significant gel strengths may develop due tothixotropic effects. Simplistically speaking this gel strength is destroyed byshear, returning the drilling mud to its dynamic yield stress (which may stillbe significant).Unlike drilling geometries, the annuli in primary cementing are relativelynarrow. If the borehole is reasonably uniform, the flow is primarily in theaxial direction and on each azimuthal section the largest shear stresses arefound at the walls of the annulus. The wall shear stresses are smaller onthe narrow side of the eccentric annulus. This means that there is a criticalminimal pressure gradient that must be exceeded in order to mobilize the1A vzr“ion of thi“ xhvptzr hv“ vppzvrzy v“: AC gou“tvziA AC Go““zlin & IC [rigvvryCqgz“iyuvl yrilling muy yuring xonyitioning of unzvzn worzholz“ in primvr– xzmzntingCpvrt F: ghzolog– vny gzomztr– zffzxt“ in non-inzrtivl ofi“C7 J. Non-Newt. Flhid Mech.A220:ML{NMA (G0F5)C145drilling fluid on the narrow side of the annulus; see e.g. [157, 225, 246].Thus, stationary residual drilling fluid is an inherent part of flowing throughirregular geometries.A different type of irregularity occurs in poorly consolidated forma-tions that may partially collapse during the drilling process. The result-ing “washouts” (see e.g. Fig. 4.1) are geometrically dependent on boththe drilling hydraulics and the local geology, hence unpredictable, althoughsometimes measured using a caliper. The effect on mud circulation of suchgeometries can however be guessed: for sufficiently deep washouts we ex-pect the drilling mud to remain static within parts of the washout during thecirculation operation. The objective of this chapter is to begin to systemati-cally study this phenomenon, with the aim of being able to offer quantitativepredictions.We have not found any work that directly studies this problem. Threedifficulties present themselves. First, the washout geometries are unpre-dictable and three-dimensional (3D), making analytical study difficult andcomputational study slow. Second, the critical feature of the yield stress inthe drilling mud must be accounted for physically and computationally in away that distinguishes slowly flowing regions from stationary regions. Forexample, slow flows past cavities of Newtonian fluids can yield arbitrarilyslowly moving fluid, see e.g. [166]. Thirdly, a range of flows are experiencedin cementing and drilling operations (see [41, 176]), ranging from the break-ing of circulation through to fully turbulent flows. Thus, at the outset itis necessary to focus in on specific flow types and make simplifications inorder to make the problem tractable while still preserving some relevance.In this chapter we study non-inertial or Stokes flow solutions (gz = 0) pasttwo-dimensional (2D) washout regions.A number of authors have studied flow of yield stress fluids past cavitiesor expansions, either in 2D or axisymmetric geometries. Mitsoulis and co-workers (e.g. [164]) have studied both planar and axisymmetric expansionflows, over a wide range of Bingham and Reynolds numbers, showing signif-icant regions of static fluid in the corner after the expansion. Flow of yieldstress fluids through an expansion-contraction has been studied experimen-146tally and computationally in [79, 174, 175]. In [79] Carbopol solutions werepumped through a sudden expansion/contraction, i.e. narrow pipe – widepipe – narrow pipe. A range of flow rates were studied and the yield surfaceswere visualised nicely by a particle seeding arrangement. Inertial effects areevident in asymmetry of the flow. For all the experiments shown, stagnantregions first appear in the corners of the expansion and for all the resultsshown there are stagnant regions. Comparisons are made between experi-mental and numerical results, which are at least qualitatively in agreement,see [79, 174]. In [175] qualitatively similar computations are carried out, butusing an elasto-viscoplastic model.Previous work has studied flow along a symmetric 2D wavy channel.Small amplitude long wavelength perturbations from a uniform channel ge-ometry have been studied in [102, 199]. The focus is on deformation andbreaking of the central plug region. Secondly, in [208] (see Chapter 2) wehave studied larger amplitude perturbations, using a numerical method sim-ilar to that here. We have derived predictions for the onset of stationaryfluid regions (fouling lvyzrs), which occurred always initially at the wallsin the widest part of the channel, quantified in terms of a critical slopz ofthe wall. The main difference between the present chapter and [208] is that,having noticed qualitative difference between smooth wavy walls and abruptexpansion-contractions, we now begin to consider geometric variability in amore systematic way, i.e. we aim to derive results that may be applicable togeneric washout shapes.An outline of this chapter is as follows. In §4.2 we derive the generalmodel and describe the four washout geometries that we study. We alsopresent an overview of the computational method, targeted here at gz = 0.For the results we first present examples of typical variations in the velocity,unyielded plug boundaries and the stress components, for the four washoutgeometries as the depth is increased; see §4.3.1. This is followed by a moresystematic study of geometry and yield stress effects in §4.3.2, focusing onlyon the plug regions. We show that the onset of stationary flow is morecomplex than suggested in [208], but in contrast for large enough amplitudesa type of similarity emerges between geometries as the flow self-selects the147CasingWashoutxˆDˆUˆHFigurz ICFO Geometry of the washout in a section along the annulus.flowing area. Section 4.3.3 describes our analysis of this self-selection andhow to predict the flowing area of the washout. In 4.3.4 we describe theeffects of the washout on the pressure drop along the channel, in terms of acorrection. The chapter closes with a summary in §4.4.ICG V simplizy fivshout moyzlPrimary cementing circulation occurs through a long narrow eccentric an-nulus. In many situations, restrictions on frictional pressures limit the flowrates to the laminar regime, in which case the main velocity components arealong the borehole axis, with azimuthal flows driven by slow axial variationsin eccentricity. Supposing in this situation that we have a washout section, areasonable simplification of the flow might be to consider a two-dimensional(2D) azimuthal-axial slice through the annulus; see Fig. 4.1. Neglecting az-imuthal flow, the channel formed has an in-gauge width Yˆ (dependent onthe inner and outer diameters and the eccentricity), being bounded on theinside by the casing and on the outside by the formation.The washout geometry is as yet unspecified, but in the 2D section de-scribed we will characterize the washout as having a length aˆ and a max-imum amplitude Hˆ. Coordinates (xˆP yˆ) are aligned along the channel and148across the annular gap, respectively; see Fig. 4.1. Fluid circulates axiallyalong the 2D section and we might assume a mean velocity jˆ0 in the xˆ-direction. As a typical width, Yˆ is in the range 3−80 mm, the drilling mudmay be considered as locally incompressible over the length of the washout,even for aˆ of the order of a few meters. A suitable model therefore is givenby the Navier-Stokes equations for an incompressible viscous fluid. Non-Newtonian effects are significant and principal amongst these is the yieldstress effect (at least for the flows we consider). The simplest rheologicalmodel containing a yield stress is the Bingham fluid. This model is also onethat is used in drilling applications, although more complex models (e.g.Herschel-Bulkley) are also used. We use the Bingham constitutive modelhere, characterizing the fluid via its density /ˆ, yield stress ˆl and plasticviscosity ˆ. Although a gross simplification, the key feature of yielding/not-yielding is captured by this model.The Navier-Stokes equations are made dimensionless using Yˆ and jˆ0 aslength and velocity scales, respectively. Time is scaled with YˆRjˆ0, the shearstresses are scaled with ˆjˆ0RYˆ and the static pressure is subtracted fromthe pressure before also scaling with ˆjˆ0RYˆ. The resulting equations are:gzYuYt= −UpUx+UUxxx +UUyxyP (4.1)gzYvYt= −UpUy+UUxxy +UUyyyP (4.2)0 =UuUx+UvUyP (4.3)where u = (uP v) is the velocity, p is the modified pressure and ij is thedeviatoric stress tensor. The constitutive laws are:ij =(1 +W˙(u))˙ij ⇐⇒  S W (4.4a)˙ij(u) = 0⇐⇒  ≤ WP (4.4b)149where˙ij(u) =UuiUxj+UujUxiP u = (uP v) = (u1P u2)P fl = (xP y) = (x1P x2)Oand ˙,  are the norms of ˙ij , ij , defined as˙ =√12∑ij˙2ij and  =√12∑ij2ij O (4.5)Two dimensionless numbers appear above. Firstly, the Reynolds num-ber, gz:gz ≡ /ˆjˆ0YˆˆP (4.6)which represents the ratio of inertial to viscous effects. Secondly, the Bing-ham number, W:W ≡ ˆl Yˆˆjˆ0P (4.7)represents the competition between yield and viscous stresses. As well asthese, two geometric dimensionless parameters result: h = HˆRYˆ, represent-ing the maximal depth of the washout, and  = YˆRaˆ, characterizing theaspect ratio of the washout.ICGCF ihz moyzl prowlzmA Re = (For the remainder of this chapter we consider only steady non-inertial flows(gz = 0), which would characterize the flow directly after the breakingof circulation, i.e. as the pressure drop is increased sufficiently for drillingmud to begin to flow. The main reason however for the neglect of inertialeffects is that our study is focused at the relative importance of viscousand yield stress effects, i.e. the potential for immobility of the mud in thewashout, and is sufficiently complicated by geometric considerations. Somesimplification is needed in order to be able to extract meaningful results. Wemight intuitively expect that more problematic cases industrially are likelyto be in wells where gz is restricted and stationary regions consequently150large, i.e. does neglect of inertia reflect a worst case situation?The equations we study therefore are:0 = −UpUx+UUxxx +UUyxyP (4.8)0 = −UpUy+UUxxy +UUyyyP (4.9)0 =UuUx+UvUyP (4.10)For a computational domain we extend the channel uniformly both upstreamand downstream of the washout section; see Fig. 4.2. The two extensionsare determined empirically (see below in §4.2.3), to be sufficient to allowthat imposition of a constant drop in pressure along the channel producesfully developed velocities at both x = −1R2−lw (entry development length)and at the exit, x = 1R2 + lx. If the washout geometry is symmetric, thendue to symmetry properties of the Stokes equations, we would expect thatlw = lx. Boundary conditions ofu = 0P (4.11)are applied at the walls (no-slip). The flow is assumed fully developed atentry and exit:xx = −p+ xx = −sP v = 0P at x = −1R2 − lwP (4.12)xx = −p+ xx = 0P v = 0P at x = 1R2 + lxO (4.13)Since we have scaled the velocity with the mean axial velocity along thechannel, the pressure drop s must be found as part of the problem. Essen-tially, s is increased until the mean velocity at the entry is 1, i.e. s satisfies:1 =∫ 10u(xP y) dyP at x = −lwO (4.14)The computational domain and boundary conditions are illustrated schemat-ically in Fig. 4.2.151x = 0.5δ -1y=1+hy=1x = -ld u=0 x = leσxx= -sv=0σxx=0v=0yxx = -0.5δ -1Figurz IC2O Dimensionless computational domain and boundary con-ditions for this study.ICGCG lvshout gzomztrizsWe consider 4 families of washout geometry in this study. For all geometriesthe maximum depth of the washout is h = HˆRYˆ and the length is −1 =aˆRYˆ, i.e. the length of that part of the channel that does not have unitwidth. For each geometry we explore a range of hP . The geometries aredescribes as follows.• Square wave: an abrupt change in channel width from 1 to 1 + h.• Triangular wave: a linear change in channel width from 1 to 1 + h,over a distance 0O5R, i.e. the angle of the washout wall, relative to theuniform channel wall is 0, where tan0 = 2h.• Sinusoidal wave: linear change in channel width from 1 to 1 + h, overa distance 0O5R following a sine wave. The upper wall of the washoutis given by: yw = 1 + 0O5h[1 + cos 4pix].• Koch snowflake: the shape is based on the Koch snowflake curve; see[242]. The typical Koch snowflake construction involves repeated sub-division of each straight edge into 3, with the addition of a triangularshape on the central 1R3 of each edge. Normally, this is carried out onan equilateral triangle. Here the initial triangle is formed only on thewashout length: x ∈ [−0O5RP 0O5R], and has depth h, i.e. as for thetriangular wave. Subsequent iterations form an isosceles triangle with152a)DˆLˆ2Hˆb) c) d)Figurz IC3O Four different categories of symmetric washout geome-try, shown for x Q 0: a) sudden expansion-contraction (squarewave); b) triangular wave; c) sinusoidal wave; d) semi-fractal,based on the Koch snow flake. All washout shapes are fullydetermined by (hP ).equal angles 0 on each central third of each straight edge. The num-ber of iterations is stopped when the straight edges being subdividedare just greater than the initial mesh size used for the computation.These 4 geometries are illustrated in Fig. 4.3 for clarity.ICGCH Computvtionvl ovzrvizfiThe main difficulty in finding solutions of Bingham fluid flows (and otheryield stress models) is due to the singularity in the effective viscosity as˙ → 0+. This limit marks the boundary of unyielded (or plug) regions, whichmay be either attached to the walls of the flow domain or moving rigidlyin the interior of the flow domain. The deviatoric stress is undeterminedin these regions, satisfying the inequality ‖‖ ≤ W. Only in simple flowsare the boundaries known and stresses calculable analytically, e.g. due tosymmetry considerations. With some pre-knowledge of the plug topology,numerical solution can be attempted with an iteration loop for the shape ofthe unyielded regions, e.g. [225]. Although effective in some cases, extensionof these methods to general flows with no initial guess for the plug shape isunclear.A common way to compute general flows is to rzgulvrizz the singularityby introducing a numerical perturbation into (4.4), to define the effective vis-cosity everywhere in the flow domain. Typically this perturbation is charac-153terized by a small parameter ϵ, that characterizes the transition to low shear-rates. Several types of regularization have been proposed, e.g. [9, 28, 186].In these schemes the computational method is exactly as for a nonlinearlyviscous (generalised Newtonian) fluid with unyizlyzy regions characterisedby very viscous fluid. The usual method to interpret unyizlyzy fluid is bythe criterion: ‖‖ ≤ W, but note that the shape of the unyielded regionsthen depends on the choice of regularization parameter and regularizationmethod. In addition, on taking ϵ→ 0 it is known that the velocity solutionconverges to that of the exact Bingham model, but the stress field will notnecessarily converge; see the discussion in [101]. This method is clearly inap-propriate for a study that seeks to distinguish static regions from non-staticand to characterize the shapes of unyielded regions. Therefore, instead wehave used the augmented Lagrangian (AL) method developed by Glowinskiand co-workers; see [99, 110, 111]. The chief advantage of AL over viscosityregularization is that the exact form of constitutive relation is used and thealgorithm gives truly unyielded regions with exactly zero strain rate.Basically, the AL method finds the velocity field which minimizes a func-tional J(u); see e.g. [92]. Because J(u) contains a non-differentiable term,conventional optimization methods cannot be used for solving the mini-mization. In the AL method J(u) is made differentiable by substitutinga strain-rate variable  into the plastic dissipation functional. Then theequality constraint of  and ˙ is relaxed, to be enforced by introducing aLagrange multiplier i . Interestingly, i will converge to the stress field inyielded regions and to an admissible stress field for unyielded regions. Theunyielded regions are easily identified by the condition  = 0 in the ALmethod. The rest of details are skipped here for conciseness, but for a goodexplanation we refer to papers by Saramito & Roquet [206, 207, 214]. Thefinal AL algorithm consists of three steps. The first step is a standard Stokesflow problem with a right-hand side term. The second and third steps aresimple point-wise updates of ij and iij fields. These steps are shown inAlgorithm 1, repeated until the acceptable convergence is achieved.Algorithm 1 shows our solution procedure for the washout flow. Basi-cally it is the same as Algorithm 1 of [199] with a modest improvement for154Vlgorithm F augmented Lagrangian algorithm (AL) for washoutrzpzvtStep 1: u minimization with unit flow constraint:−v∇O˙(uR) = −∇pR +∇O(i − v)n−1Applying pressure drop for unit flow constraint:(unP pn) = (uRP pR) +1−fRf∆p=1(u∆p=1P p∆p=1)Step 2:  minimization is explicit update at each node:nij =0P if∥∥∥in−1ij + v˙ij(un)∥∥∥ Q W1− W∥∥∥in−1ij + v˙ij(un)∥∥∥ in−1ij + v˙ij(un)1 + vP otherwiseStep 3: i minimization is explicit update at each node:inij = in−1ij + v(˙ij(un)− nij)until∥∥∥nij − ˙ij(un)∥∥∥L2≤ 10−6 or 1000≤iterationsapplying unit flow constraint. The parameter v S 0 is a constant for the al-gorithm. In step 1 of the algorithm 1 of [199] the unit flow is obtained usinga secant iteration for finding the proper overall pressure drop ∆p that givesa unit flow rate across the channel. However we note that a secant iterationis not necessary as the equation is linear with respect to ∆p. This means theflow rate f =∫inlxt uxyy also depends linearly on ∆p. Therefore, we simplyfind the solution of the system when the only right hand side comes from aunit pressure ∆p = 1, and save this solution (u∆p=1P p∆p=1P f∆p=1) denotingvelocity, pressure and flow rate, respectively. Then we can simply find thecorrect ∆p needed to get unit flow based on the flow rate fR obtained fromall other right hand side contributions except ∆p. As shown in step 1 ofalgorithm 1, we find the solution without any overall pressure drop term as(uRP pRP fR) and then correct the solution to enforce the unit flow rate. Note155that the alternative to this procedure is to enforce the flow rate constraintseparately, e.g. in an outer iterative loop, which is generally more costly.Basically, this can be done for free in step 1 due to the linearity.The finite element method is used for discretizing the Stokes problem instep one of algorithm 1. Due to symmetry of the flow and to save computa-tional time only half of the channel is considered, as illustrated in Fig. 4.3.A limited number of computations were also carried out in the code devel-opment stage with the full geometry channel, to verify the symmetry of theflow. For boundary conditions, no-slip is applied at the top and bottomwalls and symmetry conditions are used for the left/right boundaries.The AL algorithm was implemented using the ghzolzf [213] C++ FEMlibrary developed by P. Saramito and colleagues in Grenoble. The specialfeature of this implementation is an anisotropic mesh adaptation. First, thesolution is obtained on an initial mesh that is unstructured but of approxi-mately uniform size. Then a new mesh is generated that evenly distributesthe interpolation error of the dissipation field, as discussed in [214]. Theadapted mesh is refined and aligned along the boundary of unyielded re-gions. This greatly helps to improve convergence and accuracy of the steadystate results, particularly for flow applications such as here. We perform 5cycles of adaptation. The yield surfaces are clearly identified in the lastadaptation. On any computation the maximum number of mesh points waslimited to 100000, due to available hardware. BAMG performs the meshing[36, 37]. The details of mesh adaptation strategy can be found at [206]. Atypical computation is started with a nearly uniform triangular mesh with9000 elements and iterations are finished when ‖ij − ˙ij‖L2 ≤ 10−6 or af-ter a maximum of 1000 iterations. The code was validated with a uniformchannel flow, for which the analytical solution is available. Other benchmarkcomputations can be found in [199, 206].ICH gzsultsIn total we have run approximately 1000 computations for the 4 differentgeneric geometries, with the numerical procedure described above. The156ranges of parameters covered for each washout geometry are the same:1 ≤ W ≤ 100; 0O5 ≤ h ≤ 10; 0O05 ≤  ≤ 1. Examples of the computedresults are presented first in §4.3.1. Section 4.3.2 gives a qualitative descrip-tion of the parametric results for the 4 geometries, with the focus being onthe occurrence of static zones of fluid within the washout zone. In §4.3.3we uncover some of the universal aspects of stationary zones, showing thatat large h these adopt a yield surface profile that is largely independentof washout shape. This enables us to define in a meaningful way parame-ters that are useful for design of flows through channel-like geometries withwashouts.ICHCF Eflvmplzs rzsultsFor very small h and  Q 1 the flows are effectively perturbations from aplane Poiseuille flow. The central plug region initially remains intact, buteventually breaks as h is increased. The breaking process has been studiedin [102, 199] for channel flows with walls that are symmetrically perturbedand similar methods might potentially be employed here. However, this isnot the main focus of our study.Figures 4.4 and 4.5 present examples of variations in fluid velocity (speedplus streamlines) for the 4 geometries at fixed  = 0O25 and for h = 0O5P 1P 2P 4,at W = 5 and W = 100, respectively. We show half of the domain, with thefull development length extending upstream to x = −5 − 0O5R. The flowin the development section is very close to a plane Poiseuille flow near theinflow, but as the washout approaches the streamlines spread and are di-verted into the washout. This asymmetry couples with axial compressionto create deviatoric stresses that yield the fluid in the entry region to thewashout. Further along the washout (and in particular for smaller ) theflow in the main part of the channel recovers into a Poiseuille-like flow withunyielded plug region. However, the shear layers above and below the plugare different and the plug is asymmetrically positioned in the moving partof the fluid.The main interest of the flow is in the washout part of the channel, where15700.20.40.60.811.2Figurz ICIO Streamlines superimposed on colourmaps of the fluidspeed for different types of washout:  = 0O25, W = 5. Columnsare h = (0O5P 1P 2P 4) from left to right and unyielded regions areshown in gray.15800.10.20.30.40.50.60.70.80.91Figurz IC5O Streamlines superimposed on colourmaps of the fluidspeed for different types of washout:  = 0O25, W = 100.Columns are h = (0O5P 1P 2P 4) from left to right and unyieldedregions are shown in gray.159we find areas of stationary fluid growing with h. The onset of the stationaryareas is however qualitatively different, depending on geometry. First ofall, for the Koch snowflake it is common to have stationary fluid trappedin the smallest scale structures, even for relatively small h and W. Thiseffect is essentially a wall roughness effect, i.e. simulations with a repeatingroughness pattern at the wall would likely also trap fluid on small scales.Secondly, we note that the abrupt step change in channel width exhibitsthe feature that stationary fluid is first found in the corner regions; see also[79, 164, 174]. The other geometries (discounting small scale roughness)each first encounter stationary fluid zones at the widest part of the channelas h increases through a critical value.As h increases further, the stationary zone increases but the yield sur-face position in the upper part of the channel and the streamlines in theflowing region become increasingly insensitive to geometry of the washout.To the eye it appears that the shapes are broadly similar between washoutgeometries. Larger W results in more stationary fluid in the washout, asmight be expected.Typical stress distributions are explored for the 4 geometries for W = 5and W = 100, in Figs. 4.6 and 4.7, respectively (at  = 0O25, h = 4). Thecolumns (from left to right) show xx, xy, yy, p for each of the washoutgeometries. We observe broadly similar features between the different ge-ometries. The development region (just prior to entering the washout) ischaracterised by large negative xx, except close to the beginning of thewashout at the upper wall. There is a singularity in the stress gradientsaround the entry corner of the washout (square and triangle shapes), wherewe may expect relatively poor local convergence of our computations. Wealso observe an intriguing wispy strand of unyielded fluid that extends fromthe plug upwards towards the corner of the flow domain. Across this thinplug we appear to have a jump in tangential component of the deviatoricstress from positive to negative. Similar singular behaviours near the cornerof a yield surface were found in [225, 246] in studying eccentric annular ductflows; see also [199]. The shear stresses are reduced in the washout partof the flowing fluid, suggesting a significantly reduced pressure drop. As in160−6 −4 −2 0 2 4 6 8 10 12 14 −40 −30 −20 −10 0 10 −160 −140 −120 −100 −80 −60 −40 −20 0 0 20 40 60 80 100 120 140 160Figurz IC6O Components of the stress for different types of washout: = 0O25, W = 5, h = 4. Columns show xx, xy, yy, p respec-tively, from left to right. The color key for each column is sameshown at the bottom. Unyielded regions colored with gray.[208], we find that yy becomes largely independent of y in the shear layers,in the region of the washout where there is a plug.ICHCG evrvmztrix vvrivtionsFigs. 4.8 and 4.9 show a wide range of unyielded fluid regions for the 4geometries, again at W = 5 and W = 100, respectively. The Koch snowflakesexhibit the “roughness” feature at larger , but for long washouts ( =0O05) the snowflake construction does not result in sharp enough undulationsto trap fluid. Instead the general effect is of an uneven/rough washout,gradually increasing in size. The uneven wall produces interesting stressvariations in the bulk of the flowing region that result in plugs of differentsizes and shapes. However, since all these regions are flowing, the practicalimplications are minimal. For larger h than is shown, even for long washouts161−100 −50 0 50 100 150 200 250 −200 −150 −100 −50 0 50 100 −1600−1400−1200−1000 −800 −600 −400 −200 0 0 200 400 600 800 1000 1200 1400 1600Figurz ICLO Components of the stress speed for different types ofwashout:  = 0O25, W = 100, h = 4. Columns show xx, xy,yy, p respectively, from left to right. The color key for eachcolumn is same shown at the bottom. Unyielded regions coloredwith gray.the stationary zones at the wall increase to form a single connected area.For shorter washouts the upper part of the washout has stationary fluid forrelatively moderate h.The step change in channel width is interesting in that we see a two stagetransition for long washouts. Even for relatively small h we find stationaryzones in the corners. The central part of the washout is then simply a uni-form channel of width 1 + h, within which the flow develops into a uniformPoiseuille flow again. As h increases, both the corner regions and yieldeddevelopment zone at the entry to the washout appear to grow in length.Finally at a critical value of h the corner regions join and the washout hasa simply connected region of stationary fluid. After this point, the upperboundary is a yield surface and the moving plug zone becomes asymmetri-cally located in the washout, i.e. closer to the lower wall.162Figurz ICMO Unyielded regions of the flow (shown in black) for differenttypes of washout at W = 5. For each type of washout columnsshow  = (O5P O25P O1P O05) from left to right respectively and rowsare h = (O5P 1P 2P 4) from top to bottom respectively163Figurz ICNO Unyielded regions of the flow (shown in black) for differenttypes of washout at W = 100, for each type of washout columnsshow  = (O5P O25P O1P O05) from left to right respectively and rowsare h = (O5P 1P 2P 4) from top to bottom respectively164The two most similar geometries are the triangular and sinusoidal washouts.Stationary regions appear at the centre of the washout at the upper wall, atapproximately the same h, the plug regions in the flowing zone are qualita-tively very similar. The differences with larger W are as expected: a generalincrease in all unyielded regions. However, we also observe an interestingfeature for short washouts, in that the central plug region yields only in anarrow frvxturz at the start of the washout. This suggests that as we takethe washout length to zero, there is likely to be a critical  above which thecentral plug does not yield. This is perhaps analogous to a wall roughnesseffect.The above figures and our earlier observations suggest that there is lit-tle universal about the onset of stationary fluid at the upper wall, i.e. theposition of the initial fouling depends on wall roughness and the abruptnessof the angle at the end of the washout. Nevertheless, for the triangular andsinusoidal washouts the onset of fouling (stationary fluid at the wall) ap-pears unambiguously at the widest part of the washout. As in our previouswork [208], it appears that the onset occurs at a combination of (h), whichis effectively the slope of the washout wall. Figure 4.10 plots the resultsof all our computations for these 2 geometries, classified as either havingstationary or mobile fluid in the washout. The critical values of (h) appearto be primarily a function of W; see also Fig. 10 in [208].ICHCH Flofiing vrzvSince successive increases in washout size do not appear to have a large effecton the area of mobile fluid, it becomes of interest to characterize this effect.In all our computations stationary fluid has only been observed within thewashout region. We therefore define VY as the area of the washout that hasnon-zero velocity, i.e. meaning that part of the flow geometry above y = 1and between x = ∓0O5−1.The initial increase in VY scales for some washout geometries and pa-rameter ranges like hR (i.e. increasing like the washout area) and for otherslike −2 (characterized approximately by the arc of a circle). However, none1650 20 40 60 80 10010−210−1100Bhδa)0 20 40 60 80 10010−210−1100Bhδb)Figurz ICFEO Onset of stationary fluid at the upper wall for: a) tri-angular wave; b) sinusoidal wave. Dark symbols indicate sta-tionary fluid at the upper wall.of these scalings was found to be uniformly valid. For any non-zero W wehave observed that VY increases and asymptotically approaches a constantvalue as h → ∞. It was also noted in parametric studies such as those inFigs. 4.8 and 4.9, that the shape of the yield surface bordering the staticfluid, in the widest part of the channel, appeared to be similar betweengeometries, as h increases. Figure 4.11 illustrates this effect for two sets of(WP ) with both triangular and wavy washout geometries, compared againstthe square washout. The broken black line indicates the computed positionof the yield surface in the square washout. The red line indicates the yieldsurface in the wavy (or triangular) washout. As h increases we see that thered and black profiles converge in the centre of the washout. Away from thecentre of the washout the yield surfaces cannot coincide, since the wall of thewashout (for triangular and wavy geometries) lies inside the yield surfaceof the square washout. Effectively, the convergence must wait until h→∞and the washout wall progressively approaches the square washout shape.This convergence is consequently relatively slow and is slowest for the wavywashout, which is always smooth.Although the convergence is slow the trend is clear, and a simple method166Figurz ICFFO Comparison of static region for triangular and wavywashout (solid black/red) with the corresponding squarewashout (dashed). Only washout enlargement section isshown. Flowing are correction is shown in gray. Rows areh = (1P 2P 3) from top to bottom, for the two left columns(WP ) = (2P O5) and two right columns (WP ) = (10P O25)of predicting VY suggests itself. Suppose that we are able to characterize VYfor the square washout, as h → ∞ and the yield surface approaches a con-stant shape. We call this (square washout) limiting flowing area VYN∞(WP )and note that in fact the convergence to VYN∞ with h appears to be relativelyfast for the square wave. For any other shape of washout, we now define acorrection by subtracting the areas of the washout from that of the squarewashout yield surface, say VVNh(WP ). These corrections correspond to theshaded gray areas in Fig. 4.11. Note that these corrections are computedfrom the known washout geometry and the square washout yield surface,1670 2 4 6 8 100.20.30.40.50.60.70.80.911.11.2hAF0 2 4 6 8 100.511.522.533.5hAFFigurz ICF2O Corrected flowing area of wavy and triangular washoutsagainst the square washout flowing area as the washout getsdeeper (increasing h). Left plot (WP ) = (2P O5) and right plot(WP ) = (10P O25). Markers are ◦) Wavy washout and ▽) linearwashout. The dashed blue line shows VYN∞(WP ), the limitingflowing area of square washoutwithout any need to compute the actual flow. The flowing area is thenapproximated by:VY (hPWP ) ≈ VYN∞(WP )−VVNh(WP )O (4.15)To test the validity of this approach, we calculate VVNh(WP ) for the triangu-lar and wavy washouts and add the correction to the computed VY (hPWP ).Figure 4.12 plots VY (hPWP )+VVNh(WP ) for triangular and wavy washouts,comparing against VY (hPWP ) for the corresponding square washout. Wecan see that the corrected flowing areas converge to that of the squarewashout relatively well.ICHCI erzssurz yropApart from flowing area a second parameter of interest to flow design isthe effect that the washout has upon the pressure drop. Intuitively, sincethe washout section is wider we expect that the pressure drop is reducedin comparison to that experienced through a uniform channel of the same168length. We define ∆e as this reduction in pressure drop, i.e.∆e = −(aR2 + 5)UpUx|coisxuillx − [p¯(−aR2− 5)− p¯(0)] P (4.16)where SpSx |coisxuillx is the pressure gradient of the plane channel Poiseuilleflow and p¯(x) is the computed pressure, averaged with respect to y acrossthe channel. Note that xx = 0 at the symmetry plane x = 0 and xx ≈ 0 at−(aR2 + 5), due to the entry length. Thus, the drop in p¯ is essentially thatin −xx.Figure 4.13 shows typical variations in ∆e with h at fixed W, for thesquare wave washout. In each case, as h increases ∆e approaches a con-stant value that depends on both W and a (or ). This convergence to aconstant value is of course mimicked by the yield surface approaching a con-stant shape, as we have seen in the previous sections. Comparing betweenFigs. 4.13a & b we see that there is an increase in ∆e with W and eachfigure illustrates the increase in ∆e with a. At significant W we have seenpreviously that the flowing areas for all geometries converge to that for thesquare wave as h increases. Fig. 4.13b plots (red broken line) the same datafor the wavy washout for 2 values of . We can see that convergence of thepressure drops is evident but relatively slow, as we have seen for the flowingarea.Figure 4.14 shows typical variations in p¯(x), compared against the Poiseuilleflow pressure distribution. As expected the pressure drop in the washoutsection is significantly smaller than the Poiseuille flow gradient would havebeen. The gradients converge as we approach the entrance to the channel.The square washout induces a sharp change in gradient at the start of thewashout: other geometries are smoother.To quantify the significance of the pressure drop we first note that ∆earises mostly from the washout section (although there is some variation inthe development length). To compensate (partly) for the evident increasesin ∆e with a we consider instead 2∆eRa (note ∆e is for aR2 in equation(4.16)), which has an interpretation as the reduction in pressure gradient,(averaged over the washout length). We scale this quantity with SpSx |coisxuillx1690 5 10 15 20020406080100120140∆Ph  δ = 0.5δ = 0.25δ = 0.1δ = 0.05a)0 5 10 15 20050100150200250300∆Ph  δ = 0.5δ = 0.25δ = 0.1δ = 0.05b)Figurz ICF3O Pressure drop reduction (compared to uniform channel)for the square washout: a) W = 1; b) W = 10. In b the brokenred line indicates the analogous results for the wavy washout.−7 −6 −5 −4 −3 −2 −1 0050100150200250p¯xa)−10 −8 −6 −4 −2 0050100150200250300350p¯xb)Figurz ICFIO Averaged computed pressure p¯(x) (square washout)compared with the pressure in the uniform channel (brokenline): a) (hP PW) = (O5P O25P 10); (hP PW) = (O5P O1P 10).to give a dimensionless ratio, plotted in Fig. 4.15. We can see that typicallyfor shorter washouts and larger W the pressure gradient reduction is signif-icantly less that the frictional pressure gradient of the channel flow. It isonly for long washouts and smaller W that the pressure gradients becomecomparable.Although potentially from our results we could attempt to derive a pres-sure drop xorrzxtion to apply to washout geometries, we have not done so.Firstly, in oil well cementing geometries the static pressure is usually dom-1705 10 15 2000.10.20.30.40.50.60.70.80.9(2∆P/L)/(∂p/∂x) PoiseuilleL  B = 1B = 2B = 5B = 10B = 20B = 50B = 100a)0 20 40 60 80 10000.10.20.30.40.50.60.70.80.9(2∆P/L)/(∂p/∂x) PoiseuilleB  δ = 0.5δ = 0.25δ = 0.1δ = 0.05b)Figurz ICF5O Computed pressure gradient reduction for large h, com-pared to Poiseuille flow uniform pressure gradient (for thesquare washout)inant over the frictional pressure, so that the washout effect is minimal interms of total pressure. Secondly, it is unclear how such a correction shouldbe applied over a full annular geometry, where the washout geometry maychange azimuthally but the axial pressure drop does not.ICI Yisxussion vny summvryThis chapter has considered slow flows of a yield stress fluid through aplane channel with a non-uniformity on one side. This provides a modelfor circulation of drilling fluid through a narrow annulus with washouts.Washout geometries are usually not measured and may vary considerably.Although at first it seems an impossible problem to study the mechanicsin such an unknofin geometry, this proves not to be the case. Our resultshave shown that in the case that the washout is relatively deep and the yieldstress relatively large (high W), the fluid acts to self-select the flowing regionin a way that is (practically speaking) independent of the washout geometry.Practical estimates may be gained by studying the square washout, whichbecomes the limiting case for all other geometries studied, as h→∞.This chapter is of course acknowledged to be an idealisation of the in-dustrial situation. Firstly, the rheological description is incomplete. Drilling171fluids are known to be shear-thinning. Although interesting to study, e.g. aHerschel-Bulkley fluid flowing in the same situations, we cannot expect tosee qualitatively different results. Another rheological complication is thatdrilling fluids tend to thicken when left stationary in a wellbore for any ap-preciable time, i.e. they are thixotropic. Possibly, an estimate of this effectcould be accounted for by assuming that W is based on the drilling mud gelstrength, rather than dynamic yield stress. Secondly, we have simplified thestudy considerably by focusing on gz = 0. In Chapter 5 following directly,we look at inertial flows. Thirdly, there are other operational features tobe studied, e.g. the 3D geometry, the later displacement flow of the drillingmud, etc. Chapter 6 presents our initial results on the displacement flowproblem.172Chvptzr 5Inzrtiv zffzxts in thz ofi ofyizly strzss uiy in fivshoutIn this chapter we extend the work of Chapter 4 into the inertial regime.That is, we study steady inertial flows of a Bingham fluid through 2D“washout” geometries, that represent longitudinal sections of an oil andgas well under construction. The washout geometry is again characterisedby dimensionless depth h and length a = −1. The other dimensionless pa-rameters of problem are the Reynolds number gz and the Bingham numberW. The effects of increasing gz are studied both for fixed W and for fixedHedstro¨m number, Hz = gzW. The former of these represents a straightfor-ward increase in the parameter space of Chapter 4, by adding gz. However,varying gz at fixed W does not correspond to a flow rate increase, whichwould be a common parameter to vary in the cementing process. Instead,an increase in flow rate for fixed geometry and fluid properties is representedby fixing Hz and increasing gz.In both cases we observed that the variation in flowing area of thewashout (i.e. that area that is mobilised during pre-circulation) was non-monotone. Increasing gz resulted in a straightening of the streamlines pass-ing through the washout region, in the main part of the channel. For fixedHz, beyond a first critical value of gz the flowing area was observed todecrease, i.e. increasing the flow rate results in larger parts of the washout173being static, as is quite counter-intuitive and contrary to the industrial per-ception that pumping faster will circulate/condition the mud better. Thistrend persists for a significant range of gz. On passing a second critical valueof gz, we observed the onset of zones of recirculation within the washout.The flowing area thus increases but the area of washout from which fluid isactually displaced during conditioning continues to decrease. Since the re-circulating zones typically have a low velocity (and shear rate), it is unclearwhether practically speaking this is sufficient to condition the mud. Finally,we consider self-selection of the flow geometry, as observed in Chapter 4 forgz = 0. We conclude that the same phenomenon arises in inertial flowsfor a given geometry, i.e. at large enough h the flowing areas of differentlyshaped washouts with static regions become similar in shape.15CF IntroyuxtionWe continue our study of drilling mud conditioning during the pre-circulationphase of a primary cementing operation. This paper extends the study per-formed in [209] (see also Chapter 4) into inertial flow regimes. Primarycementing is a critical operation in the construction of every oil and gaswell; see [176]. The fluid mechanical aspect of this process involves the re-moval of drilling mud from the annular space between casing and formation,to be replaced by a cement slurry. When the well is drilled in poorly consol-idated formations that may partially collapse during the drilling process, anon-uniform annular space results with outer wall consisting of “washout”sections (see e.g. Fig. 5.1) that will be geometrically dependent on both thedrilling operation and the local geology.Prior to the fluid displacement phase, most service companies and opera-tors recommend to pre-circulate the well, by pumping the drilling mud fromthe bottom to the top of the well at least once (“circulating bottoms-up”)and this is the phase of the operation that we study. The circulation phasehas two main purposes. Firstly, drilled cuttings and other solids still in the1A vzr“ion of thi“ xhvptzr hv“ vppzvrzy v“: AC gou“tvzi & IC [rigvvryC qgz“iyuvlyrilling muy yuring xonyitioning of unzvzn worzholz“ in primvr– xzmzntingC evrt G: htzvy–lvminvr inzrtivl ofi“C7 J. Non-Newt. Flhid Mech.A 226:F{F5A (G0F5)C174CasingWashoutxˆDˆUˆHFigurz 5CFO Geometry of the washout in a section along the annulus.well, which may have settled as the casing is run in to the borehole, arecleared from the flowpath. Secondly, the circulation serves to shear the mudand hence condition it prior to displacement. Depending on the operationalcircumstances and the type of drilling mud, the mud may have been staticin the borehole for a period of hours before circulation. Over this time, sig-nificant gel strengths may develop due to thixotropic effects. Simplisticallyspeaking this gel strength is destroyed by shear, returning the drilling mudto its dynamic yield stress (which may still be significant).In Chapter 4 (and [209]) we simplified the flows in a number of ways.First, we considered only a two-dimensional (2D) section of the wellbore asin Fig. 5.1. Second we modeled the drilling mud as a Bingham fluid. Third,we considered only Stokes flows, neglecting inertial effects. These simplifi-cations allowed us to study a wide range of different washout geometries, interms of both shape and scale (length and depth relative to the annular gapwidth). As might be expected, yield stress fluid became trapped in sharpcorners and small scale features of the washout walls. Whereas initially thegeometric uncertainty was a daunting obstacle to understanding, our resultsshowed that for sufficiently large yield stress (ˆl ) and for sufficiently deepwashouts (Hˆ), the actual washout geometry had little effect on the amountof fluid that is mobilized. Essentially what was observed for a deep washout175of length aˆ was that the flowing fluid “self-selects” its geometry. Havingestablished a stationary region within the depths of the washout, furtherincreasing Hˆ does not significantly effect the position of the yield surface.This phenomenon of self-selection is not a new phenomenon, but is asignificant feature of many interior laminar flows of yield stress fluids. Asperhaps the simplest example of this consider a circular Couette flow, inwhich the single non-zero shear stress component ˆr decreases in magni-tude with radial distance. Depending on the torque exerted at the innercylinder and the yield stress of the fluid, there can be a stationary layerattached to the outer cylinder. As this stationary layer can be replacedwith an outer cylinder of any radius larger than the radius of the yield sur-face without affecting the moving part of the fluid, we may consider thatthe fluid has selected its own flow geometry. Yield stress fluid flows are ofcourse, mathematically speaking, free-boundary problems.One of the few classes of flow for which this self-selection phenomenahas been studied in some depth are uniaxial flows, i.e. uˆ = (0P 0P wˆ(xˆP yˆ);see [168, 169]. In [168] various results are proven concerning the criticalpressure gradient required for flow, which is a limiting case of a flow withstationary regions at the walls. More generally it is shown that the flowingdomain should have a convex boundary. Quantitative bounds on the radiusof curvature of the flow boundary are proven in [169]. Stagnant regionsclearly occur in sharp corners, but can also be found in smooth geometriessuch as an eccentric annulus, bridging between walls as in [246]. Burgoset al. [44] considered a two-dimensional anti-plane shear flow in a wedgebetween 2 plates, for which a mapping method gives the shape of the yieldsurface bounding the stationary fluid in the corner. Similar methods havebeen used by Craster [74].For more general flows, analytical prediction of the flowing area has notbeen successful to our knowledge, although there exist limiting cases of zeroflow where the flow domain is known. There are however many compu-tational studies that show self-selection of the flow domain via stationaryregions close to the wall. Mitsoulis and co-workers (e.g. [164]) have studiedboth planar and axisymmetric expansion flows, over a wide range of Bing-176ham and Reynolds numbers, showing significant regions of static fluid in thecorner after the expansion. Flow of yield stress fluids through an expansion-contraction has been studied both experimentally and computationally in[79, 174, 175]. In [79] Carbopol solutions were pumped through a suddenexpansion/contraction, i.e. narrow pipe – wide pipe – narrow pipe. A rangeof flow rates were studied and the yield surfaces were visualised nicely by aparticle seeding arrangement. Inertial effects are evident in asymmetry ofthe flow. For all the experiments shown, stagnant regions first appear in thecorners of the expansion and for all the results shown there are stagnant re-gions. Comparisons are made between experimental and numerical results,which are at least qualitatively in agreement; see [79, 174]. In [175] qualita-tively similar computations are carried out, but using an elasto-viscoplasticmodel. In [208] (see Chapter 2) we have studied large amplitude wavy walledchannel flows numerically. We have derived predictions for the onset of sta-tionary fluid regions, which occur initially at the walls in the widest part ofthe channel, unlike the abrupt expansions discussed above.Although here we focus on primary cementing as the application of inter-est, yield stress materials are common in the food and cosmetics industries.The above works may be considered as relevant to processing flow of suchmaterials through machinery and/or during mixing. The occurrence of deadzones where fluid is not circulated/mobilized can seriously affect product ho-mogeneity and have potential health consequences.In this chapter we study the effects of inertia on the results of Chapter 4(and [209]), while retaining the simplifications both of a 2D flow and aBingham fluid. The importance of including inertia lies in that varying theflow rate is one of the few methods to influence mud conditioning and hencerheology, prior to the displacement stage of the process. At very high flowrates turbulent flows are likely to mobilize the fluid efficiently. However,fully turbulent flows are not always operationally possible (due to eitherpumping limitations or frictional pressure restrictions from the pore andfrac envelope). Thus, many flows are at significant gz but are laminar. Forthese flows the effects of increasing flow rate in the laminar regime are lesspredictable. Hence the motivation for this work. The underlying motives are177that poorly conditioned mud will develop a higher yield stress and be harderto displace later on in the cementing process, even in uniform geometriese.g. [9].A brief outline of this chapter is as follows. In §5.2 we present the generalmodel and flow geometries. This is followed by §5.3 in which we outline thecomputational algorithm used, present benchmark results for validation anddetermine a suitable entry length for our flows. The main physical resultsare in §5.4, where we start with a general description of flow observations asthe Reynolds number gz is increased from zero at fixed Bingham number W.We show that the flowing area is not monotonically increased, as would beintuitive. Similar effects are observed as gz is increased at fixed Hedstro¨mnumber Hz, simulating a flow rate increase; see §5.4.1. Finally in §5.4.3 westudy the effects of varying washout geometry on flow domain self-selectionand invariance, following [209]. The chapter closes with a summary in §5.5.5CG ihz simplizy fivshoutThe flow geometry considered is similar to that in our previous work, [209].Primary cementing circulation occurs through a long narrow eccentric annu-lus for which we may assume that the main velocity components are alongthe borehole axis, with azimuthal flows driven by slow axial variations inaperture. A reasonable simplification of the flow through a washout sectionis therefore to consider a two-dimensional (2D) azimuthal-axial slice throughthe annulus; see Fig. 5.1. Neglecting azimuthal flow, the channel formed hasan in-gauge width Yˆ (dependent on the inner and outer diameters and theeccentricity), being bounded on the inside by the casing and on the outsideby the formation.In the 2D section described we will characterize the washout as havinga length aˆ and a maximum amplitude Hˆ, as illustrated. Coordinates (xˆP yˆ)are aligned along the channel and across the annular gap, respectively; seeFig. 5.1. Fluid circulates axially along the 2D section with mean velocityjˆ0 in the xˆ-direction. A typical width, Yˆ is in the range 3− 80 mm and atypical jˆ0 might be in the range 0O01−1 m/s. The drilling mud is considered178incompressible over the length of the washout, even for aˆ of the order of afew meters. Non-Newtonian effects are significant and principal amongstthese is the yield stress effect (at least for the flow effects that concern us).The simplest rheological model containing a yield stress is the Binghamfluid, as often used in drilling applications although more complex models(e.g. Herschel-Bulkley) are also used. Although a gross simplification, thekey feature of yielding/not-yielding is captured by this model.The Navier-Stokes equations are made dimensionless using Yˆ and jˆ0 aslength and velocity scales, respectively. Time is scaled with YˆRjˆ0, the shearstresses are scaled with ˆjˆ0RYˆ and the static pressure is subtracted fromthe pressure before also scaling with ˆjˆ0RYˆ, where ˆ is the plastic viscosityof the Bingham fluid. The resulting equations are:gzYuYt= −UpUx+UUxxx +UUyxyP (5.1)gzYvYt= −UpUy+UUxxy +UUyyyP (5.2)0 =UuUx+UvUyP (5.3)where u = (uP v) is the velocity, p is the modified pressure and ij is thedeviatoric stress tensor. The scaled constitutive laws are:ij =(1 +W˙(u))˙ij ⇐⇒  S W (5.4a)˙ij(u) = 0⇐⇒  ≤ WP (5.4b)where˙ij(u) =UuiUxj+UujUxiP u = (uP v) = (u1P u2)P fl = (xP y) = (x1P x2)Oand ˙,  are the norms of ˙ij , ij , defined as˙ =√12∑ij˙2ij and  =√12∑ij2ij O (5.5)179Two dimensionless numbers appear above. Firstly, the Reynolds num-ber, gz:gz ≡ /ˆjˆ0YˆˆP (5.6)which represents the ratio of inertial to viscous effects. Secondly, the Bing-ham number, W:W ≡ ˆl Yˆˆjˆ0P (5.7)which represents the competition between yield and viscous stresses. Thefluid density is denoted /ˆ, and the yield stress is denoted ˆl . As well as these,two geometric dimensionless parameters result: h = HˆRYˆ, representing themaximal depth of the washout, and  = YˆRaˆ, characterizing the aspectratio of the washout.For the computational domain we extend the channel uniformly bothupstream and downstream of the washout section; see Fig. 5.2. The twoextensions are determined empirically (see below in §5.3), to be sufficient toallow fully developed velocities at both x = −0O5−1− lw (entry developmentlength) and at the exit, x = 0O5−1 + lx. For simplicity we set lw = lx,although unlike [209] we no longer expect symmetry of the flow. No slipconditions, u = 0, are applied at the walls. The flow is assumed fullydeveloped at entry and exit, driven by a constant pressure drop:xx = −p+ xx = −sP v = 0P at x = −0O5−1 − lwP (5.8)xx = −p+ xx = 0P v = 0P at x = 0O5−1 + lxO (5.9)The pressure drop s must be found as part of the problem: s is increaseduntil the mean velocity at the entry is 1, i.e. s satisfies:1 =∫ 10u(xP y) dyP at x = −lwO (5.10)For all geometries the maximum depth of the washout is h = HˆRYˆand the length is −1 = aˆRYˆ, i.e. the length of that part of the channelthat does not have unit width. The main geometry considered is that of a180x = 0.5δ -1y=1+hy=1x = -ld u=0 x = leσxx=-sv=0σxx=0v=0yxy=0x = -0.5δ -1Figurz 5C2O Dimensionless computational domain and boundary con-ditions for this chapter.sinusoidally wavy washout: the channel width changes from 1 to 1+h, overa distance 0O5R following a sine wave. The upper wall of the washout is:yw = 1 + 0O5h[1 + cos 2.x]. Other geometries will be introduced only tocompare geometric effects.5CH Computvtionvl mzthoyOur objective is to compute steady inertial flows in the washout geometriesdiscussed. The usual difficulties of the yield stress are handled via an aug-mented Lagrangian method analogous to that used in [209]. The equationsare however changed by the addition of the inertial terms which are approx-imated using a characteristics-based method. As in [209] the saddle pointproblem is solved iteratively for updated velocity vector, strain rate andstress tensors: denoted un, n and in, respectively. The main modificationcomes in the generalized Stokes problem, which takes the form:3gz2∆tun+1 − ∇ · [v t(un+1)] +∇pn+1 = gP (5.11)∇ · un+1 = 0P (5.12)−pn+1 + v˙xx(un+1) = 0P vn+1 = 0P at x = ±0O5−1 ± lwP(5.13)un+1 = 0P at y = 0P y = yw(x)P (5.14)181where v S 0 is the augmentation parameter, ∆t is a pseudo-timestep and˙(un+1) is the strain rate tensor evaluated from un+1. The method is out-lined below as Algorithm 2.We have implemented the above algorithm using the FreeFEM++ finiteelement environment [118], which is freely available. Following [40] we haveused a backward second order (BDF2) discretization for the time derivativeWuWt , and to calculate characteristic points the velocity u˜ = 2un − un−1 isused. Note that u˜ approximates u at n + 1 and for given position fl thepositions mn(fl) and mn−1(fl) track backwards along the streamlines untiltimes n and n − 1, respectively. Thus, combining g in (5.15) with (5.11)uses a second order consistent approximation to the derivative of u alongthe streamline, i.e. WuWt . The expression un(mn) is evaluated by interpolationfrom the mesh values of un at the position mn; similarly for un−1(mn−1).Both viscous and pressure terms are implicit. As shown in [40] thisconfiguration leads to an unconditionally stable scheme which is a desiredproperty. We use Taylor-Hood (e 2−e 1) elements for velocity and pressurespaces, respectively. Having set the velocity space, it’s discrete derivativespace e 1w , consisting of discontinuous elements, is used for both strain andstress spaces: (Pi). This constraint is necessary to get full convergence ofstrain and stress fields as explained in [214].We also implemented a mesh adaptivity strategy similar to that in [209].In each adaptation cycle, first the steady solution is computed and then anew mesh is generated based on a metric that depends on the velocity field.Two metrics b1 & b2 were tested:b1 =√12˙2 +Wn˙P b2 =√b12 +gzu2OThe second metric includes inertial effects. However, in practice the flowsconsidered did not generate significant additional gradients due to inertialeffects, we didn’t see any significant improvement from b2 compared b1and thus use b1 for all results below (as for the Stokes flow in [209]).Basically this metric causes the mesh generator to refine the mesh close toyield surfaces, where the second derivative of velocity is discontinuous. We182Vlgorithm 2 Uzawa algorithm for inertial flowInitiate the variables: u−1 = u0 = 0P i0 = 0 = 0.Solve (5.11)-(5.14) with g = (1P 0)g to give (u∗P p∗).Compute the areal flow rate through the channel due to u∗:f∗ =∫ yw0u∗(xP y) dyONote that solving (5.11)-(5.14) with g = (1Rf∗P 0)g gives unit areal flowrate.for n = 0P O O O nmtx yoFBholvz the generalized Stokes problem at step n:Let u˜ = 2un − un−1Compute characteristics: mn(fl) = fl− u˜∆tP mn−1(fl) = fl− 2u˜∆tDefine g in (5.11) by:g = ∇ · (in − vn) +gz4un(mn)− un−1(mn−1)2∆tO (5.15)Solve (5.11)-(5.14) to give (un+1P pn+1).2Bboyif– the solution to give unit areal flow rate:Let fn+1 = (1−fn+1)Rf∗ where fn+1 = ∫ yw0 un+1 dyUpdate (un+1P pn+1) = (un+1P pn+1) + fn+1(u∗P p∗).3Bjpyvtz relaxed strain rate:n+1 ={0 |in + v˙(un+1)| Q W(1− U|TS+t˙(uS+1)|)TS+t˙(uS+1)1+t otherwiseIBjpyvtz stress Lagrange multiplier:in+1 = in + v(˙(un+1)− n+1)if min(‖˙(un+1)− n+1‖L2 P ‖un+1 − un‖L2)R∆t ≤ 10−6 thznStop the n loopzny ifzny for183perform 4 cycles of adaptation. A typical computation starts with a meshof 8000-10,000 points and the last adapted mesh may have up to 150,000points. Further details of the mesh adaptivity may be found in [36, 37].The modifications of (5.11)-(5.14) to fit our problem are two-fold. Firstly,since (5.11) is linear and the boundary conditions are homogeneous, we areable to enforce the fixed flow rate constraint (5.10) in step 2 of the algorithm.Secondly, we have combined time-stepping for the Uzawa iteration withthat for the time advance, as a means of accelerating convergence. ThispszuyoBtimzstzpping does not therefore compute an actual solution to anytime-dependent problem at intermediate timesteps. However, if ‖un+1 −un‖L2R∆t→ 0 the converged solution is a steady inertial flow, as required.In general, adding the yield stress increases the viscosity and is thusexpected to stabilize the flow (i.e. hydrodynamically). We expect thereforethat the velocity residual ‖un+1−un‖L2R∆t should converge more easily forlarge W at significant gz. This is indeed what we observe. Figure 5.3 showsthe decay of the velocity residual with iterations n at two different gz andfor 4 increasing values of W with (hP ) = (4P 0O25). We do not know the limitof stability for a Newtonian fluid in this geometry, but see that the velocityresidual does not decay for W = 0 or W = 1 at gz = 2000. This is evidencethat the flow is unsteady for these parameters, but as remarked above theactual computed velocity in these cases does not have physical meaning.For all results presented in this chapter the velocity residual is converged(≤ 10−6) and we may assume the flow is steady. The code has been bench-marked against the published results for the lid-driven cavity flow problemfor both Newtonian and Bingham fluids. For gz = 1000 and W = 0, Fig. 5.4apresents the comparison with the Newtonian results of Ghia et al. [107]. Ourresults also compare well with those of Syrakos et al. [224] for gz = 1000and W = 10, but not with those of Vola et al. [241] for the same parameters;see Fig. 5.4b. As a second check we have recomputed the flows using a fi-nite volume method on regular rectangular mesh, implemented in the codePELIGRIFF (see e.g. [203]). We again reproduce the results of [107, 224]but not [241].For a given geometry we have computed two types of flow. Firstly, with184103 104 10510−610−410−2100Number of iterationsVelocityresidual  a)B = 0B = 1B = 2B = 8103 104 10510−810−610−410−2100Number of iterationsVelocityresidual  b)B = 0B = 1B = 2B = 8Figurz 5C3O Decay of the velocity residual ‖un+1 − un‖L2R∆t for thewashout (hP ) = (4P 0O25) at increasing W = (0P 1P 2P 8): (a)gz = 2000; (b) gz = 500.−0.4 −0.2 0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91U (0.5, y)y  a) Ghia et .al−0.2 0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91U (0.5, y)y  b)Syrakos et .al−2014Vola et .al−2003Figurz 5CIO Validation of our code with published results from thelid-driven cavity flow. Plots show the horizontal velocity com-ponent, j(0O5P y), along the centre of the cavity. a) Comparisonwith Ghia et al. [107] for gz = 1000, W = 0. b) Compari-son with Syrakos et al. [224] and with Vola2003 et al. [241] forgz = 1000, W = 10.1850 0.2 0.4 0.6 0.8 1 1.2 1.400.10.20.30.40.50.60.70.80.91U (y)yFigurz 5C5O Convergence of the entrance velocity profile u(−0O5−1 −lxP y) to the plane Poiseuille solution as the entrance length lx =lw increases. The black dotted line shows the plane Poiseuilleflow: (gzPWP hP −1) = (200P 2P 10P 10) and entrance lengths arelx = (0P 2P 5P 10) indicated by markers (+P •P□P△), respectively.lx = lw = 0 and periodic conditions imposed on the velocity at the two endsof the channel. Inevitably these flows are strongly asymmetrical. Althoughnot relevant for the washout application, these do provide a baseline forthe degree of asymmetry in the velocity due to the geometry. The secondtype of flow is computed with the inflow/outflow conditions discussed earlierand with lx = lw increased until the entry and exit flows agree with theplane Poiseuille flow solution jcoisxuillx = j(y) to within a tolerance: ‖u−jcoisxuillx‖L∞ ≤ 0O002 at x = ±0O5−1 ± lx. An example of the evolutionin u(−0O5−1 − lxP y) with lx is shown in Fig. 5.5, with analogous results atx = 0O5−1 + lw.5CI gzsultsExample computed results are shown below in Fig. 5.6 for fixed (hP PW) =(2P 0O25P 5) and for increasing gz = 0P 50P 100P 200. We show colourmaps ofthe speed and pressure, for both the periodic geometry ( lx = lw = 0) and the186Figurz 5C6O Colourmaps of the flow speed with streamlines and pres-sure compared for periodic washout (two left columns) withwashout that has entrance length (two right columns). Param-eters are (hP PW) = (2P 0O25P 5) and gz = 0P 50P 100P 200, fromtop to bottom. The gray regions indicate unyielded fluid.fully developed inflow/outflow (lx = lw = 10). For the latter computationswe only display results within the washout length, in order to aid compar-ison. We note that the differences in velocity field between periodic andfull developed flows are relatively minor. At lower gz the streamlines tendto meander into the washout, whereas at higher gz there is a tendency tobypass the washout and the streamlines tend to straighten. The differencesin pressure are more significant, but mostly manifest outside of the washout.The net pressure drop across the washout appears to increase with gz andthe variations in pressure across the washout are more extreme in the fullydeveloped case. Neither effect is unexpected.187The most noticeable feature concerns the position and size of the staticplug region within the washout. Contrary to intuition, increases in gz ap-pear to increase the size of stationary plug initially, before eventually theincreased stresses result in a smaller stationary plug. At the same time, weobserve that although the large gz flows have mobilized the fluid within thelower part of the washout, the fluid remains largely unyielded and contin-ues to recirculate within the washout. We show a second set of results for(hP PW) = (1P 0O25P 5) (reduced h) in Fig. 5.7, again for gz = 0P 50P 100P 200but only for the fully developed washout. We see a similar trend in the sizeof the static plug and in straightening of the streamlines as gz is increasedat constant W.We see that this process also involves significant changes in the dis-tribution of stresses within the washout region. The size of extensionalstresses in the upstream part of the washout decreases with gz as the pluggrows. In [209] we saw that yy ≈ constant, within the central part of theplug/washout region at gz = 0. Here inertia results in complex distributionsof all stress components, breaking the symmetry of the gz = 0 flows.5CICF Flofi rvtz zffzxtsWe have seen that the main surprising effect above is to increase the sizeof the static plug region in the washout, with increasing gz at constant W.Typically in hydraulic flows, we interpret increasing gz as an increasing flowrate. However, here we see that increasing flow rate (hence jˆ0) would resultin decreasing W, unless one of the other parameters is also varied. Thus, wemust interpret the above results with increasing gz as either: (i) increasing/ˆ, or (ii) increasing jˆ0 and proportionately increasing ˆl . In the latter caseit is intuitive that the size of static plug layer may increase.To investigate further and to make the work more relevant to the cement-ing application, we study the effect of increasing flow rate independently. Asjˆ0 appears linearly in both gz and W, this corresponds to varying jˆ0 at con-188Figurz 5CLO Colormaps of the flow variables, restricted to the washoutregion for (hP PW) = (1P 0O25P 5). At the top of the figure thecolumns show speed, pressure and xx; the columns at the bot-tom of the figure show xy, xx and yy, from left to right. Therows show gz = 0P 50P 100P 200P 500, from top to bottom. Thegray regions indicate unyielded fluid.189stant Hedstro¨m number:Hz = gzW =/ˆˆl Yˆ2ˆ2P (5.16)i.e. as we increase gz we adjust W = HzRgz. Note that Hz depends onlyon the fluid properties and flow geometry, with no dependence on jˆ0. Tostudy flow rate effects we start with fixed Hz = 500, which is in the rangeof Hedstro¨m numbers for typical drilling muds and cementing geometries.We have conducted a parameteric study over (hP Pgz)-space with ∼ 200cases computed. The range of parameters considered was:0O25 ≤ h ≤ 40P 0O05 ≤  ≤ O5P 10 ≤ gz ≤ 500Pall for fixed Hz = 500. Smaller sets of results were then computed at differ-entHz. Figure 5.8 shows the effect of increasing flow rate at (hP ) = (1P 0O25)and Hz = 500. We again observe the straightening of the streamlines as theflow rate is increased from zero. The static plug region in the washout firstdecreases in area, then increases, up until an intermediate range of gz, thenfinally in this case disappears. This demonstrates that the previously ob-served effects are a result of changes in the flow variables and not increasesin yield stress. This of course has industrial significance as the usual percep-tion is that increasing the flow rate will be more effective at conditioning thedrilling mud. This appears to be false for a significant range of flow rates.For the same washout geometry we study in Fig. 5.9 the distribution ofpressure (averaged in y across the channel) along the channel and throughthe washout as the flow rate increases at fixed Hz = 500. Firstly, we see thatboth before and after the washout the average pressure converges relativelyquickly to a linear variation that matches that of the relevant plane Poiseuilleflow (marked with a broken line in Fig. 5.9). The pressure deviation from thePoiseuille flow is clearly asymmetric (in x) upstream and downstream of thewashout. The entry length is relatively short compared to the developmentlength (after the washout). We also observe that although at smaller gz theaverage pressure decreases monotonically through the washout, at larger gz190Figurz 5CMO Effects of increasing flow rate at (hP ) = (1P 0O25) andgz = 10P 50P 100P 200P 500 from top to bottom, at fixed Hz =500. Left: velocity magnitude and streamlines; right pressurewith unyielded regions shown as gray.191−10 −5 0 5 1002004006008001000P¯xa)−10 −5 0 5 100100200300400500600700800P¯xb)−10 −5 0 5 100100200300400500P¯xc)−10 −5 0 5 10050100150200250300350400P¯xd)Figurz 5CNO Average pressure along the channel (black) compared withthe Poiseuille flow pressure (broken line) as the flow rate in-creases. (hP −1) = (1P 4) and gz = 50P 100P 200P 500 withfixed Hz = 500. The vertical dotted lines marks the washoutstart and end.(flow rate) the pressure increases in the washout. This gain in pressuresuggests a transfer from the dynamic head to the pressure head due to theexpansion: frictional losses are insufficient to decrease the pressure.The offset between the broken and solid lines in Fig. 5.9 (at the veryleft of each plot) indicates the net effect of the washout on the frictionalpressure losses. This fivshout ∆p is positive, meaning that the frictionalpressure losses are reduced by the washout. We illustrate in Fig. 5.10 thevariation in washout ∆p with gz at fixed Hz = 500 for 2 geometries. As well1920 100 200 300 400 500020406080ReWashout∆pa)0.10.150.20.250.3Normalised∆p0 100 200 300 400 5000200400ReWashout∆pb)0.20.30.4Normalised∆pFigurz 5CFEO Pressure drop offset due to washout with increasing gzat fixed Hz = 500. The solid line shows the pressure drop andthe broken line shows the normalized value with the pressuredrop, normalized with the Poiseuille flow pressure drop overthe length of washout. a) (hP ) = (1P 0O25) and b) (hP ) =(1P 0O1).as the washout ∆p we present a normalised ∆p, for which we have scaledthe washout ∆p with the pressure drop over a uniform channel of equivalentlength to the washout. These plots show that with increasing flow rate (gz)the offset is reduced, i.e. the pressure drop gets closer to that of the planePoiseiulle flow. This might be expected phenomenologically as for large gzwe have seen that the streamlines straighten. The normalized ∆p shows amaximum at an intermediate gz.The effects of flow rate on the static plug within the washout and onconditioning are illustrated more clearly in Fig. 5.11, for 3 different washoutgeometries. In [209] we have defined the ofiing vrzv as that part of thewashout that is non-stationary. As we have seen here, even when the fluidbegins to move it does not necessarily leave the washout: typically there aresignificant parts of the washout in which the yielded fluid still recirculates.We therefore define the yisplvxzy vrzv as that part of the washout thatis actually removed from the washout. By definition the displaced area isalways less than the flowing area. From the industrial perspective, drillingmud that is flowing but not displaced would be to some extent conditioned193100 200 300 400 50000.511.52ReAreaa) 100 200 300 400 5000246810ReAreab)100 200 300 400 500012345ReAreac)Figurz 5CFFO Flowing area (black, +) and displaced area (red, ©)variations with increasing gz at fixed Hz = 500. (a) (hP ) =(1P 0O25) (b) (hP ) = (2P 0O1) (c) (hP ) = (1P 0O1)(sheared) but is not replaced during the circulation phase. Also the degree ofshearing in a recirculation zone is questionable. First we often see significantplugs within the recirculation zone. Second, the strength of the recirculationzone is generally very weak and decreases with depth into the washout ifvery deep, e.g. see the secondary vortices in [209] and the analogy with [166].We observe in Fig. 5.11a & b similar qualitative behaviour for the first2 geometries. As gz increases from zero the flowing and displaced areas194initially coincide and increase (i.e. initially increasing flow rate improvesconditioning). A first critical gz is attained when both flowing and displacedareas attain a local maximum. This appears to approximately coincide withthe onset of the pressure increase in the washout. Further increases in gzresult in decreasing flowing and displaced areas until a second critical gz isattained. At the second critical gz a part of the washout fluid begins to movein a recirculation zone. The flowing area thus increases rapidly (essentiallythere is a jump), whereas the displaced area continues to decrease with gz.Throughout this regime the streamlines are continually straightened andthe main flow appears to bypass the washout. For these two geometriesthe washout depth is modest and once recirculation starts the flowing areaquickly converges to the full area of the washout. For a deeper washout thisconvergence is not immediate, but occurs in all cases as gz increases. Thisis because at fixed Hz increasing gz means decreasing W and at low enoughW there is no static fluid in the washout.A different type of behaviour is seen in Fig. 5.11c. Here we observethat the flowing and displaced areas are constant and equal for low gz, butare also equal to the washout area. This can be understood with referenceto [208] in which the onset of static regions was studied in wavy-walledchannels. It was observed that (for gz = 0) when h Q (h)v, no static fluidwas found in the deepest part of the channel, even as W → ∞, i.e. there isa minimal slope of the washout wall required to have static fluid and therelatively shallow washout in Fig. 5.11c does not exceed the minimum.At a fixed Hz by letting gz → 0 we have W → ∞ and in this limitthere are only two possibilities: h is less or greater than the critical (h)v.For (h)v Q h we find static fluid at washout for gz = 0 (Fig. 5.11a& b). By increasing gz the shear stress over this static region increasesand the displaced area increases until the first local maximum. On theother hand, for h Q (h)v there is no static fluid at the washout wall asgz → 0 (Fig. 5.11c). For such cases, as gz increases the streamlines tendto parallelize and the size of extensional stresses in the washout area drops.Somewhat counter-intuitively, a static region appears in the washout as gzincreases! Consequently, the flowing area and displaced area decrease above195Figurz 5CF2O Effects of increasing flow rates at fixed Hz. The inertialeffects are controlled by gz = 50P 100P 200P 400 (top to bot-tom) and geometric variables are (hP ) = (1P 0O25). The grayareas denote unyielded fluid. Left column: Hz = 200, rightcolumn: Hz = 1000.196100 200 300 40000.511.52ReAreaa) 100 200 300 400 50000.20.40.60.811.21.41.61.8ReAreab)Figurz 5CF3O Flowing area (black, +) and displaced area (red, ©)variation with increasing gz for (hP ) = (1P 0O25) and differentfixed Hedstro¨m numbers: (a) Hz = 200 (b) Hz = 1000this first critical gz. With growth of the static region with gz we againreach a second critical gz, above which the fluid in the washout starts torecirculate and the flowing area rapidly increases.5CICG kvrying HeThe effects illustrated above for increasing flow rate are also found at otherHz. Figure 5.12 shows the speed colourmap and streamlines for two otherHz = 200P 1000 for increasing flow rates (gz = 50P 100P 200P 400 at fixedHz), with (hP ) = (1P 0O25). The effects are qualitatively similar to those atHz = 500. The effects on flowing and displaced areas at Hz = 200P 1000are shown in Figure 5.13 and mimic that shown above for Hz = 500. Bothcritical gz appear to be increased with increasing Hz. It is worth notingthat for the examples shown, with h = 1 the washout is not very deep. Themaximum washout area is 2 and once recirculation starts the stationary layeris effectively removed (the flowing area is approximately 2 after the secondcritical gz). In deeper washouts this is not the case at all: recirculationzones co-exist with significant stationary layers deeper into the washout.197−0.2 0 0.2 0.4 0.6 0.800.511.522.533.544.5u(−0.25δ−1, y)y  a)h = 2h = 5h = 10h = 20h = 30−0.2 0 0.2 0.4 0.600.511.522.533.544.5u(0, y)y  b)h = 2h = 5h = 10h = 20h = 30−0.2 0 0.2 0.4 0.600.511.522.533.544.5u(0.25δ−1, y)y  c)h = 2h = 5h = 10h = 20h = 30Figurz 5CFIO Variation of the x-component of velocity u(xP y) withincreasing h at a) x = −0O25−1, b) x = 0, c) x = 0O25−1 fora square washout: (gzPWP ) = (100P 5P 0O1).5CICH Effzxts of fivshout shvpzIn [209] one of the main results was the discovery at gz = 0 that for deepwashouts and at large yield stresses the flow self-selects its flowing region.The flowing region was found to become invariant as h is increased and be-tween different washout geometries. Finally, a square washout shape wasfound to have the maximal stationary regions, of all washouts consideredwith the same (hP ). In moving to inertial flows we may expect similareffects, but must be more cautious with regard to washout shape. For ex-ample, a square (or angular) washout at large gz is likely to shed vortices198−0.2 0 0.2 0.4 0.6 0.800.511.522.533.544.5u(−0.25δ−1, y)y  a)h = 2h = 5h = 10h = 20h = 30−0.2 0 0.2 0.4 0.600.511.522.533.544.5u(0, y)y  b)h = 2h = 5h = 10h = 20h = 30−0.2 0 0.2 0.4 0.6 0.800.511.522.533.544.5u(0.25δ−1, y)y  c)h = 2h = 5h = 10h = 20h = 30Figurz 5CF5O Variation of the x-component of velocity u(xP y) withincreasing h at a) x = −0O25−1, b) x = 0, c) x = 0O25−1 fora sinusoidally wavy washout: (gzPWP ) = (100P 5P 0O1).at the corners and is likely to become unsteady at lower gz than a smootherwashout. Secondly, we have seen that inertial flows transition to a recircu-latory regime occurs at a given gz. This transition and the shape of anydetached plug region within the recirculating region are likely to be sensitiveto washout shape.We first establish that as h increases, steady flows attain an invariantvelocity profile. Figure 5.14 shows the x-component of velocity u(xP y), withincreasing h at x = −0O25−1P 0P 0O25−1 for a square washout: (gzPWP ) =(100P 5P 0O1). We see that as h significantly exceeds the width of the flowing199−0.2 0 0.2 0.4 0.6 0.800.511.522.533.544.5u(−0.25δ−1, y)ya) −0.2 0 0.2 0.4 0.600.511.522.533.544.5u(0, y)yb)−0.2 0 0.2 0.4 0.600.511.522.533.544.5u(0.25δ−1, y)yc)Figurz 5CF6O comparison of x-component of velocity u(xP y) at a)x = −0O25−1, b) x = 0, c) x = 0O25−1 for (gzPWP hP ) =(100P 5P 30P 0O1), between three different washout shapes. Themarkers are: □, expansion-contraction/square washout; △,linear variation/triangular washout; ©, wavy sinusoidalwashout.2000 10 20 30020406080100hDissipationFigurz 5CFLO Plot of viscous (solid line) and plastic (broken line)dissipation functionals for (gzPWP ) = (100P 5P 0O1) and dif-ferent types of washouts. The markers are: □, expansion-contraction/square washout; △, linear variation/triangularwashout; ©, wavy sinusoidal washout.region the velocity appears to converge. Figure 5.15 shows analogous resultsbut for a wavy channel. The velocity field again converges at large h. Finally,Fig. 5.16 compares the velocity profiles at the same x-positions for threedifferent washout geometries of wavy, triangular and square. We can seethat square and triangular shapes are close, but the wavy profile has a visibledifference and needs higher h to catch up. At any rate the convergence ofthese is apparently slower than in the non-inertial flows of [209] and this isprobably due to the variations along the channel.A different way of assessing the convergence of the solutions, with large hand/or comparing between geometries, is to evaluate the viscous and plasticdissipation functionals within the washout section of the channel. Thesefunctionals include of course extensional strain rates and hence variationsalong the channel. Figure 5.17 illustrates this variation and although wecan see that the dissipation functionals are converging the convergence isrelatively slow with h, i.e. considering that the velocity is static in the upperpart of the washout at relatively small h for intermediate x. The difference201in dissipation functionals comes mainly from the entry/exit regions of thewashout which are sensitive to geometry.5C5 Yisxussion vny summvryWe have presented a largely computational study of steady inertial flows ofa Bingham fluid through 2D “washout” geometries, representing longitudi-nal sections of a narrow eccentric annulus; see Fig. 5.1. This study extendsChapter 4 (and [209]) into the inertial regime, thus giving a more compre-hensive picture of effects that occur in circulation phase of cementing of oilwells.The computations achieve steady state via a time-dependent iteration.We observed that the effect of having a yield stress was to increase the rangeof Reynolds number gz for which a steady solution could be found, hintingat an increase in stability of the flow with W, as would be expected. Thiseffect was not quantified.We investigated the effects of increasing gz on converged steady laminarflows. Some effects observed were largely predictable. Increasing gz resultsin a fore-aft asymmetry in symmetric washout shapes. Increasing gz canlead to an increase in averaged pressure in the expanding part of the washout.Entry/exit development lengths did not have a major effect on the flowwithin the washout itself.Increasing gz was studied for fixed Bingham number W and for fixedHedstro¨m number Hz = gzW. The latter represents an increase in flowrate for fixed geometry and fluid properties. In both cases we observedthat the variation in flowing area of the washout (i.e. that area that ismobilized) was non-monotone. Depending on the slope of the washout wallsh, at gz = 0 stationary fluid was either present at the walls or not (forsufficiently shallow washouts). Increasing gz resulted in a straightening ofthe streamlines through the washout region, in the main part of the channel.At fixed Hz, the flowing area initially either increased (static layerspresent at gz = 0) or remained constant (equal to the washout area: mobileat gz = 0) up until a 1st critical value of gz after which the flowing area202decreased. In this regime of decreasing gz the results are quite counterin-tuitive and contrary with the industrial perception that pumping faster willcirculate/condition the mud better. Note that in some cases increasing gzresults in static washout regions when there are none at gz = 0!At higher value of gz, on passing a second critical value, we observethe onset of zones of recirculation within the washout. The flowing areathus increases but the area of washout in which fluid is displaced duringconditioning continues to decrease. For the examples we have studied thetwo critical values of gz cover a range of 100’s (see e.g. Figs. 5.11 & 5.13),meaning that these effects are not marginal. Overall this shows that a morecareful design of the recirculation/conditioning phase of primary cementingwould be fruitful.We also studied the effects of washout geometry in a targeted way. In[209] the washout geometry was observed to have little effect on flowing areageometry for sufficiently large W and h. The same phenomena was observedagain at large h over a range of different washout geometries. We examinedconvergence of the velocity profiles and dissipation rates, for different shapesas h is increased. Thus, the self-selection of flowing area still remains validfor gz S 0 in steady laminar flows over the range of parameters considered.However, we noted that convergence was slower than for gz = 0 and retainsome caution in using reference geometries (such as the square washout in[209]) with sharp corners since these are likely to be vulnerable to inertialinstabilities induced by geometry, e.g. vortex shedding.Our attention is now focused in two future directions. Firstly, the prob-lem of fluid-fluid displacement through similar geometries as considered hererequires study. These flows are typically complicated by both density andrheology differences between the fluids. Even for iso-density displacementsit is known that static layers can result as a less viscous fluid by-passes anin-situ yield stress fluid; e.g. [9]. Including a density difference increases theflow complexity but also makes questionable the 2D assumptions made here,as buoyancy, eccentricity and inclination in geometries such as Fig. 5.1 al-most certainly will promote azimuthal displacement flows. Nevertheless, the2D study of displacement is the next logical step and our progress on this203problem is outlined next in Chapter 6. The second direction planned for thefuture is to study 3D geometries. Not only are primary cementing displace-ments likely to become more 3D, but other important cementing geometriesare also inherently 3D. An example being the setting of abandonment plugsin milled out wellbores, e.g. [46].204Chvptzr 6Yisplvxzmznt ofiIn this chapter we begin to consider the displacement of a yield stress fluidthrough washout geometries related to the primary cementing flow in oil/gaswell construction. First we review the relevant literature and formulate theproblem to solve. Then we discuss some of the computational challengesand present a set of initial results for the displacement of Bingham fluid bya Newtonian fluid in a square washout geometry.6CF IntroyuxtionTwo-fluid and displacement flows have been subject of vast interest in fluidmechanics research, both due to the many applications and to having richtheoretical aspects. For better clarity in what follows, we call the yisplvxinguiy the invvying uiy and the in situ or yisplvxzy uiy the defendingfluid.Perhaps the most well know phenomenon in displacement flows is theSaffman–Taylor instability [211] that happens when the invading fluid is lessviscous and fingers into the defending fluid. Examples include secondaryand ternary oil recovery. The Rayleigh–Taylor instability [218] occurs whenthe density difference between fluids is mechanically unstable, i.e. heavyfluid above light fluid. Examples are nuclear explosion, mushroom cloudsfrom volcanic eruptions and supernova astrophysical explosions. The Kelvin-205Helmholtz [89] is an archetypical instability that occurs at the interface oftwo fluids, moving at different velocities (e.g. in shear flow, although theorigins are inviscid), and results in astounding vortex shapes such as incloud formations, Jupiter’s Red Spot and the Sun’s corona. Thus, two-fluidflows have triggered considerable research in hydrodynamic stability theorywith many applications.One reason for the rich dynamics of two-fluid systems comes from thecontrast in physical parameters of the fluids (e.g. density, viscosity, etc...),across the intzrfvxz. For immiscible fluids surface tension is also present andcan strongly affect the interface dynamics, while for miscible fluids mixingoccurs. Due to this diversity, there is enormous amount of work in the lit-erature of two-fluid and displacement flows, that would require an extensivereview. However, with primary cementing as our principal motivation welimit our review to those displacement flows in confined geometries withthe possibility of having residual layers of defending fluid. In the primarycementing process, the (in-situ) defending fluid is a drilling mud and theinvading fluid is either a cement slurry or a spacer fluid. All these fluidsmay be considered to be shear-thinning yield stress fluids.The industrial difficulty stems from the possibility of the defending drillingfluid remaining behind in the well after displacement. In this chapter wewill consider a simplified model of the primary cementing flow in a 2D rect-angular washout. The geometry is initially filled with defending fluid andinvading fluid displaces it as it is pumped from upstream. The fraction ofthe defending fluid which is eventually displaced is called displacement ef-ficiency:  ∈ [0 1], which ideally should increase to 1 over the course ofthe displacement flow. The displacement efficiency is however a simplifiedindicator of a wide range of complex behaviours. Here we start to exploresome of these behaviours in washout geometries.One type of displacement flow with residual layers are gas displacementflows, which have application in injection molding, enhanced micro filtration[144], . . . In displacement flow of two Newtonian fluids in a uniform channels,residual wall layers can exist over very long times only when the displacingfluid is inviscid. Otherwise the residual wall layers will drain over time206due to viscous shear stresses exerted at the interface and the displacementefficiency asymptotes to 1. This means that permanent residual layer is onlypossible when the invading fluid has zero viscosity. The gas displacement ofa Newtonian fluid in a pipe was first studied experimentally by Taylor [231].As well as predictions of residual layer thickness he suggested three possibleconfigurations of the stream lines around the interface. Cox [73] gave moreexplanations and the predictions of the streamline shapes were confirmednumerically by Soares et al. [222]. Dimakopoulos and Tsamopoulos [83]studied numerically the inertial gas displacement flow in straight tubes andin a contraction. They showed that increasing the gas pressure does notaffect the thickness of deposited layer and shape of the advancing bubble instraight tubes and only makes the process faster.There are also some gas displacement studies of non-Newtonian fluids.Dimakopoulos and Tsamopoulos have numerically investigated these flowsfor several cases: displacement of viscoplastic [84] and viscoelastic [85] fluidsin straight and contraction geometries, gas displacement of viscoplastic fluidin expansion-contraction geometry [86]. De Sousa et al. [76] investigated gasdisplacement of shear-thinning and viscoplastic fluids numerically. Theyobserved that by decreasing the shear thinning index or yield stress thethickness of deposited residual layer decreases.All of the above mentioned works are for immiscible fluids and surfacetension is important. In primary cementing it is common to use a spacer fluidthat is chemically compatible with both the drilling mud and cement slurry.Commonly the fluids are water-based and miscible. Despite being miscible,the molecular diffusivity between fluids is generally very low. Hence sharpinterfaces exist in displacement flows and significant mixing occurs only if theflow becomes chaotic. This limit of zero molecular diffusion of miscible flowsis interesting as it is mathematically the same as the limit of zero surfacetension of immiscible fluids. So this type of flow stands in the mid-groundbetween miscible and immiscible displacements.One approach to modelling cementing is to consider the wellbore andcasing as two eccentrically arranged pipes separated by a thin gap, vary-ing slowly in the azimuthal and axial directions. This approach leads to207a Hele-Shaw approximation for the flow. This kind of flow has been stud-ied e.g. Alexandrou and Entov [5] who study the fingering of inviscid fluidsfor Bingham and other non-Newtonian fluids. This type of model has alsobeen extensively used for the study of primary cementing in our group; see[33, 47, 48, 153, 189]. This model gives valuable information about displace-ment flow. However, the basic flow unit considered is the flow of a singlefluid in a slowly varying channel, which therefore cannot have stationarylayers at the wall, i.e. the Hele-Shaw approach involves averaging across anarrow gap. Static fluids in this type of model only happens when the pres-sure gradient is insufficient to mobilize the entire fluid in the gap, which is adifferent mechanism from getting static fluid at the wall due to geometricalnon-uniformity.The most relevant works are those considering the displacement of vis-coplastic fluid by another fluid, with a focus on static wall layers. Alloucheet al. [9] carried out a detailed study of static layers in displacement flowof two iso-density viscoplastic fluids in plane channel. Using a lubricationmodel they give sufficient conditions to have no static wall layer, whichdepends on two dimensionless parameters: the displacing fluid Binghamnumber and the ratio of the yield stress of the two fluids. If these conditionsare not met, an upper bound (hmtx) for the static wall layer thickness canbe derived. However, comparing their theoretical estimates with 2D com-putations of the actual thickness of residual layers revealed that hmtx is tooconservative. By analyzing the streamline configuration close to the steadydisplacing front, a new recirculating thickness hvirv was derived which is inclose agreement with the 2D computations. Frigaard et al. [103] studiedsteady front displacement of iso-density fluids theoretically and showed thatfor a given shape of displacing front static layers with different thicknesscan exist. Thus they concluded that the final thickness of layers depends ontransients, not the steady solution. Frigaard et al. [104] considered two-layeraxial flows of iso-density viscoplastic fluids. They derived two variationalprinciples for these flows: strain rate minimization and stress maximizationprinciples. The strain rate minimization leads to existence and uniquenessresults for this class of flow. The stress maximization principle leads to a208number of qualitative results. For example it is shown that using stressmaximization principle, an interface shape optimization problem for max-imal wall static layer is obtained. Wielage and Frigaard [249] studied thesame problem as in [9], but with two improvements. First, they used theaugmented Lagrangian method, rather viscosity regularisation, in order tocompute (truly) static layers. Secondly, they extended their simulations forsignificantly longer times to make sure that slower transients do not changethe static layer thickness. In addition they studied the combined effects ofinertia and pulsation of the displacing flow, on the static wall layers shapeand efficiency of displacement.By looking at the work published that addresses primary cementing dis-placement flows, we see that all of the studies concerning static layers onthe walls are performed in uniform channels: the effect of non-uniformity inthe geometry has not been considered in the context of displacements. Inthis chapter we start to consider this problem.6CG boyzl prowlzmA full study of the cement-drilling mud displacement flow is extremely com-plicated. Rheologically, both fluids are usually modeled as Herschel-Bulkleyfluids. The density can be quite different and in reality the well might beinclined at any angle relative to gravity. Adding geometrical parametersto these degrees of freedom, we would end with a problem of at least 10dimensionless parameters, which is very hard to study.To simplify, we consider iso-density displacement flow of a Bingham fluidby a Newtonian. Having iso-density fluids we neglect both gravity andinclination. Further, we only use the square shape as a washout geometry.This is the simplest displacement flow for which we are likely to have staticresidual regions attached to the walls and within the washout, and for whichthe two fluids have some rheological freedom and relevance.With these simplifications, we consider the flow geometry as shown inFigure 6.1. The unsteady Navier-Stokes equations and the mass conserva-209tions govern the flow, as usual. In dimensional form these are:/ˆ(UuˆUt+ uˆO∇uˆ) = −∇pˆ+∇Oˆ P (6.1)∇Ouˆ = 0O (6.2)These equations hold for the entire flow domain of the invading/defendingfluids. The difference comes in the stress divergence term for each fluid. Forparts of the domain filled with invading fluid (cement or spacer)ˆij = ˆi ˆ˙ij P (6.3)and for parts of the domain filled with the defending fluid (drilling mud)ˆij = (ˆw +ˆyˆ˙(uˆ))ˆ˙ij ⇔ ˆ S ˆyP (6.4)ˆ˙ij = 0⇔ ˆ ≤ ˆyO (6.5)Here ˆiP ˆw are the viscosities of invading and defending fluids and ˆy is yieldstress of the defending fluid. If the fluids domains are independent, then wealso have an interface, say Fˆ (fˆlP tˆ) = 0, that is advected with the flow, viaa kinematic equation. At the interface the velocity is continuous. We alsoshould have continuous traction vector along the interface:(−pˆ+ ˆ )invtwinzOns = (−pˆ+ ˆ )wxyxnwinzOnsP (6.6)where the left and right sides are evaluated using (6.3) and equations (6.4-6.5) respectively (ns is the unit normal vector to the interface). An alternatedescription (adopted below) is to consider the fluid phases to be defined bya scalar concentration (i.e. volume fraction), that is advected and diffuses(and then consider the limit of small diffusivity).Geometrical parameters for the flow are the width Yˆ of the channel,depth Hˆ and length of washout aˆ. As with the inertial flows in Chapter 5,there is also entrance/exit development length lˆ, see Figure 6.1. To make theequations dimensionless we use these scales: mean velocity in the uniform210HˆDˆLˆlˆuˆ = 0uˆ = 0vˆ = 0uˆx = uˆNewt= 6Uˆ0yˆDˆ(1− yˆDˆ)σˆ.n = 0Figurz 6CFO Problem domain, geometrical dimensions and boundaryconditions used for displacement flowchannel jˆ0 for velocity, ˆwjˆ0RYˆ for stresses and Yˆ for lengths. With thesescales the non-dimensional equations are:gz(UuUt+ uO∇u) = −∇p+∇O P (6.7)∇Ou = 0O (6.8)The constitutive law for the invading fluid (cement slurry or spacer) be-comes:ij =b˙ij P (6.9)and for the defending fluid (drilling mud):ij = (1 +W˙(u))˙ij ⇔  S WP (6.10)˙ij = 0⇔  ≤ WO (6.11)Dimensionless dynamic parameters of the problem are the Reynolds num-ber, gz = /ˆjˆ0YˆRˆw, the Bingham number W = ˆyYˆRˆwjˆ0, and the viscosityratio b = ˆiRˆw, of invading to defending fluid. The dimensionless geomet-ric parameters are the maximum washout depth, H = HˆRYˆ, the washoutlength a = aˆRYˆ, and the development length, l = lˆRYˆ. Following Chapter 5we fix the development length to l = 10.211For boundary conditions we have no-slip on the top/bottom walls. Atthe inflow a Poiseuille velocity profile of Newtonian fluid is prescribed suchthat the non-dimensional flow rate f = 1 (as velocities scaled with meanvelocity). And at the outlet zero stress boundary condition is applied, seeFigure 6.1.We use the augmented Lagrangian method for solution of this problemas before and we do not repeat the details here. The rest of the numericalscheme, including surface tracking and Navier-Stokes problem, is explainedin the following sections.6CH cumzrixvl mzthoy for intzrfvxz trvxkingOkdFInterface tracking is the major challenge that one faces in the numericalsolution of the two fluid systems. The methods used can be mainly groupedinto avgrvngivn and Eulzrivn, with some others that use both in a hybridway.In Lagrangian methods the interface is tracked directly using a set ofpoints which are advected with the velocity field. The advantage is thatinterface tracking is accurate and simple, as we directly follow the position.Also computing the normal vector to the interface is also more accurate.However the problem is that it is hard to capture merging or break-up ofthe interface, e.g. a bubble which breaks up or two bubbles that merge.On the other hand Eulerian methods do not directly track the interface,instead a passive scalar field is used which is advected with the velocityfield on a mesh, and the interface shape is implicitly inferred from the field.Volume of fluid (VOF) [119], level set [2] and phase field [248] are exam-ples of Eulerian interface tracking methods. The main advantage is thatmerge/break-up of the interface is easily tracked. On the other hand, sincethese methods filter some information about the interface, we cannot recon-struct the shapes of the interface as accurate as with Lagrangian methods.This is a motivation for using hybrid approaches which try to mix both ofthese.212If the specific problem of study does not involve merge/break-up La-grangian method is easy and advantageous, however this is not the case forthe displacement flows that we are interested in. Here the flow phenomenato be studied are unknown at the outset. Thus we use the VOF method. Inthis method a scalar field x is defined as the fraction of the volume of eachcell in the mesh which is occupied by one fluid, see Figure 6.2. Obviouslywe have x ∈ [0 1]. The equation for evolution of x would beYxYt=UxUt+ uO∇x = 0O (6.12)By using incompressibility∇Ou = 0 we can write the equation in conservativeformUxUt+∇O(xu) = 0O (6.13)The advantage of this form is that if we use Finite Volume methods forsolution, then it is guaranteed to conserve the volume of fluids at the discretelevel.Having discrete conservation of x is good, but more is required. The otherimportant properties of x are: (a) it should preserve a sharp transition from0 to 1 around the interface; (b) The range of x should remain [0 1], as it hasno physical meaning to have x Q 0 or x S 1. This problem of conservationlaws with sharp changes has been well studied. If we use conventional fluxcomputation schemes we either get too much dissipation of x, i.e. smearing ofthe interface or often get overshoots of x to outside of the physical range. Onesolution is to use special flux computation schemes such asbjhXa, whichstands for Monotone Upstream-centred Schemes for Conservation Laws.For our problem we use the van-Leer limiter which is designed for struc-tured meshes only. Generally MUSCL schemes are much more complicatedfor unstructured meshes. Since for our problem, in this simplified geometry,using structured is adequate, we avoid those complications. We outline thescheme for a simple 1D flow in x direction as shown in Figure 6.3. Equation21310 0 0.1.2.31 .9Figurz 6C2O Volume fractions values assigned to each cell in VOF(6.13) for this flow simplifies toyxyt+y(xu)yx= 0P (6.14)where u is the velocity in positive x direction. Integration over cell i yieldsthe finite volume formyyt∫ix yx+∫iy(xu)yxyx = 0P (6.15)yyt∫ix yx+(xu)i+1P2 − (xu)i−1P2∆x= 0O (6.16)We need to compute the fluxes (xu)i+1P2P (xu)i−1P2 on faces of control volume.The 1D MUSCL scheme computes the flux Fi+1P2 on the face i+1R2 of celli asFi+1P2 = ui+1P2[xi +  (ri+1P2)(xi − xi−1)]P (6.17)ri+1P2 =xi+1 − xixi − xi−1 O (6.18)The function  is called flux limiter and ri+1P2 shows the ratio of jump214dxi i+1 i+2i-1i-2Figurz 6C3O A 1D domain and its finite volume discretization. Figurecourtesy of C. Olivier-Goochacross the boundaries of x. For the van-Leer limiter we use: (r) =0 ⇔ r ≤ 0P2r1 + r⇔ r S 0O(6.19)With this type of computing fluxes we will have both sharp boundaries andkeeping the scalar in the physical range [0 1].To extend this method for 2D we simply apply the above scheme in eachdirection separately. This is possible as we use a structured rectangulargrid. The van-Leer scheme cannot be applied to an unstructured grid inthis simple format. The 2D equation in discrete form becomesxn+1iNj − xniNjt+Fni+1P2Nj − Fni−1P2Nj∆xiNj+FniNj+1P2 − FniNj−1P2∆yiNj= 0P (6.20)where t is time step, ∆xiNj P∆yiNj are the sizes in xP y directions of cell iP j.All FkNl fluxes are computed as explained for the 1D case.One important issue with VOF that we didn’t discuss here is interfacereconstruction. A variety of ways have been proposed such as SLIC [179],PLIC [255], LVIRA [197] and others. This has applications when we want toobtain the interface normal and curvature for computation of surface tensioneffects. As we consider miscible flows no surface reconstruction is necessaryand we basically ignore these features. Also another limitation is that VOFcannot be used for systems with three or more fluids.2156CI cumzrixvl mzthoy for thz solution of thzcvvizrBhtokzs zquvtion fiith vn intzrfvxzThe VOF method for tracking of the interface was explained in the previoussection. Now to solve the displacement flow, the Navier-St