Ian Frigaard

Gas rise in viscoplastic fluids is an interesting problem both from phenomenological and industrial points of view. Unlike Newtonian fluids, gas can remain entrapped inside a viscoplastic fluid if the yield stress is high enough. In studying the rise of gas bubbles in viscoplastic fluids, it is crucial to know this critical limit beyond which there is no flow.The main motivation of this thesis stems from the problem of methanogenesis in tailings ponds, which leads to the release of a significant amount of greenhouse gas emissions into the atmosphere. Tailings ponds are large density stratified settling ponds that store the waste materials generated during the mining process. They consist of multiple layers: a water layer on the top, a sludge-like layer around the middle, and a coarse sand layer mixed with sludge close to the bottom. In this thesis, we study the gas behavior in different layers of the tailings ponds through numerical, experimental and analytical methods.This thesis contains results that use three distinct methodologies in three complementary research areas. Firstly, it investigates the static behavior of buoyant bubbles in yield-stress liquids using numerical methods, determining the minimum yield stress needed to immobilize them, and studying contributing factors such as bubble shape and surface tension. The ratio of yield stress to surface tension (yield-capillary number) was found to play a key role. In the second area, the study delves into characterizing the steady motion of bubbles that do rise, often adopting an inverted teardrop shape with a pointed tail, distinct from predictions using basic yield-stress fluid models. We explore the origin of this shape using an experimental approach. Finally, the thesis explores gas propagation in porous media filled with yield-stress fluids, utilizing a pore-throat network model and a semi-analytical approach. It evaluates the impact of variables like throat radii, fluid yield stress, network size, exit time, gas volume fraction, and network stability in response to perturbations. Understanding the mechanics of gas propagation inside viscoplastic fluids offers valuable insights into a more sustainable resource management, minimizing the environmental impacts.

View record

This thesis presents a numerical study of the flow and stoppage of a viscoplastic fluid invading into a thin, non-uniform channel. The fluid represents a cementitious sealant, and the invasion aims to regulate gas flow through the partially filled channel. The flows are driven under constant pressure and arrested due to fluid yield stress. The gas flow through the channel, modeled as a Hele-Shaw cell, is calculated using a thin gap approximation and computed before and after the invasion.Motivated by industrial applications from squeeze cementing, the first objective is to investigate the fluid interface morphology at the stoppage, i.e., how effectively the fluid blocks the gap. We present a gap-averaged framework to study the interface morphology for radial injection into a Hele-Shaw geometry, where the gap thickness variations are defined using realistic industrial data and a stochastic model. We quantify the interface morphology to investigate the effect of different parameters. Our results reveal an enormous variability resulting from the geometrical uncertainties.The second objective of the thesis is to devise a gap-averaged model for a multisource injection, where the fluid is injected into the Hele-Shaw geometries from patterned injection points. We develop a method by stacking radial injection scenarios to reconstruct the whole section. We investigate the effect of perforation patterns and rheological parameters on the leakage flow rate. We generate a probability distribution of leakage rates to estimate the mean reduction in flow leakage and compute confidence intervals. Our results suggest that higher perforation density and lower yield stress slurry lead to higher success rates.The final objective of this thesis is to study the invasion of a viscoplastic suspension into a thin channel. We consider the multiphase nature of the suspension and compute the transverse distribution evolution of liquid and solid phases during flow. The aim is to explore various phenomena, such as particle migration, jamming, and yield surface position. This allows us to validate the accuracy of gap-averaged models, i.e., how ignoring transverse flow parameters affects penetration depth estimation. It also points toward new and more complex physical models for processes such as squeeze cementing.

View record

The oil and gas industry, while offering numerous benefits, grapples with environmental and sustainability challenges such as uncontrolled gas emissions, greenhouse gas contributions, ecological damage, and risks to groundwater. This thesis focuses on a pivotal industry operation: primary cementing. It is a process aimed at mechanical stability and hydraulic isolation, achieved by placing cement slurries in the annular space between a steel casing and wellbore. Failing to meet these objectives could potentially have consequences in terms of wellbore integrity. Complications arise from the eccentricity of the annular regions and the complexities of the involved fluids.The main emphasis of this thesis lies in understanding dispersion behavior and performing classifications for displacement flows in vertical eccentric annuli. This type of flow underlies the primary cementing process which is fundamental to preventing leakage. We firstly make a comprehensive comparison between the 2D gap-averaged model (2DGA) and a 3D model. The cases show both the successes and failures of the Hele-Shaw approach. Acknowledging the main limitation of 2DGA model, we then re-derive it starting from the reduced shear flow equations by applying a different assumption regarding the distribution of fluids across the annular gap. We find significant improvement of the dispersive 2DGA (D2DGA) model in capturing the gap-scale dispersion.A series of experiments has been conducted using our lab-built setup. We start from the displacement flows of Newtonian fluids in a vertical annulus, studied from both experimental and computational aspects. The models have been validated and are found to be effective and robust in predicting different flow behaviours. We develop a set of criteria for flow classifications, applicable for both concentric and eccentric annuli. The eccentricity and buoyancy number are found to be the dominant parameters affecting the front behaviour as well as the displacement efficiency.Finally we extend our study to shear-thinning fluids, employing a combination of experimental and computational methods. The classification criteria also prove effective in discerning various flow types, aligning with the results from the Newtonian study to a degree. Additional information due to extreme viscosity ratios is provided, along with study of some intriguing shear-thinning effects.

View record

Long-chained zwitterionic surfactants form wormlike micellar (WLM) gels in solutionat room temperature and revert back to a Maxwell viscoelasticity at highertemperatures. We study the rheology and flow dynamics of such WLM gels usinga combination of rheometry and flow experiments. First, we rheologically characterizethe material and reveal details of its microstructure along with signatures ofshear-induced fracture and banding. Then we pump these WLM gels in a capillarypipe and measure the development of velocity profiles in time. Our experimentscapture flow features like strong slip, plug and shear layers along with flow-inducedheterogeneity. In the third part of the thesis we study droplet detachment and filamentthinning of our WLM gels. We show the existence of elastocapillary thinningregimes, power-law failure and discuss dependencies of said dynamics on surfactantconcentration. Our experiments reveal hitherto unknown complex dynamicseven in simple flows of WLM gels and is likely to inform research on flows whererheology and transient fluid dynamics of similar soft matter systems play a crucialrole.

View record

Polydisperse particle-laden flows are ubiquitous in both nature and industry across diverse disciplines. An in-depth understanding of the intricate interactions between the dispersed solid phase and carrier fluid phase is essential in terms of designing and optimizing the key components of various industrial processes. Due to the large separation of temporal and spatial scales across different phases in real-world problems, previous studies encountered difficulty establishing a proper two-way coupling with both satisfactory accuracy and efficiency: either the various interactions are coarse-grained and oversimplified by the average drag closures in large-scale models, or the computational costs may soar beyond affordability while fully resolving the boundary of smallest particles in industrial-level simulations. To address this challenge, we develop two deterministic models based on the particle-resolved simulation data to estimate the force and torque fluctuations within polydisperse sphere assembly, which provides a channel for bottom-up knowledge transfer to evaluate the macroscale interphase interactions via the microscale hydrodynamic behaviors. Accordingly, the primary contributions of this dissertation feature the following aspects: First, we develop a highly scalable Immersed Boundary-Lattice Boltzmann method (IB-LBM) that is implemented on the adaptive quadtree/octree grids to model the complex solid-fluid interactions in different contexts such as flows laden with fixed or moving particles. Second, we perform a series of particle-resolved simulations of flow past random arrays of moderately to strongly bidisperse and polydisperse spheres which seemed computationally impossible on uniform grids in the past. We explore the statistical distributions of hydrodynamic forces and torques exerted on individual spheres over a wide range of simulation parameters. Finally, we extend two data-driven methods to predict such force and torque distributions, namely a Microstructure-informed Probability-driven Point-particle (MPP) model and a Physics-Informed Neural Network (PINN). We demonstrate their applicability from different perspectives including prediction and generalization performance, model complexity (i.e., computational efficiency), and interpretability in the form of binary and trinary interactions. We highlight the potential of our PINN model, which appropriately balances accuracy and efficiency, to substitute the conventional average drag closures for solid-fluid coupling in Eulerian-Lagrangian simulations.

View record

In this thesis, we explore various aspects related to wellbore leakage and plug and abandonment (P&A) of gas wells in British Columbia (BC) and Alberta. The objective of this thesis is two fold, first we review current practices and trends related to P&A and secondly, we investigate leakage pathways associated with reported wellbore leakage in BC.Related to the first objective, we review available data on well architectures, industrial practices both with regards to well construction and P&A practices and highlight trends over the past few decades in British Columbia and Alberta. The data reveals a large wave of abandonments coming in the next decade and a significant increase in reported instances of surface casing vent flow, SCVF, over the past decade, which will lead to rising P&A costs. Additionally, we try to understand variability in reported instances of SCVF between operators and between provinces.Wellbore leakage, such as SCVF, is a complex issue facing many wells as shown in the first part of the thesis. Related to the second thesis objective, we present a novel approach to modelling realistic leakage along microannulus pathways of varying thickness. We use stochastic methods to calibrate leakage pathway dimensions to the SCVF leakage rates reported in BC. Results shows that representing dry microannulus thicknesses with a lognormal distribution provides a good fit for the intermediate ranges of SVCF flow rates, but that a dry microannulus alone cannot account for all instances of wellbore leakage. Additionally, we use this model of wellbore leakage to investigate the effect of a number of operational factors on well- bore leakage. These effects on leakage have not been previously investigated and our results show that each operational effect can be significant, and why. We repeatedly highlighted that reducing wellbore leakage is often localised, requiring an impermeable barrier be placed across the cap rock but that there is an element of luck regarding success of a remediation operation. This suggests that stochastic-based models that capture the variability of the well integrity can be used to support the development of risk-based plug and abandonment P&A practices.

View record

Large numbers of oil and gas wells, in Canada and worldwide, allow leakage tosurface from the reservoir. One common reason is associated with unsuccessfulmud removal during the primary cementing operation. Horizontal wellborespresent particular cementing challenges that have not been fully solved. This thesisexplores (i) the fundamental mechanisms of fluid displacements in a horizontalnarrow eccentric annulus from an experimental and computational perspective; and(ii) cementing horizontal irregular wellbores.We present results of about 300 miscible Newtonian displacement flow experimentscarried out in a dimensionally scaled laboratory setup. Annulus eccentricity,density difference and viscosity of the fluids is varied, over a wide range of laminarReynolds numbers. Comparisons with predictions from the two-dimensional gap averagedmodel of Carrasco-Teja et al. show excellent agreement in predicting theunderlying competition between buoyancy and eccentricity, which results in eithertop side or slumping flows. The main discrepancy results from a variety of dispersiveeffects that are not present in the model, e.g. dispersion within the annulargap and due to azimuthal secondary flows. We show that these features are betterpredicted on computing the flow with a three-dimensional Volume of Fluid model.Regarding cementing irregular wellbores, we present two independent studies.The first one explores the effects of local irregularities (i.e., washouts) on mudremoval in strongly inclined wellbores, experimentally and via two-dimensionalsimulations. The competition between slumping and eccentricity is affected by thewashout. In most of the experiments the mud was removed from the washout, butthe simulations showed this is not true for muds with higher yield stress than thosetested experimentally. The second study deals with the effect of borehole ovalization and large-scale irregularities (e.g., breakouts). We use a two-dimensionalgap-averaged model that can compute the displacement within eccentric and ellipticgeometries. The irregularities can induce additional azimuthal flow whencompared to uniformly circular wellbores. In cases when the displaced mud has asignificant yield stress, the displacement can leave patches of residual mud at thetop of eccentric horizontal wellbores: not usually found without ellipticity.

View record

In this thesis, we present an experimental investigation of turbulent flows of elasto-viscoplastic fluids. The motivation comes from the oil and gas industry, where turbulent flows of non-Newtonian fluids are frequently encountered. We characterize the turbulent flow of viscoplastic fluids, both under a fully turbulent flow and also when it is displaced by another fluid under turbulence in an eccentric annular geometry. Further, we investigate the fully turbulent flow of a drag reducing, elasto-viscoplastic wormlike micellar (or surfactant) solution, and compare those results to better known polymer solutions under turbulent flows.Our experiments in fully turbulent flows of viscoplastic Carbopol solutions show an enhancement of streamwise velocity fluctuations and a decrease in wall normal velocity fluctuations in comparison to water. As we increase the Reynolds numbers, the turbulence statistics approach Newtonian values. With regards to turbulent displacements of 0.15% Carbopol solutions in an eccentric annulus, we observe that the displacement is successful without the obstruction regardless of the displacing fluid. The obstruction at eccentricity e ≈ 0.5 is mostly detrimental to removal of the yield stress fluid stuck downstream of it. At high eccentricity values of e ≈ 0.7, the effect of the obstruction on the displacement of Carbopol is seen to be negligible.When wormlike micellar gels are submitted to a turbulent flow, the micellar structure near the wall appears to be mostly broken down during turbulent flow. Turbulent flows at low concentrations of surfactant show a Newtonian-like flow field throughout most of the duct, where the energy spectra shows a -5/3 power law scale with wavenumber. Conversely, energy spectra of the micellar solutions at large percentages of drag reduction show approximately a -3 power law decay. Moreover, a direct comparison of turbulent flows with flexible and rigid polymer solutions also shows similar turbulence statistics at approximately the same percentages of drag reduction. A -3 power law decay in the energy spectra is also observed with both flexible and rigid polymer solutions, and we hypothesize it may be a consequence of elasto-inertial turbulence.

View record

A three-layer model for solid-liquid flow in inclined pipes is developed. The steady-state model predicts the frictional pressure loss, critical velocity, concentration profile in the heterogeneous layer, mean heterogeneous layer and moving bed layer velocities, and bed layer heights for each set of parameters. We propose a modified correlation for the turbulent solids diffusivity, and include appropriate closures for forces and stresses attributed to the solids and liquid phases in the different layers. The proposed turbulent solids diffusivity correlation and the steady-state model predictions show a good agreement with experimentally measured results in the literature: for concentration profiles in the heterogeneous layer, pressure losses and critical (deposition) velocity, both over a wide range of parameters and for different regimes. We also define a critical Peclet number based on which, a transition boundary between bed-load and heterogeneous regimes can be found. Furthermore, we extend the three-layer model to annular geometry and utilize it for developing another model for gravel packing applications in oil & gas industry. In this operation, kilometers of sand can be successfully placed in horizontal wells, in what is called alpha-beta packing. We explain how bed height is selected via coupling between the inner and outer annuli and from the combined hydraulic relations of inner and outer annuli. We investigate the effects of important parameters such as the slurry flow rate, mean solids concentration, wash pipe diameter, leak-off rate, etc, on gravel packing flows, to give a fluid mechanics framework within which this process can be easily understood and analyzed. For improving the accuracy of the slurry flow predictions in different operating flow regimes we also develop a robust integrated method consisting artificial neural network (ANN) and support vector regression (SVR) to estimate the critical velocity, slurry flow regime change, and ultimately, the frictional pressure drop for a solid-liquid slurry flow in a horizontal pipe, covering wide ranges of flow and geometrical parameters. The prediction results of the developed integrated method show that it significantly outperforms those of the widely used existing correlations and models in the literature.

View record

The thesis presents a novel triple-layer core-annular flow method in which we purposefully position an unyielded skin of a visco-plastic fluid between the core and the lubricating fluid to eliminate the possibility of interfacial instabilities. Specifically, the skin layer is shaped which allows for lubrication force to develop as the core rises under the action of buoyancy forces. The motivation originally stems from lubricated transport of heavy viscous oils. The objective is to reduce the frictional pressure gradient while avoiding interfacial instabilities. For this aim, first, we study this methodology for a steady periodic length of established flow, to establish the feasibility for the pipelining application. Second, we address the equally important issue of how in practice to develop a triple-layer flow with a sculpted visco-plastic skin, all within a concentric manifold by control of the flow rates of the individual fluids. The axisymmetric simulation establishes that these flows may be stably established in a controlled way. We develop a long-wavelength analysis of the extensional flow to predict the minimal yield stress required to maintain the skin rigid. Third, we extend the feasibility of the method to large pipes and higher flow rates by considering the effects of inertia and turbulence in the lubricating layer. We show that the method can generate enough lubrication force for wide range of parameters if the proper wave shape is imposed on the unyielded skin. Then, three-dimensional computations are performed to capture the buoyant motion of the core to reach its equilibrium position. The study shows that development lengths (times) for the core to attain equilibrium are relatively long, meaning extensive computation. We also present a simplified analytical model using the lubrication approximation and equations of motion for the lubricant and skin layers, to quickly estimate motion to the balanced configuration for a given shape and initial conditions. Finally, we show an explicit advantage of the proposed method in producing stable core-annular flows in regimes where conventional core-annular flows are unsuitable. In summary, we establish the potential of this new method for the stable and efficient transport of highly viscous fluids along pipelines.

View record

This thesis presents a comprehensive, yet practical, two-dimensional model for the displacement of viscoplastic fluids in eccentric annuli in laminar, turbulent and mixed flow regimes. The motivations originally stem from primary cementing of oil and gas wells, as well as other types of wells such as those in Carbon Capture and Storage applications. During primary cementing, cement slurries are placed in an annular region between a steel casing and a wellbore to provide mechanical stability and hydraulic isolation. Several complications may arise due to the eccentricity of the annular region, as well as the viscoplastic nature of the fluids involved. The existing 2D and 3D models of primary cementing assume the flow is laminar, while in practice, turbulent and more importantly, mixed flow regimes are common. In this thesis, we fill this gap in knowledge. More specifically, we expand the laminar model of Bittleston et al. (2002) and develop a new formulation that includes turbulent and mixed flow regimes. This new formulation considers scaling based on the disparity of length-scales, which allows a narrow-gap averaging approach to be effective. With respect to the momentum equations, the leading-order equations correspond to a turbulent shear flow in the direction of the modified pressure gradient. With respect to the mass transport equations that model the miscible displacement, to leading-order turbulence effectively mixes the fluids. Changes in concentrations within the annular gap arise due to the combined effects of advection with the mean flow, anisotropic Taylor dispersion(along the streamlines) and turbulent diffusivity. This new extension allows us to understand the process of cementing more deeply, and resolve several questions that have been left unanswered for many years. In particular, we show that many simple statements/rules that are often employed in industry do not stand up to serious analysis. Instead, modelling approaches such as the one developed here can incorporate specific features of wells in the simulations, and therefore, yield more accurate predictions.

View record

We study downward displacement flow of buoyant miscible fluids with viscosityratio in a pipe, using experimental, numerical and mathematical approaches. Investigationof this problem is mainly motivated by the primary cementing processin oil and gas well construction. Our focus is on displacements where the degree oftransverse mixing is low-moderate and thus a two-layer, stratified flow is observed.An inertial two-layer model for stratified density-unstable displacement flowsis developed. From experiments it has been observed that these flows develop for asignificant range of parameters. Due to significant inertial effects, existing modelsare not effective for predicting these flows. The novelty of this model is that theinertia terms are retained, and the wall and interfacial stresses are modelled. Withnumerical solution of the model, back-flow, displacement efficiency and instabilityonset predictions are made for different viscosity ratios.The experiments are conducted in a long pipe, inclined at an angle whichis varied from vertical to near-horizontal. Viscosity ratio is achieved by addingxanthan gum to the fluids. At each angle, flow rate and viscosity ratio are variedat fixed density contrast. Density-unstable flows regimes are mapped in the(Fr, Re cosß/Fr)-plane, delineated in terms of interfacial instability, front dynamicsand front velocity. Amongst the many observations we find that viscosifyingthe less dense fluid tends to significantly destabilize the flow, for density-unstableconfiguration. Different instabilities develop at the interface and in the wall-layers.The results are compared to the inertial two-layer model. In density-stable experimentswe mostly focus on the effects of viscosity ratio on displacement efficiencyand stability of wall-layer. Unique instabilities appear in the case of shear-thinningdisplacements. Displacement efficiency decreases with increasing viscosity ratio, flow rate and inclination angle.Finally, a number of three-dimensional parallel numerical simulations are completedin the pipe geometry, covering both density-stable and unstable flows. UnsteadyNavier-Stokes equations are solved and the Volume of Fluid (VOF) methodis used to capture the interface between the fluids. The results give us great insightinto several features of these flows that were not available from experiments or 2Dsimulations.

View record

This thesis studies the buoyant miscible displacement flow of a Bingham fluid by a Newtonian fluid and the invasion of miscible and immiscible fluids into a yield stress fluid. The objective of the former study is to characterize the residual layer thickness and identify the flow regimes within the range of governing flow parameters. In the latter, the aim is to capture the invasion pressure of the invading fluids into a yield stress fluid, understand the actual invasion process and quantify the effect of yield stress and other influencing physical parameters.We start the first part of the thesis with density stable displacements. We show the different parametric effects on the residual layer thickness and present a novel and computationally efficient method for predicting the long-term behaviour of the residual wall layers. We then extend this study to density unstable displacement and show that static residual wall layers can exist for yield stresses below the minimum for density stable regimes. These layers are partially static and may also be thicker than the fully static layers encountered in density stable flows. We also find a range of hydrodynamic instabilities, which we map out parametrically, giving approximate onset criteria. The predictive method for density stable flows is extended to density unstable configurations and appears able to predict the occurrence of stable displacements.In the second part, we study invasion flows into a vertical column of yield stress fluid through a small hole. We first examined the invasion of water, using both experimental and computational methods. We find that the invasion pressure depends on yield stress of the fluid and height of the yield stress column. However, the invasion process is initially localised close to the hole. Similar results were found with glycerin solutions. Interfacial stress effects were then tested with a density-matched silicon oil and air, which resulted a non-local invasion. In summary, we find that miscible fluids penetrate locally at significantly lower invasion pressures than immiscible fluids.Finally, for both parts of the thesis, there are a number of useful consequences helping to understand the mechanisms leading to gas migration.

View record

A theoretical and numerical study of yield-stress fluid creeping flow about a particle is presented motivated by theoretical aspects and industrial applications. Yield stress fluids can hold rigid particles statically buoyant if the yield stress is large enough. In addressing sedimentation of rigid particles in viscoplastic fluids, we should know this critical `yield number' beyond which there is no motion. As we get close to this limit, the role of viscosity becomes negligible in comparison to the plastic contribution in the leading order, since we are approaching the zero-shear-rate limit. Admissible stress fields in this limit can be found by using the characteristics of the governing equations of perfect plasticity (i.e., the sliplines). This approach yields a lower bound of the critical plastic drag force or equivalently the critical yield number. Admissible velocity fields also can be postulated to calculate the upper bound. This analysis methodology is examined for different families of particle shapes. Numerical experiments of either resistance or mobility problems in a viscoplastic fluid validate the predictions of slipline theory and reveal interesting aspects of the flow in the yield limit. For instance, the critical limit is not unique and here we show that for the same critical limit we may have different shaped particles that are cloaked inside the same unyielded envelope. The critical limit (or critical plastic drag coefficient) is related to the unyielded envelope rather than the particle shape. We show how to calculate the unyielded envelope directly. Here we also address the case of having multiple particles, which introduces interesting new phenomena. Firstly, plug regions can appear between the particles and connect them together, depending on the proximity and yield number. This can change the yielding behaviour since the combination forms a larger (and heavier) "particle". Moreover, small particles (that cannot move alone) can be pulled/pushed by larger particles or assembly of particles. Increasing the number of particles leads to interesting chain dynamics, including breaking and reforming.

View record

We 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 investigate as much as possible. Theoretical interest arises from the self-selection phenomenon, meaning that the original flow geometry is modified by the fluid itself. This occurs due to yield stress and is accomplished via stagnant zones of the fluid attached to the boundary of original geometry. Industrial motivations stem from oil/gas well construction operations: primary and squeeze cementing and hydraulic fracturing. In all we have drilling mud, cement or a gelled fluid which exhibit yield stress. Specifically, we model a washout along the well as a non-uniform channel and extensively study flows through it. This is an enlarged segment of the well where the wellbore is washed out or collapsed. The main industrial concern is the residual mud left in the washout after primary cementing which weakens the hydraulic sealing function of the cement. Self-selection has been analytically studied for duct flows, and not much in 2D flows. Chapter 2 is a study of self-selection in wavy walled channels as a model for smooth non-uniform channels. We find similar results to duct flows, however a complete understanding eludes us. Chapter 3 looks at the flow of Bingham fluid in fractures. We study the limits of validity of Darcy approach first and then focus at the minimal pressure drop required to mobilize the fluid in fracture. We demonstrate knowing self-selection properties can greatly improve approximations here. Chapters 4-6 are step by step investigation of the flows in washout, from Stokes to inertial and finally displacement flow. In Chapter 4 we show self-selection in Stokes flow of washout and use it to estimate the residual fluid in the washout. We study the effects of inertia on it in Chapter 5, illustrating only finite amount of inertia would help in better displacement of the mud which is counter intuitive. Chapter 6 is a preliminary study of the displacement flow and we report some interesting observations.

View record

We study natural convection of viscoplastic fluids in 2D domains. A sufficiently large yield stress introduces a static solution to the Navier–Stokes equations that may not otherwise exist. We find conditions that guarantee such motionless regimes and investigate flow development between static and advective states. Considering three problems, we explore the various ways in which the yield stress modifies the hydrodynamics of steady and transient natural convection. We start by analyzing natural convection in an infinitely long rectangular cavity. Flow is driven by a constant horizontal temperature difference and a stabilizing stratification imposed on the walls. We classify different 1D flow regimes and establish that an arbitrary number of unyielded regions can exist in the domain. Secondly, considering a square cavity, we investigate conditional and unconditional stability of the stationary state. We study the transition of the fluid between conductive and advective states, revealing the possibility of temporary arrest of the flow at yield stresses less than the critical value. Finally, we study natural convection of viscoplastic fluids due to a heater of finite width positioned on the bottom wall of a cavity. We show that if the yield stress is less than the critical value, the flow starts after a finite time. We characterize transient flow and explain the processes that result in the observation of pulsing plumes at high Rayleigh. Overall, we investigate the force balance that governs the existence of steady motion, or lack thereof. When the steady regime is advective, we illustrate that depending on the boundary and initial conditions flow may start immediately or flow onset may be delayed by a finite time. We focus on problems where flow onset is due to dominance of buoyancy stresses and is not a consequence of hydrodynamic instability. In §4 we clarify the difference. Further, we explore transient flow dynamics and establish that the yield stress can intensify oscillatory transient features. This results in the dominance of different transport methods and corresponding timescales at different stages of flow development. We show that under appropriate conditions, this may lead to temporary flow arrest and create other noteworthy dynamics.

View record

This thesis studies buoyant displacement flows with two miscible fluids in pipes and 2D channels that are inclined at an angle β measured from vertical changing from 0° to 90°. The focus is on inclination angles away from nearly horizontal since these flows are previously studied in full details in the literature. Detailed experimental, analytical and computational approaches are employed in an integrated fashion.Both density stable (light fluid displacing heavy one) and density unstable (heavy fluid displacing light fluid) displacements are studied. For density stable flows the study is purely experimental in the limit of iso-viscous Newtonian fluids. The density stable configuration has been found to produce highly efficient displacements, with the bulk of the interface moving steadily at the mean velocity. The streamwise length of the stretched interface increases with the mean flow velocity, viscosity and inclination β from vertical, and decreases with density difference.The rest of the thesis deals with density unstable configuration. From experimental point of view, the pipe displacement flows are studied for iso-viscous Newtonian and also viscoplastic fluids. In the Newtonian limit, completely different regimes than nearly-horizontal case are observed. As a first order approximation, different regimes are classified in a two-dimensional (Fr; Re cosβ/Fr plane) providing leading order correlations to transitions to different regimes. Similar regimes are found for channel geometry through numerical simulations of PELICANS code. For non-Newtonian fluids we have focused on industrially interesting cases of large yield stress fluids in the pipe. The two distinct flow regimes namely central-type and slump-type first observed in nearly horizontal angles were found to also persist over other inclinations. Completely new and exotic behaviors were also observed due to the effect of inclination angle and instabilities.From mathematical and modeling point of view a two-layer weighted residual model for generalized Newtonian fluids has been developed. The model works for channel geometry and can be used to predict the displacement interface height, the front velocity and more importantly, the flow stability.

View record

This thesis studies buoyant displacement flows with two miscible fluids in pipes and 2D channels that are inclined at angles (β) close to horizontal. Detailed experimental, analytical and computational approaches are employed in an integrated fashion. The displacements are at low Atwood numbers and high Péclet numbers, so that miscibility effects are mostly observable after instability and via dispersive mixing.For iso-viscous Newtonian displacements, studying the front velocity variation as a function of the imposed flow velocity allows us to identify 3 distinct flow regimes: an exchange flow dominated regime characterized by Kelvin-Helmholtz-like instabilities, a laminarised viscous displacement regime with the front velocity linearly increasing with the mean imposed flow rate, and a fully mixed displacement regime. The transition between the first and the second regimes is found to be marked by a stationary layer of displaced fluid. In the stationary layer the displaced fluid moves in counter-current motion with zero net volumetric flux. Different lubrication/thin-film models have been used to predict the flow behaviour. We also succeed in characterising displacements as viscous or inertial, according to the absence/presence of interfacial instability and mixing. This dual characterisation allows us to define 5-6 distinct flow regimes, which we show collapse onto regions in the two-dimensional (Fr, Re cosβ/Fr)-plane. Here Fr is the densimetric Froude number and Re the Reynolds number. In each regime we have been able to offer a leading order approximation to the leading front velocity. A weighted residual method has also been used to include the effect of inertia within the lubrication modelling approach, which allows us to predict long-wave instabilities.We have extended the study to include the effects of moderate viscosity ratio and shear-thinning fluids. We see many qualitative similarities with the iso-viscous studies. Predictive models are proposed (and compared with experiments and simulations) for the viscous and inertial regimes.Having a significant yield stress in the displaced fluid leads to completely new phenomena. We identify two distinct flow regimes: a central-type displacement regime and a slump-type regime for higher density differences. In both regimes, the displaced fluid can remain completely static in residual wall layers.

View record

Interfacial instabilities of multi-layer shear flows may be eliminated by astute positioning of yield stress fluid layers that remain unyielded at the interface(s). The contribution of this thesis comes in three parts. Firstly, we have performed a computational study of these flows in the setting of a Newtonian core fluid surrounded by a Bingham lubricating fluid, within pipe and channel configurations. The simulations include an inlet geometry in the computational model and study the multi-layer flows, both as the fluids are initially injected (start up) and later the established steady flows (development lengths). Nonlinear perturbations are also studied, showing in particular that during energy decay of stable perturbations the initial rapid decay of the perturbation kinetic energy relates to reforming/breaking of the unyielded plug and is followed by slower viscous decay. For axisymmetric perturbations these flows can be stable to order unity initial perturbation amplitudes and for Re[sign omitted]10². The channel geometry allows for symmetry breaking and appears to be less stable. A number of interesting effects are explored using the channel geometry. Secondly, we focus on demonstrating whether the stable core annular flow can be achieved when lubricating a visco-elastic core fluid with a yield stress fluid. We have performed over 100 experiments using Carbopol solutions as the lubricating yield stress fluid and Polyethylene Oxide solutions as the visco-elastic fluid. Thirdly, we have applied the energy stability method to study nonlinear stability of a core-annular flow of an Oldroyd-B fluid surrounded by a Bingham fluid. Together with the experimental study, this shows that visco-elasticity is not a barrier to use of this methodology.

View record

In this thesis we study laminar displacement flows of one fluid by another in a horizontal annulus. The study comes from the primary cementing of highly deviated oil and gas wells. Highly deviated wells are those in which part of the well bore is nearly horizontal. Primary cementing is a critical process in the construction of awell. The objective is to provide zonal isolation, i.e., a hydraulic seal between the well and the surrounding rock. This is essential to protect the environment and increase the productivity of the well. Therefore, an understanding of the process is indispensable. We model primary cementing displacement flows using a Hele-Shaw approach, and provide simple scientific tools to improve the design of cementingjobs. The contribution of the thesis comes in three parts. Firstly, we analyse the displacement of one viscoplastic fluid by another in a near-horizontal eccentric annulus with a fixed inner pipe. We present examples that illustrate the differences between vertical and horizontal displacements. We then derive a 1D lubrication model whichgives analytical conditions that predict when the flow will stratify, according to the fluid properties and the annulus geometry. Secondly, we derive a 2D displacement model for Newtonian fluids which includes rotation and reciprocation of the inner cylinder. This is a common practice in the industry and not well understood. Using an asymptotic approach, we find steady-state traveling wave solutions for nearly-flat interfaces. Then we use numerical simulations to understand the flow dynamics for more elongated interfaces. In particular, we show that casing rotation can lead to local instabilities and mixing, which can shorten the length of the interface. Finally, we generalise this moving casing model to viscoplastic fluids. Using a lubrication-type model we explore the effects of casing motion, again deriving conditions for there to be steady solutions.

View record

The main focus of this work is to investigate experimentally the transition to turbulence of a yield stress shear thinning fluid in Hagen-Poiseuille flow. By combining direct high speed imaging of the flow structures with Laser Doppler Velocimetry (LDV), we provide a systematic description of the different flow regimes from laminar to fully turbulent. Each flow regime is characterized by measurements of the radial velocity, velocity fluctuations, and turbulence intensity profiles. In addition we estimate the autocorrelation, the probability distribution, and the structure functions in an attempt to further characterize transition. For all cases tested, our results indicate that transition occurs only when the Reynolds stresses of the flow equals or exceeds the yield stress of the fluid, i.e. the plug is broken before transition commences. Once in transition and when turbulent, the behavior of the yield stress fluid is somewhat similar to a (simpler) shear thinning fluid. We have also observed the shape of slugs during transition and find that their leading edges to be highly elongated and located off the central axis of the pipe, for the non-Newtonian fluids examined. Finally we present a new phenomenological approach for quantifying laminar-turbulent transition in pipe flow. This criterion is based on averaging a local Reynolds number to give ReG. Our localised parameter shows strong radial variations that are maximal at approximately the radial positions where puffs first appear during the first stages of turbulent transition.

View record