Ian Frigaard

Professor

Relevant Degree Programs

 

Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - Mar 2019)
Displacement flow of miscible fluids with density and viscosity contrast (2018)

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.

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Fluid mechanics causes of gas migration : displacement of a yield stress fluid in a channel and onset of fluid invasion into a visco-plastic fluid (2018)

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.

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Particles in a yield-stress fluid : yield limit, sedimentation and hydrodynamic interaction (2018)

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.

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Yield stress fluid flows in uneven geometries : applications to the oil & gas industry (2016)

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.

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Buoyancy-driven flow of viscoplastic fluids (2015)

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.

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Displacement flow of complex fluids in an inclined duct (2013)

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.

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From displacement to mixing in a slightly inclined duct (2012)

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.

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Multi-layer flows with yield stress fluids (2011)

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.

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Primary cementing of a highly deviated oil well (2010)

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.

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Observation of laminar-turbulent transition of a yield stress fluid in Hagen-Poiseuille flow (2009)

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.

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Master's Student Supervision (2010-2017)
Displacing visco plastic fluid with Newtonian fluid in a vertical circular pipe with buoyancy effects (2017)

In this thesis, displacement flows in a vertical pipe are studied when Newtonian fluids displace visco-plastic fluids. The density combinations between displacing and displaced fluids are varied from density unstable through iso-density to density stable, and captured dimensionlessly using Atwood numbers. In density unstable cases, three flow regimes are classified: central, mixed/turbulent and asymmetric regimes. These regimes are partially classified by a buoyancy parameter. However, we found that the buoyancy parameter has a limit in classifying the flow regimes. Once the flow enters the turbulent regime, spread of the dispersive mixed region is characterized by fitting the mean concentration changes to the solution of an 1D linear advection diffusion equation, i.e., turbulent diffusivity (or dispersivity) dominates in this regime. In iso-density cases, all flows are classified in central regime but the shapes of static layers are classified as: smooth, wavy and corrugated. We found that Re, Newtonian Reynolds number, differentiates the static layer shapes. Transitional Reynolds numbers are identified as Re = 345 for corrugated to wavy and Re = 1000 for wavy to smooth. The transitional Re for turbulent regime is identified at around 4000. Lastly, we observed that viscous fingering is common in density stable cases. Viscous fingering is observed for large effective viscosity, ratio of a viscoplastic fluid to a Newtonian fluid, and a ratio of shear stress to a yield stress of a displaced fluid ratio is small, and starts from an elongated thin layer finger. In the regime, the wall shear stress is too small to yield the visco-plastic fluid from the wall and the mobility of the displacing fluid is relatively high, so it seeks a way to channel though the visco-plastic fluid. The transitional Re for mixed/turbulent regime was not found within our experimental range. The displacement efficiency, described in the ratio of a front velocity to a mean velocity in density stable cases increases by approximately 15%, compared to density unstable and iso-density. Density unstable experiments can have better efficiency than iso-density experiments due to entering mixing regime in lower Reynolds numbers. However, the differences in the efficiency are generally small.

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Macro-Size Drop Encapsulation (2014)

No abstract available.

Interfacial effects in visco-plastic lubrication flows (2013)

Poiseuille flows with yield stress fluids produce an unyielded central plug which can act as a solid conduit surrounding central (core)flows of Newtonian or power law fluids. Effectively, the annular yield stress fluid acts as a lubricant that isolates the core flow from wall friction. Stable flows with a yield stress annular fluid and a Newtonian or power law core fluid are termed viso-plastic lubrication (VPL) flows. This study examined interfacial effects in vertical VPL Poiseuille flows using a carbopol solution as the annular (yield stress) fluid and xanthan (inelastic shear thinning fluid) or polyethlyeneoxide (PEO; an elastic shear thinning fluid) as the core fluids. Experiments with the inelastic core fluid (xanthan) involved introducing stepped (high to low) or pulsed (high to low to high) changes in the core flow to an established stable VPL flow. Step changes produced a "yield front"(narrowing of the core flow or "interfacial radius") that propagated upward at a velocity considerably greater than the velocity of the annular carbopol plug but close to the average velocity of the xanthan core flow following the step change. Pulsed changes in the core flow produced one of three outcomes depending on the magnitude of the flows preceding and following the step change: (1) a stable ("frozen in") deformation in the carbopol/xanthan interface that moved upward at the velocity of the carbopol plug,(2) no persistent deformation of the interface, or (3) a breakdown of the stable VPL flow characterized by extensive mixing of the core and annular flows. Experiments with the elastic core fluid (PEO) involved introducing multiple pulsed changes (high/low/high, high/low/high, ...) in the core flow to an established VPL flow. These pulsed changes typically produced linked multiple diamond shaped stable deformations ("diamond necklace") in the interface that moved upwards at the velocity of the carbopol plug. The frequency and amplitude (maximum radius) of the diamond deformations could be controlled by the timing of pulses and the respective flow rates, but not the diamond shape itself which appears to be a consequence of the complex rheology of the fluids.

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Transport and dispersion of particles in visco-plastic fluids (2013)

This thesis focuses on development of a model to predict “spreading” of thesolids (i.e. proppant) fraction during the fracturing operation. We developa 1D model that allows us to estimate dispersion of solid particles along avertical pipe in a fully turbulent flow of a shear thinning yield stress fluid (i.e.,visco-plastic fluid), as well as slip relative to the mean flow. In dimensionlessform, this results in a quasilinear advection-diffusion equation. Advection bythe mean flow, particle settling relative to the mean, in the direction of gravity,turbulent particle dispersivity and Taylor dispersion are the 4 main transportphenomena modelled in the 1D model. We provide a simple analysis of the1D model, suitable for spreadsheet-type field design purposes, in which weestimate “mixing lengths” due to both settling and dispersion. Secondly, weprovide an accurate numerical algorithm for solution of the 1D model andshow how pulses of proppant (i.e. slugs) may or may not interact for typicalprocess parameters.

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