4 results
Dispersion of solids in fracturing flows of yield stress fluids
- S. Hormozi, I. A. Frigaard
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- Journal:
- Journal of Fluid Mechanics / Volume 830 / 10 November 2017
- Published online by Cambridge University Press:
- 29 September 2017, pp. 93-137
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Solids dispersion is an important part of hydraulic fracturing, both in helping to understand phenomena such as tip screen-out and spreading of the pad, and in new process variations such as cyclic pumping of proppant. Whereas many frac fluids have low viscosity, e.g. slickwater, others transport proppant through increased viscosity. In this context, one method for influencing both dispersion and solids-carrying capacity is to use a yield stress fluid as the frac fluid. We propose a model framework for this scenario and analyse one of the simplifications. A key effect of including a yield stress is to focus high shear rates near the fracture walls. In typical fracturing flows this results in a large variation in shear rates across the fracture. In using shear-thinning viscous frac fluids, flows may vary significantly on the particle scale, from Stokesian behaviour to inertial behaviour across the width of the fracture. Equally, according to the flow rates, Hele-Shaw style models give way at higher Reynolds number to those in which inertia must be considered. We develop a model framework able to include this range of flows, while still representing a significant simplification over fully three-dimensional computations. In relatively straight fractures and for fluids of moderate rheology, this simplifies into a one-dimensional model that predicts the solids concentration along a streamline within the fracture. We use this model to make estimates of the streamwise dispersion in various relevant scenarios. This model framework also predicts the transverse distributions of the solid volume fraction and velocity profiles as well as their evolutions along the flow part.
Viscoplastic boundary layers
- N. J. Balmforth, R. V. Craster, D. R. Hewitt, S. Hormozi, A. Maleki
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- Journal:
- Journal of Fluid Mechanics / Volume 813 / 25 February 2017
- Published online by Cambridge University Press:
- 26 January 2017, pp. 929-954
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In the limit of a large yield stress, or equivalently at the initiation of motion, viscoplastic flows can develop narrow boundary layers that provide either surfaces of failure between rigid plugs, the lubrication between a plugged flow and a wall or buffers for regions of predominantly plastic deformation. Oldroyd (Proc. Camb. Phil. Soc., vol. 43, 1947, pp. 383–395) presented the first theoretical discussion of these viscoplastic boundary layers, offering an asymptotic reduction of the governing equations and a discussion of some model flow problems. However, the complicated nonlinear form of Oldroyd’s boundary-layer equations has evidently precluded further discussion of them. In the current paper, we revisit Oldroyd’s viscoplastic boundary-layer analysis and his canonical examples of a jet-like intrusion and flow past a thin plate. We also consider flow down channels with either sudden expansions or wavy walls. In all these examples, we verify that viscoplastic boundary layers form as envisioned by Oldroyd. For each example, we extract the dependence of the boundary-layer thickness and flow profiles on the dimensionless yield-stress parameter (Bingham number). We find that, while Oldroyd’s boundary-layer theory applies to free viscoplastic shear layers, it does not apply when the boundary layer is adjacent to a wall, as has been observed previously for two-dimensional flow around circular obstructions. Instead, the boundary-layer thickness scales in a different fashion with the Bingham number, as suggested by classical solutions for plane-parallel flows, lubrication theory and, for flow around a plate, by Piau (J. Non-Newtonian Fluid Mech., vol. 102, 2002, pp. 193–218); we rationalize this second scaling and provide an alternative boundary-layer theory.
Macro-size drop encapsulation
- A. Maleki, S. Hormozi, A. Roustaei, I. A. Frigaard
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- Journal:
- Journal of Fluid Mechanics / Volume 769 / 25 April 2015
- Published online by Cambridge University Press:
- 25 March 2015, pp. 482-521
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Viscoplastic fluids do not flow unless they are sufficiently stressed. This property can be exploited in order to produce novel flow features. One example of such flows is viscoplastically lubricated (VPL) flow, in which a viscoplastic fluid is used to stabilize the interface in a multi-layer flow, far beyond what might be expected for a typical viscous–viscous interface. Here we extend this idea by considering the encapsulation of droplets within a viscoplastic fluid, for the purpose of transportation, e.g. in pipelines. The main advantage of this method, compared to others that involve capillary forces is that significantly larger droplets may be stably encapsulated, governed by the length scale of the flow and yield stress of the encapsulating fluid. We explore this set-up both analytically and computationally. We show that sufficiently small droplets are held in the unyielded plug of a Poiseuille flow (pipe or plane channel). As the length or radius of the droplets increases, the carrier fluid eventually yields, potentially breaking the encapsulation. We study this process of breaking and give estimates for the limiting size of droplets that can be encapsulated.
Entry, start up and stability effects in visco-plastically lubricated pipe flows
- S. HORMOZI, K. WIELAGE-BURCHARD, I. A. FRIGAARD
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- Journal:
- Journal of Fluid Mechanics / Volume 673 / 25 April 2011
- Published online by Cambridge University Press:
- 10 March 2011, pp. 432-467
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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). We study the initiation, development lengths and temporal stability of such flows in the setting of a Newtonian core fluid surrounded by a Bingham lubricated fluid, within a pipe. Flow initiation is effected by starting the flow with a pipe full of stationary Bingham fluid and injecting both inner and outer fluids simultaneously. Initial instability and dispersive mixing at the front remains localised and is advected from the pipe leaving behind a stable multi-layer configuration, found for moderate Reynolds numbers (Re), for a broad range of interface radii (ri) and for different inlet diameters (Ri), whenever the base flow parameters admit a multi-layer flow with unyielded interface. The established flows have three distinct entry lengths. These relate to: (i) establishment of the first unyielded plug close to the interface (shortest); (ii) establishment of the interface radius; (iii) establishment of the velocity profile (longest). The three entry lengths increase with Re and decrease with both the Bingham number (B) and the viscosity ratio (m). Nonlinear temporal stability to axisymmetric perturbations is studied numerically, considering initial perturbations that are either localised in yielded parts of the flow or that initially break the unyielded plug regions. The aim is to understand structural aspects of the flow stability, not easily extracted from the energy stability results of Moyers-Gonzalez, Frigaard & Nouar (J. Fluid Mech., vol. 506, 2004, p. 117). The initial stages of a stable perturbed flow are characterised by a very rapid decay of the perturbation kinetic energy during which time the unyielded plug reforms (or breaks and reforms). This is followed by slower exponential decay on a viscous timescale (t ~ Re). For smaller Re and moderate initial amplitudes A, the perturbations decay to the numerical tolerance. As either Re or A is increased sufficiently, a number of interesting phenomena arise. The amount of dispersion increases, making the interfacial region increasingly diffuse and limiting the final decay. At larger Re or A, we find secondary flow structures that persist. A first example of these is when the shear stress decays below the yield stress before the velocity perturbation has decayed, leading to freezing in of the interface shape. This can lead to flows with a rigid wavy interface. Secondly, depending on the core fluid radius and thickness of the surrounding plug region, we may observe a range of dispersive structures akin to the pearls and mushrooms of d'Olce et al. (Phys. Fluids, vol. 20, 2008, art. 024104).