5 results
Three-dimensional buoyant hydraulic fractures: finite-volume release
- Andreas Möri, Brice Lecampion
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- Journal:
- Journal of Fluid Mechanics / Volume 972 / 10 October 2023
- Published online by Cambridge University Press:
- 02 October 2023, A20
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In impermeable media, a hydraulic fracture can continue expanding even without additional fluid injection if its volume exceeds the limiting volume of a hydrostatically loaded radial fracture. This limit depends on the mechanical properties of the surrounding solid and the density contrast between the fluid and the solid. We show that two dimensionless numbers characterize self-sustained fracture growth. The first is a buoyancy factor that compares the total released volume to the volume of a hydrostatically loaded radial fracture to determine whether buoyant growth occurs. The second number is the dimensionless viscosity of a radial fracture when buoyant effects become of order one. Notably, this dimensionless viscosity depends on the rate at which the fluid volume is released, indicating that both the total volume and release history impact self-sustained buoyant growth. We identify six well-defined propagation histories based on these two dimensionless numbers. Their growth evolves between distinct limiting regimes of radial and buoyant propagation, resulting in different fracture shapes. Notably, our findings reveal two growth rates depending on the dominant energy dissipation mechanism (viscous flow versus fracture creation) in the fracture head. For finite values of material toughness, the toughness-dominated limit represents a late-time solution for all fractures in growth rate and head shape (possibly reached only at a very late time). The viscosity-dominated limit can appear at intermediate times. Our three-dimensional simulations confirm the predicted scalings. This contribution highlights the importance of the entire propagation and release history for accurate analysis of buoyant hydraulic fractures.
Three-dimensional buoyant hydraulic fractures: constant release from a point source
- Andreas Möri, Brice Lecampion
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- Journal:
- Journal of Fluid Mechanics / Volume 950 / 10 November 2022
- Published online by Cambridge University Press:
- 18 October 2022, A12
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Hydraulic fractures propagating at depth are subjected to buoyant forces caused by the density contrast between fluid and solid. This paper is concerned with the analysis of the transition from an initially radial fracture towards an elongated buoyant growth – a critical topic for understanding the extent of vertical hydraulic fractures in the upper Earth crust. Using fully coupled numerical simulations and scaling arguments, we show that a single dimensionless number governs buoyant hydraulic fracture growth, namely the dimensionless viscosity of a radial hydraulic fracture at the time when buoyancy becomes of order 1. It quantifies whether the transition to buoyancy occurs when the growth of the radial hydraulic fracture is either still in the regime dominated by viscous flow dissipation or already in the regime where fracture energy dissipation dominates. A family of fracture shapes emerge at late time from finger-like (toughness regime) to inverted elongated cudgel-like (viscous regime). Three-dimensional toughness-dominated buoyant fractures exhibit a finger-like shape with a constant-volume toughness-dominated head and a viscous tail having a constant uniform horizontal breadth: there is no further horizontal growth past the onset of buoyancy. However, if the transition to buoyancy occurs while in the viscosity-dominated regime, both vertical and horizontal growths continue to match scaling arguments. As soon as the fracture toughness is not strictly zero, horizontal growth stops when the dimensionless horizontal toughness becomes of order 1. The horizontal breadth follows the predicted scaling.
Slickwater hydraulic fracture propagation: near-tip and radial geometry solutions
- Brice Lecampion, Haseeb Zia
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- Journal:
- Journal of Fluid Mechanics / Volume 880 / 10 December 2019
- Published online by Cambridge University Press:
- 10 October 2019, pp. 514-550
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We quantify the importance of turbulent flow on the propagation of hydraulic fractures (HF) accounting for the addition of friction reducing agents to the fracturing fluid (slickwater fluid). The addition in small quantities of a high molecular weight polymer to water is sufficient to drastically reduce friction of turbulent flow. The maximum drag reduction (MDR) asymptote is always reached during industrial-like injections. The energy required for pumping is thus drastically reduced, allowing for high volume high rate hydraulic fracturing operations at a reasonable cost. We investigate the propagation of a hydraulic fracture propagating in an elastic impermeable homogeneous solid under a constant (and possibly very high) injection rate accounting for laminar and turbulent flow conditions with or without the addition of friction reducers. We solve the near-tip HF problem and estimate the extent of the laminar boundary layer near the fracture tip as a function of a tip Reynolds number for slickwater. We obtain different propagation scalings and transition time scales. This allows us to easily quantify the growth of a radial HF from the early-time turbulent regime(s) to the late-time laminar regimes. Depending on the material and injection parameters, some propagation regimes may actually be bypassed. We derive both accurate and approximate solutions for the growth of radial HF in the different limiting flow regimes (turbulent smooth, rough, MDR) for the zero fracture toughness limit (corresponding to the early stage of propagation of a radial HF). We also investigate numerically the transition(s) between the early-time MDR regime to the late-time laminar regimes (viscosity and toughness) for slickwater fluid. Our results indicate that the effect of turbulent flow on high rate slickwater HF propagation is limited and matters only at early times (at most during the first minutes for industrial hydraulic fracturing operations).
A semi-infinite hydraulic fracture driven by a shear-thinning fluid
- Fatima-Ezzahra Moukhtari, Brice Lecampion
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- Journal:
- Journal of Fluid Mechanics / Volume 838 / 10 March 2018
- Published online by Cambridge University Press:
- 25 January 2018, pp. 573-605
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We use the Carreau rheological model which properly accounts for the shear-thinning behaviour between the low and high shear rate Newtonian limits to investigate the problem of a semi-infinite hydraulic fracture propagating at a constant velocity in an impermeable linearly elastic material. We show that the solution depends on four dimensionless parameters: a dimensionless toughness (function of the fracture velocity, confining stress, material and fluid parameters), a dimensionless transition shear stress (related to both fluid and material behaviour), the fluid shear-thinning index and the ratio between the high and low shear rate viscosities. We solve the complete problem numerically combining a Gauss–Chebyshev method for the discretization of the elasticity equation, the quasi-static fracture propagation condition and a finite difference scheme for the width-averaged lubrication flow. The solution exhibits a complex structure with up to four distinct asymptotic regions as one moves away from the fracture tip: a region governed by the classical linear elastic fracture mechanics behaviour near the tip, a high shear rate viscosity asymptotic and power-law asymptotic region in the intermediate field and a low shear rate viscosity asymptotic far away from the fracture tip. The occurrence and order of magnitude of the extent of these different viscous asymptotic regions are estimated analytically. Our results also quantify how shear thinning drastically reduces the size of the fluid lag compared to a Newtonian fluid. We also investigate simpler rheological models (power law, Ellis) and establish the small domain where they can properly reproduce the response obtained with the complete rheology.
Confined flow of suspensions modelled by a frictional rheology
- Brice Lecampion, Dmitry I. Garagash
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- Journal:
- Journal of Fluid Mechanics / Volume 759 / 25 November 2014
- Published online by Cambridge University Press:
- 22 October 2014, pp. 197-235
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We investigate in detail the problem of confined pressure-driven laminar flow of neutrally buoyant non-Brownian suspensions using a frictional rheology based on the recent proposal of Boyer et al. (Phys. Rev. Lett., vol. 107 (18), 2011, 188301). The friction coefficient (shear stress over particle normal stress) and solid volume fraction are taken as functions of the dimensionless viscous number
$I$ defined as the ratio between the fluid shear stress and the particle normal stress. We clarify the contributions of the contact and hydrodynamic interactions on the evolution of the friction coefficient between the dilute and dense regimes reducing the phenomenological constitutive description to three physical parameters. We also propose an extension of this constitutive framework from the flowing regime (bounded by the maximum flowing solid volume fraction) to the fully jammed state (the random close packing limit). We obtain an analytical solution of the fully developed flow in channel and pipe for the frictional suspension rheology. The result can be transposed to dry granular flow upon appropriate redefinition of the dimensionless number
$I$. The predictions are in excellent agreement with available experimental results for neutrally buoyant suspensions, when using the values of the constitutive parameters obtained independently from stress-controlled rheological measurements. In particular, the frictional rheology correctly predicts the transition from Poiseuille to plug flow and the associated particles migration with the increase of the entrance solid volume fraction. We also numerically solve for the axial development of the flow from the inlet of the channel/pipe toward the fully developed state. The available experimental data are in good agreement with our numerical predictions, when using an accepted phenomenological description of the relative phase slip obtained independently from batch-settlement experiments. The solution of the axial development of the flow notably provides a quantitative estimation of the entrance length effect in a pipe for suspensions when the continuum assumption is valid. Practically, the latter requires that the predicted width of the central (jammed) plug is wider than one particle diameter. A simple analytical expression for development length, inversely proportional to the gap-averaged diffusivity of a frictional suspension, is shown to encapsulate the numerical solution in the entire range of flow conditions from dilute to dense.
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