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The influence of turbulence and inertia in radial fracture flow

Published online by Cambridge University Press:  14 February 2024

Bruce Gee
Affiliation:
Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Robert Gracie*
Affiliation:
Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
*
Email address for correspondence: rgracie@uwaterloo.ca

Abstract

Industrial applications of flow through fractures such as geothermal energy or hydraulic stimulation involve forcing large flow rates through small fractures, thereby inducing inertial fluid behaviours and turbulence. The most common fracture flow model, Poiseuille flow (the cubic law), is incapable of capturing these phenomena and thus the impact of inertial and turbulent forces in fracture flow has remained relatively unexplored. The GG22 flow model is a newly derived fracture flow model that is capable of capturing inertial, transient and turbulent forces. In this article, we apply the GG22 flow model to hydraulic stimulation of radial fractures for the first time to determine how these phenomena manifest. We show that inertia and turbulence only manifest near the wellbore (within 30 radii) and lead to changes in fracture shape and injection pressure but have little impact on tip behaviour. Turbulence increases wellbore pressure and aperture while inertia decreases wellbore pressure and aperture. The majority of the pressure loss along the fracture occurs near the wellbore and is captured by turbulence where entrance correction factors would otherwise be needed. Using water, turbulence is the dominant mechanism that causes departures from Poiseuille flow at high $Re$. The solution departs immediately upon the manifestation of turbulence ($Re\geq 2\times 10^3$), while inertial effects manifest at higher flow rates ($Re\geq 2\times 10^4$). Using slickwater, the opposite trend is observed: inertial effects manifest first ($Re\geq 5\times 10^3$), while turbulent effects are delayed ($Re\geq 10^4$). In both cases, the threshold for departures from the Poiseuille flow solution are low and the differences are large.

Type
JFM Papers
Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Bunger, A.P. & Lecampion, B. 2017 Four critical issues for successful hydraulic fracturing applications. In Rock Mechanics and Engineering, 1st edn, vol. 5, pp. 551–593. CRC Press.CrossRefGoogle Scholar
Colebrook, C.F. 1939 Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Civil Engrs 11 (4), 133156.CrossRefGoogle Scholar
Detournay, E. 2004 Propagation regimes of fluid-driven fractures in impermeable rocks. Intl J. Geomech. 4 (1), 3545.CrossRefGoogle Scholar
Detournay, E. 2020 Slickwater hydraulic fracturing of shales. J. Fluid Mech. 886, F1.CrossRefGoogle Scholar
Dontsov, E.V. & Peirce, A.P. 2017 Modeling planar hydraulic fractures driven by laminar-to-turbulent fluid flow. Intl J. Solids Struct. 128, 7384.CrossRefGoogle Scholar
Garagash, D.I. 2006 Transient solution for a plane-strain fracture driven by a shear-thinning, power-law fluid. Intl J. Numer. Anal. Meth. Geomech. 5 (8), 14391475.CrossRefGoogle Scholar
Garagash, D.I. 2019 Cohesive-zone effects in hydraulic fracture propagation. J. Mech. Phys. Dolids 133, 103727.CrossRefGoogle Scholar
Ge, S. 1997 A governing equation for fluid flow in rough fractures. Water Resour. Res. 33 (1), 5361.CrossRefGoogle Scholar
Gee, B. & Gracie, R. 2022 Beyond Poiseuille flow: a transient energy-conserving model for flow through fractures of varying aperture. Adv. Water Resour. 164, 104192.CrossRefGoogle Scholar
Gee, B. & Gracie, R. 2023 Inertial and turbulent flow in hydro-mechanically coupled KGD-like fractures. Intl J. Numer. Anal. Meth. Geomech. 47 (16), 2925–2950.CrossRefGoogle Scholar
Habibpour, M. & Clark, P.E. 2017 Drag reduction behavior of hydrolyzed polyacrylamide/xanthan gum mixed polymer solutions. Petrol. Sci. 14 (2), 412423.Google Scholar
Konzuk, J.S. & Kueper, B.H. 2004 Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture. Water Resour. Res. 40 (2).CrossRefGoogle Scholar
Lecampion, B., Desroches, J., Jeffrey, R.G. & Bunger, A.P. 2017 Experiments versus theory for the initiation and propagation of radial hydraulic fractures in low-permeability materials. J. Geophys. Res.: Solid Earth 122 (2), 12391263.CrossRefGoogle Scholar
Lecampion, B. & Zia, H. 2019 Slickwater hydraulic fracture propagation: near-tip and radial geometry solutions. J. Fluid Mech. 880, 514550.CrossRefGoogle Scholar
Liu, D. & Lecampion, B. 2021 Propagation of a plane-strain hydraulic fracture accounting for a rough cohesive zone. J. Mech. Phys. Solids 149, 104322.CrossRefGoogle Scholar
Nilson, R.H. 1981 Gas-driven fracture propagation. J. Appl. Mech. 48 (4), 757762.CrossRefGoogle Scholar
Olufsen, M.S., Peskin, C.S., Kim, W.Y., Pedersen, E.M., Nadim, A. & Larsen, J. 2000 Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Engng 28 (11), 12811299.CrossRefGoogle ScholarPubMed
Oron, A.P. & Berkowitz, B. 1998 Flow in rock fractures: the local cubic law assumption reexamined. Water Resour. Res. 34 (11), 28112825.CrossRefGoogle Scholar
Ranjith, P.G. & Viete, D.R. 2011 Applicability of the ‘cubic law’ for non-Darcian fracture flow. J. Pet. Sci. Engng 78 (2), 321327.CrossRefGoogle Scholar
Szeri, A.Z. 1998 Fluid Film Lubrication: Theory and Design. Cambridge University Press.CrossRefGoogle Scholar
Tsai, V.C. & Rice, J.R. 2010 A model for turbulent hydraulic fracture and application to crack propagation at glacier beds. J. Geophys. Res.: Earth Surf. 115 (F3).Google Scholar
Tvergaard, V. 1990 Effect of fibre debonding in a whisker-reinforced metal. Mater. Sci. Engng A Struct. 125 (2), 203213.CrossRefGoogle Scholar
Virk, P.S. 1975 Drag reduction fundamentals. AIChE J. 21 (4), 625656.CrossRefGoogle Scholar
Wang, L., Cardenas, M.B., Slottke, D.T., Ketcham, R.A. & Sharp, J.M. Jr 2015 Modification of the local cubic law of fracture flow for weak inertia, tortuosity, and roughness. Water Resour. Res. 51 (4), 20642080.CrossRefGoogle Scholar
Witherspoon, P.A., Wang, J.S.Y., Iwai, K. & Gale, J.E. 1980 Validity of Cubic Law for fluid flow in a deformable rock fracture. Water Resour. Res. 16 (6), 10161024.CrossRefGoogle Scholar
Yu, L., Liu, R. & Jiang, Y. 2017 A review of critical conditions for the onset of nonlinear fluid flow in rock fractures. Geofluids 2017, 117.Google Scholar
Zia, H. & Lecampion, B. 2017 Propagation of a height contained hydraulic fracture in turbulent flow regimes. Intl J. Solids Struct. 110–111, 265278.CrossRefGoogle Scholar
Zimmerman, R.W., Al-Yaarubi, A., Pain, C.C. & Grattoni, C.A. 2004 Non-linear regimes of fluid flow in rock fractures. Intl J. Rock Mech. Mining Sci. 41 (3), 384384.CrossRefGoogle Scholar
Zimmerman, R.W., Kumar, S. & Bodvarsson, G.S. 1991 Lubrication theory analysis of the permeability of rough-walled fractures. Intl J. Rock Mech. Mining Sci. Geomech. Abstracts 28 (4), 325331.CrossRefGoogle Scholar
Zolfaghari, N. & Bunger, A.P. 2019 Numerical model for a penny-shaped hydraulic fracture driven by laminar/turbulent fluid in an impermeable rock. Intl J. Solids Struct. 158, 128140.CrossRefGoogle Scholar