Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T21:13:46.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Slickwater hydraulic fracture propagation: near-tip and radial geometry solutions

Published online by Cambridge University Press:  10 October 2019

Brice Lecampion Open the ORCID record for Brice Lecampion [Opens in a new window]  and
Haseeb Zia
Brice Lecampion*
Affiliation:
Geo-Energy Laboratory - Gaznat Chair on Geo-Energy, Ecole Polytechnique Fédérale de Lausanne, EPFL-ENAC-IIC-GEL, Station 18, CH-1015, Switzerland
Haseeb Zia
Affiliation:
Geo-Energy Laboratory - Gaznat Chair on Geo-Energy, Ecole Polytechnique Fédérale de Lausanne, EPFL-ENAC-IIC-GEL, Station 18, CH-1015, Switzerland
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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).

JFM classification

Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 880 , 10 December 2019 , pp. 514 - 550
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Al Hashmi, A., Al Maamari, R., Al Shabibi, I., Mansoor, A., Zaitoun, A. & Al Sharji, H. 2013 Rheology and mechanical degradation of high-molecular-weight partially hydrolyzed polyacrylamide during flow through capillaries. J. Petrol. Sci. Engng 105, 100–106.Google Scholar
Blasius, H. 1913 Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten. Springer.Google Scholar
Brandrup, J., Immergut, E. H., Grulke, E. A., Abe, A. & Bloch, D. R. 1999 Polymer Handbook, vol. 89. Wiley.Google Scholar
Cleary, M. & Wong, S. 1985 Numerical simulation of unsteady fluid flow and propagation of a circular hydraulic fracture. Intl J. Numer. Anal. Meth. Geomech. 9 (1), 1–14.Google Scholar
Desroches, J., Detournay, E., Lenoach, B., Papanastasiou, P., Pearson, J. R. A., Thiercelin, M. & Cheng, A. 1994 The crack tip region in hydraulic fracturing. Proc. R. Soc. Lond. A 447 (1929), 39–48.Google Scholar
Detournay, E. 2004 Propagation regimes of fluid-driven fractures in impermeable rocks. Intl J. Geomech. 4 (1), 35–45.Google Scholar
Detournay, E. 2016 Mechanics of hydraulic fractures. Annu. Rev. Fluid Mech. 48, 311–339.Google Scholar
Detournay, E. & Peirce, A. P. 2014 On the moving boundary conditions for a hydraulic fracture. Intl J. Engng Sci. 84, 147–155.Google Scholar
Dontsov, E. 2016a Tip region of a hydraulic fracture driven by a laminar-to-turbulent fluid flow. J. Fluid Mech. 797, R2.Google Scholar
Dontsov, E. & Peirce, A. 2017 Modeling planar hydraulic fractures driven by laminar-to-turbulent fluid flow. Intl J. Solids Struct. 128, 73–84.Google Scholar
Dontsov, E. & Peirce, A. P. 2015 A non-singular integral equation formulation to analyse multiscale behaviour in semi-infinite hydraulic fractures. J. Fluid Mech. 781, R1.Google Scholar
Dontsov, E. V. 2016b An approximate solution for a penny-shaped hydraulic fracture that accounts for fracture toughness, fluid viscosity and leak-off. R. Soc. Open Sci. 3 (12), 160737.Google Scholar
Dontsov, E. V. & Kresse, O. 2018 A semi-infinite hydraulic fracture with leak-off driven by a power-law fluid. J. Fluid Mech. 837, 210–229.Google Scholar
Emerman, S. H., Turcotte, D. & Spence, D. 1986 Transport of magma and hydrothermal solutions by laminar and turbulent fluid fracture. Phys. Earth Planet. Inter. 41 (4), 249–259.Google Scholar
Garagash, D. I. 2006 Plane-strain propagation of a fluid-driven fracture during injection and shut-in: asymptotics of large toughness. Engng Fracture Mech. 73 (4), 456–481.Google Scholar
Garagash, D. I. 2009 Scaling of physical processes in fluid-driven fracture: perspective from the tip. In IUTAM Symposium on Scaling in Solid Mechanics (ed. Borodich, F.), IUTAM Bookseries, vol. 10, pp. 91–100. Springer.Google Scholar
Garagash, D. I. & Detournay, E. 2000 The tip region of a fluid-driven fracture in an elastic medium. Trans. ASME J. Appl. Mech. 67, 183–192.Google Scholar
Garagash, D. I., Detournay, E. & Adachi, J. I. 2011 Multiscale tip asymptotics in hydraulic fracture with leak-off. J. Fluid Mech. 669, 260–297.Google Scholar
Habibpour, M. & Clark, P. E. 2017 Drag reduction behavior of hydrolyzed polyacrylamide/xanthan gum mixed polymer solutions. Petrol. Sci. 14 (2), 412–423.Google Scholar
Jones, O. C. 1976 An improvement in the calculation of turbulent friction in rectangular ducts. J. Fluids Engng 98 (2), 173–180.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Larson, R. G. & Desai, P. S. 2015 Modeling the rheology of polymer melts and solutions. Annu. Rev. Fluid Mech. 47, 47–65.Google Scholar
Lecampion, B., Bunger, A. P. & Zhang, X. 2018 Numerical methods for hydraulic fracture propagation: a review of recent trends. J. Natl Gas Sci. Engng 49, 66–83.Google Scholar
Lecampion, B. & Desroches, J. 2015 Simultaneous initiation and growth of multiple radial hydraulic fractures from a horizontal wellbore. J. Mech. Phys. Solids 82, 235–258.Google 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. 122.Google Scholar
Lister, J. 1990 Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors. J. Fluid Mech. 210, 263–280.Google Scholar
Lister, J. R. & Kerr, R. C. 1991 Fluid-mechanical models of crack propagation and their application to magma transport in dykes. J. Geophys. Res. 96 (B6), 10049–10077.Google Scholar
Madyarova, M. & Detournay, E.2004 Radial fracture driven by a power-law fluid in a permeable elastic rock. Tech. Rep. Schlumberger. Report of UMN to Modeling & Mechanics Group, EAD, Schlumberger.Google Scholar
Moukhtari, F. E. & Lecampion, B. 2018 A semi-infinite hydraulic fracture driven by a shear thinning fluid. J. Fluid Mech. 838, 573–605.Google Scholar
Nikuradse, J. 1950 Laws of Flow in Rough Pipes. National Advisory Committee for Aeronautics Washington.Google Scholar
Nilson, R. H. 1981 Gas-driven fracture propagation. J. Appl. Mech. 48 (4), 757–762.Google Scholar
Peirce, A. P. 2016 Implicit level set algorithms for modelling hydraulic fracture propagation. Phil. Trans. R. Soc. Lond. A 374 (2078), 20150423.Google Scholar
Peirce, A. P. & Detournay, E. 2008 An implicit level set method for modeling hydraulically driven fractures. Comput. Meth. Appl. Mech. Engng 197 (33–40), 2858–2885.Google Scholar
Perkins, T. K. & Kern, L. R. 1961 Widths of hydraulic fractures. J. Petrol. Tech. 222, 937–949.Google Scholar
Rice, J. R. 1972 Some remarks on elastic crack-tip stress fields. Intl J. Solids Struct. 8, 751–758.Google Scholar
Rice, J. R., Tsai, V. C., Fernandes, M. C. & Platt, J. D. 2015 Time scale for rapid draining of a surficial lake into the greenland ice sheet. J. Appl. Mech. 82 (7), 071001.Google Scholar
Savitski, A. & Detournay, E. 2002 Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions. Intl J. Solids Struct. 39 (26), 6311–6337.Google Scholar
Smith, M. B. & Montgomery, C. T. 2015 Hydraulic Fracturing. CRC Press.Google Scholar
Sneddon, I. & Lowengrub, M. 1969 Crack Problems in the Classical Theory of Elasticity. John Wiley & Sons.Google Scholar
Szeri, A. Z. 2010 Fluid Film Lubrication. Cambridge University Press.10.1017/CBO9780511782022Google Scholar
Toms, B. A. 1948 Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. In Proceedings of the 1st International Congress on Rheology, vol. 2, pp. 135–141.Google Scholar
Tsai, V. & Rice, J. R. 2010 A model for turbulent hydraulic fracture and application to crack propagation at glacier beds. J. Geophys. Res. 115, F03007.Google Scholar
Tsai, V. & Rice, J. R. 2012 Modeling turbulent hydraulic fracture near a free surface. J. Appl. Mech. 79 (3), 031003.Google Scholar
Viesca, R. C. & Garagash, D. I. 2018 Numerical methods for coupled fracture problems. J. Mech. Phys. Solids 113, 13–34.Google Scholar
Virk, P. S. 1971 Drag reduction in rough pipes. J. Fluid Mech. 45 (02), 225–246.Google Scholar
Virk, P. S. 1975 Drag reduction fundamentals. AIChE J. 21 (4), 625–656.Google Scholar
Virk, P. S., Merrill, E. W., Mickley, H. S., Smith, K. A. & Mollo-Christensen, E. L. 1967 The toms phenomenon: turbulent pipe flow of dilute polymer solutions. J. Fluid Mech. 30 (02), 305–328.Google Scholar
White, C. M. & Mungal, M. G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235–256.Google Scholar
Xing, P., Yoshioka, K., Adachi, J. I., El-Fayoumi, A. & Bunger, A. P. 2017 Laboratory measurement of tip and global behavior for zero-toughness hydraulic fractures with circular and blade-shaped (pkn) geometry. J. Mech. Phys. Solids 104, 172–186.Google Scholar
Yang, S.-Q. & Dou, G. 2010 Turbulent drag reduction with polymer additive in rough pipes. J. Fluid Mech. 642, 279–294.Google Scholar
Zia, H. & Lecampion, B. 2017 Propagation of a height contained hydraulic fracture in turbulent flow regimes. Intl J. Solids Struct. 110–111, 265–278.Google Scholar
Zia, H. & Lecampion, B. 2019 Explicit versus implicit front advancing schemes for the simulation of hydraulic fracture growth. Intl J. Numer. Anal. Meth. Geomech. 43 (6), 1300–1315.Google Scholar
Zia, H., Lecampion, B. & Zhang, W. 2018 Impact of the anisotropy of fracture toughness on the propagation of planar 3D hydraulic fracture. Intl J. Fracture 211 (1–2), 103–123.Google Scholar
Zolfaghari, N. & Bunger, A. 2018a Approximate semi-analytical solution for a penny-shaped rough-walled hydraulic fracture driven by turbulent fluid in an impermeable rock. Intl J. Solids Struct. 143, 144–154.Google Scholar
Zolfaghari, N. & Bunger, A. 2018b Numerical model for a penny-shaped hydraulic fracture driven by laminar/turbulent fluid in an impermeable rock. Intl J. Solids Struct. 158, 128–140.Google Scholar
Zolfaghari, N., Dontsov, E. & Bunger, A. 2018 Solution for a plane strain rough-walled hydraulic fracture driven by turbulent fluid through impermeable rock. Intl J. Numer. Anal. Meth. Geomech. 42 (4), 587–617.Google Scholar
Zolfaghari, N., Meyer, C. R. & Bunger, A. P. 2017 Blade-shaped hydraulic fracture driven by a turbulent fluid in an impermeable rock. J. Engng Mech. 143 (11), 04017130.Google Scholar