Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T10:15:27.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Slurry flow, gravitational settling and a proppant transport model for hydraulic fractures

Published online by Cambridge University Press:  12 November 2014

E. V. Dontsov  and
A. P. Peirce
E. V. Dontsov
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
A. P. Peirce*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The goal of this study is to analyse the steady flow of a Newtonian fluid mixed with spherical particles in a channel for the purpose of modelling proppant transport with gravitational settling in hydraulic fractures. The developments are based on a continuum constitutive model for a slurry, which is approximated by an empirical formula. It is shown that the problem under consideration features a two-dimensional flow and a boundary layer, which effectively introduces slip at the boundary and allows us to describe a transition from Poiseuille flow to Darcy’s law for high proppant concentrations. The expressions for both the outer (i.e. outside the boundary layer) and inner (i.e. within the boundary layer) solutions are obtained in terms of the particle concentration, particle velocity and fluid velocity. Unfortunately, these solutions require the numerical solution of an integral equation, and, as a result, the development of a proppant transport model for hydraulic fracturing based on these results is not practicable. To reduce the complexity of the problem, an approximate solution is introduced. To validate the use of this approximation, the error is estimated for different regimes of flow. The approximate solution is then used to calculate the expressions for the slurry flux and the proppant flux, which are the basis for a model that can be used to account for proppant transport with gravitational settling in a fully coupled hydraulic fracturing simulator.

JFM classification

Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 760 , 10 December 2014 , pp. 567 - 590
Check for updates
Copyright
© 2014 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

Adachi, J. I., Detournay, E. & Peirce, A. P. 2010 An analysis of classical pseudo-3D model for hydraulic fracture with equilibrium height growth across stress barriers. Intl J. Rock Mech. Min. Sci. 47, 625–639.CrossRefGoogle Scholar
Adachi, J., Siebrits, E., Peirce, A. & Desroches, J. 2007 Computer simulation of hydraulic fractures. Intl J. Rock Mech. Min. Sci. 44, 739–757.Google Scholar
Batchelor, G. & Green, J. 1972 The determination of the bulk stress in a suspension of spherical particles to order $c^{2}$ . J. Fluid Mech. 56, 401–427.Google Scholar
Boyer, F.2011 Suspensions concentrées: expériences originales de rhéologie. PhD thesis, Aix-Marseille Université, France.Google Scholar
Boyer, F., Guazzelli, E. & Pouliquen, O. 2011 Unifying suspension and granular rheology. Phys. Rev. Lett. 107, 188301.Google Scholar
Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103–139.CrossRefGoogle Scholar
Davis, R. H. & Acrivos, A. 1985 Sedimentation of noncolloidal particles at low Reynolds numbers. Annu. Rev. Fluid Mech. 17, 91–118.Google Scholar
Economides, M. J. & Nolte, K. G.(Eds) 2000 Reservoir Stimulation, 3rd edn. Wiley.Google Scholar
Einstein, A. 1906 A new determination of molecular dimensions. Ann. Phys. 4, 289–306.Google Scholar
Eskin, D. & Miller, M. J. 2008 A model of non-Newtonian slurry flow in a fracture. Powder Technol. 182, 313–322.Google Scholar
Garside, J. & Al-Dibouni, M. R. 1977 Velocity–voidage relationships for fluidization and sedimentation in solid–liquid systems. Ind. Eng. Chem. Process Des. Dev. 16, 206–214.CrossRefGoogle Scholar
Lecampion, B. & Garagash, D. 2014 Confined flow of suspensions modelled by a frictional rheology. J. Fluid Mech. 759, 197–235.CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.CrossRefGoogle Scholar
Lhuillier, D. 2009 Migration of rigid particles in non-Brownian viscous suspensions. Phys. Fluids 21, 023302.Google Scholar
Lister, J. R. 1990 Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors. J. Fluid Mech. 210, 263–280.Google Scholar
Matas, J.-P., Morris, J. F. & Guazzelli, E. 2004 Inertial migration of rigid spherical particles in Poiseuille flow. J. Fluid Mech. 515, 171–195.Google Scholar
Morris, J. F. & Boulay, F. 1999 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Rheol. 43, 1213–1237.CrossRefGoogle Scholar
Nott, P. R. & Brady, J. F. 1994 Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199.Google Scholar
Ouriemi, M., Aussillous, P. & Guazzelli, E. 2009 Sediment dynamics. Part 1. Bed-load transport by laminar shearing flows. J. Fluid Mech. 636, 295–319.Google Scholar
Segre, G. & Silberberg, A. 1961 Radial particle displacements in Poiseuille flow of suspensions. Nature 189, 209–210.CrossRefGoogle Scholar
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129–149.CrossRefGoogle Scholar