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

• Published in: Journal of fluid mechanics Vol. 760; pp. 567 - 590
• Main Authors: Dontsov, E. V., Peirce, A. P.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.12.2014
• Subjects: Applied sciences; Approximation; Boundary layer transition; Boundary layers; Computational fluid dynamics; Computer simulation; Crude oil, natural gas and petroleum products; Crude oil, natural gas, oil shales producing equipements and methods; Darcys law; Empirical analysis; Energy; Enhanced oil recovery methods; Exact sciences and technology; Fluid flow; Fluid mechanics; Fluids; Fractures; Fuels; Gravitation; Gravity; Hydraulic fracturing; Integral equations; Laminar flow; Mathematical models; Modelling; Newtonian fluids; Orbital velocity; Particle concentration; Prospecting and production of crude oil, natural gas, oil shales and tar sands; Reynolds number; Rheology; Sedimentation & deposition; Settling; Simulators; Slurries; Solutions; Steady flow; Studies; Transport; Two dimensional boundary layer; Two dimensional flow; Velocity; suspensions; particle/fluid flows; Two phase flow; Numerical simulation; Spherical particle; Particle suspension; Modeling; Sedimentation; Slurries; Hydraulic fracturing
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2014.606

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.