1. INTRODUCTION
Proppant transport and placement are critical parts of hydraulic fracturing treatments. Such particulate flow consists of two interacting materials i.e., proppants and the carrying fluid. Two-phase flow mechanics requires specification of stresses for each phase, plus a relation expressing the interaction of both materials.
The character of the interparticle forces in the dilute and dense regions is essentially different. The constitutive equations for dilute flow specialize the drag to include Stokes and Faxen forces, and the lift to accommodate the slip-shear force and the disturbance-shear force. While in dense proppant flow the solid fraction is high, it is more difficult for them to move with respect to one another. In the state where maximum packing has been reached, the particles are so closely interlocked that only rigid motion is possible. A locking phenomenon occurs.
Here we review briefly the approach of Passman, Nunziato and Bailey (1986) to clarify the origin of the momentum equations arising in the dense proppant flow. In addition, the constitutive equations for the momentum transfer are formulated to account for the viscous transfer arising from the highly dense particles.
In order to try to fully understand the problem, we have performed parameter studies and numerical analyses. To solve the Poiseuille flow problem in the fracture channel we have used a spectral method which is known to produce accurate results. The proppant concentration profiles, shear stress distribution, and velocity profiles for fluid and proppant are given. The explanation of the results is provided.
