We present a comprehensive theory of sand transport in thin fluid slot flow. Equations of motion are developed for each of the transport mechanisms observed in slot flow experiments: viscous drag, turbulence, and bed load transport. Viscous drag and turbulence are treated together by deriving a differential field equation for sand distribution in the slot. The terminal fall velocity and a turbulent diffusion coefficient appear as parameters. The field equation is simplified by transformation to a system of curvilinear coordinates. The coordinate lines are streamlines of sand particles in the absence of diffusion. Turbulent transport is accounted for by the diffusion of a square wave distribution at the slot entrance. The diffusion length associated with this process is a measure of the concentration of sand process is a measure of the concentration of sand transported by turbulence. We consider both natural turbulence and stimulated turbulence generated at the slot entrance. Our theoretical results show that natural turbulence produces little sand transport. Stimulated turbulence is more important, but it dies out quickly with distance. Bed load transport is treated by using a principle of virtual dissipation. We consider two kinds of dissipation in the fluidized layer. One is associated with motion of sand particles relative to the fluid. The other is due to an increase in viscosity with increasing sand concentration. We derive equations of motion for sand particles in the fluidized layer. An important result is that sand transport in the bed load does not scale up with fracture height as long as the flow velocity and the entrance concentration remain the same. This leads to the conclusion that bed load transport is a significant factor in laboratory-scale experiments, but not on a scale of field treatments. Therefore, of the three transport mechanisms observed in slot flow experiments, only viscous drag is important under hydraulic fracturing conditions. We discuss application of the complete theory to sand transport in fracturing treatments.
Proppant transport is a critical part of hydraulic fracturing technology. There is a clear division between two classes of transport behavior: that of thin fluids and that of very thick, cross-linked gels. In this paper we consider only transport in thin fluids. We exclude all cross-linked gels.
Thin fluids have been largely replaced by cross-linked gels as common fracturing fluids. Nevertheless, thin fluids still have important applications in fracturing operations. Under various conditions they have significant advantages which, in recent times, seem to have been lost sight of. One of these is the formation of a settled bank which fills the width of the fracture from the entrance outward. This characteristic assures good communication with the wellbore and avoids the problem of proppant settling ahead of fracture closure. Thin fluids favor fracture length over width, as opposed to cross-linked gels where the reverse is true. This has economic benefits in massive fracturing of tight formations where high fracture conductivity is not needed.
The loss in popularity of thin fluids probably explains why more has not been done to develop a comprehensive theory of their transport mechanics. A considerable amount of experimental work has been done, not all of which has been published. Early work by Kern et al. and by Babcock et al. introduced slot flow experiments in lucite models as the most practical way to study sand transport mechanisms. Others have used the same slot flow methods to make additional contributions to the early experimental results.
Numerical models have been developed on the basis of this experimental work.