The rheology of foams has been the subject of recent work in which the principle of volume equalization has proven effective to describe the flow behavior. This model is combined with the constant-internal-phase/constant-viscosity assumption to describe the transport of solid particles. Laminar and turbulent friction pressures, the transition criterion and the settling velocity are given in a unified framework.
During foam hydraulic fracturing treatments a significant portion of the treating fluid contains proppant. The modification of the rheological properties caused by the solids content has a direct effect on the fracture geometry, determines the proppant distribution in the created fracture through the settling velocity and affects the friction pressure in the tubular conductors. In spite of the importance of these phenomena in design, control and evaluation of the treatment, relatively little is known about solids-laden foam flow.
There are two main reasons for the limited availability of models and correlations. The first one is the lack of reliable experimental data. Conditions that act against laboratory-scale viscosity determination methods include the requirements to maintain a stable foam and uniform solids distribution simultaneously and to avoid the interaction of the instrument scale with the bubble size scale. The second is rooted in the nature of the accepted way to establish correlations. Both foam rheology and solids-laden liquid rheology are often treated by introducing an additional degree of freedom with respect to the base liquid. That is, a liquid flow curve is substituted by a family of curves. The application of this approach twice would lead to a substantial increase in experimental and modeling efforts. Therefore, it is natural to search for simplifying assumptions which may allow the applications of known results of rheology and fluid mechanics. This paper suggests a combination of such assumptions leading to a unified treatment of the different aspects of foam proppant transport.