An analysis of turbulent fluid friction was made for data of vertical flow of CO2-foam slurries. These data were obtained during a fracture stimulation in a uniquely- instrumented, 3.5-inch tubing string. Data were culled to steady state for each CO2 cut, gel concentration, sand loading, and rate. Friction gradients were computed and a new friction correlation was developed. The correlation uses the friction factor concept for high speed, single phase flow by defining appropriate mixture density, viscosity, and Reynolds number from foam slurry properties.
Two existing two-phase flow friction correlations were tried and found unsuccessful for the CO2-foam slurry data. A new friction method was developed from solid suspension and emulsion rheology concepts. Sand and CO2 bubbles are treated as internal, or suspended phases, and the gel liquid is the external, or suspending fluid phase. Assuming sand and CO2 are internal phases is consistent with Halliburton's constant internal phase concept (SPE 17532), which holds that foam slurry friction increases with increasing sand cut or CO2 cut. This paper shows how the physical concept was developed into a mathematical concept.
Internal phases create relative viscosity, a factor by which effective viscosity increases when small particles or droplets are added to the external, or suspending fluid. Relative viscosity has a power law dependence on internal phase volume fraction. Foam texture is unknown and can't be used in a field correlation. Yield stress is neglected, also. The effective viscosity of the foam slurry mixture (sand, CO2, gel) is reduced to the product of the non-Newtonian power law viscosity of the HPG gel liquid in turbulent flow, times the relative viscosity of the internal phases. This provides a meaningful mixture viscosity, thence an appropriate Reynolds number. Friction gradient is correlated by a Fanning friction factor, as in single phase flow, assuming no slip between internal and external phases. Thus, friction in a slurry with a gas foam or emulsifying liquid CO2 is reduced to an equivalent single-phase flow friction factor correlation, valid as long as the internal phases are maintained. The proposed correlation appears promising, but considerable scatter is observed if one attempts comparison to time-dependent field data or to high CO2 cuts. High CO2 cut causes somewhat unstable friction gradient. Speculation is that some loss of internal phase and/or loss of external phase drag reduction may occur at the higher CO2 cuts. Other phenomena suggested by the data appear only after close inspection and more speculation. However, postulating a depth-dependence of friction is not necessary, in contrast to the conclusions reached in ref 1.