A new kinematic wave model for interpreting cross correlation velocity measurements in vertically upward, bubbly oil-in-water flows

A new model for the speed of propagation of kinematic waves in vertical upward, bubbly oil-in-water flows has been proposed. The new kinematic wave model has been used in conjunction with (i) appropriate values for the `distribution parameter' C0, the single-droplet terminal rise velocity vt0 and an exponent n, obtained from a drift velocity model; (ii) a statistical relationship for the quantity αUh(dC0/dα); and (iii) measurements of a cross correlation velocity Ucc and the volume fraction of oil α to make predictions of the superficial velocities of the mixture, the oil and the water in vertical upward, bubbly oil-in-water flows. The systematic errors in these predicted values of the superficial velocities of the mixture, the oil and the water were 0.16, -0.04 and 0.04%, respectively. The kinematic wave model can thus be used in conjunction with a cross correlation flow meter for accurate measurement of flow rates in vertical oil wells. It was inferred from the kinematic wave model and the experimental data that, for low values of the volume fraction of oil α, the distribution parameter C0 decreases rapidly with increasing α. At higher values of α the decrease in the value of C0 with increasing α is less marked. This result is consistent with the physical explanation that, at low values of α, the oil droplets tend to preferentially accumulate in the relatively fast moving regions of the flow. As α increases the oil droplets become more uniformly distributed amongst the faster and slower moving regions of the flow.