Preferred calculation formula and buoyancy effect on capillary number

It has been well established in the literature that as the capillary number is increased, residual saturations are decreased. The concept of capillary number is very important in enhanced oil recovery. There are several definitions of the capillary number. Which equation properly describes the mechanism of residual oil reduction as the capillary number is increased? To include the effect of buoyancy, a parameter called trapping number that is similar to the capillary number is proposed, and two different equations have been derived in the literature. There is a paradox for one of these equations: It suggests that the buoyance plays a role in horizontal flow! This paper seeks answers and explanations to the question and the paradox from theoretical point of view and published experimental data. This research work from this paper also suggests which equation should be used to calculate the trapping number for the flow in an inclined core or formation, which equation should be used to calculate capillary number and how the capillary number in a multiphase flow be calculated.