Consider the displacement of oil from an oil saturated porous medium through injecting water at constant velocity. After steady state flow conditions are established a certain fraction of oil remains miroscopically trapped inside the medium. The trapped oil can be mobilized if the viscous forces overcome the capillary retention forces . Displacement experiments in a variety of porous media including micromodels show a strong correlation between the residual oil saturation and the capillary number of the waterflood [338, 203, 2, 28, 339, 340, 341, 232]. The capillary number, defined as , is the dimensionless ratio of viscous to capillary forces. Here denotes an average microscopic velocity, is the viscosity, and the surface tension of the fluid.
The experimental curves are called capillary number correlations, recovery curves or capillary desaturation curves, and they give the residual oil saturation as a function of the capillary number of the flood. All such capillary desaturation curves exhibit a critical capillary number below which the residual oil saturation remains constant. This critical capillary number marks the point where the viscous forces equal the capillary forces. Figure 25 shows a schematic drawing of the capillary desaturation curves for unconsolidated sand, sandstone and limestone (after [203, 28]).
Surprisingly, all experimentally observed values for are much smaller than 1. For unconsolidated sand is often reported to be while for sandstone and for limestone . The exceedingly small values of as well as their dependence on the type of porous medium strongly suggest that the microscopically defined capillary number cannot be an adequate measure of the balance between macroscopic viscous and macroscopic capillary forces.
The subsequent sections review recent work which relates the large discrepancy between the observed force balance and the force balance estimated from to an implicit assumption in the traditional dimensional analysis [49, 329, 330, 331]. First the microscopic equations of motion and their dimensional analysis are recalled. This leads to the familiar dimensionless numbers of fluid dynamics. Next the accepted macroscopic equations of motion are analysed. This leads to macroscopic dimensionless numbers which are then related to the traditional microscopic dimensionless groups. The results are shown to be applicable to the quantitative estimation of residual oil saturation, gravitational relaxation times and the width of the oil-water contact.