# VI.A Experimental Observations

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 S𝕆r of oil remains miroscopically trapped
inside the medium.
The trapped oil can be mobilized if the viscous forces overcome
the capillary retention forces [333].
Displacement experiments in a variety of porous media including
micromodels show a strong correlation between the residual oil
saturation S𝕆r and the capillary number Ca of the
waterflood [338, 203, 2, 28, 339, 340, 341, 232].
The capillary number, defined as Ca=μu/σ, is
the dimensionless ratio of viscous to capillary forces.
Here u denotes an average microscopic velocity, μ is
the viscosity, and σ the surface tension of the fluid.

The experimental curves S𝕆rCa 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 Cac below which the residual oil saturation remains constant.
This critical capillary number Cac 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 Cac
are much smaller than 1.
For unconsolidated sand Cac is often reported to be
Cac≈10-4 while for sandstone Cac≈3⋅10-6 and
for limestone Cac≈2⋅10-7 [28].
The exceedingly small values of Cac as well as
their dependence on the type of porous medium strongly suggest
that the microscopically defined capillary number Ca
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 Ca
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.