  Phenomenological models of the dipole cross section that enters in the
description of for instance deep inelastic scattering at very high energies
have had considerable success in describing the available small-x data in both
the saturation region and the so-called extended geometric scaling (EGS)
region. We investigate to what extent such models are compatible with the
numerical solutions of the Balitsky-Kovchegov (BK) equation which is expected
to describe the nonlinear evolution in x of the dipole cross section in these
momentum regions. We find that in the EGS region the BK equation yields results
that are qualitatively different from those of phenomenological studies. In
particular, geometric scaling around the saturation scale is only obtained at
asymptotic rapidities. We find that in this limit, the anomalous dimension
\gamma(r,x) of phenomenological models approaches a limiting function that is
universal for a large range of initial conditions. At the saturation scale,
this function equals approximately 0.44, in contrast to the value 0.628
commonly used in the models. We further investigate the dependence of these
results on the starting distribution, the small-r limit of the anomalous
dimension for fixed rapidities and the x-dependence of the saturation scale.
