  We analyze the radial distribution of HI gas for 23 disk galaxies with
unusually high HI content from the Bluedisk sample, along with a similar-sized
sample of "normal" galaxies. We propose an empirical model to fit the radial
profile of the HI surface density, an exponential function with a depression
near the center. The radial HI surface density profiles are very homogeneous in
the outer regions of the galaxy; the exponentially declining part of the
profile has a scale-length of $\sim 0.18$ R1, where R1 is the radius where the
column density of the HI is 1 M$_{\odot}$ pc$^{-2}$. This holds for all
galaxies, independent of their stellar or HI mass. The homogenous outer
profiles, combined with the limited range in HI surface density in the
non-exponential inner disk, results in the well-known tight relation between HI
size and HI mass. By comparing the radial profiles of the HI-rich galaxies with
those of the control systems, we deduce that in about half the galaxies, most
of the excess gas lies outside the stellar disk, in the exponentially declining
outer regions of the HI disk. In the other half, the excess is more centrally
peaked. We compare our results with existing smoothed-particle hydrodynamical
simulations and semi-analytic models of disk galaxy formation in a $\Lambda$
Cold Dark Matter universe. Both the hydro simulations and the semi-analytic
models reproduce the HI surface density profiles and the HI size-mass relation
without further tuning of the simulation and model inputs. In the semi-analytic
models, the universal shape of the outer HI radial profiles is a consequence of
the {\em assumption} that infalling gas is always distributed exponentially.
The conversion of atomic gas to molecular form explains the limited range of HI
surface densities in the inner disk. These two factors produce the tight HI
mass-size relation.
