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.
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