  The potential field approximation has been providing a fast, and
computationally inexpensive estimation for the solar corona's global magnetic
field geometry for several decades. In contrast, more physics-based global
magnetohydrodynamic (MHD) models have been used for a similar purpose, while
being much more computationally expensive. Here, we investigate the difference
in the field geometry between a global MHD model and the potential field source
surface model (PFSSM) by tracing individual magnetic field lines in the MHD
model from the Alfven surface (AS), through the source surface (SS), all the
way to the field line footpoint, and then back to the source surface in the
PFSSM. We also compare the flux-tube expansion at two points at the SS and the
AS along the same radial line. We study the effect of solar cycle variations,
the order of the potential field harmonic expansion, and different magnetogram
sources. We find that the flux-tube expansion factor is consistently smaller at
the AS than at the SS for solar minimum and the fast solar wind, but it is
consistently larger for solar maximum and the slow solar wind. We use the
Wang--Sheeley--Arge (WSA) model to calculate the associated wind speed for each
field line, and propagate these solar-wind speeds to 1AU. We find a more than
five hours deviation in the arrival time between the two models for 20% of the
field lines in the solar minimum case, and for 40% of the field lines in the
solar maximum case.
