  Most models for the central engine of gamma-ray bursts involve a stellar mass
black hole surrounded by a thick disk formed after the merging of a system of
compact objects or the collapse of a massive star. Energy released from the
accretion of disk material by the black hole or from the rotation of the hole
itself extracted by the Blandford-Znajek mechanism powers a relativistic wind
along the system axis. Lorentz factors of several hundreds are needed to solve
the compactness problem in the wind which implies the injection of a tremendous
power into a very small amount of matter. The Blandford-Znajek mechanism, where
the outflow follows magnetic field lines anchored to the black hole is probably
the best way to prevent baryonic pollution and can even initially produce a
purely leptonic wind. In this paper we rather study the wind emitted from the
inner part of the disk where the risk of baryonic pollution is much larger
since the outflow originates from high density regions. We show that the
baryonic load of this wind sensitively depends on the disk temperature and
magnetic field geometry and that the outflow can become ultra-relativistic
(Lorentz factor Gamma>100) under quite restrictive conditions only. Conversely,
if Gamma remains of the order of unity the dense wind emitted from the inner
disk could help to confine the central jet but may also represent a source of
baryon contamination for the Blandford-Znajek mechanism.
