  We have performed an {\it ab initio} theoretical investigation of graphene
sheet adsorbed on amorphous SiO$_2$ surface (G/a-SiO$_2$). We find that
graphene adsorbs on the a-SiO$_2$ surface through van der Waals interactions.
The inhomogeneous topology of the a-SiO$_2$ clean surface promotes a total
charge density displacement on the adsorbed graphene sheet, giving rise to
electron-rich as well as hole-rich regions on the graphene. Such anisotropic
distribution of the charge density may contribute to the reduction of the
electronic mobility in G/a-SiO$_2$ systems. Furthermore, the adsorbed graphene
sheet exhibits a net total charge density gain. In this case, the graphene
sheet becomes n-type doped, however, with no formation of chemical bonds at the
graphene--SiO$_2$ interface. The electronic charge transfer from a-SiO$_2$ to
the graphene sheet occurs upon the formation of a partially occupied level
lying above the Dirac point. We find that such partially occupied level comes
from the three-fold coordinated oxygen atoms in the a-SiO$_2$ substrate.
