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

Doping of graphene adsorbed on the a-SiO$_2$ surface