  The search for ultra-high energy downward-going and Earth-skimming cosmic
neutrinos by the Surface Detector array of the Pierre Auger Observatory (PAO)
is analyzed in the ADD model with n extra flat spatial dimensions. We assumed
that the diffuse neutrino flux dN_nu/dE_nu$ is equal to k E_nu^(-20) in the
energy range 10^(17) eV - 2.5 10^(19) eV. Taking into account that no neutrino
events where found by the PAO, we have estimated an upper bound on a value of
k. It is shown that this bound can be stronger than the upper bound on k
recently obtained by the Pierre Auger Collaboration, depending on n and
(n+4)-dimensional gravity scale M_D.
