  Inclusive bottomonium hadroproduction at the Tevatron is firstly examined in
a Monte Carlo framework with the colour-octet mechanism implemented in the
event generation. We extract some NRQCD colour-octet matrix elements relevant
for $\Upsilon(1S)$ hadroproduction. Remarkably we find a quite small
contribution (compatible with zero) from feeddown of $\chi_{bJ}$ states
produced through the colour-octet mechanism: $\Upsilon(1S)$ indirect production
via $\chi_{bJ}$ decays should be mainly ascribed to the colour-singlet model.
Finally we extrapolate to LHC energies to predict prompt $\Upsilon(1S)$
production rates.
