  Context. Millimetric observations have measured high degrees of molecular
deuteration in several species seen around low-mass protostars. The Herschel
Space Telescope, launched in 2009, is now providing new measures of the
deuterium fractionation of water, the main constituent of interstellar ices.
Aims. We aim at theoretically studying the formation and the deuteration of
water, which is believed to be formed on interstellar grain surfaces in
molecular clouds. Methods. We used our gas-grain astrochemical model GRAINOBLE,
which considers the multilayer formation of interstellar ices. We varied
several input parameters to study their impact on water deuteration. We
included the treatment of ortho- and para-states of key species, including H2,
which affects the deuterium fractionation of all molecules. The model also
includes relevant laboratory and theoretical works on the water formation and
deuteration on grain surfaces. In particular, we computed the transmission
probabilities of surface reactions using the Eckart model, and we considered
ice photodissociation following molecular dynamics simulations. Results. The
use of a multilayer approach allowed us to study the influence of various
parameters on the abundance and the deuteration of water. Deuteration of water
is found to be very sensitive to the ortho-to-para ratio of H2 and to the total
density, but it also depends on the gas/grain temperatures and the visual
extinction of the cloud. Since the deuteration is very sensitive to the
physical conditions, the comparison with sub-millimetric observation towards
the low-mass protostar IRAS 16293 allows us to suggest that water ice is formed
together with CO2 in molecular clouds with limited density, whilst formaldehyde
and methanol are mainly formed in a later phase, where the condensation becomes
denser and colder.
