  Forward Drell-Yan process at the LHC probes the proton structure at very
small Bjorken-$x$ and moderate hard scales. In this kinematical domain higher
twist effects may be significant and introduce sizeable corrections to the
standard leading twist description. We study the forward Drell-Yan process
beyond the leading twist approximation within the color dipole model framework
that incorporates multiple scattering effects. We derive the Mellin
representation of the forward Drell-Yan impact factors for fully differential
cross-sections. These results combined with the color dipole cross-section of
the saturation model are used to perform twist expansion of the Drell-Yan
structure functions at arbitrary transverse momentum $q_T$ of the Drell-Yan
pair and also of the structure functions integrated over $q_T$. We also
investigate the Lam-Tung relation, find that it is broken at twist 4 and
provide explicit estimates for the breaking term.
