  We present the science case for surveys with the Tianlai dish array
interferometer tuned to the $\left[ 1300, 1400 \right] \mathrm{MHz}$ frequency
range. Starting from a realistic generation of mock visibility data according
to the survey strategy, we reconstruct a map of the sky and perform a
foreground subtraction. We show that a survey of the North Celestial Polar cap
during a year of observing time and covering an area of $150 \, \mathrm{deg^2}$
would reach a sensitivity of $ 1.5-2 \, \mathrm{mK} $ per $1 \, \mathrm{MHz}
\times 0.25^2 \, \mathrm{deg^2 }$ voxel and be marginally impacted by
mode-mixing. Tianlai would be able to detect a handful $(\sim 10)$ of nearby
massive \HI clumps as well as a very strong cross-correlation signal of 21\,cm
intensity maps with the North Celestial Cap Survey optical galaxies. We have
also studied the performance of a mid-latitude survey, covering $\sim 1500 \,
\mathrm{deg^2}$ centered on a declination of $\delta=55^\circ$, which overlaps
the Sloan Digital Sky Survey footprint. Despite a higher noise level for the
mid-latitude survey, as well as significant distortions due to mode mixing,
Tianlai would be able to detect a highly significant cross-correlation between
the 21\,cm signal and the Sloan spectroscopic galaxy sample. Using the
extragalactic signals from either or both of these surveys, it will be possible
to assess the impact of calibration uncertainties, antenna pattern
uncertainties, sources of noise, and mode mixing for future surveys requiring
higher sensitivity.
