  Context: Although type Ia supernovae (SNe Ia) play a key role in
astrophysics, the companions of the exploding carbon-oxygen white dwarfs (CO
WDs) are still not completely identified. It has been suggested recently that a
He-rich WD (a He WD or a hybrid HeCO WD) merges with a CO WD may produce an SN
Ia. This theory was based on the double-detonation model, in which the shock
compression in the CO core caused by the surface explosion of the He-rich shell
might lead to the explosion of the whole CO WD. However, so far, very few
binary population synthesis (BPS) studies have been made on the merger scenario
of a CO WD and a He-rich WD in the context of SNe Ia. Aims: We aim to
systematically study the Galactic birthrates and delay-time distributions of
SNe Ia based on the merger scenario of a CO WD and a He-rich WD. Methods: We
performed a series of Monte Carlo BPS simulations to investigate the properties
of SNe Ia from the merging of a CO WD and a He-rich WD based on the Hurley
rapid binary evolution code. We also considered the influence of different
metallicities on the final results. Results: From our simulations, we found
that no more than 15% of all SNe Ia stem from the merger scenario of a CO WD
and a He-rich WD, and their delay times range from ~110 Myr to the Hubble time.
This scenario mainly contributes to SN Ia explosions with intermediate and long
delay times. The present work indicates that the merger scenario of a CO WD and
a He-rich WD can roughly reproduce the birthrates of SN 1991bg-like events, and
cover the range of their delay times. We also found that SN Ia birthrates from
this scenario would be higher for the cases with low metallicities.
