  We first refine the fixed concept in the literature that the usage of the
Newtonian potential in studies of black hole accretion is invalid and the
general relativistic effect must be considered. As our main results, we then
show that the energy released by neutrino annihilation in neutrino-dominated
accretion flows is sufficient for gamma-ray bursts when the contribution from
the optically thick region of the flow is included, and that in the optically
thick region advection does not necessarily dominate over neutrino cooling
because the advection factor is relevant to the geometrical depth rather than
the optical depth of the flow.
