  We present thorough analysis of the nonlinear tunneling of Bose-Einstein
condensates in static and accelerating two-dimensional lattices within the
framework of the mean-field approximation. We deal with nonseparable lattices
considering different initial atomic distributions in the highly symmetric
states. For analytical description of the condensate before instabilities are
developed, we derive several few-mode models, analyzing both essentially
nonlinear and quasi-linear regimes of tunneling. By direct numerical
simulations, we show that two-mode models provide accurate description of the
tunneling when either initially two states are populated or tunneling occurs
between two stable states. Otherwise a two-mode model may give only useful
qualitative hints for understanding tunneling but does not reproduce many
features of the phenomenon. This reflects crucial role of the instabilities
developed due to two-body interactions resulting in non-negligible population
of the higher bands. This effect becomes even more pronounced in the case of
accelerating lattices. In the latter case we show that the direction of the
acceleration is a relevant physical parameter which affects the tunneling by
changing the atomic rates at different symmetric states and by changing the
numbers of bands involved in the atomic transfer.
