  A binary neutron star merger is expected to be associated by a kilonova,
transient optical emission powered by radioactive decay of the neutron-rich
ejecta. If the post-merger remnant is a long-lived neutron star, additional
energy injection to the ejecta is possible. In this first paper of a series, we
study the dynamical evolution of the engine-fed kilonova (mergernova) ejecta in
detail. We perform a semi-analytical study of the problem by adopting a
modified mechanical blastwave model that invokes interaction between a
Poynting-flux-dominated flow and a non-magnetized massive ejecta. Shortly after
the engine is turned on, a pair of shocks would be excited. The reverse shock
quickly reaches the wind-acceleration region and disappears (in a few seconds),
whereas the forward shock soon breaks out from the ejecta (in $10^2$ - $10^3$
seconds) and continues to propagate in the surrounding interstellar medium.
Most of the energy injected into the blastwave from the engine is stored as
magnetic energy and kinetic energy. The internal energy fraction is $f_{\rm
int} < 0.3$ for an ejecta mass equal to $10^{-3}M_{\odot}$. Overall, the energy
injecting efficiency $\xi$ is at most $\sim 0.6$ and can be as small as $\sim
0.04$ at later times. Contrary to the previous assumption, efficient heating
only happens before the forward shock breaks out of the ejecta with a heating
efficiency $\xi_t \sim (0.006 - 0.3)$, which rapidly drops to $\sim 0$
afterwards. The engine-fed kilonova lightcurves will be carefully studied in
Paper II.
