  In this paper, we investigate the invasion percolation (IP) in imperfect
support in which the configuration of imperfections is considered to be
correlated. Three lattice models were engaged to realize this pattern: site
percolation, Ising model and random Coulomb potential (RCP). The first two
models are short range interaction (SRI), whereas the last one includes coulomb
like interactions which is pretty long range (long-range interactions, LRI). By
examining various dynamical observables we show that the critical exponents of
SRI IP are robust against the control parameters (temperature in the Ising
model and occupation probability in site percolation), whereas its properties
in the LRI (RCP) supports are completely different from the normal IP (i.e. on
the regular lattice). Especially the fractal dimension of the external frontier
of the largest hole converges to $1.099\pm 0.008$ for RCP IP, whereas it is
nearly $\frac{4}{3}$ for SRI IP being compatible with normal IP. Additionally a
novel dynamical crossover is seen in the RCP IP according to which the time
dependence of all of the observables is divided to three parts: the power-law
(small times), the logarithmic (mid time), and the linear (long time) regimes.
The second crossover time is shown to go to infinity in the thermodynamic
limit, whereas the first crossover time is nearly unchanged, signaling the
dominance of the logarithmic regime. The observables become nearly constant in
the thermodynamic limit for the long time, showing that it is a stationary
phase.
