  We investigate the process of invasion percolation between two sites
(injection and extraction sites) separated by a distance r in two-dimensional
lattices of size L. Our results for the non-trapping invasion percolation model
indicate that the statistics of the mass of invaded clusters is significantly
dependent on the local occupation probability (pressure) Pe at the extraction
site. For Pe=0, we show that the mass distribution of invaded clusters P(M)
follows a power-law P(M) ~ M^{-\alpha} for intermediate values of the mass M,
with an exponent \alpha=1.39. When the local pressure is set to Pe=Pc, where Pc
corresponds to the site percolation threshold of the lattice topology, the
distribution P(M) still displays a scaling region, but with an exponent
\alpha=1.02. This last behavior is consistent with previous results for the
cluster statistics in standard percolation. In spite of these discrepancies,
the results of our simulations indicate that the fractal dimension of the
invaded cluster does not depends significantly on the local pressure Pe and it
is consistent with the fractal dimension values reported for standard invasion
percolation. Finally, we perform extensive numerical simulations to determine
the effect of the lattice borders on the statistics of the invaded clusters and
also to characterize the self-organized critical behavior of the invasion
percolation process.
