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 nontrapping invasion percolation model indicate that the statistics of the mass of invaded clusters is significantly dependent on the local occupation probability (pressure) $p_e$ at the extraction site. For $p_e=0$, we show that the mass distribution of invaded clusters $P\left(M\right)$ follows a power-law $P\left(M\right)\sim M^{-\alpha }$ for intermediate values of the mass $M$, with an exponent $\alpha =1.39\pm 0.03$. When the local pressure is set to $p_e=p_c$, where $p_c$ corresponds to the site percolation threshold of the lattice topology, the distribution $P\left(M\right)$ still displays a scaling region, but with an exponent $\alpha =1.02\pm 0.03$. This last behavior is consistent with previous results for the cluster statistics in standard percolation. In spite of these differences, the results of our simulations indicate that the fractal dimension of the invaded cluster does not depend significantly on the local pressure $p_e$ 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.

• Received 14 February 2005

DOI:https://doi.org/10.1103/PhysRevE.72.041404