The Optimized Model of Multiple Invasion Percolation

When a nonviscous liquid is injected into a porous medium already filled with a viscous fluid the system can be found in two different regimes: one where the dominant forces are of capillary nature and another where the viscous forces are predominant. The theoretical description of such system is based on two models: diffusion-limited aggregation (DLA) 1 and invasion percolation . DLA describes the fast displacement situation when the viscous forces are dominant. The invasion percolation model is applied to a slow fluid flow, where the leading forces are of capillary nature. The fluid displacement follows minimum resistance paths: the smaller pores are filled or invaded first. The original invasion percolation model have been modified in order to bring it as close as possible to real world. So, the action of an external gravitational field 3,4,5 and the flux with a privileged direction 6,7 were incorporated into the model. Recently 8 the multiple invasion percolation model was proposed. In this model a certain number of lattice sites can be simultaneously invaded. It is always important to have in mind that invasion percolation is a dynamical model, that is, it intends to explain not only static properties like the clusters fractal dimensions but also how these clusters evolve in time. The lack of a meaning on associating or trying to make a correspondence between one step growth interval (of the algorithm) and one time interval separating two sucessives pore’s invasions (of the real world) is what we were concerned by the time we formulated the multiple invasion percolation model . In our models, one real time interval corresponds exactly to one growth