  Macroscopic properties of heterogeneous media are frequently modelled by
regular lattice models, which are based on a relatively small basic cluster of
lattice sites. Here, we extend one of such models to any cluster's size kxk. We
also explore its modified form. The focus is on the percolation behaviour of
the effective conductivity of random two- and three-phase systems. We consider
only the influence of geometrical features of local configurations at different
length scales k. At scales accessible numerically, we find that an increase in
the size of the basic cluster leads to characteristic displacements of the
percolation threshold. We argue that the behaviour is typical of materials,
whose conductivity is dominated by a few linear, percolation-like, conducting
paths. Such a system can be effectively treated as one-dimensional medium. We
also develop a simplified model that permits of an analysis at any scale. It is
worth mentioning that the latter approach keeps the same thresholds predicted
by the former one. We also briefly discuss a three-phase system, where the
double-thresholds paths appear on model surfaces.
