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

On multi-scale percolation behaviour of the effective conductivity for the lattice model