On the scaling properties of various invasion models

We investigate the multiscaling behaviour of usual and directed invasion percolation, with and without a power-law decay with the distance from an initial seed in the distribution of random numbers. We find universal multiscaling behaviour only in the presence of a power-law gradient in space. Invasion percolation [ 11 is a geometrical growth model that has been used to describe the penetration of a fluid into a porous medium. The clusters of penetrating fluid created by invasion percolation evolve automatically into fractals indistinguishable from the incipient infinite clusters at the critical threshold of usual percolation. For this reason invasion percolation constitutes an ideal example of self-organized criticality. One of the most interesting questions about growth models is the scaling behaviour of their growth zone. In many cases, like the Eden model [2] or epidemics [3], a new set of growth exponents is found. Even more complicated seems to be the situation in nucleation [4] or diffusion limited aggregation (DLA) [ 5 ] where recently a new type of scaling, called multiscaling, has been proposed. In this case the effective fractal dimension continuously varies radially within the growth zone. The growth of invasion percolation clusters occurs as a sequence of jumps [ 6 ] , and it is therefore likely that this behaviour reflects in the scaling properties of the growth zone. It is the purpose of this letter to investigate the radial dependence of the fractal dimension for various invasion percolation models. Besides the standard version of the model without trapping [ 13 we also consider directed invasion percolation and invasion percolation with a spatially graded distribution of random numbers. Invasion percolation is defined by placing on each site of a regular lattice a random number zi E (0,l) uncorrelated from site to site. Then one chooses in the centre of the lattice a site, the seed of the growth, and occupies this site. Finally one grows the cluster by applying over and over again the following rule: one chooses among all the sites that are nearest neighbours of occupied sites the one which has the smallest value of z, and occupies it. On a finite lattice the growth is stopped when a site on the boundary is occupied. The ensemble of occupied sites produced with this algorithm is necessarily a single cluster, i.e. each site is connected to any other site by a path of nearest-neighbouring occupied sites. It has been convincingly argued and shown numerically, but not yet rigorously proven, that the fractal dimension df of this cluster equals the fractal 0305-4470/90/170923 +06%03.50 @ 1990 IOP Publishing Ltd L923 L924 Letter to the Editor dimension of the spanning cluster at p c in standard percolation, namely df= 91/48 in two dimensions. Besides the usual invasion percolation without trapping, described above, we also study here the directed case. This implies that each bond on the lattice allows occupation in one direction only and not in the other. In our case we consider a square lattice with all horizontal bonds directed to the right and all vertical bonds directed downwards. In the description of a fluid penetrating a porous medium this means that the channels between pores have valves that allow the fluid to flow in one direction only. The algorithmic implementation of directed invasion percolation is straightforward: the site chosen to be occupied must not just be a nearest neighbour of an already occupied one, but the occupied neighbour must also be either up or to the left in the case of a square lattice. The random numbers z assigned at the beginning to the sites of the lattice are usually uniformly distributed. We study here also the effects of radially graded random numbers. The purpose is to investigate either the effect of a pressure gradient, or a radial variation in the structure of the porous medium. In order to cover a wide range of different distributions we consider a power-law decay in the value of the random numbers as function of the distance from the seed, that is