Invasion Percolation with Long-Range Correlations: First Order Phase Transitions and Nonuniversal Scaling Properties

Abstract

We present the results of extensive Monte Carlo simulations of the invasion percolation model with trapping (TIP) with long-range correlations, a problem which is relevant to multiphase flow in field-scale porous media, such as oil reservoirs and groundwater aquifers, as well as flow in rock fractures. The correlations are generated by a fractional Brownian motion characterized by a Hurst exponent H. We employ a highly efficient algorithm for simulating TIP, and a novel method for identifying the backbone of TIP clusters. Both site and bond TIP are studied. Our study indicates that the backbone of bond TIP is loopless and completely different from that of site TIP. We obtain precise estimates for the fractal dimensions of the sample-spanning cluster (SSC), the minimal path, and the backbone of site and bond TIP, and analyze the size distribution of the trapped clusters, in order to identify all the possible universality classes of TIP with long-range correlations. For site TIP with H>1/2 the SSC and its backbone are compact, indicating a first-order phase transition at the percolation threshold, while the minimal paths are essentially straigth lines. For H<1/2 the SSC, its backbone, and the minimal paths are all fractal with fractal dimensions that depend on the Hurst exponent H. The fractal dimension of the loopless backbone for bond TIP is much less than that of site TIP for any H.