Given the variety and large number of systems displaying scale invariant characteristics, it is becoming increasingly important to understand their fundamental and universal elements. Much work has attempted to apply second order phase transition mechanics due to the emergent scale invariance at the critical point. This symmetry produces much of the fractal characteristics that is shared among these systems. Percolation systems offer a simple and well developed framework that exhibit all the key features of a second order phase transition, and has been a fertile framework for exploring generalizing insights. A close variant is invasion percolation.
Invasion percolation is a model that was originally proposed to describe growing networks of fractures. Here, we first describe a loopless algorithm on random lattices, coupled with an avalanche-based model for bursts. The model reproduces the characteristic b-value seismicity and spatial distribution of bursts consistent with earthquakes resulting from hydraulic fracturing (``fracking"). We test models for both site invasion percolation (SIP) and bond invasion percolation (BIP) and characterize their density, cluster, and correlation scaling exponents $D_f$, $\tau$, and $\nu$ respectively. These models have differences on the scale of site and bond lengths $l$, but since the networks are characterized by their large scale behavior, $l \ll L$, we find only small differences between their respective scaling exponents. Though these differences are likely too small for empirical data to differentiate between models, they remain significant enough to suggest that both models belong to different universality classes.
Because we likely cannot empirically distinguish differences associated with their underlying critical characteristics, we next aim to understand the particular manifestation of criticality in the avalanche invasion percolation (AIP) system. This is a SIP based model with additional growth mechanics described through distinct, sequentially connected bursts. Instead, we find that it is a hybrid critical system with the presence of a critical Fisher type distribution, $n_s (\tau, \sigma)$, but lacks other essential features such as an order parameter and to a lesser degree hyperscaling. This suggests that we do not need a full phase transition description in order to observe scale invariant behavior. This was a positive result, since for many systems, notions of phases and critical points are both artificial and cumbersome. AIP also recasts dynamics as a slowly driven non-equilibrium proccss and provides a pathway for more suitable descriptions.
We extend our previous model, avalanche-burst invasion percolation (AIP) by introducing long-range correlations between sites described by fractional Brownian statistics. In our previous models with independent, random site strengths, we reproduced a unique set of power-laws consistent with some of the b-values observed during induced seismicity. We expand upon these models to produce a family of critical exponents which would be characterized by the local long-range correlations inherent to host sediment. Further, in previous correlated invasion percolation studies, fractal behavior was found in only a subset of the range of Hurst exponent, $H$. We find fractal behavior persists for the entire range of Hurst exponent. Additionally, we show how multiple cluster scaling power laws result from changing the generalized Hurst parameter controlling long-range site correlations, and give rise to a truly multifractal system. This emergent multifractal behavior plays a central role in allowing us to extend our model to better account for variations in the observed Gutenberg-Richter b-values of induced seismicity. Finally this provides a framework for generating scale invariant statistics without adhering to the scaling hypothesis which underlies the scale invariance of critical systems.
