A multi-scale scheme for the invasion percolation of rock fracture networks with heterogeneous fracture aperture fields is proposed. Inside fractures, fluid transport is calculated on the finest scale and found to be localized in channels as a conseq
uence of the aperture field. The channel network is characterized and reduced to a vectorized artificial channel network (ACN). Different realizations of ACNs are used to systematically calculate efficient apertures for fluid transport inside differently sized fractures as well as fracture intersection and entry properties. Typical situations in fracture networks are parameterized by fracture inclination, flow path length along the fracture and intersection lengths in the entrance and outlet zones of fractures. Using these scaling relations obtained from the finer scales, we simulate the invasion process of immiscible fluids into saturated discrete fracture networks, which were studied in previous works.
We investigate the metallic breakdown of a substrate on which highly conducting particles are adsorbed and desorbed with a probability that depends on the local electric field. We find that, by tuning the relative strength $q$ of this dependence, the
breakdown can change from continuous to explosive. Precisely, in the limit in which the adsorption probability is the same for any finite voltage drop, we can map our model exactly onto the $q$-state Potts model and thus the transition to a jump occurs at $q=4$. In another limit, where the adsorption probability becomes independent of the local field strength, the traditional bond percolation model is recovered. Our model is thus an example of a possible experimental realization exhibiting a truly discontinuous percolation transition.
Power grids, road maps, and river streams are examples of infrastructural networks which are highly vulnerable to external perturbations. An abrupt local change of load (voltage, traffic density, or water level) might propagate in a cascading way and
affect a significant fraction of the network. Almost discontinuous perturbations can be modeled by shock waves which can eventually interfere constructively and endanger the normal functionality of the infrastructure. We study their dynamics by solving the Burgers equation under random perturbations on several real and artificial directed graphs. Even for graphs with a narrow distribution of node properties (e.g., degree or betweenness), a steady state is reached exhibiting a heterogeneous load distribution, having a difference of one order of magnitude between the highest and average loads. Unexpectedly we find for the European power grid and for finite Watts-Strogatz networks a broad pronounced bimodal distribution for the loads. To identify the most vulnerable nodes, we introduce the concept of node-basin size, a purely topological property which we show to be strongly correlated to the average load of a node.
Blackouts in power grids typically result from cascading failures. The key importance of the electric power grid to society encourages further research into sustaining power system reliability and developing new methods to manage the risks of cascadi
ng blackouts. Adequate software tools are required to better analyze, understand, and assess the consequences of the cascading failures. This paper presents MATCASC, an open source MATLAB based tool to analyse cascading failures in power grids. Cascading effects due to line overload outages are considered. The applicability of the MATCASC tool is demonstrated by assessing the robustness of IEEE test systems and real-world power grids with respect to cascading failures.
Interconnected networks have been shown to be much more vulnerable to random and targeted failures than isolated ones, raising several interesting questions regarding the identification and mitigation of their risk. The paradigm to address these ques
tions is the percolation model, where the resilience of the system is quantified by the dependence of the size of the largest cluster on the number of failures. Numerically, the major challenge is the identification of this cluster and the calculation of its size. Here, we propose an efficient algorithm to tackle this problem. We show that the algorithm scales as O(N log N), where N is the number of nodes in the network, a significant improvement compared to O(N^2) for a greedy algorithm, what permits studying much larger networks. Our new strategy can be applied to any network topology and distribution of interdependencies, as well as any sequence of failures.
We performed numerical simulations of the $q$-state Potts model to compute the reduced conductivity exponent $t/ u$ for the critical Coniglio-Klein clusters in two dimensions, for values of $q$ in the range $[1;4]$. At criticality, at least for $q<4
$, the conductivity scales as $C(L) sim L^{-frac{t}{ u}}$, where $t$ and $ u$ are, respectively, the conductivity and correlation length exponents. For q=1, 2, 3, and 4, we followed two independent procedures to estimate $t / u$. First, we computed directly the conductivity at criticality and obtained $t / u$ from the size dependence. Second, using the relation between conductivity and transport properties, we obtained $t / u$ from the diffusion of a random walk on the backbone of the cluster. From both methods, we estimated $t / u$ to be $0.986 pm 0.012$, $0.877 pm 0.014$, $0.785 pm 0.015$, and $0.658 pm 0.030$, for q=1, 2, 3, and 4, respectively. We also evaluated $t / u$ for non integer values of $q$ and propose the following conjecture $40gt/ u=72+20g-3g^2$ for the dependence of the reduced conductivity exponent on $q$, in the range $ 0 leq q leq 4$, where $g$ is the Coulomb gas coupling.
Natural and technological interdependent systems have been shown to be highly vulnerable due to cascading failures and an abrupt collapse of global connectivity under initial failure. Mitigating the risk by partial disconnection endangers their funct
ionality. Here we propose a systematic strategy of selecting a minimum number of autonomous nodes that guarantee a smooth transition in robustness. Our method which is based on betweenness is tested on various examples including the famous 2003 electrical blackout of Italy. We show that, with this strategy, the necessary number of autonomous nodes can be reduced by a factor of five compared to a random choice. We also find that the transition to abrupt collapse follows tricritical scaling characterized by a set of exponents which is independent on the protection strategy.
To study epitaxial thin-film growth, a new model is introduced and extensive kinetic Monte Carlo simulations performed for a wide range of fluxes and temperatures. Varying the deposition conditions, a rich growth diagram is found. The model also repr
oduces several known regimes and in the limit of low particle mobility a new regime is defined. Finally, a relation is postulated between the temperatures of the kinetic and thermal roughening transitions.
The main purpose of this work is to simulate two-phase flow in the form of immiscible displacement through anisotropic, three-dimensional (3D) discrete fracture networks (DFN). The considered DFNs are artificially generated, based on a general distri
bution function or are conditioned on measured data from deep geological investigations. We introduce several modifications to the invasion percolation (MIP) to incorporate fracture inclinations, intersection lines, as well as the hydraulic path length inside the fractures. Additionally a trapping algorithm is implemented that forbids any advance of the invading fluid into a region, where the defending fluid is completely encircled by the invader and has no escape route. We study invasion, saturation, and flow through artificial fracture networks, with varying anisotropy and size and finally compare our findings to well studied, conditioned fracture networks.
The recent work by Achlioptas, DSouza, and Spencer opened up the possibility of obtaining a discontinuous (explosive) percolation transition by changing the stochastic rule of bond occupation. Despite the active research on this subject, several ques
tions still remain open about the leading mechanism and the properties of the system. We review the largest cluster and the Gaussian models recently introduced. We show that, to obtain a discontinuous transition it is solely necessary to control the size of the largest cluster, suppressing the growth of a cluster differing significantly, in size, from the average one. As expected for a discontinuous transition, a Gaussian cluster-size distribution and compact clusters are obtained. The surface of the clusters is fractal, with the same fractal dimension of the watershed line.
