Correlated fractal percolation and the Palis conjecture

Let F1 and F2 be independent copies of correlated fractal percolation, with Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question: does dimH(F1)+dimH(F2)>1 imply that their algebraic difference F1-F2 will contain an interval? The well known Palis conjecture states that `generically' this should be true. Recent work by Kuijvenhoven and the first author (arXiv:0811.0525) on random Cantor sets can not answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of (arXiv:0811.0525) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.Comment: 22 page

Simulating flows in multi-layered and spatially-variable permeability media via a new Gray Lattice Boltzmann model

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Percolation Thresholds in 2-Dimensional Prefractal Models of Porous Media

Considerable effort has been directed towards the application of percolation theory and fractal modeling to porous media. We combine these areas of research to investigate percolation in prefractal porous media. We estimated percolation thresholds in the pore space of homogeneous random 2-dimensional prefractals as a function of the fractal scale invariance ratio b and iteration level i. The percolation thresholds for these simulations were found to increase beyond the 0.5927... porosity expected in Bernoulli (uncorrelated) percolation networks. Percolation in prefractals occurs through large pores connected by small pores. The thresholds increase with both b (a finite size effect) and i. The results allow the prediction of the onset of percolation in models of prefractal porous media and can be used to bound modeling efforts. More fundamental applications are also possible. Only a limited range of parameters has been explored empirically but extrapolations allow the critical fractal dimension to be estimated for a large combination of b and i values. Extrapolation to infinite iterations suggests that there may be a critical fractal dimension of the solid at which the pore space percolates. The extrapolated value is close to 1.89 - the well-known fractal dimension of percolation clusters in 2-dimensional Bernoulli networks

High performance simulation of complicated fluid flow in 3D Fractured Porous Media with Permeable Material Matrix Using LBM

To analyze and depict complicated fluid behaviours in fractured porous media with various permeable material matrix across different scales, an Enhanced Heterogeneous Porous Media Computational Model is proposed based on Lattice Boltzmann method (LBM). LBM is widely employed to model basic fluid dynamics within disordered structures due to its powerful applicability to mesoscopic fluid mechanics and its potential performance of parallel computing. This paper combines with the force model, statistical material physics and the parallel algorithm to effectively describe the feature changes while fluid passes through the fractured porous media with diverse permeable material matrix of high resolution by using supercomputers. As an application example, a 3D sandstone sample is reconstructed with 36 million grids using the scanned CT images and characterized with different feature values at each lattice grid to distinguish pores, impermeable solids and permeable material matrix by stating its local physical property. The calculation and comparison results with the conventional LBM are discussed to demonstrate the advantages of our method in modeling complicated flow phenomena in fractured porous media with variable permeable material matrix across different scales, and its sound computing performance that keeps the parallel speedup linearly with the number of processors

Greater aridity increases the magnitude of urban nighttime vegetation-derived air cooling

High nighttime urban air temperatures increase health risks and economic vulnerability of people globally. While recent studies have highlighted nighttime heat mitigation effects of urban vegetation, the magnitude and variability of vegetation-derived urban nighttime cooling differs greatly among cities. We hypothesize that urban vegetation-derived nighttime air cooling is driven by vegetation density whose effect is regulated by aridity through increasing transpiration. We test this hypothesis by deploying microclimate sensors across eight United States cities and investigating relationships of nighttime air temperature and urban vegetation throughout a summer season. Urban vegetation decreased nighttime air temperature in all cities. Vegetation cooling magnitudes increased as a function of aridity, resulting in the lowest cooling magnitude of 1.4 °C in the most humid city, Miami, FL, and 5.6 °C in the most arid city, Las Vegas, NV. Consistent with the differences among cities, the cooling effect increased during heat waves in all cities. For cities that experience a summer monsoon, Phoenix and Tucson, AZ, the cooling magnitude was larger during the more arid pre-monsoon season than during the more humid monsoon period. Our results place the large differences among previous measurements of vegetation nighttime urban cooling into a coherent physiological framework dependent on plant transpiration. This work informs urban heat risk planning by providing a framework for using urban vegetation as an environmental justice tool and can help identify where and when urban vegetation has the largest effect on mitigating nighttime temperatures. © 2021 The Author(s). Published by IOP Publishing Ltd.Open access journalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]