Narrow Results

This study investigates the use of neuromorphic computing, particularly spiking neural networks (SNNs) and advanced neuromorphic hardware, to model and forecast climate patterns. Our neuromorphic system achieves high prediction accuracy, maintaining a Mean Squared Error (MSE) as low as 0.08, even with increasing data volumes. The system operates with notable energy efficiency, consuming just 0.15 J per inference at higher data loads. This efficiency, coupled with a throughput of 800 inferences per second, underscores the system’s capability to handle large-scale data effectively. The neuromorphic approach addresses key challenges in scalability and energy consumption, presenting a robust solution for real-time climate data analysis. By continuously adapting to new data inputs, the system ensures accurate and timely predictions, essential for applications in environmental monitoring and decision-making. The integration of artificial intelligence algorithms with neuromorphic architectures not only reduces computational costs but also enhances the interpretability of complex climate dynamics. These findings highlight the transformative potential of brain-like computing in environmental modeling, offering a scalable, efficient, and adaptable tool for climate prediction and analysis.

We compare the recently formulated multifractional spacetimes with field theories of quantum gravity based on the renormalization group (RG), such as asymptotic safety and Hořava–Lifshitz gravity. The change of spacetime dimensionality with the probed scale is realized in both cases by an adaptation of the measurement tools ("rods") to the scale, but in different ways. In the multifractional case, by an adaptation of the position-space measure, which can be encoded into an explicit scale dependence of effective coordinates. In the case of RG-based theories, by an adaptation of the momenta. The two pictures are mapped into each other, thus presenting the fractal structure of spacetime in RG-based theories under an alternative perspective.

A new method for estimating fractal dimension of tree crowns from digital images is presented. Three species of trees, Japanese yew (Taxus cuspidata Sieb & Zucc), Hicks yew (Taxus × media), and eastern white pine (Pinus strobus L.), were studied. Fractal dimensions of Japanese yew and Hicks yew range from 2.26 to 2.70. Fractal dimension of eastern white pine range from 2.14 to 2.43. The difference in fractal dimension between Japanese yew and eastern white pine was statistically significant at 0.05 significance level as was the difference in fractal dimension between Hicks yew and eastern white pine. On average, the greater fractal dimensions of Japanese yew and Hicks yew were possibly related to uniform foliage distribution within their tree crowns. Therefore, fractal dimension may be useful for tree crown structure classification and for indexing tree images.

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-similar two-dimensional substructures manifest themselves, ad infinitum, as the number of cells (I + 1) → ∞.

In addition to a complete gallery of time-1 characteristic functions for all 256 local rules, an accompanying table of explicit formulas is given for generating these characteristic functions directly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potential applications of these fundamental formulas, we prove rigorously that the "right-copycat" local rule is equivalent globally to the classic "left-shift" Bernoulli map. Similarly, we prove the "left-copycat" local rule is equivalent globally to the "right-shift" inverse Bernoulli map.

Various geometrical and analytical properties have been identified from each characteristic function and explained rigorously. In particular, two-level stratified subpatterns found in most characteristic functions are shown to emerge if, and only if, b₁ ≠ 0, where b₁ is the "synaptic coefficient" associated with the cell differential equation developed in Part I.

Gardens of Eden are derived from the decimal range of the characteristic function of each local rule and tabulated. Each of these binary strings has no predecessors (pre-image) and has therefore no past, but only the present and the future. Even more fascinating, many local rules are endowed with binary configurations which not only have no predecessors, but are also fixed points of the characteristic functions. To dramatize that such points have no past, and no future, they are henceforth christened "Isles of Eden". They too have been identified and tabulated.

Cumulative frequency-size distributions associated with many natural phenomena follow a power law. Self-organized criticality (SOC) models have been used to model characteristics associated with these natural systems. As originally proposed, SOC models generate event frequency-size distributions that follow a power law with a single scaling exponent. Natural systems often exhibit power law frequency-size distributions with a range of scaling exponents. We modify the forest fire SOC model to produce a range of scaling exponents. In our model, uniform energy (material) input produces events initiated on a self-similar distribution of critical grid cells. An event occurs when material is added to a critical cell, causing that material and all material in occupied non-diagonal adjacent cells to leave the grid. The scaling exponent of the resulting cumulative frequency-size distribution depends on the fractal dimension of the critical cells. Since events occur on a self-similar distribution of critical cells, we call this model Self-Similar Criticality (SSC). The SSC model may provide a link between fractal geometry in nature and observed power law frequency-size distributions for many natural systems.

The solution to a deceptively simple combinatorial problem on bit strings results in the emergence of a fractal related to the Sierpinski Gasket. The result is generalized to higher dimensions and applied to the study of global dynamics in Boolean network models of complex biological systems.

The geometric features of the square and triadic Koch snowflake drums are compared using a position entropy defined on the grid points of the discretizations (prefractals) of the two domains. Weighted graphs using the geometric quantities are created and random walks on the two prefractals are performed. The aim is to understand if the existence of narrow channels in the domain may cause the "localization" of eigenfunctions.

Stimulated reservoir volume (SRV) with large fracture networks can be generated near hydraulic fractured vertical wells (HFVWs) in tight oil reservoirs. Statistics show that natural microfractures and fracture networks stimulated by SRV were self-similar in statistical sense. Currently, various analytical models have been presented to study pressure behaviors of HFVWs in tight oil reservoirs. However, most of the existing models did not take the distribution and self-similarity of fractures into consideration. To account for stimulated characteristic and self-similarity of fractures in tight oil reservoirs, a mixed-fractal flow model was presented. In this model, there are two distinct regions, stimulated region and unstimulated region. Dual-porosity model and single porosity model were used to model stimulated and unstimulated regions, respectively. Fractal geometry is employed to describe fractal permeability and porosity relationship (FPPR) in tight oil reservoirs. Solutions for the mixed-fractal flow model were derived in the Laplace domain and were validated among range of the reservoir parameters. The pressure transient behavior and production rate derivative were used to analyze flow regimes. The type curves show that the fluid flow in HFVWs can be divided into six main flow periods. Finally, effect of fractal parameters and SRV size on flow periods were also discussed. The results show that the SRV size and fractal parameters of fracture network have great effect on the former periods and fractal parameters of matrix mainly influence the later flow periods.

According to hydraulic-fracturing practices conducted in shale reservoirs, effective stimulated reservoir volume (ESRV) significantly affects the production of hydraulic fractured well. Therefore, estimating ESRV is an important prerequisite for confirming the success of hydraulic fracturing and predicting the production of hydraulic fracturing wells in shale reservoirs. However, ESRV calculation remains a longstanding challenge in hydraulic-fracturing operation. In considering fractal characteristics of the fracture network in stimulated reservoir volume (SRV), this paper introduces a fractal random-fracture-network algorithm for converting the microseismic data into fractal geometry. Five key parameters, including bifurcation direction, generating length ($d$), deviation angle ($\alpha$), iteration times ($N$) and generating rules, are proposed to quantitatively characterize fracture geometry. Furthermore, we introduce an orthogonal-fractures coupled dual-porosity-media representation elementary volume (REV) flow model to predict the volumetric flux of gas in shale reservoirs. On the basis of the migration of adsorbed gas in porous kerogen of REV with different fracture spaces, an ESRV criterion for shale reservoirs with SRV is proposed. Eventually, combining the ESRV criterion and fractal characteristic of a fracture network, we propose a new approach for evaluating ESRV in shale reservoirs. The approach has been used in the Eagle Ford shale gas reservoir, and results show that the fracture space has a measurable influence on migration of adsorbed gas. The fracture network can contribute to enhancement of the absorbed gas recovery ratio when the fracture space is less than 0.2 m. ESRV is evaluated in this paper, and results indicate that the ESRV accounts for 27.87% of the total SRV in shale gas reservoirs. This work is important and timely for evaluating fracturing effect and predicting production of hydraulic fracturing wells in shale reservoirs.

With the purpose of researching the changing regularities of the Cantor set’s multi-fractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the Cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multi-fractal spectrums and the generalized fractal dimensions will be acquired by computer calculation. The following conclusions can be acquired. On one hand, the maximal width of the multi-fractal spectrum and the maximal vertical height of the generalized fractal dimension will become more and more narrow with getting two probability factors closer and closer. On the other hand, when two probability factors are equal to 1/2, both the multi-fractal spectrum and the generalized fractal dimension focus on the value 0.6309, which is not the value of the physical multi-fractal spectrum and the generalized fractal dimension but the mathematical Hausdorff dimension.

The Sierpinski Triangle (ST) is a fractal which has Haussdorf dimension ${log}_{2}3\approx 1.585$ that has been studied extensively. In this paper, we introduce the Sierpinski Triangle Plane (STP), an infinite extension of the ST that spans the entire real plane but is not a vector subspace or a tiling of the plane with a finite set of STs. STP is shown to be a radial fractal with many interesting and surprising properties.

It has been proved that effective stimulated reservoir volume (ESRV) is a significant area dominant to the production of the fractured well in unconventional gas reservoirs. Although ESRV properties can be estimated based on the microseismic technology and the analysis of actual fracturing data, the operations are complicated and results are inaccurate. Due to the complex structure of stimulated reservoir volume (SRV) with fractal and chaotic characteristics, a fractal evaluation model for ESRV (ESRV-FEM) of fractured wells in unconventional gas (both tight and shale gas) reservoirs is developed. Multiple gas transport mechanisms, SRV and unstimulated reservoir volume (USRV) are included. According to the pressure transient analysis (PTA), influences of multiple transport mechanisms on gas transport behaviors in ESRV are conducted. Moreover, the fractal index representing the heterogeneity degree is applied to estimate the ESRV under different inter-porosity flow coefficients and storage ratio conditions based on the ESRV-FEM. In addition, the presented ESRV-FEM is validated by an actual field case. The results show that gas adsorption has a significant effect on the radial flow duration time in SRV, and the heterogeneity makes the radial flow on PTA curves no longer show a horizontal line with the value of 0.5. Calculated ESRV sizes are compared with the assumed ones under different fractal indexes. The stronger the heterogeneity, the smaller ESRV is. The ESRV size of a fractured well in shale gas reservoirs is only 52.11% of SRV size when the fractal index equals to 0.6. The ESRV-FEM presented in this paper is expected to provide an effective method for the evaluation of the ESRV of fractured wells in unconventional gas reservoirs.

Most models for multiple fractured horizontal wells (MFHWs) in tight oil reservoirs (TORs) are based on classical simplified dual-porosity model that ignores the influence of imbibition, while the distribution of fracture system is heterogeneous, multi-scale and self-similar, which can be described by fractal dual-porosity model on fractal theory, and imbibition production is the important part of fracture system production. In this paper, a multi-linear fractal model (MFM) considering imbibition for MFHWs in TORs was established based on fractal theory and semi-analytical method. In this model, fractal theory was used to describe the heterogeneous, complex fracture network, and imbibition was considered by analogy of fluid crossflow law in fractured-porous dual media. And the approximate analytic solution was given by using the Laplace transformation and iteration method. The pressure responses in the domain of real time were obtained with Stehfest numerical inversion algorithms. The pressure transient and production rate were used to analyze, and sensitivity analysis of some related parameters were discussed. The results show that the fluid flow in MFHWs can be divided into nine main flow periods by analysis of type curves, and the fractal parameters of fracture system have great effect on the middle and later periods and imbibition influences the period of crossflow.

This paper focuses on applying fractal Julia sets to observe the topological properties related to the signs of the real and reactive electric powers. To perform this, different power combinations were used to represent the fractal diagrams with an algorithm that considers the mathematical model of Julia sets. The study considers three cases: the first study considers the change of real power when the reactive power is fixed; the second study deals with the change of the reactive power when the real power is fixed; and finally, the third study contemplates that both real and reactive powers change. Furthermore, the fractal diagrams of the power in the four quadrants of the complex plane are studied to identify the topological properties for each sign. A qualitative analysis of the diagrams helps identify that complex power loads present some fractal graphic patterns with respect to the signs considered in the different quadrants of the complex planes. The diagrams represented in the complex planes save a relation in the forms and structure with other points studied, concluding that the power is related to other figures in other quadrants. Thus, this result allows a new study of the behavior of power in an electrical circuit by showing a clear relation of the different fractal diagrams obtained by the Julia sets.

In order to research roughness of rock fracture surfaces and interior damaged status of rock, some Brazil discs of white marble and sandstone are loaded to fracture by Brazil Test and the mathematical model of fractal variation is used to distinguish the morphology characterization of fracture surface of white marble from that of sandstone. Through computing the acquired scanned data of rock fracture surfaces, the following three results are obtained. First, the roughness of fracture surface increases with increase of loading angles, furthermore, the anisotropy of white marble fracture surface is more obvious than that of sandstone one. Second, under the same loading mode, by analyzing the mean fractal variation, the fracture surface of white marble is rougher than that of sandstone. Finally, rock disc specimens loaded to 1/2 or 3/4 of mechanics of mean fracture threshold still do not fracture at the first time loading, but their interiors have been badly damaged.

The purpose of this research is to show how the complicated and irregular fractal interpolation function is represented by Fourier series. First, on the closed interval [0,1], even prolongation is operated to the fractal interpolation function generated by iterated function system constituted by affine transform and Fourier cosine series representation of fractal interpolation function is proved. Second, for fractal interpolation function, odd prolongation is done and Fourier sine series formula of fractal interpolation function is proved. Final, Fourier series expansion of fractal interpolation function on the closed interval $[-1,1]$ is proved. The result shows that complex fractal interpolation function can be represented by Fourier sine series and Fourier cosine series, so relatively simple Fourier series can be used to represent relatively complicated fractal interpolation function.

The present paper characterizes the resistive and inductive loads of an electric distribution system by Julia fractal sets, in order to discover other observations enabling the elevation of new theoretical approaches. The result shows that indeed the electrical load reflects a clear graphic pattern in the fractal space of the Julia sets. This result, then, is a new contribution that extends the universal knowledge about fractal geometry.

The multiphase flow behavior in shale porous media is known to be affected by multiscale pore size, dual surface wettability, and nanoscale transport mechanisms. However, it has not been fully understood so far. In this study, fractal model of gas–water relative permeabilities (RP) in dual-wettability shale porous media for both injected water spontaneous imbibition and the flow back process are proposed using fractal geometry. The shale pore structure is described as tortuous with different pore sizes and morphologies including slit pore, equilateral triangle, circular pore and square pore. The proportion of each pore morphology can be obtained from SEM/FIB-SEM pore structure characterization results. Injected water spontaneous imbibition after hydraulic fracturing is modeled as the capillary force dominated process and injected water flow back is modeled as a non-wetting gas phase drainage process in inorganic matter. The organic pores are deemed to be not accessible by injected water. The boundary slip of water and free gas flow in the inorganic matrix are considered while both free gas flow and adsorbed gas flow are modeled in organic matter. The proposed gas–water RP fractal model is verified via comparisons with the available experimental data and is discussed in detail. Study results reveal that gas phase RP increases with increasing pore fractal dimensions and tortuosity fractal dimensions, whereas it decreases with increasing Total Organic Carbon (TOC) volumes. Water phase RP decreases with increasing of pore fractal dimensions and tortuosity fractal dimensions, whereas it increases with increasing TOC volumes.

Fluid transport in shales is complex due to the various storage spaces and multiple transport mechanisms, especially for multiphase transport during flowback and early stage of production. This study proposes a gas-water relative permeability fractal model during a gas displacing water process in shale gas reservoirs, with incorporations of (1) real gas transport controlled by Knudsen Number (Kn) and second-order slip boundary, (2) slip length for water phase transport, (3) a mobile water film with varying thickness due to rock–fluid interaction and (4) stress-dependence. Specially, the varying thickness of water film is determined according to the extended Derjaguin–Landau–Verwey–Overbeek (DLVO) theory through van der Waals, electrostatic and structural force during a drainage process. Moreover, the organic matter (OM) and inorganic matter (IOM) pore structures are considered with individual pore/tortuosity fractal dimensions. The proposed model is verified by comparing with an analytical model and experimental data. Results show that the decreasing pore pressure during depressurization brings a decline in gas relative permeability, while the decreasing pore pressure has little impact on water relative permeability. The impact of pore and tortuosity fractal dimensions of OM can be ignored compared with that of IOM. Furthermore, neglecting the mobile water film with varying thickness during a gas drainage process leads to an overestimation of gas relative permeability, especially at smaller pore sizes. This work presents a comprehensive model to determine gas-water relative permeability in shales by considering fluids/reservoir properties and rock–fluid interaction in full, which reveals multiphase transport mechanisms in the unconventional reservoirs.

Accurate characterization of pore-scale structures of porous media is necessary for studying their transport mechanisms and properties. An analytical model for pore and capillary structures of porous media is developed based on fractal theory in this study. The pore and tortuosity fractal dimensions are introduced to characterize the pore size distribution and tortuous flow paths. A power law scaling between fractal probability function and pore diameter is proposed, which can be applied to determine the pore fractal dimension. The explicit expression for tortuosity fractal dimension is derived based on exactly self-similar fractal set and fractal capillary bundle model. The present fractal model has been validated by comparison with that of experiments and numerical simulations as well as theoretical models. The results show that the tortuosity fractal dimension decreases as porosity and pore fractal dimension increase, it increases with the increment of tortuosity. Both the particle shape and pore size range take important effect on the tortuosity fractal dimension under certain porosity. The proposed pore-scale model can present a conceptual tool to study the transport mechanisms of porous media and may provide useful guideline for oil and gas exploitation, hydraulic resource development, geotechnical engineering and chemical engineering.