Fractal Network Research Articles

A smart productivity evaluation method for shale gas wells based on 3D fractal fracture network model

The generation method of three-dimensional fractal discrete fracture network (FDFN) based on multiplicative cascade process was developed. The complex multi-scale fracture system in shale after fracturing was characterized by coupling the artificial fracture model and the natural fracture model. Based on an assisted history matching (AHM) using multiple-proxy-based Markov chain Monte Carlo algorithm (MCMC), an embedded discrete fracture modeling (EDFM) incorporated with reservoir simulator was used to predict productivity of shale gas well. When using the natural fracture generation method, the distribution of natural fracture network can be controlled by fractal parameters, and the natural fracture network generated coupling with artificial fractures can characterize the complex system of different- scale fractures in shale after fracturing. The EDFM, with fewer grids and less computation time consumption, can characterize the attributes of natural fractures and artificial fractures flexibly, and simulate the details of mass transfer between matrix cells and fractures while reducing computation significantly. The combination of AMH and EDFM can lower the uncertainty of reservoir and fracture parameters, and realize effective inversion of key reservoir and fracture parameters and the productivity forecast of shale gas wells. Application demonstrates the results from the proposed productivity prediction model integrating FDFN, EDFM and AHM have high credibility.

FRACTAL PROPAGATION AND IMMUNITY ON NETWORK

In this paper, the epidemic spreading was investigated from the point of view of fractal. Firstly, the real network was abstracted as a fully connected fractal network. Based on the fractal network, the fractal spreading process was studied. The fractal spreading process was simulated to analyze the change of infection density during transmission. The results showed that the infection density presented an upward trend of the ladder-shaped, and a jumping change in infection density occurred during a certain time. As an illustration, the pandemic of COVID-19 was analyzed, the results indicated that the proposed method was valid. Experiments and analyses have shown that intervention at key jump points in virus propagation can effectively control the spread of the virus.

Quantum transport in fractal networks

Quantum transport in fractal networks

Towards point cloud completion: Point Rank Sampling and Cross-Cascade Graph CNN

The Point Fractal Network (PF-Net) is a seminal work with capability of completing the missing regions of point clouds. However, the multi-resolution structure of PF-Net neglects effective feature fusion between each resolution, which makes the resulting shape lack local geometric details. To tackle this problem, we design a novel shape completion network named PRSCN. We first present Point Rank Sampling to rate and sample feature points more objectively through local outline form. In this way, the sampled points can facilitate the downstream tasks. Subsequently, considering the correlation between features from different scales, we design a Cross-Cascade Module to combine features hierarchically. Moreover, we propose Leap-type EdgeConv to enlarge the receptive field while maintain the kernel size unchanged. These improvements together make our CD error 3% lower than that of state-of-the-art method on ShapeNet-part dataset.

Diffusional and Thermal Resistances of Substomatal Cavity and Its Application on Wick

Microstructures of wicks have been widely investigated on their performances of heat and mass transfer and various structures have been attempted. As have been optimized by the nature, the microstructures inside plants may also be applied on the wick structure. In this paper, four substomatal cavity structures are chosen to investigate the influence of the structural parameters on the diffusional resistance by using Fick's first law of diffusion and the equation of mass conservation comprehensively. The result indicates that both the shape and parameters affect the diffusion, and the sphere cavity structure presents the lowest diffusional resistance because of its smoothest surface. Based on the result of the comparison, the sphere is chosen as the optimal structure of the substomatal cavity and applied on the design of the microstructure of wick. In order to analyze the performances of the substomatal cavity inspired wick structure with fractal network microchannels and sphere cavity, comparisons with the leaf vein inspired wick structure with fractal network microchannels and micro fin-pins are conducted. It is obvious to find that the substomatal cavity inspired structure is promising in the application on wick structure because of its good performances of heat and mass transfer.

Combined extrusion-printed and laser-induced graphene enabled self-sensing composites with a strategic roadmap toward optimization of piezoresistivity

Laser-induced graphene (LIG) has endowed the assembly and integration of smart devices/structures in next-generation multifunctional composites with unique processing characteristics. Current LIG hybridized composites rely on precedent formation of graphene before integration process, restricting in situ structural design and performance regulation toward multi-scenario sensing applications. Here, an all-computer-aided protocol is creatively proposed with combined extrusion-printing (EP) and LIG for guaranteeing the graphene conversion on filled or pre-patterned polyimide layer, which is effective for balancing manufacturing cost and structural complexity. On the basis of unique dual-modal processing, piezoresistivity of LIG composites is optimized by a roadmap study through sequential and/or combinational control of lasing conditions, levels of pre-deformation treatment, and formats of multi-scaled fractal network. It is discovered that multi-scribing plus pre-bending with specific micro-contact structural design could regularly improve the gauge factor of LIG/composites from 0.67 to 648.8 to assure accurate strain mapping.

Average path length and degree distribution of networks generated by random sequence

Considering that many real networks do not have strict self-similarity property, compared with deterministic evolutionary fractal networks, networks with random sequence structure may be more in accordance with the properties of real networks. In this paper, we generate a hierarchical network by a random sequence based on BRV model. Using the encoding method, we present a way to judge whether two nodes are neighbors and calculate the total path length of the network. We get the degree distribution and limit formula of the average path length of a class of networks, which are obtained by analytical method and iterative calculation.

An Extended Correlation Dimension of Complex Networks.

Fractal and self-similarity are important characteristics of complex networks. The correlation dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimension is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects.

Discrete fracture network model analysis of the effects of fluid transport on the morphology of a cluster of activated fractures.

Convective transport in low-permeability rocks can be enhanced by the injection of a pressurized fluid to activate preexisting weak planes (fractures). These fractures are initially closed, but fluid-pressure-induced slippage creates void space that allows for fluid flow. The Coulomb-Mohr criterion yields a critical pressure required to open each of the fractures. Due to the intrinsic porosity of the rock, the injected fluid can flow from the fractures' surfaces to the rock matrix through a process referred to as leakoff. Following this activation mechanism, the connectivity of the cluster of activated fractures is strongly dependent on the ratio F_{N} of the standard deviation of the critical pressures to the viscous pressure drop over a fracture's length. Recently, we proposed a continuum model to predict the effects of fluid transport on the morphology of the cluster of activated fractures formed by this process over a specific intermediate range of values of F_{N} [Alhashim and Koch, J. Fluid Mech. 847, 286 (2018)JFLSA70022-112010.1017/jfm.2018.313]. In this paper, the activation process of a discrete well-connected network of preexisting fractures embedded in a highly heterogeneous rock is modeled to analyze the effects of fluid transport on the resulting cluster's morphology for a wider range of F_{N} and show how the idealized averaged equation solution arises in a discrete system. We derive a length scale ξ_{ch}, which is a function of F_{N} above which the viscous pressure drop is important. This length scale, along with the radius of the cluster R and the average separation between the preexisting fractures ξ_{0}, can be used to define distinct growth regimes where different models can be used to describe the growth dynamics and predict the connectivity of the active network. When ξ_{0}∼ξ_{ch}≪R, the cluster is well connected and a linear pressure diffusion equation can be used to describe the cluster's growth. When ξ_{ch}≫R≫ξ_{0}, a fractal network is formed by an invasion percolation process. In an intermediate regime ξ_{0}≪ξ_{ch}≪R, percolation theory relates the porosity and permeability of the network to the local fluid pressure. For this regime, we validate the predictions of the continuum theory we recently developed to describe the cluster growth on length scales larger than ξ_{ch}.

Gas–Liquid Taylor Flow Characteristics in a Fractal Microchannel Network during Numbering-up and Sizing-up

In the present study, the characteristics of gas-liquid Taylor flow in a fractal microchannel network were investigated by numerical modeling and experiments. With perpendicular gas intake, the fractal design methodology resulted in uniform flow distribution. Unstable Taylor flow and saddle-shaped velocity profile were observed in a high-level branch with 0.3 mm width. The liquid slug exhibited different flow fields during splitting in the bifurcations of different levels. The significant size effects were further investigated in four channels representing different sizes (0.3, 0.6, 1.2, and 2.4 mm) via a μ-PIV and high-speed camera. The normalized slug length (LS/W) increased with the increasing gas-liquid flow rate ratio (jG/jL), while the bubble length (LB/W) followed an opposite trend. The stability of the gas-liquid Taylor flow pattern became worse when the gas velocity increased beyond a certain value (Reynolds number Re > 220) under the microscale effect. Two typical counter rotating vortices were observed in the liquid slug, and the swirling strength increased exponentially with the jG/jL. It was found that both the velocity profile in the 0.3 mm channel deviated from the laminar flow. Particularly, a flow field similar to turbulence was observed in the 0.3 mm channel when the jG/jL reached 4. This work quantified the significant size effect in micro- and millichannels, providing theoretical basis for the effective scale-up of the microreactor.

Numerical Simulation of Nonequilibrium Foamy Oil for Cyclic Solvent Injection in Reservoirs after Cold Heavy Oil Production with Sand

SummaryNonequilibrium foamy oil behavior and solvent transport are two important recovery mechanisms for cyclic solvent injection (CSI) processes in post-cold heavy oil production with sand (CHOPS) reservoirs. The nonequilibrium solvent exsolution and gas bubbles generated during the pressure depletion stage have the typical characteristics of foamy oil flow. In this paper, a field-scale post-CHOPS model is constructed and upscaled from a core model, which was calibrated against detailed experimental data involving various propane (C3H8)-based and carbon dioxide (CO2)-based solvent mixtures. The field model is upscaled from the core model to analyze the impacts of simulation scales, heterogeneous wormholes, and the operating schedules on foamy oil behavior of different solvent systems.Reaction kinetics are implemented to represent the nonequilibrium gas dissolution and exsolution for foamy oil flow. A fractal wormhole network is modeled. To analyze the impacts of pressure depletion strategies, single-stage pressure depletion involving three oil solvent systems, as well as two cycles of CSI production processes, are examined. Detailed sensitivity analyses involving different solvent compositions are discussed.The results illustrate that both C3H8-based and CO2-based solvents exhibit significant nonequilibrium foamy oil characteristics, enabling the oil viscosity to remain close to its value with dissolved solvent during the pressure depletion process. However, the amount of nonequilibrium foamy oil flow is strongly dependent on the pressure depletion rate: A faster depletion rate is beneficial for higher oil recovery. The core model results are more sensitive to the solvent types, whereas the field-scale simulations show comparable recovery performance for both C3H8-based and CO2-based solvents. This observation highlights the significance of domain size, time scale, and wormhole heterogeneities on the ensuing foamy oil behavior.Although several post-CHOPS models were developed in the past, detailed field-scale models that simulate nonequilibrium foamy oil kinetics in a realistic wormhole network are lacking. The simulation model developed here has been calibrated against detailed experimental measurements and upscaled from a core-scale model. Improving our understanding of solvent dissolution/exsolution would aid in the design of operating strategies (e.g., pressure depletion and solvent injection schemes) for enhanced solvent/oil mixing and transport.

MEAN TIME TO ABSORPTION ON THE JOINT SIERPINSKI GASKET

As a kind of classical fractal network models, Sierpinski Gasket and its extended network models have attracted the attention of many scholars because of their many applications in real-world networks. Based on the classical Sierpinski Gasket, this paper proposes a joint Sierpinski Gasket model, which can be used not only to study the topology and dynamic properties of Sierpinski Gasket itself, but also to study the Sierpinski Gasket under special conditions, such as the residual Sierpinski Gasket after damage. Since the mean time to absorption (MTA) on the network is a basic dynamic property related to the random walk, in this paper, we study the expression of MTA on the proposed general joint Sierpinski Gasket model, and find that the MTA is determined by the number of iterations [Formula: see text] and two variables determined by the mode variable [Formula: see text], which can be solved by matrix algorithm. Therefore, using the expression of MTA on the general joint Sierpinski Gasket model, we calculate the analytical expressions of the MTA on a general example of the joint Sierpinski Gasket model and three Sierpinski Gaskets with varying degrees of damage and do the corresponding numerical simulation to verify the correctness of the conclusion.

Ultrawide Photonic Bandgap and Ultrastrong Photonic Localization Produced by Series of Periodic Networks

AbstractA series of periodic networks is designed, in which all of the unit cells construct a series of fractal triangles, and their extraordinary optical characteristics for producing ultrawide photonic bandgap (PBG) and ultrastrong photonic localization are investigated. The results indicate that with increasing fractal generation, the number of equivalent triangular loops increases exponentially, and consequently the gap–midgap ratio of the PBG produced by the networks with the integer waveguide length ratio of 1:2 enlarges rapidly and tends to the limit at 200%. For the networks with the noninteger waveguide length ratio of (where ), an extremely narrow passband occurs in the middle of the wide PBG. Furthermore, the ultrastrong photonic localization is generated when the electromagnetic wave with the frequency of the all‐transmission peak in the ultranarrow passband propagates in the networks. This type of fractal triangular networks may have applications to design all‐optical devices, such as highly sensitive optical filters, highly sensitive all‐optical switches, and high‐performance photonic energy‐storage devices.

Fractal networks with Sturmian structure

Many real complex networks behave similarly with the scale-free and the small-world properties. In this paper, we create a special hierarchical network with the Fibonacci word. This network is self-similar on some scales due to Fibonacci word’s recurrence property. Based on the construction of the graph, we study the clustering coefficient, the average path length and the cumulative degree distribution of our network. These results show the scale-free and the small-world effects of the evolving network. Moreover, the monotony of the standardized average path length and the average clustering coefficient is coincided with that of the Fibonacci word by applying the balance property of the Sturmian word. We think the Sturmian word can be an useful tool to describe some complex networks.

Plasticity-on-Chip Design: Exploiting Self-Similarity for Data Communications

With the increasing demand for distributed big data analytics and data-intensive programs which contribute to large volumes of packets among processing elements (PEs) and memory banks, we witness a pressing need for new mathematical models and algorithms that can engineer a brain-inspired plasticity into the computing platforms by mining the topological complexity of high-level programs (HLPs) and exploiting their self-similar and fractal characteristics for designing reconfigurable domain-specific computing architectures. In this article, we present Plasticity-on-Chip (PoC) by engineering plasticity into ”artificial brains” to mine and exploit the self-similarity of HLPs. First, we present a communication modeling of HLPs (e.g., C/C++ implementations of various applications) that relies on static and dynamic compiler analysis of programs with varying input seeds, performing comprehensive program analysis of all traces, and representing the HLPs as weighted directed acyclic graphs while capturing the intrinsic timing constraints and data/control flow requirements. Second, we propose a rigorous mathematical framework for determining the optimal parallel degree of executing a set of interacting HLPs (by partitioning them into clusters of densely interconnected supernodes - tasks) which helps us decide the number of available heterogeneous PEs, the amount of required memory and the structure of the synthesized deadlock-free irregular NoC topology that offers an efficient communication medium. These clusters serve as abstract models of computation for the synthesized PEs within the parallel execution model. Finally, exploiting the fractal and complex networks concepts, we extract in-depth features from graphs that serve as inputs for distributed reinforcement learning. Our experimental results on synthesized PEs and NoCs show performance improvements as high as 7.61x when compared to the traditional NoC and 2.6x compared to gem5-Aladdin.

Analysis of blood flow characteristics in fractal vascular network based on the time fractional order

Fractional calculus has shown good adaptability in describing the mechanical properties of viscoelastic materials. Based on fractional order theory, the characteristics of blood flow in the fractal network of blood vessels are analyzed, and experiments are carried out by using a mixed solution to prove the results of simulation analysis that contains glycerol, gelatin, sodium chloride, etc. Under different time of fractional order conditions, the effects of Reynolds number, vascular network fractal dimension, and bifurcation series conditions, the blood flow characteristics in fractal vascular network are analyzed. The analysis results show that the blood flow increases with the increase in time fractional order and the flow resistance decreases with the increase in time fractional order. The blood flow law curve based on the equivalent Casson fluid theory has the same trend between the orders α = 0.9 and α = 1.0 of the time fractional order curve, so the order of time fractional order can describe the blood flow characteristics of equivalent Casson fluid. The experimental data are distributed on both sides of the theoretical calculation curve and the relative error is small, which is in good agreement with the effect of vascular fractal parameters obtained by integer order on blood.

Multi-objective optimizations on thermal and hydraulic performance of symmetric and asymmetric bionic Y-shaped fractal networks by genetic algorithm coupled with CFD simulation

With the high energy output of electronic devices, effective heat dissipation with low energy consumption has drawn great attention. Based on the construction theory, the flow resistance of the generalized bionic Y-shaped fractal networks is theoretically analyzed. It is found that the symmetrical width has a positive effect on reducing the hydraulic resistance, while the symmetrical length plays a negative influence. Then, the optimal designs of symmetrical and asymmetrical bionic Y-shaped fractal heat sinks are obtained by multi-objective optimization with genetic algorithm (GA). The pressure loss in the asymmetrical bionic Y-shaped fractal heat sink is always lower than that of the symmetrical bionic Y-shaped fractal heat sink at the same fractal number. As the constraint of thermal resistance reduces from 0.045 K/W to 0.042 K/W, the pumping power difference between symmetrical and asymmetrical bionic Y-shaped fractal heat sinks increases from 70.1 mW to 99.1 mW. Furthermore, according to the CFD investigation, it is noted that the symmetrical heat sink exhibits a lower temperature difference and more uniform temperature distribution.

Comparison of DFN Modelled Microfracture Systems with Petrophysical Data in Excavation Damaged Zone

Physical and petrographic properties of drill core specimens were determined as a part of investigations into excavation damage in the dedicated study area in the ONKALO® research facility in Olkiluoto, Western Finland. Microfractures in 16 specimens from two drillholes were analysed and used as a basis for fractal geometry-based discrete fracture network (DFN) modelling. It was concluded that the difference in resistivity between pegmatoid granite (PGR) and veined gneiss (VGN) specimens of similar porosity was likely due to differences in the types of microfractures. This hypothesis was confirmed from microfracture analysis and simulation: fractures in gneiss were short and mostly in one preferred orientation, whereas the fractures in granite were longer and had two preferred orientations. This may be due to microstructure differences of the rock types or could suggests that gneiss and granite may suffer different types of excavation damage. No dependencies on depth from the excavated surface were observed in the geometric parameters of the microfractures. This suggests that the excavation damaged zone cannot be identified based on the changes in the parameters of the microfracture networks, and that the disturbed layer observed by geophysical methods may be caused by macro-scale fractures.

THE POWER FUNCTION HIDDEN IN THE VULNERABILITY OF FRACTAL COMPLEX NETWORKS

As a special kind of complex network, fractal complex networks have their unique characteristics such as fractal dimension, scale invariance, percolation threshold, etc. There have been some researches on the topology of fractal network, among which the calculation of fractal dimension, the construction of fractal model, the phase transition and the robustness of fractal network were the hotspots. But unlike the studies of the scale-free and small-world networks, there was little research on the relationship between the stability and the fractal feature, especially between the vulnerability and fractal dimension of the fractal networks. In this paper, the relationship between vulnerability and fractal dimension was discussed based on two kinds of fractal growth models and different vulnerability measurement methods. The same power functional relationships were obtained and a universal expression was proposed. Then, the effect of fractal feature on the vulnerability was exploring through the fitting error when fractal dimensions were calculated. Finally, the vulnerability of the real networks and their similarities to the fractal models in term of vulnerability showed the rationality of the fractal network models in establishing stable models. The results of this study have important reference for creating structurally stable artificial neural networks, transportation networks and information networks.

A machine learning algorithm for retrieving the geometrical characteristic parameters of soot fractal aggregates from polarized light signal

Polarized light measurement technology is provided to study geometric characteristics of soot fractal aggregates. The inverse problem is solved by two mature applications of machine learning, i.e. stacked ensemble model and deep network. A new Fractal Aggregates Generative Adversarial Network (FAGAN) is proposed to generate synthetic data to solve the class imbalance problem. The results show that compared with the stacked ensemble model which is sensitive to noise, the performance of the deep network is still satisfactory even under 10 % Gaussian noise, and the maximum mean absolute percentage error is not more than 3.87 %, which means that the synthetic data generated by FAGAN is very similar to the real data. As a whole, the combination of deep network and FAGAN has important guiding significance for complex and time-consuming experiments and engineering applications where noise is difficult to control or continuous and regular data cannot be obtained.