Fractal Prediction Model Research Articles

A Fractal Model of Fracture Permeability Considering Morphology and Spatial Distribution

Summary In fractured reservoirs, the fracture system is considered to be the main channel for fluid flow. To better investigate the impacts of fracture morphology (tortuosity and roughness) and spatial distribution on the flow capacity, a fractal model of fracture permeability was developed. Based on micro-computed tomography (CT) images, the 3D structure of the fracture was reconstructed, and the fractal characteristics were systematically analyzed. Finally, the control of permeability by fracture morphology and spatial distribution in different fractured reservoirs was identified. The results demonstrate that the complexity of the fracture distribution in 2D slices can represent the nature of the fracture distribution in 3D space. The permeability fractal prediction model was developed based on porosity (φ), spatial distribution fractal dimension (Df), tortuosity fractal dimension (DT), and opening fractal dimension of the maximum width fracture (Db). The permeability prediction results of the fractal model for Samples L-01 (limestone), BD-01 (coal), BD-02 (coal), S-01 (sandstone), M-01 (mudstone), and C-01 (coal) are 0.011 md, 0.239 md, 0.134 md, 0.119 md, 1.429 md, and 27.444 md, respectively. For different types of rocks, the results predicted by the model show good agreement with numerical simulations (with an average relative error of 2.51%). The factors controlling the permeability of fractured reservoirs were analyzed through the application of the mathematical model. The permeability is positively exponentially correlated with the fractal dimension of spatial distribution and negatively exponentially correlated with the fractal dimension of morphology. When Df < 2.25, the fracture spatial structure is simple, and the morphology and spatial distribution jointly control the seepage capacity of fractured reservoirs. When Df > 2.25, the fracture spatial structure is complex, and the impact of morphology on seepage capacity can be disregarded. This work can effectively lay the foundation for the study of fluid permeability in fractured reservoirs by investigating the effects of fracture morphology (tortuosity and roughness) and spatial distribution on flow capacity.

Threshold pressure gradient in fractured reservoirs: Experimental simulation and mathematical modeling

Aiming to solve the problems of unclear fluid flow laws and controlling factors in fractured reservoirs, as well as the lack of a mathematical model of threshold pressure gradient (TPG) considering fracture structural parameters, research on the structural characterization of fractured reservoirs, physical simulation of fluid flow, and TPG prediction models has been carried out. Physical simulations demonstrate that fluid flow in fractured reservoirs under stress is nonlinear and exhibits TPG. The TPG displays a significant positive exponential relationship with confining pressure and water saturation. As the confining pressure or water saturation increases, the effective opening of fractures decreases. The TPG will increase dramatically when the fracture opening decreases to the critical thickness for the rapid increase in TPG. On the basis of the fluid flow law study, the TPG fractal prediction model is further developed based on fractal dimension of fracture length distribution (Df), fractal dimension of tortuosity (DT), water saturation (Sw), maximum fracture width (wmax), limit shear stress (η0), and sample characteristic length (L0). The results predicted by the model show good agreement with physical simulations (with average relative error of 7.56%). Experimental simulation and mathematical modeling of TPG in fractured reservoirs can reveal the fluid flow mechanism and improve the accurate evaluation of production capacity during different development processes.

A Fractal Prediction Method for Contact Stiffness of Helical Gear Considering Asperity Lateral Contact and Interaction

The normal contact stiffness (NCS) on rough surfaces has a significant impact on the dynamic characteristics of helical gear. Aiming at the problem of inaccurate calculation of the NCS model under the traditional Hertz theory of smooth surfaces, a fractal prediction model of helical gear contact stiffness considering asperity lateral contact and interaction between asperities is proposed in this paper. The variation formula of asperity and the correction coefficient of a tooth contact surface under asperity lateral contact and interaction are derived, and the influence of micro-elements on normal load and NCS is qualitatively analyzed. The results show that the NCS of considering the interaction and lateral contact of asperity is closer to the experimental results; the contact surface correction coefficient increases with the increase of curvature radius and load. The NCS of a tooth surface increases with the increase in fractal dimension D or the decrease in roughness amplitude G. The influence of asperity lateral contact and interaction decreases with the increase in D and the decrease in G. The NCS of the helical gear decreases under the lateral contact and interaction of the asperity, which is critical for exact estimation of the NCS of contact surfaces in gear.

Fractal prediction of frictional force against the interior surface of forming channel coupled with temperature in a ring die pellet machine

For the biomass ring die pellet machine, the frictional force against the interior surface of the forming channel is the main cause for its frictional wear and also is key to the research of wear mechanism as well as its prediction. In this study, four ring die samples were used to measure and obtain data on their surface morphology. The fractal dimension D and fractal feature G were calculated using the Yardstick method, and lastly a fractal prediction model of sliding frictional force against the interior surface of forming channel was built, which was coupled with a fractal model of temperature distribution over friction surface. Numerical simulation, as well as friction-wear test were conducted to verify the accuracy of the model. The result showed that: when Ar < Arc, the slope of F was larger, which means the frictional force increased more rapidly, and the larger slope of FD represented a rapidly decreasing unit of frictional force. When true contact area Ar = 3.93%, Aa, FT, and FTD increased with the increase in temperature; FT increased rapidly at first and then gradually slowed down. When Ar was small, FTD increased sharply with the increase in temperature.

Unloading contact mechanics analysis of elastic–plastic fractal surface

At present, there are many researches on the loading between contact surfaces, but it is also very important to predict the unloading process between rough surfaces. In this paper, based on the three-dimensional fractal loading model we built earlier and the unloading finite element results of the single asperity by Etsion et al., the unloading fractal prediction model between rough contact surfaces is established. The model takes into account the friction factor between surfaces, the three-dimensional fractal characteristics of rough surfaces and the elastic–plastic deformation mechanism of asperities. The model is compared with the unloading model based on the two-dimensional fractal curve established by Miao et al. and the finite element results. The results show that the unloading process between rough surfaces depends on the final loading state of the surface; the loading and unloading curves form a hysteresis loop, which represents the energy dissipation in a contact cycle; both the friction factor and fractal dimension of the contact surface have important effects on the unloading model.

Endpoint-based optimal fractal interpolation for predicting BDS-3 system time offsets

In order to improve the interoperability within the Global Navigation Satellite System (GNSS), the International Committee on Global Navigation Satellite Systems published a joint statement in December 2019 that stated that all GNSS providers agree to monitor and broadcast the time offsets between each system timescale and the Universal Time Coordinated (UTC) or the rapid realization of UTC (UTCr). This commitment requires the study of precise prediction models for system time offsets. The prediction model of system time offsets is different from that of the atomic clock because of the control of the system timescale. The offsets between the system time of the Beidou Satellite Navigation System-3 (BDS-3) and the National Time Service Center (NTSC), called [UTC(NTSC)-BDT], have two main periods of 12 h and 24 h, according to the Fast Fourier Transform analysis. The rescaled range (R/S) analysis demonstrates that it has long memory, making it a fractal time series with a memory period of about 10.4 h. While using the fractal interpolation method to predict the [UTC(NTSC)-BDT] series, we found that the prediction error reaches its minimum value if adding disturbance on the estimated endpoint of the forecasted interval. After verifying the correlation between the estimated endpoints with the minimal interpolation error and minimal prediction error and proving the existence and uniqueness of the estimated endpoint with the minimal interpolation error, we established the endpoint-based optimal fractal interpolation prediction method. The experimental results indicate that the average prediction accuracy of the proposed prediction model is improved by 57.90% and 39.26% compared to that of a quadratic model and standard fractal prediction model, respectively. The accuracy analysis results of numerical tests indicate that the proposed prediction model can restrain the divergence of prediction error. Finally, we transform the [UTC(NTSC)-BDT] into [UTCr -BDT] using the [UTCr-UTC(NTSC)] published by the Bureau International des Poids et Mesures (BIPM) to meet the requirement of GNSS interoperability. The prediction accuracies of daily [UTCr-BDT] using the proposed prediction model are no more than 1.5 ns with uncertainty about 6 ns.

Fractal Prediction Model of Normal Contact Stiffness of Micro-pitting Gear

Fractal Prediction Model of Normal Contact Stiffness of Micro-pitting Gear

Study on Fractal Multistep Forecast for the Prediction of Driving Behavior

The application and development of new technology make it possible to acquire real-time data of vehicles. Based on these real-time data, the behavior of vehicles can be analyzed. The prediction of vehicle behavior provides data support for the fine management of traffic. This paper proposes speed and acceleration have fractal features by R/S analysis of the time series data of speed and acceleration. Based on the characteristic analysis of microscopic parameters, the characteristic indexes of parameters are quantified, the fractal multistep prediction model of microparameters is established, and the BP (back propagation neural networks) model is established to estimate predictable step of fractal prediction model. The fractal multistep prediction model is used to predict speed acceleration in the predictable step. NGSIM trajectory data are used to test the multistep prediction model. The results show that the proposed fractal multistep prediction model can effectively realize the multistep prediction of vehicle speed.

Research on building energy consumption prediction model based on fractal theory

ABSTRACT Nowadays, the energy problem is becoming more and more serious, and the actual energy consumption of the building is one of the important links in the field of building energy conservation. At present, most prediction algorithms fail to fully consider the complex characteristics of building energy consumption, resulting in unsatisfactory prediction results. Fractal theory can directly analyze some rules of abstract composite complex nonlinear things and then analyze and predict them correctly. Therefore, it is also a new way to analyze fractal theory and solve the problem of large-scale public construction energy consumption prediction. Taking a building as the object, an energy consumption prediction model using the fractal collage principle and fractal interpolation algorithm is proposed. In order to verify the validity of the model, a prediction model of traditional mature BP neural network is established, and the experimental results of the two models were compared. Mean relative error (MRE) and root mean square error (RMSE) basis are used to evaluate the performance of the model on the daily. The results show that the fractal prediction model has good prediction effect and accuracy. The energy prediction data provided by the model can provide a scientific basis for energy management and energy conservation control of such buildings.

Fractal behavior of BDS-2 satellite clock offsets and its application to real-time clock offsets prediction

The satellite clocks of the BeiDou Navigation Satellite System (BDS) have characteristic differences compared to those of the global positioning system and Galileo. Therefore, the satellite clock offsets prediction method of BDS is different from those of these two systems. The basis for establishing a more appropriate prediction model is to clarify the statistical characteristics of the BDS satellite clock offsets, which can be reflected by fitting residuals of precise clock errors. Fractal behavior is generally not considered in existing studies. In this study, the rescaled range analysis method is improved by using the moving-average method in order to verify the fractal behavior of the BDS clock offsets. The computation results of the Hurst exponent show that the BDS clock offsets are fractal series with long-term memory, and the memory spans are obtained by V-statistic calculation. The quadratic polynomial fitting residuals of BDS clock offsets are fitted by using the periodic model and fractal interpolation model, where the latter approach has a higher accuracy. In the predictive modeling process, the quadratic polynomial forecasting model is improved by using the memory span, so that the forecasting model not only inherits the overall clock offsets trend but also considers the local memory trend. The fractal interpolation prediction model of the clock offsets is established by using the extension method of the affine iteration function system. The experimental results show that the average prediction accuracy of the fractal model in 3, 6, 12, and 24 h can reach 1.4890, 2.0222, 3.1609, and 4.9278 ns, which is 57.74%, 50.20%, 52.66%, and 49.42% higher than the products from the China iGMAS ultra-rapid prediction, respectively.

Building a shear crushing specific energy consumption prediction model for manufactured sand aggregate

In the field of concrete materials in the construction industry, manufactured sand gradually replaces natural sand in order to cope with the shortage of natural sand resources, thus leading to the problem of energy consumption in producing manufactured sand. In this paper, single particle shear crushing test of three kinds of manufactured sand aggregates of granite, limestone and diabase were taken as the research object. The energy consumption in single particle shear crushing was measured and the fractal theory was introduced to analyze the mass distribution of crushed particles. With the help of crushing specific energy consumption, a fractal prediction model of crushing energy consumption was established, and R-R model was used to correct the model. Finally, this paper verified the effectiveness of the corrected model by a series of tests that compare actual measured values, predictive values from the uncorrected model and predictive values from the corrected model. The method is well worth considering and learning from when it comes to energy saving in producing manufactured sand.

A Fractal Prediction Model for Evaluating Normal Contact Stiffness of Micro-pitting Gear

Estimating normal contact stiffness of micro-pitting is one important subject to study the dynamic fault characteristics of gear system. Combined with the fractal parameters of surface morphology and the distribution area of contact points on the microscopic scale, a fractal model of joint surface with normal contact stiffness was established by investigated the elastic-plastic contact mechanical properties of micro-convex. The estimated fractal parameters for micro-pitting characteristic was established with the macroscopic physical dimension of involute gear and microscopic physical cause of micro-pitting, to investigate normal contact stiffness of involute gear under different machining technology of tooth surface and different levels of micro-pitting. The result shows that the surface microscopic elements have different effects on normal load and normal contact stiffness. The surface contact coefficients are different along with the engagement point of the tooth profile, and improving the accuracy of the tooth surface (reducing the roughness) can improve the normal contact stiffness of meshing tooth surface. The normal contact stiffness of tooth is lower significantly with micro-pitting. The simulation results are consistent with the actual engineering. The proposed mathematical model of micro-pitting can be used as a basis for an analysis of dynamic response and expansion of micro-pitting.

A loading fractal prediction model developed for dry-friction rough joint surfaces considering elastic–plastic contact

In the microscopic view, the contact between two solid surfaces is reckoned to be the contact among asperities indeed. These asperities loaded which contact with each other are to experience three stages of deformation in succession: elastic deformation, elastic–plastic deformation, and plastic deformation, not the direct transition from elastic deformation to plastic deformation. In addition, these asperities should be distributed on the three-dimensional surface topography, and the corresponding fractal dimension is $$2<D<3$$ . In this study, firstly, based on the fractal theory and Hertz contact theory, the loading fractal model of rough dry-friction joint surfaces is established in consideration of elastic–plastic deformation of asperities and three-dimensional surface topography. Secondly, two important fractal parameters, fractal dimension D and fractal roughness G, can be calculated according to sampled surface topography of 45 steel and power spectrum density function method. Then, the effects of friction factor $$\mu $$ , fractal dimension D, and fractal roughness G on the loading fractal model are analyzed by numerical simulation. Finally, the present model is compared with experimental data and other models, which indicates that their trends are the same. The elastic–plastic loading modeling provides a certain theoretical reference for the accurate solution of contact among rough surfaces.

A normal contact stiffness fractal prediction model of dry-friction rough surface and experimental verification

Abstract The normal contact stiffness is one of important dynamic parameters and the accurate construction of its model is conducive to subsequent modeling and analysis of dynamics on whole machine. However, in most of previous normal contact stiffness models, the influence of friction factor of rough surfaces is neglected. Given that the friction factor of rough surfaces has a significant influence on the structural dynamic characteristics, in this paper, a normal contact stiffness fractal prediction model of dry-friction rough surface considering friction factor based on three-dimensional fractal function is established. And the given (power spectrum density) PSD function is used to identify two important fractal parameters , three-dimensional fractal dimension D and the fractal roughness G . The analysis of numerical simulation indicates that the normal contact stiffness monotonically increases and decreases respectively with the three-dimensional fractal dimension and fractal roughness increasing; it increases with the actual contact area and the maximum contact area of asperity increasing; it tends to decrease on the whole with the friction factor increasing, which turns out to be linear decrease firstly and then exponential decrease. In addition, the friction factor has an important influence on natural frequency of the whole structure. In this paper, the validity of the established model is verified by experiment and the normal contact stiffness of surface should be considered in the dynamics modeling of the whole machine.

Fractal Prediction Model During the Wear Process Based on Archard Formula

It is very important for the tribological design of the mechanical system to develop a mathematical model to predict the rate of wear depth in a wear process. Archard formula is the basement of wear calculation, it is very important to estimate the adhesive wear life. This paper uses the fractal parameters with scale-independence to characterize surface topography in the wear process by introducing the fractal theory, and takes into account the different shape of asperities during the running-in process and the stable wear process, the revised Archard wear calculation formula is developed. The numerical simulation results show that the rate of wear depth and the wear depth changing with time and the fractal parameters are consistent with that described by classical theories, it will provide a theoretical support for accurate prediction of the friction pairs' wear life.

Covert Voice over Internet Protocol Communications with Packet Loss Based on Fractal Interpolation

The last few years have witnessed an explosive growth in the research of information hiding in multimedia objects, but few studies have taken into account packet loss in multimedia networks. As one of the most popular real-time services in the Internet, Voice over Internet Protocol (VoIP) contributes to a large part of network traffic for its advantages of real time, high flow, and low cost. So packet loss is inevitable in multimedia networks and affects the performance of VoIP communications. In this study, a fractal-based VoIP steganographic approach was proposed to realize covert VoIP communications in the presence of packet loss. In the proposed scheme, secret data to be hidden were divided into blocks after being encrypted with the block cipher, and each block of the secret data was then embedded into VoIP streaming packets. The VoIP packets went through a packet-loss system based on Gilbert model which simulates a real network situation. And a prediction model based on fractal interpolation was built to decide whether a VoIP packet was suitable for data hiding. The experimental results indicated that the speech quality degradation increased with the escalating packet-loss level. The average variance of speech quality metrics (PESQ score) between the “no-embedding” speech samples and the “with-embedding” stego-speech samples was about 0.717, and the variances narrowed with the increasing packet-loss level. Both the average PESQ scores and the SNR values of stego-speech samples and the data-retrieving rates had almost the same varying trends when the packet-loss level increased, indicating that the success rate of the fractal prediction model played an important role in the performance of covert VoIP communications.

Fast fractal prediction of surge over underwater vehicle

In order to realize accurate control of underwater vehicles, this paper proposed a fractal prediction model of surge. Firstly, we introduce the current state-of-the-art of fractal prediction; secondly, we complete a fractal analysis with the R_S method, calculating the fractal dimension of surge using the short-time dimension method; thirdly, we correct the vertical scaling factor of the Iterated Function System (IFS) using the fractal dimension. Then the predictive value can be calculated through anti-iteration according to the statistical significantly IFS and proposing a fractal prediction model. Finally, we complete a simulation analysis based on the actual data and a comparison to another prediction algorithm. The result shows that the fractal prediction algorithm of this article has better comprehensive performance.

An improved fractal prediction model for forecasting mine slope deformation using GM (1, 1)

The forecasting slope deformation potential is required to evaluate slope safety during open-pit mining, allowing us to formulate and promote effective emergency strategies in advance to prevent slope failure disasters. Although fractal models have been used to predict slope deformation, such limitations as low prediction accuracy, poor stability and the requirement for large amounts of data must be overcome. This article proposes an improved fractal model to forecast mine slope deformation using the grey system theory. The GM (1, 1) model is used in the improved fractal model to optimize the fitting function of the fractal dimension because of its high computational efficiency and strong fitting ability. Data sequences spanning 13 days from 11 global positioning system monitoring stations in the Jinduicheng open-pit mine in Shaanxi Province, China, were applied to forecast the slope deformation. The results from both the traditional fractal model and the improved fractal model can accurately forecast the slope deformation value fairly close to the actual field monitoring value, but the latter can make a more accurate prediction than the former. There is a significant relationship between the prediction accuracy and the data sequence dispersion. Further analysis revealed that our improved fractal model is more capable of resisting the volatility existing in the data sequences than the traditional fractal model. These findings assist in understanding the applicability of prediction models and the deformation trends of open-pit mine slopes.

Fractal prediction model and simulation of contact stiffness considering friction factors between joint surfaces

In order to accurately consider the influence of friction on the joint surfaces, the fractal prediction model of the contact stiffness of joint surfaces is established based on tribology theory and fractal contact theory. Then the kinetic model of joint surfaces is improved to a certain extent. The influence of actual contact area, contact force and friction coefficient on the contact stiffness is revealed with the research of numerical simulation. Simulation results indicate that normal contact stiffness of joint surfaces increased with the increase of actual contact area, increased with the increase of contact force. And when the friction coefficient is less than 0.3, the normal contact stiffness appears linear attenuation with the increase of the friction coefficient. And when the friction coefficient is larger than 0.3, normal contact stiffness appears exponential attenuation with the increase of the friction coefficient. Tangential stiffness appears to be the similar rules with different parameters.

Fractal Prediction Model for Normal Contact Damping of Joint Surfaces considering Friction Factors and Its Simulation

To research the dynamic characteristics of the joint surfaces from the microperspective, the fractal prediction model of the normal contact damping of joint surfaces is established based on the “solid-gap-solid” contact model which was proposed by the author, tribology theory, and contact fractal theory. And the kinetic model of joint surfaces is improved to a certain extent. The influences of actual contacting area, friction coefficient, and the fractal dimension on the normal contact damping are revealed by the research of numerical simulation. Simulation results show that the normal contact damping of joint surfaces decreases to a constant value with the increase of actual contacting area, increases with the increase of friction coefficient, and decreases with the increase of fractal dimension. Actual contacting area influences the location of the turning point of the C n*-μ (critical friction coefficient), and the critical friction coefficient increases with the increase of actual contacting area. Normal damping coefficient of joint surfaces continuously decreases with the increase of fractal dimension and the difference of magnitude exists between normal contact damping coefficients with different fractal dimension values.