A semi-analytical model for coupled flow in stress-sensitive multi-scale shale reservoirs with fractal characteristics

A large number of nanopores and complex fracture structures in shale reservoirs results in multi-scale flow of oil. With the development of shale oil reservoirs, the permeability of multi-scale media undergoes changes due to stress sensitivity, which plays a crucial role in controlling pressure propagation and oil flow. This paper proposes a multi-scale coupled flow mathematical model of matrix nanopores, induced fractures, and hydraulic fractures. In this model, the micro-scale effects of shale oil flow in fractal nanopores, fractal induced fracture network, and stress sensitivity of multi-scale media are considered. We solved the model iteratively using Pedrosa transform, semi-analytic Segmented Bessel function, Laplace transform. The results of this model exhibit good agreement with the numerical solution and field production data, confirming the high accuracy of the model. As well, the influence of stress sensitivity on permeability, pressure and production is analyzed. It is shown that the permeability and production decrease significantly when induced fractures are weakly supported. Closed induced fractures can inhibit interporosity flow in the stimulated reservoir volume (SRV). It has been shown in sensitivity analysis that hydraulic fractures are beneficial to early production, and induced fractures in SRV are beneficial to middle production. The model can characterize multi-scale flow characteristics of shale oil, providing theoretical guidance for rapid productivity evaluation.

VIBRATION ANALYSIS OF DIESEL'S COMBU SION BASED ON FRACTAL THEORY

Self-similarity of diesel's combustion process is analyzed.Vibration signals of combustion are picked up from the cylinder wall,and correlation fractal dimension D is calculated by using G - P algorithm in fractal theory. Parameters which influence D are investigated.At the same time, D can present cylinder's combustion condition to some extent.Thus,D can be used as a diagnostic parameter and used in diesel's fault diagnosis.

Application Study of Fractal Theory in Mechanical Transmission

Mechanical transmissions are applied widely in various electrical and mechanical products, but some qualities of some high-end products can＇t meet people＇s demand, and need to be improved with some new methods or theories. The fractal theory is a new mathematic tool, which provides a new approach for the further study in the area of the mechanical transmission, and helps to solve some problems. The basic contents of the fractal theory are introduced firstly, especially the two important concepts, the self-similar fractal and the fractal dimension. Then, the deferent application of the fractal theory in this area are given to display how to further the study and improve some important characteristics of the mechanical transmission, such as contact surfaces, manufacturing precise, friction and wear, stiffness, strength, dynamics, fault diagnosis, etc. Finally, the problems of the fractal theory and its application are discussed, and some weaknesses, such as the calculation capacity of the fractal theory is not strong, are pointed out. Some new solutions are suggested, such as combining the fractal theory with the fuzzy theory, the chaos theory and so on. The new application fields of the fractal theory in the area of the mechanical transmission are proposed.

Elastoplastic Contact Mechanics Model of Rough Surface Based on Fractal Theory

Because the result of the MB fractal model contradicts with the classical contact mechanics, a revised elastoplastic contact model of a single asperity is developed based on fractal theory. The critical areas of a single asperity are scale dependent, with an increase in the contact load and contact area, a transition from elastic, elastoplastic to full plastic deformation takes place in this order. In considering the size distribution function, analytic expression between the total contact load and the real contact area on the contact surface is obtained. The elastic, elastoplastic and full plastic contact load are obtained by the critical elastic contact area of the biggest asperity and maximun contact area of a single asperity. The results show that a rough surface is firstly in elastic deformation. As the load increases, elastoplastic or full plastic deformation takes place. For constant characteristic length scale G, the slope of load-area relation is proportional to fractal dimension D. For constant fractal dimension D, the slope of load-area relation is inversely proportional to G. For constant D and G, the slope of load-area relation is inversely proportional to property of the material ~b, namely with the same load, the material of rough surface is softer, and the total contact area is larger. The contact mechanics model provides a foundation for study of the friction, wear and seal performance of rough surfaces.

Advances in applied studies of fractal theory in geography in China

The fractal theory put forward by American mathematician B B Mandelbrot (1967) supplies an effective approach to solve complex problems. The complex problems in geography have become the main positive study field of fractal theory. Based on the works of China's geographers and the summarization of contents of fractal theory, the authors comment on the present situation of its applications to almost every branch of geography and discuss the related problems and the prospects of fractal study in geography.

Fractal evolution mechanism of rock fracture in undersea metal mining

Through rock mechanics test, similar simulation experiment, borehole photographic observation of rock fissure, numerical simulation calculation of plastic zone distribution and deformation monitoring of rock mass during undersea mining, the fractal evolution mechanisms of rock fracture in undersea metallic deposits of Sanshandao Gold Mine were studied by fractal theory. The experimental researches on granite mechanics test in undersea deposit indicate that with the increase of load, the granite deformation energy and the fractal dimension of acoustic emission(FDAE) increase gradually. However, after reaching the peak stress of specimen, the fractal dimensions of acoustic emission(FDAEs) decrease and the granite specimen fails. Therefore, the fractal dimension evolution of rock failure can be divided into four stages, which are fissure inoculation stage, fissure growth stage, fissure expansion stage and fracture instability stage, respectively. By calculating and analyzing the damage photographs of rock specimens in Sanshandao Gold Mine, the fractal dimension of rock fissure is 1.4514, which is close to the average value of FDAE during granite destruction, i.e., 1.4693. Similar simulation experiments of undersea mining show that with the excavation proceeding, the FDAE in rock stratum increases gradually, and when the thickness of the isolation roof is less than 40 m, the FDAE begins to decrease, and meanwhile the sign of water inrush emerges. The numerical simulation researches on the plastic zone distribution of undersea mining in Sanshandao Gold Mine indicate that the fractal dimension of plastic zone(FDPZ) where the failure characteristics occur is 1.4598, close to the result of similar simulation experiment of 1.4364, which shows the sign of water inrush. Meanwhile, the thickness of the isolation roof for undersea mining should be more than 40 m, which is consistent with the results of similar simulation experiment. In Sanshandao Gold Mine, the rock fissures in undersea mining were observed by borehole photography and the rock mass deformation was monitored by multi-point displacement meters, and at the same time the fractal dimensions of strata borehole fissure distribution and energy release ratio(ERR) of rock mass were calculated by fractal principle, which are 1.2328 and 1.2685, respectively. The results demonstrate that rock deformation and fissure propagation are both in the second stage of fissure growth, and have not reached the fourth stage of fracture instability. Therefore, the conclusion can be obtained that the undersea mining in Sanshandao Gold Mine is safe at present.

Use of digital image analysis combined with fractal theory to determine particle morphology and surface texture of quartz sands

The particle morphology and surface texture play a major role in influencing mechanical and hydraulic behaviors of sandy soils. This paper presents the use of digital image analysis combined with fractal theory as a tool to quantify the particle morphology and surface texture of two types of quartz sands widely used in the region of Vitória, Espírito Santo, southeast of Brazil. The two investigated sands are sampled from different locations. The purpose of this paper is to present a simple, straightforward,reliable and reproducible methodology that can identify representative sandy soil texture parameters.The test results of the soil samples of the two sands separated by sieving into six size fractions are presented and discussed. The main advantages of the adopted methodology are its simplicity, reliability of the results, and relatively low cost. The results show that sands from the coastal spit(BS) have a greater degree of roundness and a smoother surface texture than river sands(RS). The values obtained in the test are statistically analyzed, and again it is confirmed that the BS sand has a slightly greater degree of sphericity than that of the RS sand. Moreover, the RS sand with rough surface texture has larger specific surface area values than the similar BS sand, which agree with the obtained roughness fractal dimensions. The consistent experimental results demonstrate that image analysis combined with fractal theory is an accurate and efficient method to quantify the differences in particle morphology and surface texture of quartz sands.

Multi-scale pore fractal characteristics of differently ranked coal and its impact on gas adsorption

Well-developed pores and cracks in coal reservoirs are the main venues for gas storage and migration.To investigate the multi-scale pore fractal characteristics,six coal samples of different rankings were studied using high-pressure mercury injection(HPMI),low-pressure nitrogen adsorption(LPGA-N2),and scanning electron microscopy(SEM)test methods.Based on the Frankel,Halsey and Hill(FHH)fractal theory,the Menger sponge model,Pores and Cracks Analysis System(PCAS),pore volume complexity(D_(v)),coal surface irregularity(Ds)and pore distribution heterogeneity(D_(p))were studied and evaluated,respectively.The effect of three fractal dimensions on the gas adsorption ability was also analyzed with high-pressure isothermal gas adsorption experiments.Results show that pore structures within these coal samples have obvious fractal characteristics.A noticeable segmentation effect appears in the Dv1and Dv2fitting process,with the boundary size ranging from 36.00 to 182.95 nm,which helps differentiate diffusion pores and seepage fractures.The D values show an asymmetric U-shaped trend as the coal metamorphism increases,demonstrating that coalification greatly affects the pore fractal dimensions.The three fractal dimensions can characterize the difference in coal microstructure and reflect their influence on gas adsorption ability.Langmuir volume(V_(L))has an evident and positive correlation with Dsvalues,whereas Langmuir pressure(P_(L))is mainly affected by the combined action of Dvand Dp.This study will provide valuable knowledge for the appraisal of coal seam gas reservoirs of differently ranked coals.

Application of Fractal Contact Model in Dynamic Performance Analysis of Gas Face Seals

Fractal theory provides scale?independent asperity contact loads and assumes variable curvature radii in the contact analyses of rough surfaces, the current research for which mainly focuses on the mechanism study. The present study introduces the fractal theory into the dynamic research of gas face seals under face?contacting conditions. Structure?Function method is adopted to handle the surface profiles of typical carbon?graphite rings, proving the fractal con?tact model can be used in the field of gas face seals. Using a numerical model established for the dynamic analyses of a spiral groove gas face seal with a flexibly mounted stator, a comparison of dynamic performance between the Majumdar?Bhushan(MB) fractal model and the Chang?Etsion?Bogy(CEB) statistical model is performed. The result shows that the two approaches induce differences in terms of the occurrence and the level of face contact. Although the approach distinctions in film thickness and leakage rate can be tiny, the distinctions in contact mechanism and end face damage are obvious. An investigation of fractal parameters D and G shows that a proper D(nearly 1.5) and a small G are helpful in raising the proportion of elastic deformation to weaken the adhesive wear in the sealing dynamic performance. The proposed research provides a fractal approach to design gas face seals.

Application of the fractal theory for evaluating effects of coal comminution by waterjet

Comminution of coal to ultrafine sizes by high-pressure waterjet provides a novel method for preparation of coal-water fuels for next generation,near-zero emission electric power generation.The particle size distribution(PSD)of ground coal is a key parameter in the preparation of slurries as it determines the settling behavior of the particles and viscosity of the coal-water mixture.There are several methods available for representation and evaluation of particle size analysis data.However,fractal theory provides a means by which the entire PSD of comminuted materials can be quantified by using of a specific and exact value.In this paper,a volume-based fractal model was deduced to characterize the PSD of the coal which is ground in a specially designed comminution cell.During the size reduction process,the inlet pressures up to 276 MPa were used.

Fractal analysis of thermal conduction of loose coal

For deep mining engineering, heat transfer of coal mass is a vital factor in the thermal environment of coal mines. In order to study the thermal conduction mechanism, we obtained gray images of coal mass microstructure by scanning samples with a digital microscope. With the use of Matlab, these gray images were transformed into binary images, which were then transformed into a corresponding matrix consisting only of the values 0 and 1. According to the calculation method of box-counting dimension, we calculated the fractal dimension of the loose coal to be approximately 1.86. The thermal conductivity expressions of loose coal were derived based on the simulation method of thermal resistance. We calculated the thermal conductivity of loose coal by using a fractal model and compared the calculated values with our experimental data. The results show that the test data show an encourag-ing agreement with the calculated values. Hence fractal theory is a feasible method for studying thermal conductivity of loose coal.

Dynamic development model for long gap discharge streamer-leader system based on fractal theory

Establishing a long air gap discharge model considering the streamer-leader transition and randomness of the discharge path is of great signiflcance to improve the accuracy of discharge characteristic prediction and optimize external insulation design.Based on fractal theory and thermal ionization theory of streamer-leader transition,this work establishes a dynamic development model for the long air gap discharge streamer-leader system,which includes streamer inception,streamer development,leader inception,development of streamer-leader system and flnal jump.The positive discharge process of a 3 m rod plate is simulated to obtain the fractal distribution of the discharge path and the law of leader development for comparison with the discharge test results.The results show that the simulation model is similar to test results in the development characteristics of leader path distribution,each stage time and leader velocity.Finally,a simulation calculation of a 50%breakdown voltage of the rod-plate gap and ball-plate gap is carried out,with results fairly consistent with test data,proving the effectiveness and practicality of the model.

Contact Model of Revolute Joint with Clearance Based on Fractal Theory

The contact sti ness of a mechanical bonding surface is an important parameter in determining the normal and radial contact force. To improve the calculation accuracy of the contact force model, the surface roughness of the bonding surface and the energy loss that necessarily occurs during the impact process should be considered com?prehensively. To study the normal contact force of a revolute joint with clearance more accurately in the case of dry friction, a nonlinear sti ness coe cient model considering the surface roughness was established based on fractal theory, which considers the elastic, elastic?plastic, and plastic deformations of the asperities of the contact surface during the contact process. On this basis, a modified nonlinear spring damping model was established based on the Lankarani–Nikravesh contact force model. The laws influencing the surface roughness, recovery coe cient, initial velocity, and clearance size on the impact force were revealed, and were compared with the Lankarani–Nikravesh model and a hybrid model using MATLAB. The maximum impact force was obtained using a modified contact force model under di erent initial velocities, di erent clearances, and di erent degrees of surface roughness, and the calculated results were then compared with the experiment results. This study indicates that the modified model can be used more widely than other models, and is suitable for both large and small clearances. In particular, the modified model is more accurate when calculating the contact force of a revolute joint with a small clearance.

Novel welding image processing method based on fractal theory

Computer vision has come into used in the fields of welding process control and automation. In order to improve precision and rapidity of welding image processing, a novel method based on fractal theory has been put forward in this paper. Compared with traditional methods, the image is preliminarily processed in the macroscopic regions then thoroughly analyzed in the microscopic regions in the new method. With which, an image is divided up to some regions according to the different fractal characters of image edge, and the fuzzy regions including image edges are detected out, then image edges are identified with Sobel operator and curved by LSM (Lease Square Method). Since the data to be processed have been decreased and the noise of image has been reduced, it has been testified through experiments that edges of weld seam or weld pool could be recognized correctly and quickly.

Fractal Evolving Theory and Growing Model of Olefin Polymerization Process

The surface morphology of Ti-Mg supported catalyst and the polyethyleneparticles are studied using scanning electron microscope(SEM) technology. The results show thateithen the catalyst's surface or polymer particle's surface is irregular and has fractalcharacteristics, which can be described by fractal parameter. The more interesting discovery is thatthe surface fractal dimension values of the polymer particles vary periodically with thepolymerization time. We call this phenomenon fractal evolution, which can be divided into the'revolution' stage and the 'evolution' stage. And then we present polymerization fractal growingmodel (PFGM), and successfully describe and/or predict the whole evolving process of thepolyethylene particle morphology under the different slurry polymerization (includingpre-polymerization) conditions without H_2.

The Application of Fractal to Design Knitted Sweaters

Pattern design is very important for knitting industry. Fractal theory is adopted to create image automatically, which is applied to design knitted sweaters. Based on Mandelbrot set(M-set), image could be generated automatically by visual basic (VB) program. Plenty of wonderful patterns are created by the following methods: selecting any parts of the image and magnifying, extending generalized M-set, editing the selected images with photoshop. The edited pattern is used to set on any parts of clothes to show fashion effect. Actual knitted structure is simulated by knitting CAD software. The results show that the fractal theory has greater potential applications in knitted pattern design, and it deserves further research.

Improve Fractal Compression Encoding Speed Using Feature Extraction and Self-organization Network

Image compression consists of two main parts: encoding and decoding. One of the important problems of the fractal theory is the long encoding implementation time, which hindered the acceptance of fractal image compression as a practical method. The long encoding time results from the need to perform a large number of domain-range matches, the total encoding time is the product of the number of matches and the time to perform each match. In order to improve encoding speed, a hybrid method combining features extraction and self-organization network has been provided, which is based on the feature extraction approach the comparison pixels by pixels between the feature of range blocks and domains blocks. The efficiency of the new method was been proved by examples.

Application of Fractal Theory in Simulation of Ferromagnetic Elements' Hysteresis Loop in Transformer

The simulation of the transformer transient is one of the indispensable qualifications for improving the performance of transformer protection, the key technique of the transformer's transient simulation is the treatment of ferromagnetic elements' loop. Thus the shapes of the primary hysteresis loop and each internal secondary hysteresis loop in the identical magnetism conducting are analyzed, and then it is proposed that there are some fractal characteristics in the relation between them. The fractal phenomenon of the ferromagnetic elements' hysteresis loop in the transformer's transient simulation is first brought forward, the mutuality between the ferromagnetic elements' primary hysteresis loop and its secondary hysteresis loops is revealed in mechanism by using the fractal theory. According to the iterated function system of fractal theory, the secondary hysteresis loops can be generated by the iterative calculation of the primary loop. The simulation results show the validity of this idea.

The Research of ECG Signal Automatic Segmentation Algorithm Based on Fractal Dimension Trajectory

In this paper a kind of ECG signal automatic segmentation algorithm based on ECG fractal dimension trajectory is put forward.First,the ECG signal will be analyzed,then constructing the fractal dimension trajectory of ECG signal according to the fractal dimension trajectory constructing algorithm,finally,obtaining ECG signal feature points and ECG automatic segmentation will be realized by the feature of ECG signal fractal dimension trajectory and the feature of ECG frequency domain characteristics.Through Matlab simulation of the algorithm,the results showed that by constructing the ECG fractal dimension trajectory enables ECG location of each component displayed clearly and obtains high success rate of sub-ECG,providing a basis to identify the various components of ECG signal accurately.

Two Phase Flow Simulation of Fractal Oil Reservoir Based on Meshless Method

The reservoir is the networked rock skeleton of an oil and gas trap,as well as the generic term for the fluid contained within pore fractures and karst caves.Heterogeneity and a complex internal pore structure characterize the reservoir rock.By introducing the fractal permeability formula,this paper establishes a fractal mathematical model of oil-water two-phase flow in an oil reservoir with heterogeneity characteristics and numerically solves the mathematical model using the weighted least squares meshless method.Additionally,the method’s correctness is verified by comparison to the exact solution.The numerical results demonstrate that the fractal oil-water two-phase flow mathematical model developed using the meshless method is capable of more accurately and efficiently describing the flow characteristics of the oil-water two-phase migration process.In comparison to the conventional numerical model,this method achieves a greater degree of convergence and stability.This paper examines the effect of varying the initial viscosity of the oil,the initial formation pressure,and the production and injection ratios on daily oil production per well,water cut in the block,and accumulated oil in the block.For 10 and 60 cp initial crude oil viscosities,the water cut can be 0.62 and 0.80,with 3100 and 1900 m^(3)cumulative oil production.Initial pressures have little effect on production.In this case,the daily oil production of well PRO1 is 1.7 m^(3)at 7 and 10 MPa initial pressure.Block cumulative oil production is 3465.4 and 2149.9m^(3)when the production injection ratio is 1.4 and 0.8.The two-phase meshless method described in this paper is essential for a rational and effective study of production dynamics patterns in complex reservoirs and the development of reservoir simulations of oil-water flow in heterogeneous reservoirs.