Fractal model of spontaneous imbibition in low-permeability reservoirs coupled with heterogeneity of pore seepage channels and threshold pressure

Spontaneous imbibition(SI)is an important mechanism for enhancing oil recovery in low-permeability reservoirs.Due to the strong heterogeneity,and the non-Darcy flow,the construction of SI model for lowpermeability reservoirs is extremely challenging.Commonly,traditional SI models based on single or averaged capillary tortuosity ignore the influence of heterogeneity of pore seepage channels and the threshold pressure(TP)on imbibition.Therefore,in this work,based on capillary model and fractal theory,a mathematical model of characterizing SI considering heterogeneity of pore seepage channels is established.On this basis,the threshold pressure was introduced to determine the pore radius at which the wetted phase can displace oil.The proposed new SI model was verified by imbibition experimental data.The study shows that for weakly heterogeneous cores with permeability of 0-1 m D,the traditional SI model can characterize the imbibition process relatively accurately,and the new imbibition model can increase the coefficient of determination by 1.05 times.However,traditional model has serious deviations in predicting the imbibition recovery for cores with permeability of 10-50 m D.The new SI model coupling with heterogeneity of pore seepage channels and threshold pressure effectively solves this problem,and the determination coefficient is increased from 0.344 to 0.922,which is increased by2.68 times.For low-permeability reservoirs,the production of the oil in transitional pores(0.01-0.1μm)and mesopores(0.1-1μm)significantly affects the imbibition recovery,as the research shows that when the heterogeneity of pore seepage channels is ignored,the oil recovery in transitional pores and mesopores decreases by 7.54%and 4.26%,respectively.Sensitivity analysis shows that increasing interfacial tension,decreasing contact angle,oil-water viscosity ratio and threshold pressure will increase imbibition recovery.In addition,there are critical values for the influence of these factors on the imbibition recovery,which provides theoretical support for surfactant optimization.

Delineation of Geochemical Anomalies Based on Cu by the Boxplot as an Exploratory Data Analysis (EDA) Method and Concentration-Volume (C-V) Fractal Modeling in Mesgaran Mining Area, Eastern Iran

The target in this investigation is separation and delineation of geochemical anomalies for the single element Cu in Mesgaran mining area, eastern Iran. Mesgaran mining area is located in south part of Sarbishe county with about 29 Km distance to the county center. This region is part of an Ophiolite sequence and the copper anomalies seem to be related to a volcanic massive sulfide (VMS) deposit whose main part (massive sulfide Lens) has been eroded. In order to delineate Cu anomalies, the boxplot as an Exploratory Data Analysis (EDA) method and concentration-volume (C-V) Fractal modeling are employed. Both of the methods reveal low-deep anomalies which are highly correlated with geological and geophysical studies. As the main result of this study we show that Fractal modeling in spite of the Boxplot, is not recommended for complex geological settings. The proved shallow anomalies recorded by geophysical studies and defined by the used methods are in accordance to the stringer zone of a volcanic massive sulfide (VMS) deposit in Mesgaran mining area which means this region is the bottom of a VMS deposit and geochemical anomalies are related to the remained parts of the deposit.

Fractal Modeling of Sphalerite Banding in Jinding Pb-Zn Deposit, Yunnan, Southwestern China

Sphalerite banding is a common texture in Jinding （金顶） Pb-Zn deposit, Yunnan （云南）, southwestern China. The frequency distribution and irregularity of sphalerite grains observed in the bandings are characterized quantitatively by fractal models. Fractal dimensions calculated by several fractal models including box-counting model, perimeter-area （P-A） model, and number-area （N-A） model show the gradual change from outer banding to inner banding, indicating a decrease in area percentage, in irregularity, in shape and in grain size, and an increase in the numbers of grains. These results may imply an inward growth of sphalerite during mineralization, and self-organization properties are involved in the nonlinear process of mineralization.

Applicability of Fractal Models in Estimating Soil Water Retention Characteristics from Particle-Size Distribution Data

Soil water retention characteristics are the key information required in hydrological modeling. Frac-tal models provide a practical alternative for indirectly estimating soil water retention characteristics fromparticle-size distribution data. Predictive capabilities of three fractal models, i.e, Tyler-Wheatcraft model,Rieu-Sposito model, and Brooks-Corey model, were fully evaluated in this work using experimental datafrom an international database and literature. Particle-size distribution data were firstly interpolated into20 classes using a van Genuchten-type equation. Fractal dimensions of the tortuous pore wall and the poresurface were then calculated from the detailed particle-size distribution and incorporated as a parameter infractal water retention models. Comparisons between measured and model-estimated water retention cha-racteristics indicated that these three models were applicable to relatively different soil textures and pressurehead ranges. Tyler-Wheatcraft and Brooks-Corey models led to reasonable agreements for both coarse- andmedium-textured soils, while the latter showed applicability to a broader texture range. In contrast, Rieu-Sposito model was more suitable for fine-textured soils. Fractal models produced a better estimation of watercontents at low pressure heads than at high pressure heads.

A FRACTAL MODEL AND ENERGY DISSIPATION FOR EN ECHELON FRACTURES

observations from the field and the taboratory show that en echelon fractures withinfracture zones have a foede1 within Ricdel structure’ The tensile fallure mecbanism of en echelonfractures can be described by the pile-ups of sbear crack-dislocations. A fractal model can be used tosimulate the Riedel within Xiedel geometry, allowing the direct rneasarement of tbe ftactal dimen sions of en echelon fractare systems. The energy dissipation of tbe en ccbe1on fracture system canbe deduced using a fracil damage evo1ution model which exptalns tbe evo1ution process of en eche lon fracture svstems. The fractal nature of the fractures can be used to dcrive an accurate estimateof total energy dissipation.

Study for the Gating Kinetics of Potassium Channel in Clonal Pheochromocytoma Cells Based On Fractal Model

To study the gating kinetics of voltage dependent K + channel in clonal pheochromocytoma (PC12) cells, the inward currents of K channel in PC12 cells were recorded using cell attached patch clamp technique. The fractal features of the open and closed time distributions were determined. The fractal dimensions and kinetic setpoints for the open and closed durations were estimated. It was found that the fractal dimension D was independent of voltage and the logarithm logA of the setpoint A was inversely proportional to the pipette potential (Vp) for the closed durations. While for the open durations the fractal dimension D was inversely proportional to the pipette potential and the logarithm logA of the setpoint A was independent of voltage. Thus, for this channel the open and closed durations and voltage dependence of the gating could be well described by the fractal model.

Fractal analysis of major faults and fractal dimension of lineaments in the Indo-Gangetic Plain on a regional scale

The Indo-Gangetic Plain(IGP)is one of the most seismically vulnerable areas due to its proximity to the Himalayas.Geographic information system(GIS)-based seismic characterization of the IGP was performed based on the degree of deformation and fractal dimension.The zone between the Main Boundary Thrust(MBT)and the Main Central Thrust(MCT)in the Himalayan Mountain Range(HMR)experienced large variations in earthquake magnitude,which were identified by Number-Size(NS)fractal modeling.The central IGP zone experienced only moderate to low mainshock levels.Fractal analysis of earthquake epicenters reveals a large scattering of earthquake epicenters in the HMR and central IGP zones.Similarly,the fault fractal analysis identifies the HMR,central IGP,and south-western IGP zones as having more faults.Overall,the seismicity of the study region is strong in the central IGP,south-western IGP,and HMR zones,moderate in the western and southern IGP,and low in the northern,eastern,and south-eastern IGP zones.

Mapping Spatial Distribution Characteristics of Lineaments Extracted from Remote Sensing Image Using Fractal and Multifractal Models

Mapping mineral prospectivity in vegetated areas is a challenge. For this reason, we aimed to map spatial distribution characteristics of linear structures detected in remote sensing images using fractal and multifractal models. The selected study area was the Pinghe District of the Fujian Province（China）, located in the Shanghang-Yunxiao polymetallic and alunite ore belt（within the Wuyishan polymetallic belt）, where mineral resources such as copper, molybdenum, gold, silver, iron, lead, zinc, alunite and pyrophyllite have been discovered. The results of our study showed that：（1） the values of fractal dimension for all lineaments, NW-trending lineaments, and NE-trending lineaments, are 1.36, 1.32, and 1.23, respectively, indicating that these lineaments are statistically self-similar;（2） the fractal dimensions of the spatial distribution of the linear structures in the four selected hydrothermal-type ore deposits of the Pinghe District, named Zhongteng, Panchi, Xiaofanshan and Fanshan, are 1.43, 1.52, 1.37 and 1.37, respectively, which are higher than the mean value in South China;（3） the spatial distribution of the linear structures extracted from the remote sensing image and displayed by the contour map of fractal dimensions, correlates well with the known hydrothermal ore deposits; and（4） the results of the anomaly map decomposed by the spectrum-area（S-A） multifractal model is much better than the original fractal dimension contour map, which showed most of the known hydrothermal-type deposits occur in the high anomalous area. It is suggested that a high step tendency possibly matches with the boundary of the volcanic edifice and the deep fault controlling the development of the rock mass and the volcanic edifice. The complexity of the spatial distribution of mapped lineations（faults） in the Pinghe District, characterized by high values in the anomaly map, may be associated with the hydrothermal polymetallic ore mineralization in the study area.

Two categories of fractal models of rock and soil expressing volume and size-distribution of pores and grains

Based on the Sierpinski carpet and Menger sponge models, two categories of fractal models of rock and soil which are composed of the volume fractal model of pores, the volume fractal model of grains, pore-size or particle-size distribution fractal models are established and their relations are clarified in this paper. Through comparison and analysis, it is found that previous models can be unified by the two categories of fractal models, so the unified fractal models are formed. Experimental results presented by Katz indicate that the first category of fractal models can be used to express the fractal behavior of sandstone. A scanning electron microscope (SEM) will be used to study the microstructure of soft clay and it will be testified that the fractal behavior of soft clay suits the second category of fractal models.

STUDIES ON FRACTAL MODELS OF THE MICROFRACTURE OF MARBLE

The fracture of rock is dependent strongly on its microstructures and the loading situations. The rough and irregular fracture is affected by these and other complicated factors. But if the fracture is examined by means of mathematical method, only a geometry of irregularity appears. B. B. Mandelbrot proposed a new geometry—fractal geometry, and he applied it to various fields to describe irregular phenomena in nature.

Performance-based fractal fracture model for complex fracture network simulation

The paper presents a novel hydraulic fracturing model for the characterization and simulation of the complex fracture network in shale gas reservoirs. We go beyond the existing method that uses planar or orthogonal conjugate fractures for representing the ＇＇complexity＇＇ of the network. Bifurcation of fractures is performed utilizing the Lindenmayer system based on fractal geometry to describe the fracture propagation pattern, density and network connectivity. Four controlling parameters are proposed to describe the details of complex fractures and stimulated reservoir volume（SRV）. The results show that due to the multilevel feature of fractal fractures, the model could provide a simple method for contributing reservoir volume calibration. The primary-and second-stage fracture networks across the overall SRV are the main contributions to the production, while the induced fracture network just contributes another 20% in the late producing period. We also conduct simulation with respect to different refracturing cases and find that increasing the complexity of the fracture network provides better performance than only enhancing the fracture conductivity.

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.

A Research on Fractal Dimension and Nonlinear Dynamic Model of Xintan Landslide，China

The dynamic research of landslide is one of the key Points in landslidology. In the paper,from the view point of nonlinear dynamic theory. some features of Xintan landslide, such as the distribution regularities of spatial and temporal fractal dimensions and their corresponding relationships to landslide occurring are researched. The accumulative principles of fractal dimension reduction is exploratively pointed out. The nonlinear dynamic equation of the landslide is built by analyzing the relationship between the correlation dimension and the phase space. Finally, the forecasted results and error analysis are listed. The research results are satisfactory.

Fractal Evolving Theory and Growing Model of Olefin Polymerization Process

The surface morphology of Ti-Mg supported catalyst and the polyethylene particles are studied using scanning electron microscope(SEM) technology.The results show that either the catalyst's surface or polymer particle's surface is irregular and has fractal characteristics,which can be described by fractal parameter.The more interesting discovery is that the surface fractal dimension values of the polymer particles vary periodically with the polymerization 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 growing model(PFGM),and successfully describe and /or predict the whole evolving process of the polyethylene particle morphology under the different slurry polymerization(including pre-polymerization) conditions without H2.

Fractal Prediction Model of Thermal Contact Conductance of Rough Surfaces

The thermal contact conductance problem is an important issue in studying the heat transfer of engineering surfaces, which has been widely studied since last few decades, and for predicting which many theoretical models have been established. However, the models which have been existed are lack of objectivity due to that they are mostly studied based on the statistical methodology characterization for rough surfaces and simple partition for the deformation formats of contact asperity. In this paper, a fractal prediction model is developed for the thermal contact conductance between two rough surfaces based on the rough surface being described by three-dimensional Weierstrass and Mandelbrot fractal function and assuming that there are three kinds of asperity deformation modes： elastic, elastoplastic and fully plastic. Influences of contact load and contact area as well as fractal parameters and material properties on the thermal contact conductance are investigated by using the presented model. The investigation results show that the thermal contact conductance increases with the increasing of the contact load and contact area. The larger the fractal dimension, or the smaller the fractal roughness, the larger the thermal contact conductance is. The thermal contact conductance increases with decreasing the ratio of Young＇s elastic modulus to the microhardness. The results obtained indicate that the proposed model can effectively predict the thermal contact conductance at the interface, which provide certain reference to the further study on the issue of heat transfer between contact surfaces.

Fractal model of lightning channel for simulating lightning strikes to transmission lines

How to accurately evaluate the direct-strike lightning protection is one of the key issues in the design of transmission lines. In this paper, three important issues in applying the fractal simulation to the lightning protection of transmission lines were discussed, including the criteria and implementation of upward leader inception, the connection with the magnitude of lightning current, and the calculation and control of fractal dimensions. Then we conducted the simulation iterately, leading to statistical results, which indicate that even if the transmission line satisfies the perfect shielding condition, shielding failure fault remains possible. Furthermore, we calculated the shielding failure fault rates of an EHV line with different ground obliquities and distribution of strike points over the interval between two neighboring towers along a UHV-DC line to find out the weak point of transmission-line lightning protection. This work provides a promising approach for improving the lightning protection property of transmission lines by optimizing the configuration of shielding wires and phase or pole conductors.

A two scale nonlinear fractal sea surface model in a one dimensional deep sea

Using the theory of nonlinear interactions between long and short waves, a nonlinear fractal sea surface model is presented for a one dimensional deep sea. Numerical simulation results show that spectra intensity changes at different locations （in both the wave number domain and temporal-frequency domain）, and the system obeys the energy conservation principle. Finally, a method to limit the fractal parameters is also presented to ensure that the model system does not become ill-posed,

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.

Numerical Modelling of Ore-forming Dynamics of Fractal Dispersive Fluid Systems

Based on an analysis of the fractal structures and mass transport mechanism of typical shear-fluid-ore formation system, the fractal dispersion theory of the fluid system was used in the dynamic study of the ore formation system. The model of point-source diffusive illuviation of the shear-fluid-ore formation system was constructed, and the numerical simulation of dynamics of the ore formation system was finished. The result shows that: (1) The metallogenic system have nested fractal structure. Different fractal dimension values in different systems show unbalance and inhomogeneity of ore-forming processes in the geohistory. It is an important parameter to symbolize the process of remobilization and accumulation of ore-forming materials. Also it can indicate the dynamics of the metallogenic system quantitatively to some extent. (2) In essence, the fractal dispersive ore-forming dynamics is a combination of multi-processes dominated by fluid dynamics and supplemented by molecule dispersion in fluids and fluid-rock interaction. It changes components and physico-chemical properties of primary rocks and fluids, favouring deposition and mineralization of ore-forming materials. (3) Gold ore-forming processes in different types of shear zones are quite different. (1) In a metallogenic system with inhomogeneous volumetric change and inhomogeneous shear, mineralization occurs in structural barriers in the centre of a shear zone and in geochemical barriers in the shear zone near its boundaries. But there is little possibility of mineralization out of the shear zone. (2) As to a metallogenic system with inhomogeneous volumetric change and simple shear, mineralization may occur only in structural barriers near the centre of the shear zone. (3) In a metallogenic system with homogeneous volumetric change and inhomogeneous shear, mineralization may occur in geochemical barriers both within and out of the shear zone.

The Influence of the Model Fractal Parameters on the Electromagnetic Response in Environment with Cylindrical Layers

The induced polarization response of an environment presenting cylindrical layers was obtained. The fractal model for complex resistivity was employed as an intrinsic property of the polarizable layers. The influence of the model fractal parameters on the electromagnetic response was investigated. The results demonstrated that the fractal parameters dominate the apparent resistivity phase response;measurements of the induced polarization data allow for the determination of the fractal properties of the environment without noticeable electromagnetic coupling effects at frequencies below 104 Hz.