Modeling Soil Water Retention Curve with a Fractal Method

Many empirical models have been developed to describe the soil water retention curve (SWRC). In this study, a fractal model for SWRC was derived with a specially constructed Menger sponge to describe the fractal scaling behavior of soil; relationships were established among the fractal dimension of SWRC, the fractal dimension of soil mass, and soil texture; and the model was used to estimate SWRC with the estimated results being compared to experimental data for verification. The derived fractal model was in a power-law form, similar to the Brooks-Corey and Campbell empirical functions. Experimental data of particle size distribution (PSD), texture, and soil water retention for 10 soils collected at different places in China were used to estimate the fractal dimension of SWRC and the mass fractal dimension. The fractal dimension of SWRC and the mass fractal dimension were linearly related. Also, both of the fractal dimensions were dependent on soil texture, i.e., clay and sand contents. Expressions were proposed to quantify the relationships. Based on the relationships, four methods were used to determine the fractal dimension of SWRC and the model was applied to estimate soil water content at a wide range of tension values. The estimated results compared well with the measured data having relative errors less than 10% for over 60% of the measurements. Thus, this model, estimating the fractal dimension using soil textural data, offered an alternative for predicting SWRC.

Solution and Type Curve Analysis of Fluid Flow Model for Fractal Reservoir

Conventional pressure-transient models have been developed under the assumption of homogeneous reservoir. However, core, log and outcrop data indicate this assumption is not realistic in most cases. But in many cases, the homogeneous models are still applied to obtain an effective permeability corresponding to fictitious homogeneous reservoirs. This approach seems reasonable if the permeability variation is sufficiently small. In this paper, fractal dimension and fractal index are introduced into the seepage flow mechanism to establish the fluid flow models in fractal reservoir under three outer-boundary conditions. Exact dimensionless solutions are obtained by using the Laplace transformation assuming the well is producing at a constant rate. Combining the Stehfest’s inversion with the Vongvuthipornchai’s method, the new type curves are obtained. The sensitivities of the curve shape to fractal dimension (θ) and fractal index (d) are analyzed;the curves don’t change too much when θ is a constant and d change. For a closed reservoir, the up-curving has little to do with θ when d is a constant;but when θ is a constant, the slope of the up-curving section almost remains the same, only the pressure at the starting point decreases with the increase of d;and when d = 2 and θ = 0, the solutions and curves become those of the conventional reservoirs, the application of this solution has also been introduced at the end of this paper.

Fractal Reconstruction of Log Curve

Sedimentary cyclothems at different scales show formations’ fractal structure which can be reflected on logs. The slope of the power spectrum of log is related to the fractal dimension of formations. The fractal dimensions from two logs with similar vertical resolutions are the same. Using fractal interpolating algorithm density log can be reconstructed. The reconstructed log can be compared with core density in washout intervals.

FRACTAL PROPERTIES OF HYPERBOLIC CURVES

In this paper, we study the fractal properties of the hyperbolic curve introduced by J. Belair ([Be]). We obtain some conditions of nowhere-differentiability of this kind of curves and its Bouligand dimension, and find a class of curves which are almost everywhere differertiable and have Bouligand dimensions being greater than one simultaneously.

THE UPPER BOUND OF BOX DIMENSION OF THE WEYL-MARCHAUD DERIVATIVE OF SELF-AFFINE CURVES

A certain type of self-affine curves can be transferred into fractal functions. The upper bound of fractal dimensions of the Weyl-Marchaud derivative of these functions has been investigated, and further questions have been put forwarded.

Hölder Derivative of the Koch Curve

In this paper, we introduce a K Hölder p-adic derivative that can be applied to fractal curves with different Hölder exponent K. We will show that the Koch curve satisfies the Hölder condition with exponent and has a 4-adic arithmetic-analytic representation. We will prove that the Koch curve has exact -Hölder 4-adic derivative.

Research on the Solidification Mechanism Based on the Water Characteristic Curve of Solidified Dredged Sediment

The fractal model about water characteristics of solidified sediment was built according to the granular metric analysis curve of solidified dredged sediment, the measured value during the low-suction stage of the curing process was used for fitting parameters in the model to obtain the complete water characteristic curve of solidified dredged sediment. Then, the quantitative calculation model of capillary water, attached water, evaporated water and bound water was built by the water characteristic curve and from the view of quantitative angle, the paper analyzed the solidification mechanism of solidified dredged sediment. The result showed that： the model can realize the quantitative calculation about different tapes of water during the curing process, the evaporated water during the curing process mainly came from the capillary water, and the generated bound water during the curing reaction came from the attached water.

STUDY ON IMAGE EDGE PROPERTY LOCATION BASED ON FRACTAL THEORY

A novel approach of printed circuit board(PCB)image locating is presented Based on the rectangle mark image edge of PCB,the featur es is used to describe the image edge and the fractal properby of image edge is analyzed It is proved that the rectangle mark image edge of PCB has some fractal features A method of deleting unordinary curve noise and compensating the length of the fractal curve is put forward,which can get the fractal dimension value from one complex edge fractal property curve The relation between the dim ension of the fractal curve and the turning angle of image can be acquired from an equation,as a result,the angle value of the PCB image is got exactly A real image edge analysis result confirms that the method based on the fractal theory is a new powerful tool for angle locating on PCB and related image area.

Two Fractal Regimes of the Soil Hydraulic Properties

A fractal analysis of the soil retention and hydraulic conductivity curves is presented. The retention process is modeled by a two fractal regimes: one pertaining to high water content values, and another accounting for the low water content data. This significantly improves the physical insight of the retention process as compared with the case of one-fractal models. The fractal dimensions characterizing the two regimes are estimated by fitting the retention curve model upon real data, and subsequently they are used to determine the hydraulic conductivity which for the retention curve models of Mualem and Burdine, is obtained in closed form. The reliability of the model is tested against independent conductivity data collected in a field-scale campaign.

ON STAR PRODUCTFRACTAL SURFACES AND THEIR DIMENSIONS

In this paper, by using fractal curves, a family of fractal surfaces are defined. Each fractal surface of this family is called Star Product Fractal Surface (SPFS). A theorem of the dimensions of the SPFS is strictly proved. The relationship between the dimensions of the SPFS and the dimensions of the fractal curves constructing the SPFS is obtained.

Study of Fractal Characteristics of the Cementation Index in Shale Gas

The description of pores and fracture structures is a consistently important issue and certainly a difficult problem, especially for shale or tight rocks. However, the exploitation of so-called unconventional energy, such as shale methane and tight-oil, has become more and more dependent on an understanding of the inner structure of these unconventional reservoirs. The inner structure of porous rocks is very difficult to describe quantitatively using normal mathematics, but fractal geometry, which is a powerful mathematical tool for describing irregularly-shaped objects, can be applied to these rocks. To some degree, the cementation index and tortuosity can be used to describe the complexity of these structures. The cementation index can be acquired through electro-lithology experiments, but, until now, tortuosity could not be quantitatively depicted. This research used the well-logging curves of a gas shale formation to reflect the characteristics of the rock formations, and the changes in the curves to indicate the changes of the rock matrix, the pores, the connections among the pores, the permeability, and the fluid type. The curves that are affected most by the rock lithology, such as gamma ray, acoustic logging, and deep resistivity curves, can provide significant information about the micro-or nanostructure of the rocks. If the rock structures have fractal characteristics, the logging curves will also have fractal properties. Based on the definition of a fractal dimension and the Hausdorff dimension, this paper presents a new methodology for calculating the fractal dimensions of logging curves. This paper also reveals the deep meaning of the rock cementation index, m, through the Hausdorff dimension, and provides a new equation to calculate this parameter through the resistivity and porosity of the formation. Although it represents a very important relationship between the saturation of hydrocarbons with pores and resistivity, the Archie formula was not available for shale and tight rock. The major reason for this was an incorrect understanding of the cementation index, and the calculation of saturation used a single m value from the bottom to the top of the well. Unfortunately, this processing method is clearly inappropriate for the intensely heterogeneous material that is shale and tight rock. This paper proposes a method of calculating m through well-logging curves based on a fractal geometry that can change with different lithologies, so that it would have more agreement with in situ scenarios than traditional methods.

Fractal Interpolation Functions: A Short Survey

The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.

Self-Similarities of Pulmonary Arterial Tree and a New Integrated Model of Pulmonary Circulation with the Name of Fractal Phasic Perfusion (FPP) Model

Pulmonary arterial hypertension (PAH) has become an important topic of basic and clinical research in recent years. Morphologic researches have shown that specific PAH-lesions are located in the lobular small muscular arteries and correlate with hemodynamic measurements. However, it still remains to be shown how pathological changes of the small arteries in the lobule develop to PAH. Based on both fractal properties of pulmonary arterial tree and asynchronous phasic contractions of lobular arterial muscles under the evenness of the pulmonary capillary pressure (PCP) in the lung, the author has constructed an integrated model of pulmonary circulation which has produced a mathematical relationship between the mean pulmonary arterial pressure (MPAP) and the cardiac output (CO). By use of the expression between MPAP and CO, it has been able to explain the pathogenesis of PAH in terms of statistical changes among regional and temporal perfusions in the lung. In order to detect clinically the early stage of PAH, the author has suggested that it is important to establish the pulmonary functional imaging of regional and temporal perfusions.

Upper Bound and Lower Bound Estimate of Monotone Increasing Fractal Function

Mass distribution principle is one of important tools in studying Hausdorff dimension and Hausdorff measure. In this paper we will give a numerical approximate method of upper bound and lower bound of mass distribution function f（x）（it is a monotone increasing fractal function） and its some applications.

一类级数边界的Fractal曲线

本文讨论一类Ｄｉｒｉｃｈｌｅｔ级数的一些有趣而又奇怪的边界性质，文中定理１，２指明这种级数的边界曲线能覆盖某个圆盘，定理３指出它是拓扑维数是１，熵维数是３的Ｆｒａｃｔａｌ曲线。

A Study on Application of Fractal Theory in Examination of Striation Marks

Use scissors and cutting pliers to produce some striation marks. The data collection apparatus is used to collect the surface data of such marks produced by scissors and cutting pliers, and then get the profile curve that is vertical to the surface of striation marks. In tha application of fractal theory, the fractal dimension of such a profile curve is then calculated, and further studies are made on its fractal characteristics. As an exploration on tool types and individual identification, this is aimed to provide a new theory and approach to examination and identification of striation tool marks.

Lift-Off Effect of Koch and Circular Differential Pickup Eddy Current Probes

A flexible or planar eddy current probe with a differential structure can suppress the lift-off noise during the inspection of defects.However,the extent of the lift-off effect on differential probes,including different coil structures,varies.In this study,two planar eddy current probes with differential pickup structures and the same size,Koch and circular probes,were used to compare lift-off effects.The eddy current distributions of the probes perturbed by 0°and 90°cracks were obtained by finite element analysis.The analysis results show that the 90°crack can impede the eddy current induced by the Koch probe even further at relatively low lift-off distance.The peak-to-peak values of the signal output from the two probes were compared at different lift-off distances using finite element analysis and experimental methods.In addition,the effects of different frequencies on the lift-off were studied experimentally.The results show that the signal peak-to-peak value of the Koch probe for the inspection of cracks in 90°orientation is larger than that of the circular probe when the lift-off distance is smaller than 1.2 mm.In addition,the influence of the lift-off distance on the peak-to-peak signal value of the two probes was studied via normalization.This indicates that the influence becomes more evident with an increase in excitation frequency.This research discloses the lift-off effect of differential planar eddy current probes with different coil shapes and proves the detection merit of the Koch probe for 90°cracks at low lift-off distances.

Estimation of Soil Water Retention Curve: An Asymmetrical Pore-Solid Fractal Model

The soil water retention curve is an important hydraulic function for the study of flow transport processes in unsaturated soils. The objective of this study was to develop a soil water retention function using a generalized fractal approach. The model exhibits asymmetry between the solid phase and pore phase, which is in marked contrast to the symmetry between phases present in a conventional fractal model. The retention function includes 4 parameters： the saturated water content θs, the air entry value ha, the fractal dimension Df, and an empirical parameter β, characterizing the complicated soil pore structures. Sixty one data sets, covering a wide range of soil structure and textural properties, were used to evaluate the applicability of the proposed soil water retention function. The retention function is shown to be a general model, which incorporates several existing retention models. The values of β/θs and （θs-θr ）/β were used as indexes to quantify the relationships between the proposed retention function and the existing retention models. The proposed function fits all the data very well, whereas other tested models only match about 16%-48% of the soil retention data.

Fractal approximation of the stress-strain curve of frozen soil

A method to approach the stress-strain curve of frozen soil is presented based on the fact that the stressstrain curve of frozen soil has fractal property. First, a linear hyperbolic iterated function system (LHIFS) in which the perpendicular contraction factors are regarded as parameters is established using fractal geometry theories. Secondly, a method to calculate the best point which makes the attractor of the LHIFS an optimal approximation of the stress-strain curve of frozen soil is presented. Then, a method for calculating the fractal dimension of the stress-strain curve of frozen soil is obtained. Finally, a simple example is provided. The method presented in this paper provides a new method for simulating the stress-strain curve and calculating its fractal dimension of geomaterials that have the fractal feature by using computer.