Fractal Analysis on the Spatial Structure of Land Use Patterns in a Non-Point Source Pollu

fractal and fractional佩普学术-回复Fractal and Fractional PEP AcademicIntroduction:Fractals and fractions are two mathematical concepts that have significant applications in various fields, including physics, computer graphics, and finance. This article aims to provide a comprehensive understanding of fractals and fractions, exploring their basic definitions, properties, and real-life applications.I. Fractals:1. Definition:Fractals are geometric shapes that exhibit self-similarity, meaning that they contain smaller copies of themselves in a never-ending pattern. They can be generated through a mathematical process called recursion. Examples of well-known fractals include the Mandelbrot set and the Sierpinski triangle.2. Properties:Fractals possess several distinctive properties, including infinite complexity, fractional dimension, and non-integer scaling. These properties contribute to their unique visual appearance and make them applicable in various fields, such as computer graphics and image compression.3. Applications:Fractals find applications in many practical areas. In computer graphics, they are used for creating realistic landscapes, textures, and natural objects. Fractal-based algorithms are also employed in image compression techniques, enabling efficient storage and transmission of digital images. Additionally, fractal analysis is utilized in medical imaging, financial forecasting, and weather prediction.II. Fractions:1. Definition:Fractions are numerical expressions representing a part or parts ofa whole. They consist of a numerator and a denominator, with the numerator representing the number of parts involved and the denominator indicating the total number of equal parts that make up the whole. For example, 3/4 represents three parts out of four equal parts.2. Properties:Fractions possess various properties, including equivalence, addition, subtraction, multiplication, and division. Equivalent fractions represent the same part-to-whole ratio, while adding, subtracting, multiplying, or dividing fractions follow specific rules and algorithms.3. Applications:Fractions have numerous real-life applications. In cooking and baking, fractions are used to determine ingredient quantities accurately. In finances, fractions are utilized to calculate interest rates, percentages, and financial ratios. Moreover, fractions play a significant role in measurements, allowing precise representations of lengths, weights, and volumes.III. Fractals and Fractions:1. Fractional Crystals:Fractional crystals are a special type of fractal pattern that combines the concepts of fractals and fractions. They are formed by repeatedly replacing parts of a shape with smaller copies. Each iteration involves dividing the shape into fractions of the original size and replacing them with smaller-scale copies.2. Applications:Fractional crystals offer an effective way to represent complex structures with fractional dimensions. They find applications in physics, chemistry, and materials science. For instance, they are used to model the behavior of polymers, the structure of porous materials, and the properties of amorphous solids.Conclusion:Fractals and fractions are fundamental mathematical concepts withsignificant practical applications. Fractals exhibit self-similarity and possess unique properties, making them useful in computer graphics, image compression, and numerous scientific fields. Fractions, on the other hand, represent parts of a whole and find applications in cooking, finance, and measurements. The combination of fractals and fractions leads to the concept of fractional crystals, enabling the representation of complex structures with fractional dimensions. Understanding these concepts is essential for anyone interested in mathematics or its various applications.。

第39卷第6期中南大学学报(自然科学版) V ol.39No.6 2008年12月J. Cent. South Univ. (Science and Technology) Dec. 2008双剪统一强度理论计算塑性金属材料强度的唯一性刘光连(中南大学机电工程学院，湖南长沙，410083)摘要：研究了双剪统一强度理论及其有关关系式，得到塑性金属材料双剪统一强度理论参数b的计算式，对某一塑性金属材料，其值为常数，取值范围是b＞−1；运用双剪统一强度理论计算某一塑性金属材料的强度时有唯一确定的强度计算值，而不是多个值，不能得出塑性金属材料的τs/σs；分析材料的拉压强度比为1时双剪统一强度理论的2组等价变换式，对b＜0的材料，双剪统一强度理论计算表明，材料的破坏不是由中间主剪应力引起，这与双剪统一强度理论的假设矛盾；该理论不适用于三向等值拉应力状态的计算。

关键词：双剪统一强度理论；屈服；塑性；金属中图分类号：TH114 文献标识码：A 文章编号：1672−7207(2008)06−1280−05Uniqueness in calculating strength of plastic metals based ontwin-shear unified strength theoryLIU Guang-lian(School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China)Abstract: Twin-shear unified strength theory and its related functions were studied. The formula of parameter b in the twin-shear unified strength theory for a certain plastic metal was developed and its value was determined to be constant for a certain plastic metal and its value range is b＞−1; only a certain value can be obtained when the strength of a plastic metal based on the twin-shear unified strength theory was calculated and the value of τs/σs could not be obtained. The results show that based on the analysis of the two equivalent equations of the twin-shear unified strength theory when the ratio of tensile strength to compressive strength equals 1, for materials with b＜0, the fracture of materials is not caused by the intermediate principal shear stress, which does not agree with the hypothesis of the twin-shear unified strength theory, and this theory can not be applied in stress state such as equitriaxial tension.Key words: twin-shear unified strength theory; yield; plastic; metalYu于1961年提出了双剪应力屈服准则[1]，1985 年发展为双剪应力强度理论[2]，1991年提出了新的双剪统一强度理论[3]。

叶片有限元分析中弹塑性过渡区应力奇异产生原因及解决方法仲继泽;徐自力;方宇;范小平;赵世全【摘要】采用弹塑性有限元法、借助大型商业有限元软件对汽轮机叶片进行应力分析时,弹塑性过渡区应力的计算值有时会高于塑性区应力的计算值,即会产生应力奇异现象.为分析产生这一现象的原因,以8节点六面体单元为例,研究了有限元法计算应力的过程,并在理想弹塑性的条件下,采用有限元法和解析法计算了弹塑性过渡区单元节点应力.研究发现,有限元法通常采用高斯积分点应力值外推插值法得到单元节点应力,当单元一部分位于弹性区、另一部分位于塑性区时,这种外插算法会导致节点应力计算值高于结构的实际应力,甚至超出理想弹塑性材料的屈服极限,从而造成应力奇异.研究表明,在叶片弹塑性的有限元分析中,采用相邻高斯积分点应力加权平均的方法计算单元节点应力,可有效避免弹塑性过渡区应力产生奇异的现象.【期刊名称】《西安交通大学学报》【年(卷),期】2015(049)009【总页数】5页(P47-51)【关键词】叶片;有限元;弹塑性过渡区;应力奇异【作者】仲继泽;徐自力;方宇;范小平;赵世全【作者单位】西安交通大学航天航空学院,710049,西安;西安交通大学机械结构强度与振动国家重点实验室,710049,西安;西安交通大学航天航空学院,710049,西安;西安交通大学机械结构强度与振动国家重点实验室,710049,西安;东方汽轮机有限公司,618000,四川德阳;东方汽轮机有限公司,618000,四川德阳;东方汽轮机有限公司,618000,四川德阳【正文语种】中文【中图分类】TK263.3;TB125叶片在高温、高压蒸汽的推动下驱动汽轮机转子转动,将蒸汽的热能转化为机械能。

为了提高单机的出力和机组的效率,长叶片和超长叶片不断应用于汽轮机末级,例如:国内研发的全转速钢制1 092 mm叶片已应用在1 000 MW汽轮机中[1],并开发出了全转速钢制1 200 mm叶片;日本三菱重工开发了转速为3 000 r/min的1 524 mm钢制长叶片[2]。

基于分形方法的水力压裂起裂模型的研究李统中;樊军【摘要】Hydraulic fracturing is a measure used to increase oil production. To accurately predict the pressure of fracturing, which is the key of the successful implementation of the hydraulic fracturing. Through derivingthe burst pressure of homogeneous formation a-gain based on the fractal theory, the breakdown pressure calculation model is established. Based on Fractal theory, this paper analy-zes the relationship between fracture toughness of rock along the fractal crack and straight crack with the helpof matlab. The theoret-ical analysis shows that the fracture extending toughness of rock along the fractal crack is greater than that of the straight line model. that’ s why the current theory calculation value of hydraulic fracturing construction pressure is always less than one of the actual con-struction of pressure. Furthermore, it’ s also proved that the computational model is closer to the actual situation in this paper.%水力压裂是油井增产通常采用的一种措施，准确地预测压裂井的破裂压力是水力压裂成功实施的关键所在。

第24卷　第4期2009年8月实　验　力　学J OU RNAL OF EXPERIM EN TAL M ECHANICSVol.24　No.4Aug.2009文章编号:100124888(2009)0420327207混凝土断裂能测试方法研究3杨松森,徐菁,赵铁军(青岛理工大学土木工程学院,山东266033)摘要:基于局部断裂能分布的双直线模型,推导出混凝土真实断裂能和受尺寸影响的断裂能的计算公式;并进一步通过分析四组不同尺寸试件的楔形劈裂试验数据,得出了不受尺寸影响的混凝土真实断裂能。

本文为确定混凝土的断裂能提供了一种实践可行的测试方法。

通过实验数据拟合,给出了断裂能非均匀分布的外部区域长度与试件尺寸的关系表达式。

这对于在规范中规定测试断裂能的标准尺寸试件是有意义的。

关键词:断裂能;测试方法;断裂韧带;尺寸效应;楔形劈裂试验中图分类号:TU313.2 文献标识码:A0　引言自从1961年Kaplan[1]将断裂力学的概念应用于混凝土并进行材料的断裂韧度试验以来,已经有四十多年的历史。

人们很快发现,将线弹性断裂力学应用于混凝土并不成功。

许多学者致力于这方面的改进,发展了多种非线性断裂力学理论。

在这些理论中,Hillerborg教授在1976年提出的虚拟裂缝模型(Fictitious Crack Model,简称FCM)受到广泛重视。

虚拟裂缝模型比线弹性断裂力学更好地揭示了混凝土裂缝萌生和扩展规律,同时可以利用它计算出断裂区长度及裂缝失稳前的亚临界扩展长度。

1985年,RIL EM(国际结构与材料研究所联合会)采纳Hillerborg教授所提出的用三点弯曲试验测试混凝土断裂能的实验方法为标准方法[2]。

在使用虚拟裂缝模型进行结构的有限元分析时,混凝土的断裂能是所采用的本构关系中必不可少的物理量。

实际上,现在几乎每一非线性断裂模型的提出都依赖于这一参数。

另外,在普通混凝土构件的设计中,如缺少抗剪钢筋的梁和板的剪切破坏、素混凝土管道的拉裂破坏等,断裂能亦是重要的力学性能指标,其重要性就如同常规的混凝土强度指标一样。

Fractal GeometryYale UniversityMichael Frame, Benoit Mandelbrot, and Nial NegerJuly 4, 2009An Escheresque fractal by Peter Raedschelders."I find the ideas in the fractals, both as a body of knowledge and as a metaphor, an incredibly important way of looking at the world." Vice President Al Gore, New York Times, Wednesday, June 21, 2000, discussing some of the "big think" questions that intrigue him.This is a collection of pages meant to support a first course in fractal geometry for students without especially strong mathematical preparation, or any particular interest in science.Each of the topics contains examples of fractals in the arts, humanities, or social sciences; these and other examples are collected in the panorama.Fractal geometry is a new way of looking at the world; we have been surrounded by natural patterns, unsuspected but easily recognized after only an hour's training.On these pages new windows are spawned by a simple JavaScript program. Some popup blockers disable these windows.Comments or questions should be directed to michael.frame@This material is based upon work supported by the National Science Foundation under Grant No. 0203203.Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.1. Introduction to FractalsHere we introduce some basic geometry of fractals, with emphasis on the Iterated Function System (IFS) formalism for generating fractals.In addition, we explore the application of IFS to detect patterns, and also several examples of architectural fractals.First, though, we review familiar symmetries of nature, preparing us for the new kind of symmetry that fractals exhibit.smaller copies of itself. The copies are similar tois the simplest methoddatingmechanicsfractalsto. Thisversionof Johnny Carson's "Karnak the Magnificent" routine.is another way to render fractalotherIFS, but apply them to a single point, one at a time,a variation on the Random Algorithm tointhemecommon to many cultures, developed independently soFinally, fractals seem to be a an easy concept for1. A. Self-SimilarityThe simplest fractals are constructed by iteration. For example, start with a filled-in triangle and iterate this process:For every filled-in triangle, connect the midpoints of the sides and remove the middle triangle. Iterating this process produces, in the limit, the Sierpinski Gasket.The gasket is self-similar. That is, it is made up of smaller copies of itself.each1/2 as tall and 1/2 as wide as the original. But notecopiestheeach 1/4 by 1/4 of theUsually,"Big gaskets are made of little gaskets,The bits into which we slice 'em.And little gaskets are made of lesser gasketsAnd so ad infinitum."This implies fractals possess a scale invariance.Return to Introduction to fractals.1. B. More Examples ofSelf-SimilarityHere we survey more examples of self-similarity and its variations.Return to Introduction to fractals.1.C. Initiators and GeneratorsOne way to guarantee self-similarity is to build a shape by applying the same process over smaller and smaller scales. This idea can be realized with a process called initiators and generators.The initiator is the starting shape.The generator is a collection of scaled copies of the initiator.The rule is this: in the generator, replace each copy of the initiator with a scaled copy of the generator (specifying orientations where necessary).Examples1. D. Geometry of PlaneTransformationsTo generate all but the simplest fractals, we need to understand the geometry of plane transformations. Here we describe and illustrate the four features of plane transformations, and show how they are encoded in our software.We adopt this convention:scalings first,reflections second,rotations third, andtranslations last.(This is implied by the matrix formulation.)Emphasizing this order, the components of a transformation are encoded in tables of this formWith this encoding of transformations of the plane, we can now make fractals using the method called Iterated Function Systems (IFS).Now we apply these ideas to generate fractals in 1. E. Iterated Function Systems.1.E. Iterated Function SystemsGenerating fractals by iterating a collection of transformations is the Iterated Function System(IFS) method, popularized by Barnsley, based on theoretical work by Hutchinson and Dekking. We use a simple example to see how it works.Toset ofThe gasket rules leave the gasketgasket is the only shape (of finite extent) leftunchanged by these rules.Now we consider theFinally,1.F. The Inverse ProblemGiven a fractal F, the Inverse problem is to find affine transformations T1, ..., T n for whichF = T1(F) U ... U T n(F)Here we present a method to solve this problem, as well as two implementations of the method.1.G. The Random IFS AlgorithmGiven IFS rules, the Deterministic Algorithm renders a picture of the fractal by1. applying all the rules to any (compact) initial picture,2. then applying all the rules to the resulting picture,3. and continuing this process.Regardless of the starting shape, this sequence of pictures converges to a unique limiting shape, the only (compact) set invariant under simultaneous application of all the rules.The Random Algorithm is another method of rendering the fractal determined by a given set of rules, T1, ..., T N. In this section we explore this method for producing fractals.What happens if the IFS rules are not applied in a random order? To find the answer, continue to 1.H. Driven IFS1.H. Driven IFS and Data AnalysisA natural question, as far as we know first posed by Ian Stewart, isWhat picture does the Random Algorithm generate if the driving sequenceis not random?After studying examples of the pictures produced by different sorts of nonrandom sequences, we use this rough visual vocabulary to detect patterns in real data.Because we use a data sequence to select the order in which the transformations are applied, we call this approach driven IFS. The data drive the order in which the IFS rules are applied.Fractals in Architecture Architecture is mostly about building places for us to live and work.Because manufacturing is very good at producing (approximately) Euclidean shapes - bricks, boards, girders - it is no surprise that buildings have Euclidean aspects.On the other hand, some architectural styles are informed by Nature, and much of Nature is manifestly fractal.So perhaps we should not be so surprised to find fractal architecture.As we shall see, fractals appear in architecture for reasons other than mimicking patterns in Nature.To emphasize that fractal architecture arose naturally in different cultures, we divide our examples into three categories.2. Natural Fractals and DimensionsPictured below are the Koch curve and three of its relatives. From top left to bottom right, these pictures become increasingly "fuzzy." Can we find a way to quantify the difference in these pictures? Perhaps such a method could be used to distinguish the coastline of Norway from that of Italy, for example, or the beating of a healthy heart from that of a diseased heart, or the closing prices of a conservative stock from those of a more risky stock, or a text of Shakespeare from one of Bacon. Let us see.Contents of this page:to fractals. Arguing by analogy with Euclidean dimension, wecomputingscaledindication. Among several variants, we study thescales structures, dust clumps and natural sponges, for example,. Fractal curves that enclosesubtlefrom other fractals, how is the dimension of the whole relatedsurvey a few examples, and of the physical and biological industry, and note examples of how dimension directs some2. A. Ineffective Ways to MeasureA familiar method of measuring the length of a curve is toapproximate the curve by straight line segments; add the lengths of the line segments.Smaller line segments should give a better approximation, andwe look for a limiting value of the lengths of the line segments, as we use smaller and smaller line segments.In more detail,Our eyes tell the story: as the segments are replaced with smaller segments, distinguishing the curve from the collection of segments becomes more difficult.Length is a one-dimensional measure; area a two-dimensional measure. Neither is a useful measure for the Koch curve, so the Koch curve is somehow more than one-dimensional and less than two-dimensional. What is it?2. B. Box-Counting DimensionWe have seen that trying to measure the length of the Koch curve gives infinity, while trying to measure the area of the Koch curve gives zero.Neither is a useful result. Here we shall introdce a more general measure that leads to the idea of box-counting dimension.Wechanges with the size of the boxes.If the object is 1-dimensional, such as the unitsegmentas the squares get smaller, more will be needed to coverthe object.)IfsquareFor1/r may be aThisFor the1.58996 ... . The gasket is more than 1-dimensional, butless than 2-dimensional.For the1.26186 ... . The Koch curve is more than 1-dimensional,but less than 2-dimensional.What happens when we measure an object in thedimensionTodimension in familiar cases, consider the filled-intriangleNow we compute the box-counting dimension of theMiddle Thirds Setand of a combination of theand of a combination of theHerebox-counting dimension.Here are someFinally,dimensions.In Similarity Dimension we shall see many of these computations can be done in a much simpler way.However, the box-counting dimension also can be computed for many natural fractals.2. C. Similarity DimensionFor exactly self-similar fractals, with all pieces scaled by the same factor, calculation of the box-counting dimension can be simplified greatly.The Moran equation generalizes this computation to self-similar fractals where different pieces are scaled by different factors.2. D. The Moran EquationThe similarity dimension equation can be applied only when the all the pieces are scaled by the same amount.Yet many self-similar fractals are made of pieces scaled by different amounts. Here we learn to compute the similarity dimension of these more general self-similar fractals.2. E. Other DimensionsBox-counting and similarity dimension are two of a multitude of variations on the notion of dimension.Another, the mass dimension, is based on the idea of how the mass of an object scales with the object's size (assuming unchanged density).The mass dimension, d m, of an object is defined bylocating a point P inside the object (near the middle) anddenoting by M(r) the amount of mass of the object inside the circle (sphere if the object lies in space instead of in the plane)of radius r and centered at P.If this power law relationM(r) = k r dholds over some range of r values, then the mass dimension d m = d.Assuming the shape has finite extent, the power relation holds only for a range of r values.When r becomes too large, the entire object is contained inside the circle of radius r and M(r) no longer changes as r increases.When r becomes too small, we are in the realm of atoms, quarks, superstrings, who knows what? There is no reason to exect the power law relation to hold on scales smaller than the forces that sculpt the object.Several examples are in power laws.There are other dimensions - Hausdorff, packing, Minkowski, ... many more. Each has advantages and disadvantages. For simple sets, all agree. But beyond these, the subject becomes a thicket difficult to penetrate.Area-Perimeter Relationobtained by the comparing the perimeter of the curve and.fractalIf the dimension, d, of the curve satisfies d > 1, thenfinite.formtheperimetersSome Algebra of Dimensionsof A and B relatedto the dimensions A and B? Typically, the dimensiontothe dimensions A and B? Typically, the dimension ofHow is the dimension of A related to the dimension ofof A to B? Think of the projection asthe shadow A casts on B. For typical directions ofofthe projection of A plus the dimension of the typicalPanorama of Fractals and Their UsesNature and FractalsThe Cantor set, the Koch curve, The Sierpinski gasket and carpet, the Menger sponge, Julia sets, Brownian motion trails, ... these mathematical constructions that now are examples of fractals, have been known for a long time.Part of Benoit Mandelbrot's brilliance lay in organizing these ideas into a coherent field, but perhaps a larger contribution lay in recognizingthese ideas constituted a powerful organizing principle for natural phenomena.Before the development of fractal geometry, typicallyNature was regarded as noisy Euclidean geometry.For example, a mountain is primarily a roughened cone. The clearest statement of this view may have been given in Paul Cezanne's instructions to young painters:"Everything in Nature can be viewed in terms of cones, cylinders, andspheres."In contrast to this, Mandelbrot asserts,"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straightline."Here are a few examples, among many, Many, MANY more.are extinct relatives of the nautilus. The suturesaretherecognized.tointhefocusthemup and weahtering breaking them down. Little surprise theyare good examples of natural fractals, because of theiroff branches) andpictures taken from variousManufactured FractalsNature is filled with fractals, but the manufactured world is mostly Euclidean. Technology developed without exploiting iteration across many scales. Yet the lessons of blind physical forces, and of billions of years of organic evolution, assert the importance of fractals. Recently, some industries have started exploiting fractal forms. Here are a few examples.: a multitude of scales: multiple length scales ofexcitations - only a part of the drumheadmoves - and to very efficient absorptionoptical fibers make waveguides with verylungs, two fluids can be thoroughly mixedvariationusedPerhaps the real blossoming of industrial fractals will come once nanotechnology has developed cell-sized constructors. Swarms of these could organize themselves hierarchically to build fractals, with results different from those suggested by Lem, we hope.3. The Mandelbrot Set and Julia SetsClick to magnify Click to magnifyThese images come from the Mandelbrot set gallery of Frank Roussel, http://graffiti.u-bordeaux.fr/MAPBX/roussel/fractals.html.Applets to explore Julia sets and the Mandelbrot set, and other fractal topics, can be found at Bob Devaney's Dynamical Systems and Technology Project website.Scarcely twenty years old, the Mandelbrot set may well be the most familiar image produced by the mathematics of the last century. Its status as a cultural icon needs no support.From a philosophical perspective, the Mandelbrot set challenges familiar notions of simplicity and complexity: how could such a simple formula, involving only multiplication and addition, produce a shape of great organic beauty and infinite subtle variation?Also, deep mathematics underlies the Mandelbrot set. Despite years of study by brilliant mathematicians (three of whom won Fields Medals), some natural and simple-to-state questions remain unanswered. Much of the rebirth of interest in complex dynamics was motivated by efforts to understand the stunning images of the Mandelbrot set.Also, as we shall see, hidden within it are metaphors (and more) for some of the richness of contemporary literature and music.Finally, some instances are just plain entertaining, in one way or another.arithmetic:. For a complex number c, the filled-inTheset.. The Mandelbrot set is the setstartingfrom z = 0, does not diverge to infinity. Julia setsinfinitely many points. The Mandelbrot set is thosewith each disc and cardioid of the Mandelbrot set isofa feature to those of nearby features. From this we. Themany copies of the Mandelbrot set. In fact, as closeinfinitely many little Mandelbrots. The boundary issetforarisemethodcopiesnothing.century?3. The Mandelbrot Set and Julia Sets3. A. Complex IterationUsing complex numbers, the process iterated to generate the Mandelbrot set and the Julia sets takes a very simple form:z -> z2 + cwhere z and c are complex numbers.To iterate the process, pick a complex number c and a complex number z0. Then generate the sequence of complex numbers z1, z2, z3, ... byz1 = z02 + cz2 = z12 + cz3 = z22 + cand in generalz n+1 = z n2 + cHow this process generates the Mandelbrot set and Julia sets is the subject of Julia Sets and the Mandelbrot Set. Here we review the mechanics of z -> z2 + c.Return to the Mandelbrot set and Julia sets.3. The Mandelbrot Set and Julia Sets3. B. Julia SetsFirst we define the filled-in Julia set, K c, for each complex number c.For each point z0 of the plane, generate a sequence z1, z2, z3, ... by the basic iteration rulez n+1 = z n2 + cIf the sequence does not run away to infinity, then the point z0 belongs to K c;Return to the Mandelbrot set and Julia sets.3. The Mandelbrot Set and Julia Sets3. C. The Mandelbrot SetReturn to the Mandelbrot set and Julia sets.3. The Mandelbrot Set and Julia Sets3. D. Combinatorics in the Mandelbrot SetReturn to the Mandelbrot set and Julia sets.3. The Mandelbrot set and Julia Sets3. E. The boundary of the Mandelbrot setIn many ways, the most interesting part of the Mandelbrot set is its boundary. First, what is the boundary of a set S in the plane?.Return to the Mandelbrot set and Julia sets.3. The Mandelbrot Set and Julia Sets3. F.Scalings in the Mandelbrot SetReturn to the Mandelbrot set and Julia sets.3. Julia Sets and the Mandelbrot Set3. G. Complex Newton's MethodNext we shall see how applying this method to a family of cubic polynomials led to the discovery that the Mandelbrot is universal, in a sense ubiquitous.Return to the Mandelbrot set and Julia sets3. Julia Sets and the Mandelbrot Set3. H. Universality of the Mandelbrot setCurry, Garnett, and Sullivan studied ways in which Newton's method fails to converge to a root, and found a surprise. Let us see what they found.Return to the Mandelbrot set and Julia sets.THEMANDELBROTMONKby Ray GirvanUntil recently, Udo ofAachen occupied a sideline in the history books as a minor poet, copyist and theological essayist. Even his birth and death dates of this mediaeval Benedictine monk are unknown, though he probably lived from around 1200-1270 AD. [*1] A new study of his work, however, has led to his recognition as an outstandingly original and talented mathematician.While Udo himself is little-known, one of his works is far more familiar. This 13th century German monk was the author of a poem called Fortuna Imperatrix Mundi (Luck, Empress of the World) in the collection of mediaeval underground verses now known as the Carmina Burana. [*2] Orchestrated by composer Carl Orff in 1937, Udo's poem is now widespread as the choral work, O Fortuna, which has been used by the media many times, from incidental music to the film Excalibur to the backing forafter-shave lotion advertisements.The first clue to Udo's undiscovered skills was found by mathematician Bob Schipke, a retired professor of combinatorics. On a holiday visit to Aachen cathedral, the burial place of Charlemagne, Schipke sawsomething that amazed him. In atiny nativity sceneilluminating the manuscript of a13th century carol, O froehlicheWeihnacht, he noticed that theStar of Bethlehem looked odd. Onexamining it in detail, he sawthat the gilded image seemed tobe a representation of theMandelbrot set, one of the iconsof the computer age. [*3]Discovered in 1976 by IBMresearcher Benoit Mandelbrot, the Mandelbrot set is the most famous fractal (a mathematical object with the property of infinite detail). Only the advent of fast computers made feasible the repeated calculations involved - or so it was thought. [*4]"I was stunned," Schipke says. "It was like finding a picture of Bill Gates in the Dead Sea Scrolls. The colophon [the title page] named the copyist as Udo of Aachen, and I just had to find out more about this guy."Schipke visited Bavaria, where the poems, Cantiones profanae (now the Carmina Burana), were discovered in 1837. Written by wandering scholars and monks in the 13th century, they were collected as an antholog y in the Benedictine monastery at Beuron, near Munich, and Schipke began his search there. With the help of historian Dr Antje Eberhardt at the University of Munich, Schipke gained access to ecclesiastical archives, where he found a document called the Codex Udolphus. Written in illuminated Latin, with informal marginalia in Greek, the Codex bore the signature of Udo himself."Although it had been discovered in the 19th century, it had promptly been filed away again," Schipke says. "The local historian who found it was clearly no mathematician, and dismissed it as obscure theology. But it yielded several major surprises."In a recent paper, Schipke and Eberhardt report on Udo's discoveries. [*5] The first chapter, Astragali (Dice) was originally thought to be a discourse on the evils of gambling. It turned out to be Udo's research into what we now would call probability theory. He derived simple rules to add and multiply probabilities, and thus devised strategies for several card and dice games.The second part, Fortuna et Orbis (Luck and a Circle) describes Udo's determination of the value of pi by scattering equal sticks on a ruled surface, and counting what proportion lie across the lines. This was an anticipation of the Buffon's Needle technique, named after the 18th century mathematician normally credited with its discovery. [*6] This is a very laborious method, but Udo managed to get a respectable - but very lucky - approximation of 866/275 (3.1418...) and had enough confidence in it to dispute the value of pi=3 implied in the Bible. [*7] (I say 'lucky' because Buffon's Method converges extremely badly, and it's well possible that Udo achieved this good result by choosing his stopping point judiciously - perhaps influenced by the 3.1418 quoted by his contemporary, Leonard of Pisa, otherwise known as Fibonacci).Schipke continues: "What was interesting at this point was that we looked back at the words of O Fortuna, and suddenly they fell into place. Verse two - Luck / like the moon / changeable in state / We are cast down / like straws upon a ploughed field / Our fates measuring / the eternal circle- is very clearly an allusion to the Buffon's Needle method." [*8]More was to come. In the final and longest chapter, Salus (Salvation), Schipke uncovered the most radical work. Udo had, it seemed, investigated the Mandelbrot set, seven centuries before Mandelbrot.Initially,Udo's aim wasto devise amethod fordeterminingwho wouldreach heaven.He assumedeach person'ssoul wascomposed ofindependentparts hecalled"profanus"(profane) and"animi"(spiritual), and represented these parts by a pair of numbers. Then he devised rules for drawing and manipulating these number pairs. In effect, he devised the rules for complex arithmetic, the spiritual and profan e parts corresponding to the real and imaginary numbers of modern mathematics.In Salus, Udo describes how he used these numbers: "Each person's soul undergoes trials through each of the threescore years and ten of allotted life, [encompassing?] its own nature and diminished or elevated in stature by others [it] encounters, wavering between good and evil until [it is] either cast into outer darkness or drawn forever to God."When Schipke saw the translation, at once he saw it for what it was: an allegorical description of the iterative process for calculating the Mandelbrot. In mathematical terms, Udo's system was to start with a complex number z, then iterate it up to 70 times by the rule z -> z*z+ c, until z either diverged or was caught in an orbit. [*4]Below the description was drawn the first crude plot of the Mandelbrot, which Udo called the "Divinitas" ("Godhead"). He set it out in a 120x120 frame he termed a "columbarium" (i.e. a dovecote, which has a similar grid of niches) and records that it took him nine years to calculate, even with the newly imported technique of ‘algorism', calculation with Arabic numerals rather than abacus."It tends to be taken for granted," Schipke says, "That the Mandelbrot is too calculation-intensive to be done without computers. What we have to remember is the sheer devotion of the monastic life. This was a labour of faith, and Udo was prepared to work for years. Some slowly-converging pixels must have taken weeks."Why did the work of this gifted mathematician go unnoticed for so long? Schipke blames, in part, specialisation. "When the Codex was unearthed in 1879, only a non-mathematician got to see it, and he didn't know what he was looking at. It's a common enough story. Take Hildegard of Bingen, whose accounts of her visions were taken as pure mysticism, but neurologist Oliver Sacks instantly recognised them as accurate descriptions of migraine symptoms. Likewise, literary critics dismissed Edgar Allan Poe's final work, Eureka, as alcoholic ravings. But now scientists are finding valid insights in it, such as Poe's correct solution of the Olbers paradox in astronomy, or his coining of the classic Einsteinian phrase, 'Space and duration are one'." [*9, *10]"But there were also contemporary reasons why Udo's knowledge didn't make it into the mainstream. His basic belief - that salvation and damnation could be determined in advance - was heretical, and his use of Arabic numerals was thought a bit of a black art. And there was the disagreement with Thelonius."Despite the borderlinenature of his work, Udoimpressed his abbot at themonastery of Sankt Umbertusnear Aachen. Life for a 13thcentury monk wasn'tnecessarily austere: thescurrilous Cantionesprofanae poems record thedelights of sex, eating,drinking and gambling. In afootnote to Astragali, Udowrites: "My enumeration ofthe ways [of dice] helped mylord abbot to win thirty-twoflorins and a fine new cloakfrom the Burgermeister atIrrendorf, and he haspromised me a helper for mywork".But Udo and his helper, Thelonius, ran into instant disagreement. Udo had always interpreted the Mandelbrot as signifying God. Thelonius took the opposite view: that it represented the Devil. Numbers that escaped to infinity, he argued, were souls flying free to heaven, and those caught in an orbit had fallen into the pit of Hell. Like many theological collaborations, they had a schism on their hands.Udo noted that their differences brought all work to a halt, and finally the two were reprimanded by the abbot for coming to blows in the refectory. "Sadly I write," says Udo on the last page of the Codex Udolphus, "that on pain of excommunication I must lay down my dice and my numbers. I have seen into a realm of heavenly complexity, and my heart is heavy that the door is now closed."Bob Schipke comments: "It's a pity that personal differences ended research that could have moved mathematics forward by centuries. But fortunately, Udo couldn't leave the subject alone. By dropping clues into the Cantiones profanae and the manuscripts he illuminated later in his life, he ensured that we were able to recover his work and give him the recognition that he deserves."。

A fractal analysis of permeability for fracturedrocksTongjun Miao a ,b ,Boming Yu a ,⇑,Yonggang Duan c ,Quantang Fang caSchool of Physics,Huazhong University of Science and Technology,1037Luoyu Road,Wuhan 430074,Hubei,PR ChinabDepartment of Electrical and Mechanical Engineering,Xinxiang Vocational and Technical College,Xinxiang 453007,Henan,PR China cState Key of Oil and Gas Reservoir Geology and Exploitation,Southwest Petroleum University,8Xindu Road,Chengdu 610500,Sichuan,PR Chinaa r t i c l e i n f o Article history:Received 6September 2013Received in revised form 28March 2014Accepted 5October 2014Keywords:Permeability Rock Fractal FracturesFracture networksa b s t r a c tRocks with shear fractures or faults widely exist in nature such as oil/gas reservoirs,and hot dry rocks,etc.In this work,the fractal scaling law for length distribution of fractures and the relationship among the fractal dimension for fracture length distribution,fracture area porosity and the ratio of the maxi-mum length to the minimum length of fractures are proposed.Then,a fractal model for permeability for fractured rocks is derived based on the fractal geometry theory and the famous cubic law for laminar flow in fractures.It is found that the analytical expression for permeability of fractured rocks is a function of the fractal dimension D f for fracture area,area porosity /,fracture density D ,the maximum fracture length l max ,aperture a ,the facture azimuth a and facture dip angle h .Furthermore,a novel analytical expression for the fracture density is also proposed based on the fractal geometry theory for porous media.The validity of the fractal model is verified by comparing the model predictions with the available numerical simulations.Ó2014Elsevier Ltd.All rights reserved.1.IntroductionFractured media and rocks with shear fractures or faults widely exist in nature such as oil/gas reservoirs,and hot dry rocks,ually,the fractures are embedded in porous matrix with micro pores,which play negligible effect on the seepage characteristic,and randomly distributed fractures dominate the seepage charac-teristic in the media.The randomly distributed fractures are often connected to form irregular networks,and the seepage character-istic of the fracture networks has the significant influence on nuclear waste disposal [1],oil or gas exploitation [2],and geother-mal energy extraction [3].In this work,we focus our attention on the seepage characteristics of fracture networks in fractured rocks and ignore the seepage performance from micro pores in porous matrix.Over the past four decades,many investigators studied the seepage characteristics of fracture networks/rocks and proposed several models.Snow [4]developed an analytical method for per-meability of fracture networks according to parallel plane model.Kranzz et al.[5]studied the permeability of whole jointed granite and tested the parallel plane model by experiments.Koudina et al.[6]investigated the permeability of fracture networks with numer-ical simulation method in the three-dimensional space,they assumed that fracture network consists of polygonal shape frac-tures and fluid flow in each fracture meets the Darcy’s law.Dreuzy et al.[7]studied the permeability of randomly fractured networks by numerical and theoretical methods in two dimensions,and they verified the validity of the model by comparing to naturally frac-tured networks.Klimczak et al.[8]obtained the permeability of a single fracture by parallel plate model with the fracture length and aperture satisfying power-law and verified by the numerical simulation.However,these models did not provide a quantitative relationship among the permeability of fracture networks,poros-ity,fracture density and microstructure parameters of fractures,such as fracture length,aperture,inclination,orientation etc.Fractures in rocks are usually random and disorder and they have been shown to have the statistically self-similar and fractal characteristic [3,9–13].Chang and Yortsos [10]studied the single phase fluid flow in the fractal fracture networks.Watanabe and Takahashi [3]investigated the permeability of fracture networks and heat extraction in hot dry rock by using fractal method.But,they did not propose an expression of permeability with micro-scopic parameters included.Jafari and Babadagli [14]obtained the permeability expression with multiple regression analysis of random fractures by the fractal geometry theory according to observed data in the well logging.In addition,their expression with several empirical constants does not include the orientation factor and microstructure parameters of fracture networks.The tree-like fractal branching networks were often considered as/10.1016/j.ijheatmasstransfer.2014.10.0100017-9310/Ó2014Elsevier Ltd.All rights reserved.⇑Corresponding author.E-mail address:yubm_2012@ (B.Yu).fracture networks by many investigators.Xu et al.[15,16]studied the seepage and heat transfer characteristics of fractal-like tree networks.Recently,Wang et al.[17]studied the starting pressure gradient for Binghamfluid in a special dual porosity medium with randomly distributed fractal-like tree network embedded in matrix porous media.Most recently,Zheng and Yu[18]investigated gas flow characteristics in the dual porosity medium with randomly distributed fractal-like tree networks.However,the fractal-like tree network is a kind of ideal and symmetrical network.The purpose of the present work is to derive an analytical expression and establish a model for permeability of fracture rocks/media based on the parallel plane model(cubic law)and frac-tal geometry theory.The proposed permeability and the predicted fracture density will be compared with the numerical simulations.2.Fractal characteristics for fracture networksMany investigators[3,9–13,19–23]reported that the relation-ships between the length and the number of fractures exhibit the power-law,exponential and log-normal types.Torabi and Berg [19]made a comprehensive review on fault dimensions and their scaling laws,and they summarized several types of scaling laws such as the length distributions for faults and fractures in siliciclas-tic rocks from different scales and tectonic settings.The power-law exponents of the scaling-law between the fault length and the number of faults were found to be in the range of1.02–2.04and are probably influenced by factors such as stress regime,linkage of faults,sampling bias,and size of the dataset.Interested readers may consult Refs.[3,9–13,19–23]for detail.In addition,the self-similar fractal structures of fracture net-works were extensively studied[22,23],and the application in complex rock structures with the fractal technique was recently reviewed by Kruhl[24].Velde et al.[25]and Vignes-Adler et al.[26]studied the data at several length scales with fractal method and found that the fracture networks are fractal.Barton and Zoback [27]analyzed the2D maps of the trace length of fractures spanning ten orders,ranging from micro to large scale fractures and found that D f=1.3–1.7.The width between two plates/walls of a fracture,i.e.the paral-lel plate model is used to represent the effective aperture of a frac-ture.Generally,the relationship between the effective aperture a and the fracture length l is given by[28,29]a¼b l nð1Þwhere b and n are the proportionality coefficient and a constant according to fracture scales,respectively.The value of n=1is important,which indicates a linear scaling law,and the fracture network is self-similarity and fractal[19,29].Thus,in the current work the value of n=1is chosen for fractures with fractal characteristic.Thus,Eq.(1)can be rewritten asa¼b lð2ÞEq.(2)will be used in this work.It is well-known that the cumulative size distribution of islands on the Earth’s surface obeys the fractal scaling law[30]NðS>sÞ/sÀD=2ð3aÞwhere N is the total number of island of area S greater than s,and D is the fractal dimension for the size distribution of islands.The equality in Eq.(3a)can be invoked by using s max to represent the largest island on Earth to yield[31]NðS>sÞ¼s maxsD=2ð3bÞEq.(3b)implies that there is only one largest island on the Earth’s surface,and Majumdar and Bhushan[31]used this power-law equation to describe the contact spots on engineering surfaces,where s max¼g k2max(the maximum spot area)and s¼g k2(a spot area),with k being the diameter of a spot and g being a geometry factor.It has been shown that the length distribution of fractures sat-isfies the fractal scaling law[3,9–13,19,22,23,32],hence,Eq.(3b) for description of islands on the Earth’s surface and spots on engi-neering surfaces can be extended to describe the area distribution of fractures on a fractured surface,i.e.NðS!sÞ¼a max l maxalD f=2ð3cÞwhere a max l max represents the maximum fracture area with a max and l max respectively being the maximum aperture and maximum fracture length,and al refers to a fracture area with the aperture and length being a and l,respectively.Inserting Eq.(2)into Eq.(3c),we obtainNðS!sÞ¼b l2maxb l!D f=2ð3dÞThen,from Eq.(3d),the cumulative number of fractures whose length are greater than or equal to l can be expressed by the follow-ing scaling law:NðL!lÞ¼l maxD fð4Þwhere D f is the fractal dimension for fracture lengths,0<D f<2(or 3)in two(or three)dimensions;and Eq.(4)implies that there is only one fracture with the maximum length.Some investigators [3,9–13,19,32]reported that the length distribution of fractures in rocks has the self-similarity and the fractal scaling law can be described by N/ClÀD f,where C is afitting constant,D f is the fractal dimension for the length(l)distribution of fractures and N is the number of fractures,and this fractal scaling law is similar to Eq.(4).Eq.(4)is also the base of the box-counting method[33]for mea-suring the fractal dimension of fracture lengths in fracture net-works,and Chelidze and Guguen[9]applied the box-counting method and found that the fractal dimension of fracture network (described by Nolen-Hoeksema and Gordon[34])in a2D cross sec-tion is1.6.Since there usually are numerous fractures in fracture net-works,Eq.(4)can be considered as a continuous and differentiable function.So,differentiating Eq.(4)with respect to l,we can get the number of fractures whose lengths are in the infinitesimal rang l to l+dl:ÀdNðlÞ¼D f l D fmaxlÀðD fþ1Þdlð5ÞEq.(5)indicates that the number of fractures decreases with the increase of fracture length andÀdN(l)>0.The relationship among the fractal dimension,porosity and the ratio k max=k min for porous media was derived based on the assump-tion that pores in porous media are in the form of squares with self-similarity in sizes in the self similarity range from the mini-mum size k min to the maximum size k max,i.e.[35]D f¼d Eþln emax minð6Þwhere e is the effective porosity of a fractal porous medium,d E is the Euclid dimension,and d E=2and3respectively in two and three dimensions.It has been shown that Eq.(6)is valid not only for exactly self-similar fractals such as Sierpinski carpet and Sierpinski gasket but also for statistically self-similar fractal porous media.Fractures in rocks or in fractured media are analogous to pores in porous media.Therefore,Eq.(6)can be extended to describe the76T.Miao et al./International Journal of Heat and Mass Transfer81(2015)75–80relationship among the fractal dimension for length distribution, porosity of fractures and the ratio l max/l min of fractures in rocks,i.e.D f¼d Eþln/lnðl max=l minÞð7Þwhere l max and l min are the maximum and the minimum fracture lengths,respectively,and/is the effective porosity of fractures in a rock.The area porosity/of fractures is defined as/¼A PAð8Þwhere A is the area of a unit cell,A P is the total area of all fractures in the unite cell.Based on Eq.(5),the total area of all fractures in the unite cell can be obtained asA p¼ÀZ l maxl min aÁlÁdNðlÞ¼b D f l2max2ÀD f1Àl minl max2ÀD f"#ð9ÞInserting Eq.(7)into Eq.(9)yieldsA p¼b D f l2max2ÀD f1À/ðÞð10Þwhere porosity/is applied in Eq.(7)in two dimensions,i.e.d E=2is used.3.Relationship between fracture density and fractal dimensionThe total fracture lengths in a unit cell of area A can be obtained byl total¼ÀZ l maxl min lÁdNðlÞ¼D f l max1ÀD f1Àl minl max1ÀD f"#ð11ÞThe fracture density is defined by[36]D¼l totalð12Þwhere l total is the total length of all fractures(not a single fracture) which may be connected to form a network in the unit cell.Inserting Eqs.(7),(8)and(11)into Eq.(12)results in the fracture densityD¼ð2ÀD fÞ/1Àl minl max1ÀD fð1ÀD fÞb l max1Àl minmax2ÀD fð13aÞInserting Eq.(7)into Eq.(13a),the fracture density can also bewritten asD¼ð2ÀD fÞ1À/ðÞ1ÀD ff"#/ð1ÀD fÞb l max1À/ðÞð13bÞIt is evident that the fracture density D of fractures is a functionof the fractal dimension D f for fracture area,area porosity/,proportionality coefficient b and l max.Fig.2compares the predictions by the present fractal model(Eq.(13a))with numerical simulations of four groups of randomfracture networks by Zhang and Sanderson[36],who proposed anew numerical method for producing the self-avoiding randomgenerations,and the parameters such as the lengths of fracturescan be controlled.In their simulations,the lengths of fractures liefrom0.0005to1.5m,and the averaged fractal dimension D f is1.3.So,in this work we take the maximum length and minimumlength of fractures are1.5m and0.0005m,respectively,and theaveraged fractal dimension D f=1.3.The average porosity/is0.018calculated by Eq.(7).It can be seen from Fig.2that the pre-dictions are in good agreement with the numerical simulations.Fig.2clearly indicates that the fracture density increases withthe increase of the fractal dimension,and this is consistent withpractical situation.Fig.3presents the fracture density versus porosity of fracturenetworks as l max=1.5m,b=0.01.It can be seen from Fig.3thatthe fracture density increases with porosity.This can be explainedthat the pore area of fractures increasing with porosity means thatthe fracture density increases with porosity.This result is in agree-ment with the Monte Carlo simulations by Yazdi et al.[37].4.Fractal model for permeability of fractured rocksThe orientation of each fracture in fracture networks is definedby two angles,the fracture azimuth and fracture dip angle,whichsignificantly affect theflow and transport properties.The orienta-tions of fractures in a fracture network are non-uniform,but usu-ally with a preferred orientation[38,39].In general,the numberof fractures in fracture networks is very large.Based on generalpractice,the fracture azimuths of all fractures are taken as aver-aged/mean angle,for instance,Massart et al.[40]showed a meandip angle of70°,mean N–S(North–South)orientation from thetotal number of1878fractures.In this work,the mean dip angleof fractures between fracture orientations andfluidflow direction,and the mean azimuth of fractures perpendicular tofluidflowdirection are assumed to be h and a,respectively(see Fig.1(a)).(a)(b)T.Miao et al./International Journal of Heat and Mass Transfer81(2015)75–8077Therefore,the scalar quantity of permeability alongflow direction needs to be calculated.Iffluidflow through fractures is assumed to be laminarflow,the flow rate along theflow direction through a fracture can be described by the famous cubic law[41,42]qðlÞ¼a3l12lD PL0ð14Þwhere L0is the length of the structural unit,l is the fracture trace length,a is the fracture aperture,and D P is the pressure drop across a fracture alongflow direction.If the single fracture forms an angle with theflow direction,due to the projection on theflow direction of the fracture,theflow rate through the fracture can be written by[43,44]qðlÞ¼a3l1Àcos2a sin2h12lD PL0ð15Þwhere a and h are respectively the mean facture azimuth and facture dip angle.When a=0,Eq.(15)is reduced toqðlÞ¼a3l cos2h12lD PL0ð16ÞThis is the famous Parsons’model.See Fig.1(b)[43,44].The totalflow rate through all the fractures can be obtained by integrating Eq.(16)from the minimum length to the maximum length in a unit cross section,i.e.Q¼ÀZ l maxl minqðlÞdNðlÞ¼b3128lD f1Àcos2a sin2h4ÀD fD PL0l4max1Àl minl max4ÀD f"#ð17Þwhere D f represents the fractal dimension for the length distribu-tion of fractures.In general,l min<<l max.Since0<D f<2in two dimensions,andðl min=l maxÞ4ÀD f<<1,so that Eq.(17)can be simplified as:Q¼b3128lD f1Àcos2a sin2h4ÀD fD PL0l4maxð18ÞEq.(18)indicate that the totalflow rate through the fracture net-work is related to the fractal dimension D f of the fracture lengths, the facture azimuth a and facture dip angle h.Eq.(18)also indicates that theflow rate is very sensitive to the maximum fracture length l max.Darcy’s law for Newtonianfluidflow in porous media is given byQ¼KAlD PL0ð19ÞComparing Eq.(18)to Eq.(19),we can obtain the permeability for Newtonianfluidflow through the fracture networks asK¼b3128AD f1Àcos2a sin2h4ÀD fl4maxð20ÞInserting Eqs.(12)and(13b)into Eq.(20),the permeability for Newtonianfluidflow through fracture networks can be written asK¼b3D1281ÀD f4ÀD fl3max1Àcos2a sin2h1À/ðÞ1ÀD ff"#ð21ÞEq.(21)shows that the permeability is a function of the fractal dimension D f for the fracture length distribution,the structural parameters(maximum fracture length l max,fracture density D,fac-ture azimuth a and facture dip angle h)and fracture porosity/of fracture networks.Eq.(21)also reveals that the permeability strongly depends on the maximum fracture length l max,and the longer fracture with wider apertures conduct the higher volume offluid and higher permeability.As a result,the present fractal model can well reveal the mechanisms of seepage characteristics78T.Miao et al./International Journal of Heat and Mass Transfer81(2015)75–80in fracture networks than conventional methods.For example, many investigators proposed fracture network models by assum-ing that the media have ideal structures such as the parallel frac-ture network[4,5,8,45],the orthogonal plane network cracks [46,47],alternate level matrix layer and fractures[48]etc.The frac-ture network permeability was often expressed as K=/a2/12, where/is fracture porosity and a is fracture aperture.Recently, Jafari and Babadagli[14]obtained an expression(with several empirical constants)by fractal geometry for fracture networks according to the well logging and observation data.Therefore,it is clear that Eq.(21)has the obvious advantages over the conven-tional models/methods.5.Results and discussionIn this section,the model predictions will be compared with the simulated data and the effects of model parameters on the perme-ability will be discussed.The procedures for determination of the relevant parameters in Eq.(21)are as follows:(1)Given the fracture network parameters(such as l max,/,a,hand b)based on a real sample.(2)Find the fractal dimension D f of fracture lengths in a fracturenetwork by the box-counting method or by Eq.(7).(3)Determine the fracture density D by Eq.(13b).(4)Finally,calculate the permeability by Eq.(21).Jafari and Babadagli[49]obtained the fractal dimensions D f of2D maps from22different nature fracture networks by box-counting method,and then they calculated the equivalent fracture network permeability by a3D model with a block size of100Â100Â10m simulated/constructed.The maximum fracture length was taken to be2m and dip angle h=0.In comparison,the fracture density D and permeability are calculated by procedures3and4,respec-tively.Fig.4shows that the present model predictions are in good agreement with the simulation results[49].Fig.5depicts the permeability for Newtonianfluid through frac-ture networks against porosity of fracture networks at different dip angles at l max=10mm,b=0.01.It is seen from Fig.5that the per-meability for fracture networks increases with porosity.This is consistent with practical situation.From Fig.5,we can also see that the permeability decreases as the fracture plane dip angle increases.This can be explained that a higher fracture plane dip angle leads to an increase of theflow resistance.Fig.6plots the permeability versus the fracture density of the fracture networks at l max=10mm,a=0,h=p/4and b=0.01.It suggests that the permeability of the fracture networks increases with the increases of fracture density.The reason is that when the fracture density D increases,the area of fracture networks increases and thus results in increasing the permeability.This result agrees with the numerical simulation results in Ref.[50]. 6.ConclusionsIn this paper,the fractal geometry theory has been applied to describe the fractal fracture system,and the fractal scaling law for length distribution of fractures and the relationship among the fractal dimension for fracture length distribution,fracture area porosity and the ratio of the maximum length to the minimum length of fractures have been proposed.Then,a model for perme-ability of fractured rocks has been derived based on the famous cubic law,fractal geometry theory and technique.A novel expres-sion for the fracture density has also been proposed based on the fractal scaling law of length distribution of fractures.The present results show that the permeability of fracture networks increases with the increases of porosity and fracture density.Our results agree well the available numerical simulations.This verifies the validity of the proposed models.It should be point out that the percolation and critical behavior are not involved in this work.In this paper,we focus on the perme-ability that all fractures are assumed to be connected to form frac-ture network,which contributes the permeability of the fracture system.This means that we have ignored the interaction between fractures.The permeability after including the interaction and con-nectivity between fractures and critical behavior of fractures near the threshold undoubtedly is an interesting topic,and this may be our next workConflict of interestNone declared.AcknowledgmentThis work was supported by the National Natural Science Foundation of China through Grant Number10932010. 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