Development Of General Fractal Topography And Characterization Modeling Of Geoscience Phenomena

The micro-fractures and micro-pores in coal reservoir are the sites for the migration and storage of coal-bed methane.A detailed description of pore-fracture structure of the reservoir is helpful for understanding the transport and storage of coal-bed methane from a mechanistic level.While the pore-fracture structure of coal reservoirs is extremely complex with scale-invariance fractal feature.The application of fractal theory to realize the detailed description and quantitative characterization of the pore-fracture structure in coal-rock medium,has become a hot spot in the study of porous media in coal-rock reservoirs.Fractal geometry is used to describe the irregular and fragment objects,which is different from the traditional Euclidean geometry.The complexity of fractal uniquely quantified by fractal dimension D.Previously,a great deal of work has been done on the determination of fractal dimensions,however these methods are only used to determine other than define fractal dimension.Fractal geometry is a set enclosing two kind of complexities,the original complexity from the scaling object G and behavior complexity.This paper focuses on the study of complexity of fractal behavior.In recent work,we propose the concept of fractal topology ?(P,F),which is described mathematically by defining two dimensionless parameters:Scaling lacunarity(P)and Scaling Coverage(F).However,?(P,F)can not describe the stochastic,heterogeneous,self-affine and anisotropic of fractal.Therefore we called it“Special Fractal Topography".For that,we extend ?(P,F)mathematically to establish a "General Fractal Topography"?(P,F),where P is the general scaling lacunarity set accounting for direction-dependent scaling behaviors,and F is the scaling coverage set considering stochastic and heterogeneous effects.And a scale-invariant definition of space fractal dimension (?) is proposed as (?)=d×Dx/?i=1dHix.The relationship between (?) and Ds is obtained,which is (?)=d×Ds.To ease the quantification of a desired scale-invariant property of the pore structure,G is further divided into G composed of regime of fractal phase G_and that of the composition of determined phases G+ which wraps the original complexity.Based on the scale-invarnant definition of space fractal dimension,general fractal topography and G,a mathematical model F3S(?,G,L)is therefore proposed as an open mathematical framework to unify the definition of deterministic or statistical,self-similar or self-affine,single-or multi-phase/scale properties of scaling object and fractal behavior.For simple,we define these desired properties P.A generalized calculation model PM is proposed to quantify the probability density distribution of P.Then,some modeling methods are developed based on F3S to simulate the pore structure in natural reservoris.A random self-similar,self-affine fractal curve is constructed for linear fractal objects,and the curve length calculation model M(l)is deduced and verified.Meanwhile,the relationship between the geometric tortuosity and length fractal dimension is obtained,and the quantitative characterization model of the geometric tortuosity of pores and fractures is also presented;Together with the definition ofD,we propose a general multi-scale W-M function(GWMF)to analyze the influence of fractal geometric on fractal dimension with different scaling parameters P,<F>,Hyx;A self-affine bead-based model was constructed based on the PSF model.The scaling parameters of different bead-based model were brought into the classical number-size model to verify its applicability.Meanwhile,the calculation model of area fractal dimension Ds was verified;A multi-phase self-affine fractal network modeling wasproposed to characterize the randomness,heterogeneity,scale-invariant,low-permeability but high connectivity of the fractal porous media,and the applicability of the fractal network is verified by our porosity calculation model proposed previously.Our investigations indicate that the "General Fractal Topography" not only reduces modeling complexity significantly,but also eases the quantifications of scale-invariant properties in an arbitrary fractal set.Most importantly,the definition of fractal behavior is scale-invariant and independent of scaling object,which makes the general fractal topography open to admit scaling objects and fractal behaviors as complex as possible.