SOME ASYMPTOTIC PROPERTIES OF THE CONVOLUTION TRANSFORMS OF FRACTAL MEASURES

We study the asymptotic behavior near the boundary of u（x, y） = Ky ＊ μ （x）, defined on the half-space R^＋ x RN by the convolution of an approximate identity Ky （.） （y 〉 0） and a measure μ on IRN. The Poisson and the heat kernel are unified as special cases in our setting. We are mainly interested in the relationship between the rate of growth at boundary of u and the s-density of a singular measure μ. Then a boundary limit theorem of Fatou＇s type for singular measures is proved. Meanwhile, the asymptotic behavior of a quotient of Kμ and Ku is also studied, then the corresponding Fatou-Doob＇s boundary relative limit is obtained. In particular, some results about the singular boundary behavior of harmonic and heat functions can be deduced simultaneously from ours. At the end, an application in fractal geometry is given.

Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes

Integral and differentiation are two mathematical operations in modern calculus and analysis which have been commonly applied in many fields of science.Integration and differentiation are associated and linked as inverse operation by the fundamental theorem of calculus.Both integral and differentiation are defined based on the concept of additive Lebesgue measure although various generations have been developed with different forms and notations.Fractals can be considered as geometry with fractal dimension(e.g.,non-integer)which no longer possesses Lebesgue additive property.Accordingly,the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity.This paper introduces a recently developed concept of fractal differentiation and integral operations.These operations are expressed using the similar notations of the ordinary operations except the measures are defined in fractal space or measures with fractal dimension.The calculus operations can be used to describe the new concept of fractal density,the density with fractal dimension or density of matter with fractal dimension.The concept and methods are also applied to interpret the Bouguer anomaly over the mid-ocean ridges.The results show that the Bouguer gravity anomaly depicts singularity over the mid-ocean ridges.The development of new calculus operations can significantly improve the accuracy of geodynamic models.

A hybrid simulation approach to evaluate stimulation micro-seismic events and production potential of fractured geothermal reservoirs

This study presents an integrated approach to simulate fluid flow and to predict the micro seismic eventsduring stimulation and circulation of cold water over a longer term in geothermal reservoirs. The integrated approach based on new three dimensional fully coupled thermo-poroelastic numerical model forevaluation of energy recoverable. In the presented approach, the fracture aperture due to fractureslippage is calculated by shear and dilation. The shear slippage is controlled by the concept of shearfailure using linear Mohr-Coulomb criterion. The numerical model is validated against an analytical Oda’smodel for permeability tensor calculation and against an analytical solution for thermo-poroelasticmodel. The heat transfer between the rock and fluid is modelled by using the conductive heat transferwithin the reservoir rock and convective heat transfer in discrete fractures. The thermal stress changesare included in the model to be studied by using roughness induced shear displacement principle in aporo-thermo-elastic environment. The fracture aperture changes are estimated by using an analyticalmodel based on the distributed dislocation technique. The roughness of fracture surfaces is used in thecalculation of residual fracture aperture. The presented approach is used to study the potential ofpermeability enhancement for Habanero geothermal reservoir at a depth of 3600 m. The result show thatthe increasing in tensile effective stress tend to increase the fracture aperture within the zone of cooling.This increasing in fracture aperture led to significant changes in pressure distribution (decrease inimpedance) and hence, increase in the flow rate.