CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / In the last years, the fractal geometry has become widely studied. Its application in several areas increased substantially, particularly in materials engineering and science, aiming the analysis of failures and the study of the mechanical properties of materials. Some studies have shown the relationship between the fracture surfaces and their mechanical properties using the fractal geometry and its properties of fractal dimension and selfsimilarity. In this research, the fracture surface of duplex stainless steel, which was obtained by the Charpy impact test, was studied applying the fractal geometry. Considering the 475ÂC embrittlement, the steel was submitted to thermal aging to obtain the fracture surfaces, whose images were captured by the scanning electron microscope (SEM). In the fractal analysis, a study was made applying the island method and profile analysis through the digitalization of the images and the application of image analyzing software. Emphasis was given on the calculation of the fractal dimension (FD) of the surface, on the energy absorbed during the impact test, on the involved fracture mechanisms and as well on the relationship between FD and thermal aging. In order to better understanding the subject, it was done the review about fracture mechanics, of duplex stainless steel and of fractal geometry. Finishing the research, the obtained fracture surface, the absorbed energy and the obtained values of FD were analyzed. The obtained results demonstrated a relationship between the fractal dimension, the size of the dimples in fracture surfaces and the impact energy to obtain them. / A geometria dos fractais nos Ãltimos anos tem se tornado bastante difundida no meio cientÃfico. O seu emprego em diversas Ãreas aumentou substancialmente, em particular na engenharia e ciÃncia dos materiais, com o objetivo de analisar falhas e estudar as propriedades mecÃnicas dos materiais. Alguns estudos tÃm mostrado a relaÃÃo entre as propriedades mecÃnicas de superfÃcies de fratura com a geometria dos fractais e suas propriedades de dimensÃo fractal e auto-similaridade. Nesta pesquisa, foi estudada, com base na geometria dos fractais, a superfÃcie de fratura do aÃo inoxidÃvel duplex obtida atravÃs do ensaio de impacto Charpy. Considerando a fragilizaÃÃo a 475C, o aÃo foi submetido ao tratamento tÃrmico de envelhecimento para a obtenÃÃo das superfÃcies de fraturas cujas imagens foram captadas no microscÃpio eletrÃnico de varredura (MEV). Na anÃlise fractal foi feito um estudo aplicando os mÃtodos das ilhas e anÃlise de perfil atravÃs da digitalizaÃÃo das imagens e aplicaÃÃo de softwares de anÃlise de imagem. Foi dada Ãnfase na anÃlise do cÃlculo da dimensÃo fractal (Df) da superfÃcie, na energia absorvida no ensaio de impacto, nos mecanismos de fratura envolvidos, bem como na relaÃÃo entre Df e o tratamento tÃrmico de envelhecimento. Para o melhor entendimento do trabalho foi feita uma revisÃo bibliogrÃfica sobre a mecÃnica da fratura, o aÃo inoxidÃvel duplex e a geometria dos fractais. Para finalizar a pesquisa, foi feita a anÃlise da superfÃcie da fratura obtida, da energia absorvia e de valores de Df alcanÃados. Os resultados obtidos demonstraram uma relaÃÃo entre a dimensÃo fractal, o tamanho dos dimples em superfÃcies de fratura e a energia de impacto para a obtenÃÃo das mesmas.

FundaÃÃo de Amparo à Pesquisa do Estado do Cearà / CoordenaÃÃo de AperfeiÃoamento de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho propomos dois modelos para a geraÃÃo de fraturas em substratos regulares. No primeiro modelo, empregamos iterativamente o conceito de caminho crÃtico para determinar sistematicamente o elemento de menor âcondutividadeâ da rede. Estes elementos sÃo entÃo identificados como âfalhasâ e removidos permanentemente da estrutura atà que uma fratura macroscÃpica destrua a conectividade global da rede. Uma vez detectada, esta fratura à caracterizada topologicamente como uma estrutura auto-similar de dimensÃo fractal Dp ≈ 1.21. No segundo modelo, empregamos iterativamente o algoritmo de Dijkstra para determinar o caminho mÃnimo em uma paisagem aleatÃria, retirando sistematicamente desta estrutura o elemento de maior energia. Como no modelo anterior, estes elementos sÃo identificados como âfalhasâ atà que um conjunto conecto deles resulte em uma fratura macroscÃpica. A mÃdia realizada sobre vÃrias amostras de fraturas em diferentes tamanhos de substratos revela a presenÃa de uma estrutura auto-similar de dimensÃo fractal Df ≈ 1.21. A semelhanÃa numÃrica entre os expoentes Dp e Df sugere que os dois modelos pertencem à mesma classe de universalidade. / In this work we propose two models for fracture generation in regular substrates. In the first model, we iteratively apply the concept of critical path to systematically determine the lower âconductivityâ element in the connected spanning network. At each iteration, once these elements are identified as local âcracks ́ ́, they are permanently removed from the structure up to the point in which a macroscopic fracture can destroy the global network connectivity. This fracture is then topologically characterized as self-similar with fractal dimension Dp ≈ 1.21. In the second model, we employ the algorithm of Dijkstra to determine the minimal path in a random energy landscape and remove its highest energy element. As in the previous model, these elements are considered to be local âcracks ́ ́ till a subset of them can be identified as a macroscopic fracture. The average over many samples of fractures calculated for different system sizes reveals the presence of a self-similar structure with fractal dimension Df ≈ 1.21. The resemblance between the two exponents Dp e Df suggests that the two models belong to the same universality class.

The exploitation of unconventional reservoirs goes hand in hand with the practice of hydraulic fracturing and, with an ever increasing demand in energy, this practice is set to experience significant growth in the coming years. Sophisticated analytic models are needed to accurately describe fluid flow in a hydraulic fracture and the problem has been approached from different directions in the past 3 decades - starting with the use of line-source functions for the infinite conductivity case, followed by the application of Laplace Transforms and the Boundary-Element Method for the finite-conductivity case. This topic remains an active area of research and, for the more complicated physical scenarios such as multiple transverse fractures in ultra-tight reservoirs, answers are presently being sought. Fractal theory has been successfully applied to pressure transient testing, albeit with an emphasis on the effects of natural fractures in pressure-rate behavior. In this work, we begin by performing a rigorous analytical and numerical study of the Fractal Diffusivity Equation and we show that it is more fundamental than the classic linear and radial diffusivity equations. Subsequently, we combine the Fractal Diffusivity Equation with the Trilinear Flow Model, culminating in a new semi-analytic solution for flow in a finite-conductivity vertical fracture which we name the "Fractal-Fracture Solution". This new solution is instantaneous and has an overall accuracy of 99.7%, thus making it comparable to the Trilinear Pseudoradial Solution for practical purposes. It may be used for pressure transient testing and reservoir characterization of hydrocarbon reservoirs being produced by a vertically fractured well. Additionally, this is the first time that fractal theory is used in fluid flow in porous media to address a problem not related to reservoir heterogeneity. Ultimately, this work is a demonstration of the untapped potential of fractal theory; our approach is very flexible and we believe that the same methodology may be extended to develop new reservoir flow solutions for pressing problems that the industry currently faces.