Damage Characteristics Analysis and Fractal Study of Shale With Prefabricated Fractures under Thermal-mechanical Coupling

To study the damage and failure of shale with different fracture inclination angles under uniaxial compression loading,in this work,RFPA2D-Thermal,a two-dimensional real failure process analysis software,was used for numerical simulation.Numerical simulation results show that quartz in shale mainly affects the tensile and compressive strength of shale by increasing rock brittleness.The coupling of temperature and pressure will cause lateral and volume destruction of shale,which enables the shale body to be more easily broken.Fracture inclination is the key factor affecting shale damage patterns.The failure mode of shale with low-and high-angle fractures is mainly shear failure,and the compressive strength does not vary with crack inclination.The damage mode of obliquely intersecting fractured shale is slip damage along the fracture face,the compressive strength decreases and then increases with the fracture inclination,and a minimum value exists.The acoustic emission simulation results of the damage process effectively reflect the accumulated internal damage and macroscopic crack appearance until fracture instability when the prefabricated fractured shale is subjected to uniaxial compressive loading.The crack inclinations of 0°and 120℃ corresponds to the most complex"N"shape damage mode.The crack inclinations of 30°and 60°,and the damage mode is an inverted"λ"shape.

Experimental investigation on coal pore-fracture variation and fractal characteristics synergistically affected by solvents for improving clean gas extraction

Chemical solvents instead of pure water being as hydraulic fracturing fluid could effectively increase permeability and improve clean methane extraction efficiency.However,pore-fracture variation features of lean coal synergistically affected by solvents have not been fully understood.Ultrasonic testing,nuclear magnetic resonance analysis,liquid phase mass spectrometry was adopted to comprehensively analyze pore-fracture change characteristics of lean coal treated by combined solvent(NMP and CS_(2)).Meanwhile,quantitative characterization of above changing properties was conducted using geometric fractal theory.Relationship model between permeability,fractal dimension and porosity were established.Results indicate that the end face fractures of coal are well developed after CS2and combined solvent treatments,of which,end face box-counting fractal dimensions range from 1.1227 to 1.4767.Maximum decreases in ultrasonic longitudinal wave velocity of coal affected by NMP,CS_(2)and combined solvent are 2.700%,20.521%,22.454%,respectively.Solvent treatments could lead to increasing amount of both mesopores and macropores.Decrease ratio of fractal dimension Dsis 0.259%–2.159%,while permeability increases ratio of NMR ranges from 0.1904 to 6.4486.Meanwhile,combined solvent could dissolve coal polar and non-polar small molecules and expand flow space.Results could provide reference for solvent selection and parameter optimization of permeability-enhancement technology.

Effect of shear-induced contact area and aperture variations on nonlinear flow behaviors in fractal rock fractures

This study experimentally analyzes the nonlinear flow characteristics and channelization of fluid through rough-walled fractures during the shear process using a shear-flow-visualization apparatus.A series of fluid flow and visualization tests is performed on four transparent fracture specimens with various shear displacements of 1 mm,3 mm,5 mm,7 mm and 10 mm under a normal stress of 0.5 MPa.Four granite fractures with different roughnesses are selected and quantified using variogram fractal dimensions.The obtained results show that the critical Reynolds number tends to increase with increasing shear displacement but decrease with increasing roughness of fracture surface.The flow paths are more tortuous at the beginning of shear because of the wide distribution of small contact spots.As the shear displacement continues to increase,preferential flow paths are more distinctly observed due to the decrease in the number of contact spots caused by shear dilation;yet the area of single contacts in-creases.Based on the experimental results,an empirical mathematical equation is proposed to quantify the critical Reynolds number using the contact area ratio and fractal dimension.

Fractal Feature of Western Fracture Zone in Xikuangshan Antimony Mine and its Geological Significance

In Xikuangshan antimony ore field, the western fracture zone is a composite of major fault, F 75 , and its secondary faults, such as F 71 , F 72 and F 3 etc.. On plane, the fracture zone scatters from southwest to northeast, and concentrates from upper to deeper level on profile. All ore bodies exist in the carbonate of footwall of the major fault or that of the footwall of its secondary faults. From 480 m and 320 m to 120 m level, the fractal dimensional number of the fault system decreases from 1 482 2 and 1 448 6 to 1 339 2, which indicates the form of fracture zone becoming more simple at deeper level. And in five sub ranges, the Ⅲ and Ⅳ sub ranges are the known area, and the Ⅰ, Ⅱ and Ⅴ sub ranges are unknown. The fractal studies of the western fracture zone in these sub ranges show that the fractal dimensional numbers of the Ⅰ and Ⅱ, being 1 201 5 and 1 278 0, respectively, are smaller than that of the Ⅲ and Ⅳ, being 1 475 9 and 1 576 9, respectively; and that of the Ⅴ, being 1 571 2, keeps with that of the Ⅲ, Ⅳ sub ranges. So mineralization is not well in Ⅰ and Ⅱ sub ranges, and Ⅴ sub range is the best to benefit mineralization.

Fractal lacunarity of trabecular bone and magnetic resonance imaging:New perspectives for osteoporotic fracture risk assessment

Osteoporosis represents one major health condition for our growing elderly population. It accounts for severe morbidity and increased mortality in postmenopausal women and it is becoming an emerging health concern even in aging men. Screening of the population at risk for bone degeneration and treatment assessment of osteoporotic patients to prevent bone fragility fractures represent useful tools to improve quality of life in the elderly and to lighten the related socio-economic impact. Bone mineral density(BMD) estimate by means of dual-energy X-ray absorptiometry is normally used in clinical practice for osteoporosis diagnosis. Nevertheless, BMD alone does not represent a good predictor of fracture risk. From a clinical point of view, bone microarchitecture seems to be an intriguing aspect to characterize bone alteration patterns in aging and pathology. The widening into clinical practice of medical imaging techniques and the impressive advances in information technologies together with enhanced capacity of power calculation have promoted proliferation of new methods to assess changes of trabecular bone architecture(TBA) during aging and osteoporosis. Magnetic resonance imaging(MRI) has recently arisen as a useful tool to measure bone structure in vivo. In particular, high-resolution MRI techniques have introduced new perspectives for TBA characterization by non-invasive non-ionizing methods. However, texture analysis methods have not found favor with clinicians as they produce quite a few parameters whose interpretation is difficult. The introduction in biomedical field of paradigms, such as theory of complexity, chaos, and fractals, suggests new approaches and provides innovative tools to develop computerized methods that, by producing a limited number of parameters sensitive to pathology onset and progression, would speed up their application into clinical practice. Complexity of living beings and fractality of several physio-anatomic structures suggest fractal analysis as a promising approach to quantify morphofunctional changes in both aging and pathology. In this particular context, fractal lacunarity seems to be the proper tool to characterize TBA texture as it is able to describe both discontinuity of bone network and sizes of bone marrow spaces, whose changes are an index of bone fracture risk. In this paper, an original method of MRI texture analysis, based on TBA fractal lacunarity is described and discussed in the light of new perspectives for early diagnosis of osteoporotic fractures.

Fractal and Multifractal Propertiesof the Spatial Distribution of Natural Fractures——Analyses and Applications

The Cantor's dust theory is applied to investigate the scaling properites of the spatial distribution of natural fractures obtained from detailed scanline surveys of 27 field sites in the Appalachian Plateau of western New York, USA. The results obtained in this study indicate: 1) fracture spacing is characterized by fractal and multifractal properties. On small scales analyses yield an average fractal dimension of 0.15, which suggests a very high degree of clustering. In contrast, on large scales, fractures tend to be more regular and evenly distributed with an average fracture dimension of 0.52; 2) fractal dimension varies with different sets in different orientations, which can be attributed to interactions between pre-existing fractures and younger ones, as well as variations of the intensity of the stresses under which the fractures were formed; 3) a time sequence of fracture set formation can be proposed based on fractal and multifractal analyses, which consists of (from old to young): N-S, NW, ENE, and NE-striking sets. This time sequence is confirmed by the study of the abutting relationships of different fracture sets observed in the field.

Fractal Model for Estimating the Roughness of Rock Fracture Surfaces

The joint roughness coefficient (JRC) is an important geometric parameter in rock mechanics, seismo-geology, engineering geology, and structural geology. In recent years, the fractol geometry has been widely applied in researching JRC In this paper, a simpler fractal model for estimating JRC was developed, whick can be applied in simulating rock discontinuity profiles, aud the rougher the surface, the greater the fractal dimension D The epirical equation of JRC and D was established according to the model’ The JRC values of six joint profiles were estimated according to Eq (10),which are ratker approximate to the typical JRC values; so the method may be applied to estimate the roughness of the fractured surfaces of natural rocks For a give profile the estimates of D and JRC can be obtained from Eqs (5) and (l0) with only one parameter e’ (the average angle of the profile fluctuation) being needed.

Proppant transport in rough fractures of unconventional oil and gas reservoirs

A method to generate fractures with rough surfaces was proposed according to the fractal interpolation theory.Considering the particle-particle,particle-wall and particle-fluid interactions,a proppant-fracturing fluid two-phase flow model based on computational fluid dynamics(CFD)-discrete element method(DEM)coupling was established.The simulation results were verified with relevant experimental data.It was proved that the model can match transport and accumulation of proppants in rough fractures well.Several cases of numerical simulations were carried out.Compared with proppant transport in smooth flat fractures,bulge on the rough fracture wall affects transport and settlement of proppants significantly in proppant transportation in rough fractures.The higher the roughness of fracture,the faster the settlement of proppant particles near the fracture inlet,the shorter the horizontal transport distance,and the more likely to accumulate near the fracture inlet to form a sand plugging in a short time.Fracture wall roughness could control the migration path of fracturing fluid to a certain degree and change the path of proppant filling in the fracture.On the one hand,the rough wall bulge raises the proppant transport path and the proppants flow out of the fracture,reducing the proppant sweep area.On the other hand,the sand-carrying fluid is prone to change flow direction near the contact point of bulge,thus expanding the proppant sweep area.

Performance-based fractal fracture model for complex fracture network simulation

The paper presents a novel hydraulic fracturing model for the characterization and simulation of the complex fracture network in shale gas reservoirs. We go beyond the existing method that uses planar or orthogonal conjugate fractures for representing the ＇＇complexity＇＇ of the network. Bifurcation of fractures is performed utilizing the Lindenmayer system based on fractal geometry to describe the fracture propagation pattern, density and network connectivity. Four controlling parameters are proposed to describe the details of complex fractures and stimulated reservoir volume（SRV）. The results show that due to the multilevel feature of fractal fractures, the model could provide a simple method for contributing reservoir volume calibration. The primary-and second-stage fracture networks across the overall SRV are the main contributions to the production, while the induced fracture network just contributes another 20% in the late producing period. We also conduct simulation with respect to different refracturing cases and find that increasing the complexity of the fracture network provides better performance than only enhancing the fracture conductivity.

Ti(C, N) BASED CERAMETS AND ITS FRACTAL ANALYSIS OF FRACTURE SURFACES

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Relationship between Fracture Toughness and Fracture Surface Fractal Dimension in AISI 4340 Steel

This study analyzes the relationship between fracture toughness and the fracture surface fractal dimension for a set of twenty-four CT-type AISI 4340 steel specimens heat treated to a variety of tensile strengths. Specimens were tested in accordance with ASTM E 399. Their respective fracture surfaces were plated, polished, photographed under an SEM in BSE mode and measured according to the Richardson method to obtain fractal dimensions. For brittle materials the limited results are consistent with previous literature: increasing fractal dimension with increasing toughness. For partially or fully ductile materials the results indicate a decrease in fractal dimension with an increase in fracture toughness. The data are modeled using a variation of the function applied to ceramics. Fracture in a ductile mode is characterized by the formation of dimples which exhibit fractal characteristics. The results are discussed in terms of the micromechanisms of fracture.

Drained rock volume around hydraulic fractures in porous media:planar fractures versus fractal networks

This study applies the Lindenmayer system based on fractal theory to generate synthetic fracture networks in hydraulically fractured wells.The applied flow model is based on complex analysis methods,which can quantify the flow near the fractures,and being gridless,is computationally faster than traditional discrete volume simulations.The representation of hydraulic fractures as fractals is a more realistic representation than planar bi-wing fractures used in most reservoir models.Fluid withdrawal from the reservoir with evenly spaced hydraulic fractures may leave dead zones between planar fractures.Complex fractal networks will drain the reservoir matrix more effectively,due to the mitigation of stagnation flow zones.The flow velocities,pressure response,and drained rock volume(DRV)are visualized for a variety of fractal fracture networks in a single-fracture treatment stage.The major advancement of this study is the improved representation of hydraulic fractures as complex fractals rather than restricting to planar fracture geometries.Our models indicate that when the complexity of hydraulic fracture networks increases,this will suppress the occurrence of dead flow zones.In order to increase the DRV and improve ultimate recovery,our flow models suggest that fracture treatment programs must find ways to create more complex fracture networks.

Multirange Fractals in Materials: Applications to Fracture and Mechanical Alloying under Ball Milling

A new model of multirange fractals is proposed to explain the experimental results observed on the fractal dimensions of the fractured surfaces in materials. A new explanation to the Williford's multifractal curve on the relationship of fractal dimension with fracture properties in materials has been given. It shows the importance of fractorizing out the effect of fractal structure from other physical causes and separating the appropriate range of scale from multirange fractals. Mechanical alloying process under ball milling as a non-equilibrium dynamical system has been also analyzed.

A smart productivity evaluation method for shale gas wells based on 3D fractal fracture network model

The generation method of three-dimensional fractal discrete fracture network(FDFN)based on multiplicative cascade process was developed.The complex multi-scale fracture system in shale after fracturing was characterized by coupling the artificial fracture model and the natural fracture model.Based on an assisted history matching(AHM)using multiple-proxy-based Markov chain Monte Carlo algorithm(MCMC),an embedded discrete fracture modeling(EDFM)incorporated with reservoir simulator was used to predict productivity of shale gas well.When using the natural fracture generation method,the distribution of natural fracture network can be controlled by fractal parameters,and the natural fracture network generated coupling with artificial fractures can characterize the complex system of different-scale fractures in shale after fracturing.The EDFM,with fewer grids and less computation time consumption,can characterize the attributes of natural fractures and artificial fractures flexibly,and simulate the details of mass transfer between matrix cells and fractures while reducing computation significantly.The combination of AMH and EDFM can lower the uncertainty of reservoir and fracture parameters,and realize effective inversion of key reservoir and fracture parameters and the productivity forecast of shale gas wells.Application demonstrates the results from the proposed productivity prediction model integrating FDFN,EDFM and AHM have high credibility.

Universality and Specificity of Fractal Dimension of Fractured Surfaces in Materials

After calculation on the fracture angles under various conditions of specific surface energies with different symmetry operations of rotation, the complicated behavior of dependence of fractal dimension on the structure of crystal is shown. It is found that the crack propagates along the weakest crystal plane no matter what the direction of the maximum stress is if the anisotropy is sufficiently strong; and then, the fractal dimension of the fractured surfaces might be determined by the approximate fractal structure already existed in the material. Specificity of the fractal dimension of fractured surfaces would be easy to appear in this case. Reversely, the crack propagates along the direction of the maximum stress no matter what direction of the weakest crystal plane is if the anisotropy is sufficiently weak. Universality of the fractal dimension of fractured surfaces would be possible to appear in this case. In many real materials, universality and specificity of the materials are associated. The fractal dimension measured may more or less be influenced by the structure of materials and it shows its universality through the specificity of materials.

FRACTAL ANALYSIS OF IMPACT FRACTURE IN STEEL

By means of fractal analysis,the fractal dimension with relation to the impact toughness,im- pact fracture characteristics and mierostructures of steel under low temperature impact have been investigated.Experimental results show that the logarithms of impact toughness in- creases monotonically with the increase of fractal dimension.The microstruetures have a great influence on fraetal dimension.There is an appropriate correlation between fraetal dimension and fracture mechanism.

Fractal Analysis of Fracture Surfaces in Refractories

Fractal analysis of fracture surfaces in different kinds of refractories including magnesia-carbon and magnesia-chrome refractories was investigated.The fractal dimensions of fracture surfaces for three points bending test were determined by slit island method.The results show that the cold modulus of rupture increases with fractal dimension increasing due to the change in fracture mode depending on the bonding mechanism for a certain refractory material.

The Fracture Density and Fractal Dimension Prediction Based on Support Vector Machine

The key of effective development for the fractured reservoir is to describe the distribution of the fracture and build the fracture geological model. To acquire more optimal exploration and development of the oilfield, objective geologic model of reservoir fractures is needed for further knowledge of the spatial distribution condition of fractures. Adopting well-logging and seismic techniques can be expensive and usually yield multiple solutions, yet resolution will disturb the seismic method, making it difficult to acquire accurate and sound fracture parameters from seismic data. In this paper, the theoretical foundation for support vector machine, fractal geometry, combined the various information in geology, logging, well core, seism and field outcrop about the fracture and calculate the parameters of the fracture (fracture density and fractal dimension), and the good foundation is established for fractured reservoir description of discrete fracture network model. Based on analysis of conventional prediction methods of development indices and factor influencing the parameters of the fracture, a support vector machine method is established to predict the parameters of the fracture. The new support vector machine method is based on time series analysis to select the kernel function. Trains and tests the support vector machine network with historical data to construct the support vector regression prediction model. A case was fit into the model to test and analyse its reliability, the results suggested that the model had a high prediction performance, and could be well applied to the prediction of fracture parameters.

Fracture and Damage Behaviors of Concrete in the Fractal Space

The fracture toughness, the driving force and the fracture energy for an infinite plate with a fractal crack are investigated in the fractal space in this work. The perimeter-area relation is adopted to derive the transforma-tion rule between damage variables in the fractal space and Euclidean space. A plasticity yield criterion is introduced and a damage variable tensor is decomposed into tensile and compressive components to describe the distinct behaviors in tension and compression. A plastic damage constitutive model for concrete in the Euclidean space is developed and generalized to fractal case according to the transformation rule of damage variables. Numerical calculations of the present model with and without fractal are conducted and compared with experimental data to verify the efficiency of this model and show the necessity of considering the fractal effect in the constitutive model of concrete. The structural response and mesh sensitivity of a notched unre-inforced concrete beam under 3-point bending test are theoretical studied and show good agreement with the experimental data.

FRACTAL DIMENSION AS MORPHOLOGY AND SIZE PARAMETERS OF FRACTURED PARTICLES OF ROCK

Semi-automatic image analysis system was used for the surface fractal dimension measurement of 15 kinds of rock-fractured particles. The results indicate that the surfaces of the particles that have a size of about 200 mesh are fractal, and that the fractal dimension is in the range of 2.07～2.11. Therefore, some definitions of particle size such as surface diameter, equivalent volumetricly surface diameter are equivocal without giving the step length and the drag diameter, free-fall diameter, and stokes’diameter may be modified by the fractal dimension.