A new mixed type crack propagation criterion in shale reservoirs

Hydraulic fracturing is a mainstream technology for unconventional oil and gas reservoirs development all over the world.How to use this technology to achieve high-level oil and gas resource extraction and how to form complex fracture networks as hydrocarbon transportation channels in tight reservoirs,which depends to a large extent on the interaction between hydraulic and pre-existing cracks.For hydraulic fracturing of fractured reservoirs,the impact of natural fractures,perforation direction,stress disturbances,faults and other influencing factors will produce a mixed I&II mode hydraulic fracture.To forecast whether hydraulic fractures cross pre-existing fractures,according to elastic mechanics and fracture mechanics,a stress state of cracks under the combination of tensile(I)and shear(II)is presented.A simple mixed-mode I&II hydraulic fracture's crossing judgment criterion is established,and the propagation of hydraulic fractures after encountering natural fractures is analyzed.The results show that for a given approaching angle there exists a certain range of stress ratio when crossing occurs.Under high approaching angle and large stress ratio,it is likely that hydraulic cracks will go directly through pre-existing cracks.The reinitiated angle is always controlled within the range of approximately 30among the main direction of penetration.

Towards fully automatic modelling of the fracture process in quasi-brittle and ductile materials:a unified crack growth criterion

Fully automatic finite element(FE) modelling of the fracture process in quasi-brittle materials such as concrete and rocks and ductile materials such as metals and alloys,is of great significance in assessing structural integrity and presents tre-mendous challenges to the engineering community. One challenge lies in the adoption of an objective and effective crack propagation criterion. This paper proposes a crack propagation criterion based on the principle of energy conservation and the cohesive zone model(CZM) . The virtual crack extension technique is used to calculate the differential terms in the criterion. A fully-automatic discrete crack modelling methodology,integrating the developed criterion,the CZM to model the crack,a simple remeshing procedure to accommodate crack propagation,the J2 flow theory implemented within the incremental plasticity framework to model the ductile materials,and a local arc-length solver to the nonlinear equation system,is developed and im-plemented in an in-house program. Three examples,i.e.,a plain concrete beam with a single shear crack,a reinforced concrete(RC) beam with multiple cracks and a compact-tension steel specimen,are simulated. Good agreement between numerical predictions and experimental data is found,which demonstrates the applicability of the criterion to both quasi-brittle and ductile materials.

Fracture initiation and propagation in intact rock—A review

The initiation and propagation of failure in intact rock are a matter of fundamental importance in rock engineering. At low confining pressures, tensile fracturing initiates in samples at 40%-60% of the uniaxial compressive strength and as loading continues, and these tensile fractures increase in density, ultimately coalescing and leading to strain localization and macro-scale shear failure of the samples. The Griffith theory of brittle failure provides a simplified model and a useful basis for discussion of this process. The Hoek-Brown failure criterion provides an acceptable estimate of the peak strength for shear failure but a cutoff has been added for tensile conditions. However, neither of these criteria adequately explains the progressive coalition of tensile cracks and the final shearing of the specimens at higher confining stresses. Grain-based numerical models, in which the grain size distributions as well as the physical properties of the component grains of the rock are incorporated, have proved to be very useful in studying these more complex fracture processes.

Find Better Boolean Functions in the Affine Equivalence Class

The Boolean functions in an affine equivalence class are of the same algebraicdegree and nonlinearity, but may satisfy different order of correlation immunity and propa-gation criterion. A method is presented in this paper to find Boolean functions with higherorder correlation immunity or satisfying higher order propagation criterion in an affine equiv-alence class. 8 AES s-box functions are not better Boolean functions in their affine equiva-lence class.

NEW CONCEPT FOR DETERMINING THE ORDERS OF SAC AND PC

The autocorrelation of a Boolean function possesses the capability to reflect such characteristics as linear structure, Strict Avalanche Criterion(SAC) and Propagation Criterion(PC)of degree k. But it can do nothing in determining the order of SAC or PC. A calculating table for the autocorrelation is constructed in this paper so as to show what is beyond the autocorrelation and how the three cryptographic characteristics are exhibited. A deeper study on the calculating table in a similar way has helped us to develop a new concept, named as the general autocorrelation, to address efficiently the problem how to determine the orders of SAC and PC. The application on the Advanced Encryption Standard(AES) shows the SAC and PC characteristics of Boolean functions of AES S-box.

THE CRITERION FOR PROPAGATION OF THE POSITIVE STREAMER AND ITS APPLICATION

Through analysing the upward leader from a tall structure under storm condition,it is concluded,in this paper,that the intense electric field in a limited,small region provides the necessary condition for the natural lightning discharge to occur.Such a “local intense field” could make it possible that a weak streamer grows and continues to travel in ambient field of only several tens of kilovolts per meter,which otherwise,could propagate only in the ambient field of several hundreds of kilovolts per meter. Based on the positive streamer theory and energy-balance limitation,we have deduced a criterion for de- tcrmining whether the positive streamer propagates continuously or not. Using this criterion,we derive the striking distance for tall structures,and the electric environment for ar- tificially triggering lightning by rocket launching with grounding wire.Both results agree with observations.