Extended finite element-based cohesive zone method for modeling simultaneous hydraulic fracture height growth in layered reservoirs

In this study,a fully coupled hydromechanical model within the extended finite element method(XFEM)-based cohesive zone method(CZM)is employed to investigate the simultaneous height growth behavior of multi-cluster hydraulic fractures in layered porous reservoirs with modulus contrast.The coupled hydromechanical model is first verified against an analytical solution and a laboratory experiment.Then,the fracture geometry(e.g.height,aperture,and area)and fluid pressure evolutions of multiple hydraulic fractures placed in a porous reservoir interbedded with alternating stiff and soft layers are investigated using the model.The stress and pore pressure distributions within the layered reservoir during fluid injection are also presented.The simulation results reveal that stress umbrellas are easily to form among multiple hydraulic fractures’tips when propagating in soft layers,which impedes the simultaneous height growth.It is also observed that the impediment effect of soft layer is much more significant in the fractures suppressed by the preferential growth of adjoining fractures.After that,the combined effect of in situ stress ratio and fracturing spacing on the multi-fracture height growth is presented,and the results elucidate the influence of in situ stress ratio on the height growth behavior depending on the fracture spacing.Finally,it is found that the inclusion of soft layers changes the aperture distribution of outmost and interior hydraulic fractures.The results obtained from this study may provide some insights on the understanding of hydraulic fracture height containment observed in filed.

Numerical analysis of rock fracturing by gas pressure using the extended finite element method

High energy gas fracturing is a simple approach of applying high pressure gas to stimulate wells by gen- erating several radial cracks without creating any other damages to the wells. In this paper, a numerical algorithm is proposed to quantitatively simulate propagation of these fractures around a pressurized hole as a quasi-static phenomenon. The gas flow through the cracks is assumed as a one-dimensional transient flow, governed by equations of conservation of mass and momentum. The fractured medium is modeled with the extended finite element method, and the stress intensity factor is calculated by the simple, though sufficiently accurate, displacement ex- trapolation method. To evaluate the proposed algorithm, two field tests are simulated and the unknown parameters are determined through calibration. Sensitivity analyses are performed on the main effective parameters. Considering that the level of uncertainty is very high in these types of engineering problems, the results show a good agreement with the experimental data. They are also consistent with the theory that the final crack length is mainly determined by the gas pressure rather than the initial crack length produced by the stress waves.

Fracture properties of epoxy asphalt mixture based on extended finite element method

Crack is found to be a major distress that affects the performance of the epoxy asphalt pavement.An extended finite element method was proposed for investigating the fracture properties of the epoxy asphalt mixture.Firstly,the single-edge notched beam test was used to analyze the temperature effect and calculate the material parameters.Then,the mechanical responses were studied using numerical analysis.It is concluded that 5℃ can be selected as the critical temperature that affects the fracture properties,and numerical simulations indicate that crack propagation is found to significantly affect the stress state of the epoxy asphalt mixture.The maximum principal stress at the crack surface exhibits different trends at various temperatures.Numerical solution of stress intensity factor can well meet the theoretical solution,especially when the temperature is lower than 5℃.

Evaluation of mixed-mode stress intensity factors by extended finite element method

Extended finite element method （XFEM） implementation of the interaction integral methodology for evaluating the stress intensity factors （SIF） of the mixed-mode crack problem is presented. A discontinuous function and the near-tip asymptotic function are added to the classic finite element approximation to model the crack behavior. Two-state integral by the superposition of actual and auxiliary fields is derived to calculate the SIFs. Applications of the proposed technique to the inclined centre crack plate with inclined angle from 0° to 90° and slant edge crack plate with slant angle 45°, 67.5° and 90° are presented, and comparisons are made with closed form solutions. The results show that the proposed method is convenient, accurate and computationallv efficient.

Extended multiscale finite element method for mechanical analysis of heterogeneous materials

An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis.On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus,different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures.Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale,especially under the periodic boundary conditions.

Thermoelastic analysis of multiple defects with the extended finite element method

In this paper, the extended finite element method (XFEM) is adopted to analyze the interaction between a single macroscopic inclusion and a single macroscopic crack as well as that between multiple macroscopic or microscopic defects under thermal/mechanical load. The effects of different shapes of multiple inclusions on the material thermomechanical response are investigated, and the level set method is coupled with XFEM to analyze the interaction of multiple defects. Further, the discretized extended finite element approximations in relation to thermoelastic problems of multiple defects under displacement or temperature field are given. Also, the interfaces of cracks or materials are represented by level set functions, which allow the mesh assignment not to conform to crack or material interfaces. Moreover, stress intensity factors of cracks are obtained by the interaction integral method or the M-integral method, and the stress/strain/stiffness fields are simulated in the case of multiple cracks or multiple inclusions. Finally, some numerical examples are provided to demonstrate the accuracy of our proposed method.

Simulation of three-dimensional tension-induced cracks based on cracking potential function-incorporated extended finite element method

In the finite element method,the numerical simulation of three-dimensional crack propagation is relatively rare,and it is often realized by commercial programs.In addition to the geometric complexity,the determination of the cracking direction constitutes a great challenge.In most cases,the local stress state provides the fundamental criterion to judge the presence of cracks and the direction of crack propagation.However,in the case of three-dimensional analysis,the coordination relationship between grid elements due to occurrence of cracks becomes a difficult problem for this method.In this paper,based on the extended finite element method,the stress-related function field is introduced into the calculation domain,and then the boundary value problem of the function is solved.Subsequently,the envelope surface of all propagation directions can be obtained at one time.At last,the possible surface can be selected as the direction of crack development.Based on the aforementioned procedure,such method greatly reduces the programming complexity of tracking the crack propagation.As a suitable method for simulating tension-induced failure,it can simulate multiple cracks simultaneously.

Cracking Propagation of Hardening Concrete Based on the Extended Finite Element Method

Self-deformation cracking is the cracking caused by thermal deformation, autogenous volume deformation or shrinkage deformation. In this paper, an extended finite element calculation method was deduced for concrete crack propagation under a constant hydration and hardening condition during the construction period, and a corresponding programming code was developed. The experimental investigation shows that initial crack propagation caused by self-deformation loads can be analyzed by this program. This improved algorithm was a preliminary application of the XFEM to the problem of the concrete self-deformation cracking during the hydration and hardening period. However, room for improvement exists for this algorithm in terms of matching calculation programs with mass concrete temperature fields containing cooling pipes and the influence of creep or damage on crack propagation.

Mesoscopic characterization and modeling of microcracking in cementitious materials by the extended finite element method

This study develops a mesoscopic framework and methodology for the modeling of microcracks in concrete. A new algorithm is first proposed for the generation of random concrete meso-structure including microcracks and then coupled with the extended finite element method to simulate the heterogeneities and discontinuities present in the meso-structure of concrete. The proposed procedure is verified and exemplified by a series of numerical simulations. The simulation results show that microcracks can exert considerable impact on the fracture performance of concrete. More broadly, this work provides valuable insight into the initiation and propagation mechanism of microcracks in concrete and helps to foster a better understanding of the micro-mechanical behavior of cementitious materials.

Factors Influencing Fracture Propagation in Collaborative Fracturing ofMultiple HorizontalWells

Horizontal well-stimulation is the key to unconventional resource exploration and development.The development mode of the well plant helps increase the stimulated reservoir volume.Nevertheless,fracture interference between wells reduces the fracturing effect.Here,a 2D hydro-mechanical coupling model describing hydraulic fracture(HF)propagation is established with the extended finite element method,and the effects of several factors on HF propagation during multiple wells fracturing are analyzed.The results show that with an increase in elastic modulus,horizontal principal stress difference and injection fluid displacement,the total fracture area and the reservoir stimulation efficiency are both improved in all three fracturing technologies.After a comparison of the three technologies,the method of improved zipper fracturing is proposed,which avoids mutual interference between HFs,and the reservoir stimulation effect is improved significantly.The study provides guidance for optimizing the fracturing technology of multiple horizontal wells.

Fixed-length roof cutting with vertical hydraulic fracture based on the stress shadow effect:A case study

Pre-driven longwall retracement roadway(PLRR)is commonly used in large mine shaft.The support crushing disasters occur frequently during the retracement,and roof management is necessary.Taking the 31107 panel as research background,the roof breaking structure of PLRR is analyzed.It is concluded that the roof cutting with vertical hydraulic fracture(HF)at a specified position,that is,fixed-length roof cutting,can reduce support load and keep immediate roof intact.The extended finite element method(XFEM)is applied to simulate hydraulic fracturing.The results show that both the axial and transverse hydraulic fracturing cannot effectively create vertical HFs.Therefore,a novel construction method of vertical HF based on the stress shadow effect(SSE)is proposed.The stress reversal region and HF orientation caused by the prefabricated hydraulic fracture(PF)are verified in simulation.The sub-vertical HFs are obtained between two PFs,the vertical extension range of which is much larger than that of directional hydraulic fracturing.The new construction method was used to determine the field plan for fixed-length roof cutting.The roof formed a stable suspended structure and deformation of the main PLRR was improved after hydraulic fracturing.

Numerical computation of central crack growth in an active particle of electrodes influenced by multiple factors

Mechanical degradation, especially fractures in active particles in an electrode, is a major reason why the capacity of lithiumion batteries fades. This paper proposes a model that couples Li-ion diffusion, stress evolution, and damage mechanics to simulate the growth of central cracks in cathode particles（Li Mn_2 O_4） by an extended finite element method by considering the influence of multiple factors. The simulation shows that particles are likely to crack at a high discharge rate, when the particle radius is large, or when the initial central crack is longer. It also shows that the maximum principal tensile stress decreases and cracking becomes more difficult when the influence of crack surface diffusion is considered. The fracturing process occurs according to the following stages： no crack growth, stable crack growth, and unstable crack growth. Changing the charge/discharge strategy before unstable crack growth sets in is beneficial to prevent further capacity fading during electrochemical cycling.

Simulation of PFZ on intergranular fracture based on XFEM and CPFEM

A unit cell including the matrix, precipitation free zone(PFZ) and grain boundary was prepared, and the crystal plasticity finite element method(CPFEM) and extended finite element method(XFEM) were used to simulate the propagation of cracks at grain boundary. Simulation results show that the crystallographic orientation of PFZ has significant influence on crack propagation, which includes the crack growth direction and crack growth velocity. The fracture strain of soft orientation is larger than that of hard orientation due to the role of reducing the stress intensity at grain boundary in intergranular brittle fracture. But in intergranular ductile fracture, the fracture strain of soft orientation may be smaller than that of hard orientation due to the roles of deformation localization.

Fatigue Crack Propagation Analysis of Orthotropic Steel Bridge with Crack Tip Elastoplastic Consideration

Due to the complex structure and dense weld of the orthotropic steel bridge deck(OSBD),fatigue cracks are prone to occur in the typical welding details.Welding residual stress(WRS)will cause a plastic zone at the crack tip.In this paper,an elastoplastic constitutive model based on the Chaboche kinematic hardening model was introduced,and the extended finite element method(XFEM)was used to study the influence of material elastoplasticity and crack tip plastic zone on the law of fatigue crack propagation.By judging the stress state of the residual stress field at the crack tip and selecting different crack propagation rate models to investigate the crack propagation law when plastic deformation was considered,the propagation path and propagation rate of fatigue crack of the OSBD were obtained.The results show that,whether the residual stress field is considered or not,the plastic deformation at the crack tip will not cause the obvious closure of the fatigue crack at the U-rib toe during the crack propagation process,but will significantly affect the crack propagation path.When material plasticity is considered,the propagation angle of fatigue crack at the U-rib toe basically remains unchanged along the short-axis direction of the initial crack,but is going up along the long-axis direction,and the crack tip plastic zone inhibits the propagation of the crack tip on one side.Compared with linear elastic materials,the crack propagation law considering material plasticity is more consistent with that in actual bridge engineering.In terms of the propagation rate,if the residual stress field is not considered,the fatigue crack propagation rate at U-rib toe with plasticity considered is slightly higher than that without plasticity considered,because plastic deformation will affect the amplitude of energy release rate.When considering the WRS field,the fatigue crack propagation rate at U-rib toe is increased due to the combined actions of plastic deformation and stress ratio R.

Numerical simulation of fracture propagation in Russia carbonate reservoirs during refracturing

Refracturing treatment is often performed on Russian carbonate reservoirs because of the quick production decline of reservoirs.The traditional refracturing model assumes that a refracture initiates in the normal direction relative to the initial hydro-fracture.This assumption is inconsistent with oilfield measurements of refracture propagation trajectories.Indeed,the existing model is not based on an indepth understanding of initiation and propagation mechanisms of the second hydraulic fractures during refracturing.In this study,we use the extended finite element method to investigate refracture propagation paths at different initiation angles.Both the enriched function approach and phantom mode technique are incorporated into the refracturing model,thereby ensuring that the refracture can freely extend on the structured mesh without any refinement near the crack tips.Key factors including production time,stress anisotropy and initiation angle,and the propped mechanical effect are analyzed in detail.This study provides new insight into the mechanism of refracture propagation in unconventional reservoirs.

A Numerical Investigation of an Abnormal Phenomenon of Stress Intensity Factor(SIF)in a Cracked T-Butt Joint Accounting for Welding Effect

Industry design standards such as BS 7910 deployed some empirical formulas for the prediction of stress intensity factor(SIF) based on simulation results from traditional finite element method(FEM).However,such FEM simulation occasionally failed to convince people due to the large discrepancies compared with engineering practice.As a consequence,inaccuracy predictions via such formulas in engineering standards inevitably occur,which will compromise the safety of structures.In our previous research work,an abnormal phenomenon of SIF in a cracked T-butt joint accounting for welding effect has been observed.Compared with BS 7910,the calculation results of SIF at the surface points of welded specimens cannot be well predicted,with a large discrepancy appearing.In order to explore such problem with an abnormal increase at the surface points of cracked welded specimens,a numerical investigation in terms of SIF among BS 7910,XFEM,and FEM is performed in this paper.Numerical models on both a simple cracked plate without welding effect and a cracked T-butt joint with welding effect are developed through ABAQUS.Parametric studies in terms of the effects of varied crack depth to thickness ratio(a/T) and the effects of crack depth to crack half-length ratio(a/c) are carried out.Empirical solutions from BS 7910 are used for comparison.It is found that the XFEM can provide predictions of SIF at both the crack deepest point and crack surface point of a simple cracked plate as accurate as FEM.For a T-butt joint with a transverse stiffener,a large discrepancy in terms of the weld magnification factors(Mk) occurs at the crack surface point compared with empirical predictions.An exceptional increase of von Mises stress gradient in regions close to the weld-toe is found through the simulation of FEM,whereas a constant stress gradient is obtained through XFEM.The comparison results indicate an inappropriate prediction of SIF by the utilization of the empirical formulas in BS 7910.A more reasonable prediction of the SIF at the surface point of a crack is obtained by the XFEM.Therefore,further updating of the empirical solutions in BS7910 for SIF accounting for welding effect is recommended.

Fracture Behavior of Epoxy Asphalt Pavement on Steel Bridges based on Optical Fiber Sensing Technology and Numerical Simulation

The distributed optical fiber sensing technology was used to investigate the fracture behavior of the Epoxy Asphalt Mixture. The spatial distribution and variation of the strain development with crack propagation were acquired using the brillouin optical time-domain reflectometer through the loading experiments of the composite beam structure. In addition, a finite element model of the composite beam structure was developed to analyze the mechanical responses of the epoxy asphalt mixture using the extended finite element method. The experimental results show that the development of crack propagation becomes instable with the increase of the load, and larger loads will generate deeper cracks. Moreover, the numerical results show that the mechanical response of the crack tip changes with the crack propagation, and the worst areas that subjected to crack damage are located on both sides of the composite beam structure.

Stable Fatigue Crack Propagation of 16MnR Steel

A research on the stable fatigue crack propagation of 16MnR steel is investigated systematically in this paper.First,control experiments of 16MnR with compact tension specimen is conducted to study the effect of R-ratios,specimen thickness and notch sizes.The experiments show that the fatigue crack growth(FCG)rate in stable propagation was insensitive to these factors.Then,the stress intensity factor(SIF)is computed and compared by displacement interpolation method,J integral and interaction integral method respectively.The simulation shows that optimization on the mesh density and the angle of singular element improved the computational efficiency and accuracy of SIF and the interaction integral method has an obvious advantage on stability.Finally,the FCG rate is modeled by the Jiang fatigue damage criterion and the extended finite element method(XFEM)respectively.The simulation results of FCG rate are in line with experiments data and indicate that XFEM method is more accurate than Jiang fatigue damage method.

Numerical modeling of thermally-induced fractures in a large rock salt mass

Numerical modeling of thermally-induced fractures is a concern for many geo-structures including deep underground energy storage caverns. In this paper, we present the numerical simulation of a large-scale cooling experiment performed in an underground rock salt mine. The theory of fracture mechanics was embedded in the extended finite element code used. The results provide reliable information on fracture location and fracture geometry. Moreover, the timing of the fracture onset, as well as the stress redis- tribution due to fracture propagation, is highlighted. The conclusions of this numerical approach can be used to improve the design of rock salt caverns in order to guarantee their integrity in terms of both their tightness and stability.

A novel enriched CB shell element method for simulating arbitrary crack growth in pipes

In this work, a novel numerical method is developed for simulating arbitrary crack growth in pipes with the idea of enriched shape functions which can represent the discontinuity independent of the mesh. The concept of the enriched shape functions is introduced into the continuum-based (CB) shell element. Due to the advantage of CB shell element, the shell thickness varia- tion and surface connection can be concerned during the deformation. The stress intensity factors of the crack in the CB shell element are calculated by using the 'equivalent domain integral' method for 3D arbitrary non-planar crack. The maximum en- ergy release rate is used as a propagation criterion. This method is proved able to capture arbitrary crack growth path in pipes which is independent of the element mesh. Numerical examples of different fracture patterns in pipes are presented here.